Analog channel estimation techniques for beamformer design in massive MIMO systems

Abstract

Novel channel estimation techniques that can be performed solely in the analog domain are provided. The techniques use continuous analog channel estimation (CACE), periodic analog channel estimation (PACE), and multiantenna frequency shift reference (MAFSR). These techniques provide sufficient channel knowledge to enable analog beamforming at the receiver while significantly reducing the estimation overhead. These schemes involve transmission of a reference tone of a known frequency, either continuously along with the data (CACE, MAFSR) or in a separate phase from data (PACE). Analysis of the methods show that sufficient receiver beamforming gain can be achieved in sparse channels by building the beamformer using just the amplitude and phase estimates of the reference tone from each receiver antenna.

Claims

1. A MIMO system that applies continuous analog channel estimation, the MIMO system comprising: a transmitter (TX) that transmits a transmitted signal that includes a data signal combined with a predetermined reference signal; and a receiver (RX) including: a plurality of antennas wherein each antenna receives the transmitted signal and outputs an associated received signal; a baseband conversion processor that either includes an independent oscillator or recovers the predetermined reference signal including either or both of signal amplitude and phase, each associated received signal being multiplied with an independent oscillator signal or a recovered reference signal and with its quadrature component in the analog domain, resulting in processor output signals that are low-pass signals with at least partially compensated inter-antenna phase shift; an amplitude and phase compensation processor that adjusts outputs from the baseband conversion processor via analog phase shifters; an analog adder that receives output signals from the amplitude and phase compensation processor, the output signals including an in-phase-derived control signal and a quadrature output signal for each antenna in the plurality of antennas, the analog adder configured to emulate signal combining and beamforming by performing a weighted sum of the output signals from the amplitude and phase compensation processor to form a summed signal output, the analog adder including a first component configured to sum in-phase-derived output signals and a second part configured to sum quadrature output signals, wherein the summed signal output includes a real component Re(ω.sub.LPF(t)) and an imaginary component Im(ω.sub.LPF(t); and an analog to digital converter that samples the real component Re(ω.sub.LPF(t)) and imaginary component Im(ω.sub.LPF(t) of the summed signal output.

2. The MIMO system of claim 1 wherein the amplitude and phase compensation processor applies a baseband reference signal in the outputs from the baseband conversion processor as control signals to the analog phase shifters to further compensate for inter-antenna phase-shifts in the outputs from the baseband conversion processor.

3. The MIMO system of claim 1 wherein the transmitter transmits the transmitted signal by orthogonal frequency division multiplexing (OFDM).

4. The MIMO system of claim 1 wherein the analog adder applies a weight sum when summing the processor output signals.

5. The MIMO system of claim 1 wherein the baseband conversion processor includes a local oscillator instead of a signal recovery circuit.

6. The MIMO system of claim 1 wherein the baseband conversion processor includes: at least one reference signal recovery circuit; a plurality of first mixers with each antenna having an associated first mixer; and a plurality of second mixers with each antenna having a second mixer.

7. The MIMO system of claim 6 wherein each antenna has an associated signal recovery circuit.

8. The MIMO system of claim 6 wherein the reference signal recovery circuit includes a secondary phase locked loop array that receives signals-from a subset of the plurality of antennas and wherein outputs from each secondary phase locked loop are summed and fed as control signals to a primary phase locked loop.

9. The MIMO system of claim 8 wherein the outputs from each phase locked loop are summed as a weighted sum.

10. The MIMO system of claim 6 wherein: an associated reference signal recovery circuit of each antenna isolates and recovers the predetermined reference signal as an associated isolated reference signal from the associated received signal; the associated first mixer of each antenna multiplies the associated isolated reference signal with the associated received signal to produce an associated in-phase-derived output signal; and an associated second mixer of each antenna multiplies the quadrature component of the associated isolated reference signal with the associated received signal to produce a quadrature-derived output signal such that the associated in-phase-derived output signal and the quadrature-derived output signal are the processor output signals.

11. The MIMO system of claim 10 further comprising a plurality of low noise amplifiers positioned between each antenna and the baseband conversion processor.

12. The MIMO system of claim 10 wherein each associated reference signal recovery circuit includes an injection locked oscillator, a phase locked loop, or a bandpass filter.

13. The MIMO system of claim 10 wherein sufficient frequency separation may be provided between the predetermined reference signal and the data signal to enable each reference recovery circuit.

14. The MIMO system of claim 10 wherein a plurality of filters extract a baseband signal and filter out noise from the associated in-phase-derived output signal outputted by the plurality of first mixers and a second plurality of low pass filters extract associated quadrature-derived signals outputted by the plurality of second mixers.

15. The MIMO system of claim 14 further comprising a low pass filter and/or down converter that receives the summed signal output and outputs an analog baseband signal.

16. The MIMO system of claim 10 wherein a sparse nature of wireless channels is exploited to ensure a large beamforming gain after the summed signal output.

17. The MIMO system of claim 12 that can perform receive beamforming without digital channel estimation.

18. The MIMO system of claim 12 further comprising an analog to digital converter that converts an analog baseband signal to digital baseband signal and a demodulator that demodulates the digital baseband signal.

19. The MIMO system of claim 1 comprising the amplitude and phase compensation processor which includes associated phase shifters, each phase shifter receiving in-phase-derived control signal and a quadrature-derived control signal such that outputs from the baseband conversion processor are phase adjusted.

20. The MIMO system of claim 19, wherein the amplitude and phase compensation processor includes lowpass filters that filter in-phase and quadrature output signals from the baseband conversion processor to generate the in-phase-derived control signal and the quadrature-derived control signal to be used for the associated phase shifters, the baseband conversion processor including: at least one reference signal recovery circuit; a plurality of first mixers with each antenna having an associated first mixer; and a plurality of second mixers with each antenna having a second mixer.

21. The MIMO system of claim 1 wherein the baseband conversion processor recovers the predetermined reference signal.

22. The MIMO system of claim 1 wherein signal combining and beamforming is emulated without the receiver applying explicit channel estimation.

23. A MIMO system that applies continuous analog channel estimation, the MIMO system comprising: a transmitter (TX) that transmits a transmitted signal that includes a data signal combined with a predetermined reference signal; and a receiver (RX) including: at least three antennas wherein each antenna receives the transmitted signal and outputs an associated received signal; a baseband conversion processor that either includes an independent oscillator or recovers the predetermined reference signal including either or both of signal amplitude and phase, each associated received signal being multiplied with an independent oscillator signal or a recovered reference signal and with its quadrature component in the analog domain, resulting in processor output signals that are low-pass signals with at least partially compensated inter-antenna phase shift; and an analog adder configured to emulate signal combining and beamforming by performing a weighted sum of output signals from the baseband conversion processor to form a summed signal output, the output signals including an in-phase-derived output signal for each antenna and the summed signal output having a real component Re(ω.sub.LPF(t)) and imaginary component Im(ω.sub.LPF(t); and an analog to digital converter that samples the summed signal output.

24. The MIMO system of claim 23 wherein the transmitter transmits the transmitted signal by orthogonal frequency division multiplexing (OFDM).

25. The MIMO system of claim 23 wherein the analog adder applies a weighted sum when summing the processor output signals.

26. The MIMO system of claim 23 wherein the baseband conversion processor includes a local oscillator instead of a signal recovery circuit.

27. The MIMO system of claim 23 wherein the baseband conversion processor includes: at least one reference signal recovery circuit; a plurality of first mixers with each antenna having an associated first mixer; and a plurality of second mixers with each antenna having a second mixer.

28. The MIMO system of claim 27 wherein each antenna has an associated signal recovery circuit.

29. The MIMO system of claim 27 wherein the reference signal recovery circuit includes a secondary phase locked loop array that receives signals from a subset of the at least three antennas and wherein outputs from each secondary phase locked loop are summed and fed as control signals to a primary phase locked loop.

30. The MIMO system of claim 29 wherein the outputs from each phase locked loop are summed as a weighted sum.

31. The MIMO system of claim 26 wherein: an associated reference signal recovery circuit of each antenna isolates and recovers the predetermined reference signal as an associated isolated reference signal from the associated received signal; an associated first mixer of each antenna multiplies the associated isolated reference signal with the associated received signal to produce an associated in-phase-derived output signal; and an associated second mixer of each antenna multiplies the quadrature component of the associated isolated reference signal with the associated received signal to produce a quadrature-derived output signal such that the associated in-phase-derived output signal and the quadrature-derived output signal are the processor output signals.

32. The MIMO system of claim 31 further comprising a plurality of low noise amplifiers positioned between each antenna and the baseband conversion processor.

33. The MIMO system of claim 31 wherein each associated reference signal recovery circuit includes an injection locked oscillator, a phase locked loop, or a bandpass filter.

34. The MIMO system of claim 31 wherein sufficient frequency separation may be provided between the predetermined reference signal and the data signal to enable each reference recovery circuit.

35. The MIMO system of claim 32 wherein a plurality of filters extract a baseband signal and filter out noise from the associated in-phase-derived output signal outputted by a plurality of first mixers and a second plurality of low pass filters extract associated quadrature-derived signals outputted by a plurality of second mixers.

36. The MIMO system of claim 35 further comprising a low pass filter and/or down converter that receives the summed signal output and outputs an analog baseband signal.

37. The MIMO system of claim 31 wherein a sparse nature of wireless channels is exploited to ensure a large beamforming gain after the summed signal output.

38. The MIMO system of claim 33 that can perform receive beamforming without digital channel estimation.

39. The MIMO system of claim 33 further comprising an analog to digital converter that converts an analog baseband signal to digital baseband signal and a demodulator that demodulates the digital baseband signal.

40. The MIMO system of claim 23 wherein signal combining and beamforming is emulated without the receiver applying explicit channel estimation.

41. A MIMO system that applies continuous analog channel estimation, the MIMO system comprising: a transmitter (TX) that transmits a transmitted signal that includes a data signal combined with a predetermined reference signal; and a receiver (RX) including: a plurality of antennas wherein each antenna receives the transmitted signal and outputs an associated received signal and wherein each antenna has an associated signal recovery circuit; a baseband conversion processor that either includes an independent oscillator or recovers the predetermined reference signal including either or both of signal amplitude and phase, each associated received signal being multiplied with an independent oscillator signal or a recovered reference signal and with its quadrature component in the analog domain, resulting in processor output signals that are low-pass signals with at least partially compensated inter-antenna phase shift, the baseband conversion processor further including at least one reference signal recovery circuit, a plurality of first mixers with each antenna having an associated first mixer, and a plurality of second mixers with each antenna having a second mixer, wherein the reference signal recovery circuit includes a secondary phase locked loop array that receives signals-from a subset of the plurality of antennas and wherein outputs from each secondary phase locked loop are summed and fed as control signals to a primary phase locked loop; an amplitude and phase compensation processor that adjusts outputs from the baseband conversion processor via analog phase shifters; an analog adder that receives and sums output signals from the amplitude and phase compensation processor as a summed signal output thereby emulating signal combining and beamforming, the output signals including an in-phase-derived output signal for each antenna in the plurality of antennas, the summed signal output having a real component Re(ω.sub.LPF(t)) and imaginary component Im(ω.sub.LPF(t); and an analog to digital converter that samples a real component Re(ω.sub.LPF(t)) and imaginary component Im(ω.sub.LPF(t) of the summed signal output.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1: Block diagram for a MIMO system with a multi-antenna CACE receiver.

(2) FIG. 2: Block diagram for a MIMO system with a multi-antenna CACE receiver

(3) FIG. 3: Block diagram for a MIMO system with a multi-antenna PACE receiver.

(4) FIG. 4: Block diagram of weighted carrier arraying for reference tone recovery.

(5) FIG. 5: Block diagram for a MIMO system with a multi-antenna frequency shift reference (MA-FSR) receiver.

(6) FIGS. 6A and 6B: An illustration of the transmit and receive signals with CACE aided beamforming. A) Transmit signal at TX antenna m; B) Baseband signal at RX antenna m.

(7) FIG. 7: Comparison of analytical (from Lemma 3.1) and simulated statistics of the nDFT coefficients of a sample RX phase-noise process with T.sub.s=1 μs, K.sub.1=K.sub.2+1=512 and σ.sub.θ=1/√{square root over (T.sub.s)}. Simulations averaged over 10.sup.6 realizations.

(8) FIG. 8: Comparison of analytical SERs (from (1.28) with/without Remark 3.1) to simulated results, for different sub-carriers of a CACE receiver with Quadrature Phase Shift Keying. Simulations consider σ.sub.θ.sup.2=1/T.sub.s (−93 dBc at 10 MHz offset), mean RX oscillator frequency of f.sub.c and f.sub.c+5 MHz, and are obtained by averaging over 10.sup.6 realizations.

(9) FIG. 9: Comparison of approximate capacity (from (29)) versus ĝ, with optimal choice of E.sup.(r) and E.sup.(r) chosen via (32), respectively, for σ.sub.θ.sup.2=1/T.sub.s and β.sub.maxE.sub.s/KN.sub.0=−3,3 dB.

(10) FIGS. 10A and 10B: Throughput of ACE schemes (PACE, CACE, MA-FSR) and of digital CE with either perfect rCSI or nested array sampling versus SNR and L. For PACE, the arrayed PLL from [51 V. V. Ratnam and A. F. Molisch, “Periodic analog channel estimation aided beamforming for massive MIMO systems,” IEEE Transactions on Wireless Communications, 2019 . . . ] is used with identical parameters, and RX beamformer design phase lasts 6 symbols. For nested array sampling, horizontal(4,4) and vertical (2,2) nested arrays are used. A) sparse channels B) dense stochastic channels.

(11) FIG. 11: An illustrative transmission block structure for the PACE scheme.

(12) FIG. 12: Block diagram of the PLL at antenna 1 for reference recovery, and a sample illustration of its output.

(13) FIGS. 13A and 138: Accuracy of the analytical approximation for the filter, sample and hold outputs in (15) versus SNR for the one PLL and weighted arraying architectures. In FIG. 13A, we plot

(14) $1-𝔼.Math.{\int }_{T_1}^{T_2}\frac{e^{-j\theta \left(t\right)}}{D_2{T}_{cs}}\mathrm{dt}.Math.$
for simulations and

(15) $1-e^{-\frac{\mathrm{Var}\left({\theta }_1\right)}{2}}$
for analytic approximation. We assume A.sub.1.sup.(r)=1, A.sub.5.sup.(r)=0.7e.sup.jπ/3, A.sub.15.sup.(r)=0.5e.sup.−jπ/3 and the remaining parameters are from Table 1.

(16) FIGS. 14A and 14B: Comparison of iSE for PACE based beamforming and other schemes versus SNR and number of MPCs. Here E.sup.(r)=20E.sub.cs/K, E.sub.k.sup.(d)=E.sub.cs/K∀k∈ and the PLL parameters are from Table 1. For FIG. 14B we use

(17) $\frac{\beta \left(0,0\right)E_{\mathrm{cs}}}{{\mathrm{KN}}_0}=1.$

(18) FIG. 15: SER for data streams k=50,74,94 of an MA-FSR RX with QAM modulation (K=50, E.sub.r=E.sub.s/2, E.sub.d.sup.(k)=E.sub.s/(2||), K=50, g=5).

(19) FIG. 16: Comparison of the iSR (without channel estimation overhead) of MA-FSR and analog beamforming (a) For MA-FSR, E.sub.r=E.sub.s/2, K=128 and g=5 (b) For analog beamforming, we only use the subcarriers {0, . . . , K−g−1}.

(20) FIGS. 17A and 17B: MA-FSR designs with improved performance. A) with noise suppression. B) with narrow band pass filter.

DETAILED DESCRIPTION

(21) Reference will now be made in detail to presently preferred compositions, embodiments and methods of the present invention, which constitute the best modes of practicing the invention presently known to the inventors. The Figures are not necessarily to scale. However, it is to be understood that the disclosed embodiments are merely exemplary of the invention that may be embodied in various and alternative forms. Therefore, specific details disclosed herein are not to be interpreted as limiting, but merely as a representative basis for any aspect of the invention and/or as a representative basis for teaching one skilled in the art to variously employ the present invention.

(22) It is also to be understood that this invention is not limited to the specific embodiments and methods described below, as specific components and/or conditions may, of course, vary. Furthermore, the terminology used herein is used only for the purpose of describing particular embodiments of the present invention and is not intended to be limiting in any way.

(23) It must also be noted that, as used in the specification and the appended claims, the singular form “a,” “an,” and “the” comprise plural referents unless the context clearly indicates otherwise. For example, reference to a component in the singular is intended to comprise a plurality of components.

(24) The term “comprising” is synonymous with “including,” “having,” “containing,” or “characterized by.” These terms are inclusive and open-ended and do not exclude additional, unrecited elements or method steps.

(25) The phrase “consisting of” excludes any element, step, or ingredient not specified in the claim. When this phrase appears in a clause of the body of a claim, rather than immediately following the preamble, it limits only the element set forth in that clause; other elements are not excluded from the claim as a whole.

(26) The phrase “consisting essentially of” limits the scope of a claim to the specified materials or steps, plus those that do not materially affect the basic and novel characteristic(s) of the claimed subject matter.

(27) With respect to the terms “comprising,” “consisting of,” and “consisting essentially of,” where one of these three terms is used herein, the presently disclosed and claimed subject matter can include the use of either of the other two terms.

(28) Throughout this application, where publications are referenced, the disclosures of these publications in their entireties are hereby incorporated by reference into this application to more fully describe the state of the art to which this invention pertains.

Abbreviations

(29) “ACE” means analog channel estimation. “aCSI” means average channel state information. “ADC” means analog-to-digital converters. “BS” means base-station. “DAC” means digital-to-analog converters. “IA” means initial access. “MA-FSR” means multi-antenna frequency shift reference “MIMO” means multiple-input-multiple-output. “MPC” means multi-path components. “OFDM” means orthogonal frequency division multiplexing. “CACE” means continuous analog channel estimation “PACE” means periodic analog channel estimation. “rCSI” means required channel state information. “RTAT” means reference tone aided transmission. “SNR” means signal-to-noise ratio. “TX” means transmitter. “RX” means receiver. “UE” means user equipment. “aBD” mean analog beamformer design phase

(30) With reference to FIGS. 1 and 2, a MIMO system and in particular a CACE system implementing continuous analog channel estimation aided beamforming is provided. MIMO system 10 or 10′ includes transmitter 12 that transmits a transmitted signal that includes a data signal and a continuously transmitted reference signal. Typically, the reference signal is, but is not restricted to be, a sinusoidal reference tone having a predetermined frequency. In a refinement, transmitter 12 transmits the transmitted signal by orthogonal frequency division multiplexing. Receiver 14 includes a plurality of antennas 18.sup.i where i is an integer from 1 to M.sub.rx the number of antennas. Each antenna 18.sup.i receives the transmitted signal and outputs an associated received signal s.sub.rx,i(t). The baseband conversion processor 20 either contains an independent oscillator or recovers the transmitted reference signal including either or both of the signal amplitude and phase, and then multiplies each associated received signal with the independent oscillator/or the recovered reference signal and with its quadrature component in the analog domain, resulting in processor output signals. The baseband conversion processor 20 output signals are essentially lowpass signals with at least partially compensated inter-antenna phase shifts. The RX may also contain an optional amplitude and phase compensation processor 22 for phase and amplitude adjustment of the signals outputted from baseband conversion processor 20 via the use of analog phase shifters. The amplitude and phase compensation processor 22 may utilize the baseband reference signal in the outputs from the baseband conversion processor 20 as control signals to the phase-shifters, to further compensate the inter-antenna phase-shifts in the outputs from the baseband conversion processor. Analog adder 24 sums the baseband conversion processor 20's output signals or the optional amplitude and phase compensation processor 22's output signals as a summed signal output (i.e., Re(ω.sub.LPF(t)) and Im(ω.sub.LPF(t))) thereby emulating signal combining and beamforming without the receiver applying explicit channel estimation. In a refinement, analog adder 24 applies a weight sum when summing the processor output signals. Advantageously, the sparse nature of wireless channels is exploited to ensure a large beamforming gain after the summed signal output. Moreover, MIMO system 10 or 10′ can suppress the phase noise of transmit and receive oscillators.

(31) Still referring to FIGS. 1 and 2, in a variation, MIMO system 10 or 10′ includes a plurality of low noise amplifiers 26 with each antenna 18.sup.i having an associated low noise amplifier 26.sup.i that outputs the associated received signal. The associated low noise amplifier 26.sup.i of each antenna 18.sup.i receives and amplifies associated received signal to form an amplified associated received signal s.sub.rx,i(t). Baseband conversion processor 20 may either include one or more reference signal recovery circuits 28 (in FIG. 1) or may include an RX local oscillator 28′ (in FIG. 2). In a refinement illustrated in FIG. 1, each antenna 18.sup.i has an associated reference recovery circuit 28.sup.i. The associated reference recovery circuit 28.sup.i of each antenna 18.sup.i isolates and recovers the received reference signal as an associated isolated reference signal from the associated amplified received signal. In a refinement illustrated in FIG. 1, each associated reference recovery circuit 28.sup.i includes a narrow band pass filter and/or an injection locked oscillator and/or a phase locked loop followed by an optional variable gain amplifier. Sufficient frequency separation may be provided between the transmitted reference signal and data signals to enable the reference recovery. Baseband conversion processor 20 also includes a plurality of first mixers 30 with each antenna 18.sup.i having an associated first mixer 30.sup.i. The associated first mixer 30.sup.i of each antenna 18.sup.i multiplies the associated isolated reference signal (in FIG. 1) or the local oscillator output 28′ (in FIG. 2) with the associated received signal to produce an associated in-phase-derived output signal. Baseband conversion processor 20 also includes a plurality of second mixers 32.sup.i with each antenna 18.sup.i having an associated second mixer 32.sup.i. The associated second mixer 32.sup.i of each antenna multiplies a quadrature component of the associated isolated reference signal or local oscillator output with the associated received signal to produce a quadrature-derived output signal. In a refinement, the quadrature component of the associated isolated reference signal can be formed by 90° phase shifters, illustrated as 34.sup.i in FIGS. 1 and 34 in FIG. 2. Characteristically, the in-phase-derived output signal and the quadrature-derived output signal are the processor output signals.

(32) In the variation depicted in FIG. 1, MIMO system 10 includes a low pass filter and/or downconverter 40 that receives the in-phase-derived output signal and the quadrature-derived output signal and outputs a baseband signal (which is a low pass filtered summed signal output). The baseband signal is sampled by Analog to digital converter 36 and then demodulated by OFDM demodulator 38 that demodulates the bandpass signal.

(33) In the variation depicted in FIG. 2, low pass filters 42.sup.i filter outputs from mixers 30.sup.i while low pass filters 44.sup.i filters outputs from mixers 32.sup.i prior to signals being provided to amplitude and phase compensation 22. Filters 42.sup.i and filters 44.sup.i extract the baseband signal and filter out noise from the associated in-phase-derived output signals and the associated quadrature-derived output signals from the mixer outputs 30.sup.i and 32.sup.i. In particular, the amplitude and phase compensation processor includes these lowpass filters to filter the in-phase and quadrature output signals from a baseband processor to generate the in-phase derived and quadrature-derived control signals to be used for the associated phase shifters. Amplitude and phase compensation circuit 22 provides phase and amplitude adjustment of the plurality of in-phase-derived output signals and the plurality of quadrature-derived output signals from baseband conversion processor 20. The amplitude and phase compensation circuit 22 involves a low pass filter 50.sup.i at each antenna i, that receives an in-phase-derived output signal and outputs an in-phase-derived control signal. Similarly, low pass filter 52.sup.i receives a quadrature-derived output signal and outputs a quadrature-derived control signal. These control signals correspond to the baseband reference signals in the baseband outputs from filters 42.sup.i and 44.sup.i. Amplitude and phase compensation 22 also includes a plurality of variable gain phase-shifters 56 with each antenna 18.sup.i having an associated variable gain phase-shifter 56.sup.i. The associated variable gain phase-shifter 56.sup.i of each antenna 18.sup.i receives from filters 50.sup.i and 52.sup.i the associated in-phase-derived control signal and the associated quadrature-derived control signal, through which the data signal is processed during a data transmission phase. Analog adder 24 sums outputs from the plurality of the variable gain phase-shifters as a summed signal output (i.e., baseband signal with Re(ω.sub.LPF(t)) and Im(ω.sub.LPF(t))).

(34) With reference to FIG. 3, a schematic illustration of a system a MIMO system implementing periodic analog channel estimation (PACE) aided beamforming is provided. MIMO system 60 includes transmitter 62 that transmits a transmitted signal that includes a data signal during a data transmission phase and includes a reference signal during the beamformer design phase. Typically, the reference signal is, but is not restricted to be, a sinusoidal tone having a predetermined frequency. Moreover, this system does not require continuous transmission of the reference signal. In a refinement, transmitter 62 transmits the transmitted signal by orthogonal frequency division multiplexing. Receiver 64 includes a plurality of antennas 68.sup.i where i is an integer from 1 to M.sub.rx the number of antennas. Each antenna 68.sup.i receives the transmitted signal and outputs an associated received signal. Phase and amplitude estimator 70 recovers the reference signal during a beamformer design phase and then multiplies each associated received signal with the recovered reference tone and a quadrature component of the reference tone in the analog domain to form a plurality of in-phase-derived baseband control signals and a plurality of quadrature-derived baseband control signals with each antenna having an associated in-phase-derived control signal and an associated quadrature-derived control signal. Receiver 64 also includes a plurality of variable gain phase-shifters 76 with each antenna 68.sup.i having an associated variable gain phase-shifter 76.sup.i. The associated gain phase-shifter 76.sup.i of each antenna 68.sup.i receives the associated in-phase-derived control signal and the associated quadrature-derived control signal through which the data signal is processed during a data transmission phase. An analog adder 80 sums outputs from the plurality of the variable gain phase-shifters as a summed signal output. In a refinement, analog adder 80 applies a weight sum when summing the processor output signals. Advantageously, the sparse nature of wireless channels is exploited to ensure a large beamforming gain after the summed signal output.

(35) Still referring to FIG. 3, in a variation, MIMO system 60 includes a plurality of low noise amplifiers 78 with each antenna 68.sup.i having an associated low noise amplifier 78.sup.i. The associated low noise amplifier 78.sup.i of each antenna 68.sup.i receives and amplifies associated received signal to form an amplified associated received signal.

(36) Still referring to FIG. 3, in a variation, the phase and amplitude estimator 70 includes a reference tone recovery circuit 82 that recovers and isolates the reference signal as an isolated reference signal from the associated received signal of a single antenna (e.g., the first antenna 68.sup.i) during a beamformer design phase. In another variant, the reference recovery circuit 82 may recover the isolated reference signal from received signals from a plurality of antennas (see FIG. 4). In a refinement, reference recovery circuit 82 includes a narrow band pass filter and/or an injection locked oscillator and/or a phase locked loop followed by an optional variable gain amplifier. In the example depicted in FIG. 3, the reference tone recovery circuit 82 is a phase locked loop. The phase and amplitude estimator 70 also includes a plurality of first mixers 84 with each antenna having an associated first mixer 84.sup.i. The associated first mixer 84.sup.i of each antenna multiplies the isolated reference tone with the associated received signal s.sub.rx,i(t) to produce an associated in-phase-derived output signal. The phase and amplitude estimator 70 also includes a plurality of second mixers 86 with each antenna having an associated second mixer 86.sup.i. The associated second mixer 86.sup.i of each antenna 68.sup.i multiplies a quadrature component of the isolated reference signal with the associated received signal s.sub.rx,i(t) to produce an associated quadrature-derived output signal. The phase and amplitude estimator 70 also includes plurality of first ‘filter, sample and hold’ circuits 90 with each antenna having an associated first filter, sample and hold circuit 90.sup.i. The associated first filter, sample and hold circuit 90.sup.i of each antenna 68.sup.i receives the associated in-phase-derived output signal and outputs the associated in-phase-derived control signal for phase shifter 76.sup.i. The phase and amplitude estimator 70 also has a plurality of second ‘filter, sample and hold’ circuits with each antenna having an associated second filter, sample and hold circuit 92.sup.i. The associated second filter, sample and hold circuit 92.sup.i of each antenna 68.sup.i receives the associated quadrature-derived output signal and outputs the associated quadrature-derived control signal. The filter sample and hold circuits, low pass filter the outputs from mixers 84.sup.i and 86.sup.i, and sample the corresponding filtered outputs. In one variant, such filter, sample and hold circuits can be implemented by a low pass filter followed by a low sampling rate ADC. In another variant, the filter, sample and hold circuits can be implemented by an integrate and hold circuit. The data signals received at each antenna 68.sup.i during the data transmission phase are phase shifted by the phase shifters 76.sup.i, whose control signals are determined by the filter, sample and hold circuit 90.sup.i and 92.sup.i outputs from the preceding beamformer design phase. Advantageously, the system of FIG. 3 can perform receive beamforming without digital channel estimation.

(37) In a refinement, the reference signal can be recovered from one antenna or a plurality of antennas. FIG. 4 depicts a weighted carrier array subsystem 110 for reference tone recovery from a subset of the plurality of antennas. Each antenna of the subset is designated by 68.sup.j where j is a label for the antennas in the subset. Subsystem 110 includes a plurality of first mixers 105, with each antenna 68.sup.j connected to subsystem 110 having an associated first mixer 105.sup.j. Subsystem 110 also includes a primary voltage controlled oscillator (VCO) 117, with a nominal oscillation frequency that may be same or different from the reference signal frequency. The first mixer 105.sup.j at antenna 68.sup.j multiplies the received signals from antenna 68.sup.j with the output from the VCO 117 to derive an associated intermediate frequency signal. Subsystem 100 also includes a plurality of first low pass filters 106, with each antenna 68.sup.j connected to subsystem 110 having an associated first low-pass filter 106 that filters the intermediate frequency output signal from mixer 105.sup.j from noise and other high frequency components. Subsystem 110 also includes a plurality of secondary phase locked loops, with each antenna 68 connected to subsystem 110 having an associated secondary phase locked loop 114.sup.j that locks to the intermediate frequency output signal associated with first low pass filter 106.sup.j. In one refinement depicted in FIG. 4, each secondary phase locked loop 114.sup.j includes a secondary variable gain amplifier, a secondary mixer and a secondary VCO with nominal oscillation frequency approximately equal to the intermediate frequency signal. The secondary PLLs convert the intermediate frequency output signals at the associated first low pass filter outputs to base-band with at least partially compensated inter-antenna phase shift. The outputs from the secondary PLLs are further weighted by a plurality of variable gain amplifiers, with each secondary VCO having an associated variable gain amplifier 115.sup.j. The weighted base-band signals from the secondary PLLs are combined using an adder 116 to obtain a base-band combined control signal with enhanced signal to noise ratio. The base-band control signal from the adder 116 is used as a control signal to control the oscillation frequency of the primary VCO 117. Due to the enhanced signal to noise ratio, the primary VCO output may recover the transmitted reference signal with a lower accumulation of channel noise and phase noise. Subsystem 110 may also include an optional second mixer 118 that takes as an input the output of the primary VCO 117, and provides an output signal a frequency shifted version of the primary VCO output, to better match the reference signal frequency.

(38) In a variation, MIMO system 60 includes a low pass filter and/or downconverter 100 that receives as input the summed signal output and outputs a baseband signal. The baseband signal is sampled by Analog to digital converter 102 to form a digital baseband signal. The digital baseband signal is then demodulated by OFDM demodulator 104 that demodulates the bandpass signal.

(39) With reference to FIG. 5, a non-coherent MIMO system using a multiantenna frequency shift reference (MA-FSR) receiver is provided. MIMO system 120 includes transmitter 122 that transmits a transmitted signal that includes a data signal and a reference signal. Typically, the reference signal is, but is not restricted to be, a sinusoidal tone having a predetermined frequency. In a refinement, transmitter 122 transmits the transmitted signal by orthogonal frequency division multiplexing. Receiver 124 includes a plurality of antennas 128.sup.i where i is an integer from 1 to M.sub.rx the number of antennas. Each antenna 128.sup.i receives the transmitted signal and outputs an associated received signal which passes through band pass filter 130.sup.i that leaves the associated received signal un-distorted while suppressing out-of-band noise. Squaring circuit 132.sup.i squares each associated received signal to form associated squared received signals. The squared outputs involve, among other signals, a product between the reference signal and data signals with the inter-antenna phase shift compensated. Analog adder 134 adds the squared received signals to produce a summed signal r.sub.sq(t). Low pass filter 136 filters the summed signal to form a low pass filtered signal. The low pass filtered signal is sampled by ADC 140 and OFDM demodulation by OFDM demodulator 144 to extract the output signal corresponding to the product between the reference signal and data signals. Advantageously, the sparse nature of wireless channels is exploited to ensure a large beamforming gain after the summed signal output. The system of this embodiment can suppress the phase noise of the transmit oscillator. Advantageously, this embodiment, does not require an oscillator at the receiver.

(40) With reference to FIG. 5, the reference signal and data signals are designed such that the product of the data signal with itself does not cause interference to the product between the reference signal and data signal at the outputs of the squaring circuit 132.sup.i. In one variant, this is implemented by allowing the reference signal to occupy an even numbered OFDM sub-carrier, while all data signals occupy odd numbered OFDM sub-carriers or vice versa.

(41) With respect to the systems of FIGS. 1-5, the occurrence and potential suppression of phase noise should be considered. Phase noise is an oscillator impairment, that causes an oscillator output instantaneous frequency to waver randomly. Therefore, in a conventional MIMO receiver, when the received signal is converted to base band by an oscillator, phase noise distorts the resulting base-band signal, causing degradation in performance. In CACE and MA-FSR based receivers, while this phase-noise induced distortion is random, the same distortion is applied to both the data and reference signals, since the data and reference signals are transmitted simultaneously. Thus, if the base-band reference signal is used to control the phase shifters 56 in the CACE system of FIG. 2, the distortion caused by the phase noise is implicitly undone at the phase shifter outputs. In a similar way, the baseband conversion circuit 20 for the CACE receiver in FIG. 1 and the squaring circuits 132.sup.i in the MA-FSR receiver in FIG. 5, multiply the received data signal with the reference signal, prior to combining and demodulation. Since the phase noise distorts both the reference and data signals identically, their multiplication undoes the impact of the phase noise at the baseband conversion circuit output 20 in FIG. 1 and at the squaring circuit 132.sup.i outputs in FIG. 5. Thus CACE and MA-FSR receivers provide resilience to oscillator phase noise.

(42) The following examples illustrate the various embodiments of the present invention. Those skilled in the art will recognize many variations that are within the spirit of the present invention and scope of the claims.

(43) 1. Continuous Analog Channel Estimation Aided Beamforming for Massive MIMO Systems

(44) I. Introduction

(45) In this embodiment, a more generalized ACE approach for RX beamforming, called continuous ACE (CACE) is explored, that does not require carrier recovery at the RX, mitigates oscillator phase noise and works in multi-path channels. The latter is accomplished by exploiting not only phase of the carrier signal at the RX but also its amplitude. In CACE, a reference tone, i.e. a sinusoidal tone at a known frequency, is continuously transmitted along with the data by the TX as illustrated in FIG. 2. At the RX, the received signal at each antenna is converted to baseband by a bank of mixers and a local oscillator that is tuned (approximately) to the reference frequency, as illustrated in FIG. 2. The in-phase (I) and quadrature-phase (Q) components of the resulting baseband signal at each antenna are low-pass filtered, to extract the received signals corresponding to the reference, as illustrated in FIG. 1. These filtered outputs are then used to feed variable gain, baseband analog phase-shifters which generate the RX analog beam. The un-filtered baseband received signals at each antenna are processed by these phase-shifters, added and fed to a single ADC for demodulation. As shall be shown, this process emulates using the received signal for the reference as a matched filter for the received data signals, and it achieves a large RX beamforming gain in sparse, wide-band massive MIMO channels. This is because while the reference and data signals may have different frequencies and thus experience different channel responses, such channel responses exhibit a strong coupling across frequency. Furthermore since oscillator phase-noise affects both the reference and data similarly, the match filtering helps mitigate the phase-noise from the demodulation outputs. Unlike conventional analog beamforming, CACE aided beamforming also improves diversity against MPC blocking by combining the received signal power from multiple channel MPCs. Finally, no dedicated pilot symbols are required to update the RX analog beam, unlike with digital CE. The phase shifts from the receiver circuit can also be utilized for transmit beamforming on the reverse link. Furthermore, by providing an option for digitally controlling these phase-shifter inputs, the proposed architecture can also support conventional RX beamforming approaches when required. On the flip side, CACE may require additional analog hardware in comparison to conventional digital CE, including 2M.sub.rx mixers and low-pass filters. Additionally, the accumulation of power from multiple MPCs, while improving diversity, may cause performance degradation in frequency selective channels, as shall be shown. Finally, the proposed approach in its suggested form does not support reception of multiple spatial data streams and can only be used for beamforming at one end of a communication link. This architecture is therefore more suitable for use at the user equipment (UEs). The possible extensions to multiple spatial stream reception shall be explored in future work. A different ACE technique that does not require continuous transmission of the reference, called PACE, is also described herein. While PACE prevents wastage of transmit resources on the reference, it suffers from an exponential degradation of performance due to phase-noise, unlike CACE. A third type of non-coherent ACE technique, called MA-FSR, that uses square law components at the RX was explored by us in [52]. While being resilient to phase-noise like CACE and having a low hardware cost, MA-FSR is a non-coherent technique that suffers from a poor bandwidth efficiency of 50%. It should be emphasized that CACE, PACE and MA-FSR are three different ACE schemes to help reduce CE overhead in massive MIMO systems, each having their unique advantages and RX architectures, and each requiring separate performance analysis techniques. The detailed analysis presented in this paper for CACE, in combination with the analysis of PACE and MA-FSR (also described herein), shall aid in a detailed comparison of these schemes for a specific application. The contributions of the present embodiment include: 1. A novel transmission technique called CACE and a corresponding RX architecture are proposed that enable RX beamforming without dedicated pilot symbol transmissions. 2. Analytically characterization of the achievable system throughput with CACE aided beamforming in a wide-band channel. 3. In the process, the impact of the system phase-noise and the ability of CACE to suppress it are also analyzed. 4. Simulations under practically relevant channel models are presented to support the analytical results.

(46) Notation: scalars are represented by light-case letters; vectors by bold-case letters; and sets by calligraphic letters. Additionally, j=√{square root over (−1)}, a* is the complex conjugate of a complex scalar a, |a| represents the .sub.2-norm of a vector a, A.sup.T is the transpose of a matrix A and A.sup.† is the conjugate transpose of a complex matrix A. Finally, .sub.a is an a×a identity matrix, .sub.a,b is the a×b all zeros matrix, { } represents the expectation operator, represents equality in distribution, Re{.Math.}/lm{.Math.} refer to the real/imaginary component, respectively, and (a, B) represents a circularly symmetric complex Gaussian vector with mean a and covariance matrix B.

(47) II. General Assumptions and System Model

(48) We consider a single cell system in downlink, where a M.sub.tx antenna base-station (BS) transmits data to multiple UEs simultaneously via spatial multiplexing. Since we mainly focus on the downlink, we shall use the terms BS/TX and UE/RX interchangeably. Each UE is assumed to have a hybrid architecture, with M.sub.rx antennas and one down-conversion chain, and it performs CACE aided RX beamforming. On the other hand, the BS may have an arbitrary architecture and it transmits a single spatial data-stream to each scheduled UE. For convenience, we consider the use of noise-less and perfectly linear antennas, filters, amplifiers and mixers at both the BS and the representative UE. We assume the downlink BS-UE communication to be divided into three stages: (i) Initial Access (IA)(ii) TX beamformer design—where the TX acquires rCSI for all the UEs and uses it to perform UE scheduling, TX beamforming and power allocation and (iii) Data transmission—wherein the BS transmits data signals and the scheduled UEs use CACE to adapt the RX beams and receive the data. Through a major portion of this paper, we assume that the IA and TX beamformer design have been performed apriori and shall focus on the data transmission stage. However in Section 5, we shall also discuss how CACE beamforming can help in stages (i) and (ii).

(49) In stage (iii), we assume the BS to transmit spatially orthogonal signals to the scheduled UEs to mitigate inter-user interference. This can be achieved, for example, by careful UE scheduling and/or via avoiding transmission to common channel scatterers (A. Adhikary, E. A. Safadi, M. K. Samimi, R. Wang, G. Caire, T. S. Rappaport, and A. F. Molisch, “Joint spatial division and multiplexing for mm-wave channels,” IEEE Journal on Selected Areas in Communications, vol. 32, pp. 1239-1255, June 2014). For this system model and for a given TX beamformer and power allocation, we shall restrict the analysis to a single representative UE without loss of generality. The BS is assumed to transmit orthogonal frequency division multiplexing (OFDM) symbols to the representative UE, with K=K.sub.1+K.sub.2+1 sub-carriers indexed as ={−K.sub.1, . . . , K.sub.2}. The 0-th sub-carrier is used as a reference tone, i.e., a pure sinusoidal signal with a pre-determined frequency, while data is transmitted on the K.sub.1−g lower and K.sub.2−g higher sub-carriers, represented by the index set {K.sub.1, . . . , −g−1, g+1, . . . , K.sub.2}. The 2g sub-carriers indexed as {−g, . . . , −1, 1, . . . , g} are blanked to act as a guard band between the reference and data sub-carriers as illustrated in FIG. 6A, with g being a design parameter. Since the BS can afford an accurate oscillator, by ignoring its phase-noise, the complex equivalent transmit signal for the 0-th OFDM symbol of stage (iii) can then be expressed as:

(50) $\begin{array}{cc}{\tilde{s}}_{\mathrm{tx}}\left(t\right)=\sqrt{\frac{2}{T_s}}t\left[\sqrt{E^{\left(r\right)}}+{.Math.}_{k\in 𝒦\backslash 𝒢}{x}_k{e}^{j2\pi f_{k}t}\right]e^{j2\pi f_{c}t},& \left(1.1\right)\end{array}$
for −T.sub.cp≤t≤T.sub.s, where t is the M.sub.tx×1 unit-norm TX beamforming vector for this UE (designed apriori in stage (ii)), E.sup.(r) is the energy-per-symbol allocated to the reference tone, ={−g, . . . , 0, . . . g} defines the non-data sub-carriers, x.sub.k is the data signal on the k-th sub-carrier, f.sub.c is the reference frequency, f.sub.k=k/T.sub.s represents the frequency offset of the k-th sub-carrier and T.sub.s, T.sub.cp are the symbol duration and the cyclic prefix duration, respectively. Here we define the complex equivalent signal such that the actual (real) transmit signal is given by s.sub.tx(t)=Re{{tilde over (s)}.sub.tx(t)}. For the data sub-carriers (k∈\), we assume the use of independent data streams with equal power allocation, and circularly symmetric Gaussian signaling, i.e., x.sub.k˜(0, E.sup.(d)). The transmit power constraint is then given by E.sup.(r)+(K−||)E.sup.(d)≤E.sub.s, where E.sub.s is the total OFDM symbol energy (excluding the cyclic prefix). The channel to the representative UE is assumed to have L MPCs with the M.sub.rx×M.sub.tx channel impulse response matrix and its Fourier transform, respectively, given as (M. Akdeniz, Y. Liu, M. Samimi, S. Sun, S. Rangan, T. Rappaport, and E. Erkip, “Millimeter wave channel modeling and cellular capacity evaluation,” IEEE Journal on Selected Areas in Communications, vol. 32, pp. 1164-1179, June 2014.19):
H(t)=a.sub.rx()a.sub.tx().sup.†δ(t−),  (1.2a)
(f)=a.sub.rx()a.sub.tx().sup.†,  (1.2b)
where α.sub. is the complex amplitude and τ.sub. is the delay and a.sub.tx(), a.sub.rx() are the M.sub.tx×1 TX and M.sub.rx×1 RX array response vectors, respectively, of the -th MPC. As an illustration, the -th RX array response vector for a uniform planar array with M.sub.H horizontal and M.sub.V vertical elements (M.sub.rx=M.sub.HM.sub.V) is given by a.sub.rx()=ā.sub.rx(ψ.sub.azi.sup.rx(), ψ.sub.ele.sup.rx()), where

(51) $\begin{array}{cc}{\left[{\stackrel{\_}{a}}_{\mathrm{rx}}\left({\psi }_{\mathrm{azi}}^{\mathrm{rx}},{\psi }_{\mathrm{ele}}^{\mathrm{rx}}\right)\right]}_{M_{V}h+v}=\exp \left\{j2\pi \frac{\begin{array}{c}{\Delta }_{H}h\sin \left({\psi }_{\mathrm{azi}}^{\mathrm{rx}}\right)\sin \left({\psi }_{\mathrm{ele}}^{\mathrm{rx}}\right)+\\ {\Delta }_V\left(v-1\right)\cos \left({\psi }_{\mathrm{ele}}^{\mathrm{rx}}\right)\end{array}}{\lambda }\right\},& \left(1.3\right)\end{array}$
for h∈{0, . . . , M.sub.H−1} and ν∈(1, . . . , M.sub.V), ψ.sub.azi.sup.rx(), ψ.sub.ele.sup.rx() are the azimuth and elevation angles of arrival for the -th MPC, Δ.sub.H, Δ.sub.V are the horizontal and vertical antenna spacings and λ is the carrier wavelength. Expressions for a.sub.tx() can be obtained similarly. Note that in (1.2) we implicitly ignore the frequency variation of individual MPC amplitudes (α.sub.0, . . . , α.sub.L−1) and the beam squinting effects (S. K. Garakoui, E. A. M. Klumperink, B. Nauta, and F. E. van Vliet, “Phased-array antenna beam squinting related to frequency dependency of delay circuits,” in European Microwave Conference, pp. 1304-1307, October 2011.53), which are reasonable assumptions for moderate system bandwidths. It is emphasized that the complete channel response (including all MPCs) however still experiences frequency selective fading. To prevent inter-symbol interference, we let the cyclic prefix be longer than the maximum channel delay: T.sub.cp>τ.sub.L−1. We also consider a generic temporal variation model, where the time for which MPC parameters {,a.sub.tx(),a.sub.rx(),} stay approximately constant is much larger than the symbol duration T.sub.s. Finally, we do not assume any distribution prior or side information on {,a.sub.tx(),a.sub.rx(),}.

(52) Each RX antenna front-end is assumed to have a low noise amplifier (LNA) followed by a band-pass filter (BPF) that leaves the desired signal un-distorted but suppresses the out-of-band noise. The filtered signal is then converted to baseband using the in-phase and quadrature-phase components of an RX local oscillator, as depicted in FIG. 2. This oscillator is assumed to be independently generated at the RX (i.e., without locking to the received reference). While we model the RX oscillator to suffer from phase-noise, for ease of theoretical analysis we assume the mean RX oscillator frequency to be equal to the reference frequency f.sub.c. This assumption shall be relaxed later in the simulation results in Section 6. Then, from (1.1)-(1.2), the received baseband signal for the 0-th OFDM symbol can be expressed as:

(53) $\begin{array}{cc}{\tilde{s}}_{\mathrm{rx},\mathrm{BB}}\left(t\right)=\mathrm{LPF}\left\{{.Math.}_{\ell =0}^{L-1}{\alpha }_{\ell }{a}_{\mathrm{rx}}\left(\ell \right){a_{\mathrm{tx}}\left(\ell \right)}^{†}{s}_{\mathrm{tx}}\left(t-{\tau }_{\ell }\right)\sqrt{2}{e}^{-j\left[2\pi f_{c}t+\theta \left(t\right)\right\}}\right\}+\tilde{w}\left(t\right)=\frac{1}{\sqrt{T_s}}\left[\mathscr{H}\left(0\right)t\sqrt{E^{\left(r\right)}}+{.Math.}_{k\in 𝒦\backslash 𝒢}\mathscr{H}\left(f_k\right){\mathrm{tx}}_k{e}^{j2\pi f_{k}t}\right]e^{-j\theta \left(t\right)}+\tilde{w}\left(t\right)& \left(1.4\right)\end{array}$
for 0≤t≤T.sub.s, where the Re/Im parts of {tilde over (s)}.sub.rx,BB(t) are the outputs corresponding to the in-phase and quadrature-phase components of the RX oscillator, θ(t) is the phase-noise process of the RX oscillator and {tilde over (w)}(t) is the M.sub.rx×1 complex equivalent, baseband, stationary, additive, vector Gaussian noise process, with individual entries being circularly symmetric, independent and identically distributed (i.i.d.), and having a power spectral density: .sub.w(f)=N.sub.0 for −f.sub.K.sub.1≤f≤f.sub.K.sub.2. Note that while (1.4) is obtained by assuming no TX phase-noise, the results can be generalized under some mild constraints by treating the TX phase-noise as a part of θ(t) (P. Mathecken, T. Riihonen, S. Werner, and R. Wichman, “Performance analysis of OFDM with wiener phase noise and frequency selective fading channel,” IEEE Transactions on Communications, vol. 59, pp. 1321-1331, May 2011. We model the RX phase-noise θ(t) as a Wiener process, which is representative of a free running oscillator [L. Piazzo and P. Mandarini, “Analysis of phase noise effects in OFDM modems,” IEEE Transactions on Communications, vol. 50, pp. 1696-1705, oct 2002; S. Wu, P. Liu, and Y. Bar-Ness, “Phase noise estimation and mitigation for OFDM systems.” IEEE Transactions on Wireless Communications, vol. 5, pp. 3616-3625, December 2006; D. Petrovic, W. Rave, and G. Fettweis, “Effects of phase noise on OFDM systems with and without PLL: Characterization and compensation,” IEEE Transactions on Communications, vol. 55, pp. 1607-1616, August 2007). In Appendix 1C, we also discuss how the results can be extended to phase-noise modeled as an Ornstein Uhlenbeck (OU) process, which is representative of an oscillator either locked to the received reference, or synthesized from a stable low frequency source (D. Petrovic, W. Rave, and G. Fettweis, “Effects of phase noise on OFDM systems with and without PLL: Characterization and compensation,” IEEE Transactions on Communications, vol. 55, pp. 1607-1616, August 2007; A. Mehrotra, “Noise analysis of phase-locked loops,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 49, pp. 1309-1316, September 2002). For the Wiener model, θ(t) is a non-stationary Gaussian process which satisfies:

(54) $\begin{array}{cc}\frac{d\theta \left(t\right)}{\mathrm{dt}}=w_{\theta }\left(t\right),& \left(1.5\right)\end{array}$
where w.sub.θ(t) is a real white Gaussian process with variance σ.sub.θ.sup.2. We assume the RX to have apriori knowledge of σ.sub.θ. As illustrated in FIG. 2, the baseband signal {tilde over (s)}.sub.rx,m(t) at each RX antenna m is passed through a low-pass filter to isolate the received reference signal. For convenience, this filter LPF.sub.ĝ is assumed to be an ideal rectangular filter having a cut-off frequency of f.sub.ĝ, where ĝ≤g/2. Neglecting the contribution of the data sub-carriers to the filtered outputs (which is accurate for low phase-noise i.e., σ.sub.θ.sup.2«g/T.sub.s), these outputs can be expressed for 0≤t≤T.sub.s as:

(55) $\begin{array}{cc}{\hat{s}}_{\mathrm{rx},\mathrm{BB}}\left(t\right)=\frac{1}{\sqrt{T_s}}\mathscr{H}\left(0\right)t\sqrt{E^{\left(r\right)}}A\left(t\right)+\hat{w}\left(t\right),& \left(1.6\right)\end{array}$
where we define A(t)LPF.sub.ĝ{e.sup.−jθ(t)} and ŵ(t) is the M.sub.rx×1 filtered Gaussian noise with power spectral density: .sub.w(f)=N.sub.0 for −f.sub.ĝ≤f≤f.sub.ĝ. An illustration of this filtering operation is provided in FIG. 6B. These filtered signals ŝ.sub.rx,BB(t) are used as the control signals to a variable gain phase-shifter array, through which the baseband received signal vector ŝ.sub.rx,BB(t) is processed, as shown in FIG. 2. We assume the filter cut-off frequency ĝ/T.sub.s to be small enough to allow the phase-shifters to respond to the slowly time varying control signals ŝ.sub.rx,BB(t). A more detailed discussion about ĝ is considered in Sections 4 and 6. The phase-shifted outputs are then summed up and fed to an ADC that samples at K/T.sub.s samples/sec. This sampled output for the 0-th OFDM symbol can be expressed as:

(56) $\begin{array}{cc}\begin{array}{c}y\left[n\right]={{\hat{s}}_{\mathrm{rx},\mathrm{BB}}\left({\mathrm{nT}}_s/K\right)}^{†}{\tilde{s}}_{\mathrm{rx},\mathrm{BB}}\left({\mathrm{nT}}_s/K\right)\\ =\frac{1}{T_s}{t}^{†}{\mathscr{H}\left(0\right)}^{†}\sqrt{E^{\left(r\right)}}\left[\mathscr{H}\left(0\right)t\sqrt{E^{\left(r\right)}}+{.Math.}_{k\in 𝒦\backslash 𝒢}\mathscr{H}\left(f_k\right){\mathrm{tx}}_k{e}^{j2\pi k\frac{n}{K}}\right]\\ A^{*}\left[n\right]e^{-j\theta \left[n\right]}+\sqrt{\frac{1}{T_s}}{w\left[n\right]}^{†}\\ \left[\mathscr{H}\left(0\right)t\sqrt{E^{\left(r\right)}}+{.Math.}_{k\in 𝒦\backslash 𝒢}\mathscr{H}\left(f_k\right){\mathrm{tx}}_k{e}^{j2\pi k\frac{n}{K}}\right]e^{-j\theta \left[n\right]}+\\ \sqrt{\frac{1}{T_s}}{t}^{†}{\mathscr{H}\left(0\right)}^{†}\sqrt{E^{\left(r\right)}}\tilde{w}\left[n\right]A^{*}\left[n\right]+{\hat{w}\left[n\right]}^{†}\tilde{w}\left[n\right],\end{array}& \left(1.7\right)\end{array}$
where we define

(57) 0$A\left[n\right]\stackrel{\Delta }{=}A\left(\frac{{\mathrm{nT}}_s}{K}\right),\theta \left[n\right]\stackrel{\Delta }{=}\theta \left(\frac{{\mathrm{nT}}_s}{K}\right),\tilde{w}\left[n\right]\stackrel{\Delta }{=}\tilde{w}\left(\frac{{\mathrm{nT}}_s}{K}\right)\mathrm{and}\hat{w}\left[n\right]\stackrel{\Delta }{=}\hat{w}\left(\frac{{\mathrm{nT}}_s}{K}\right).$
Conventional OFDM demodulation follows on y[n] to obtain the different OFDM sub-carrier outputs, as analyzed in Section 3.

(58) III. Analysis of the Demodulation Outputs

(59) Without loss of generality, we shall restrict the analysis to the representative 0-th OFDM symbol and thus, we only focus on {y[n]|0≤n<K}. Note that the sampled, band-limited additive noise {tilde over (w)}[n] and the sampled RX phase-noise e.sup.−jθ[n] for 0≤n<K can be expressed using their normalized Discrete Fourier Transform (nDFT) expansions as:
{tilde over (w)}[n]=W[k]e.sup.j2πkn/K,  (1.8a)
e.sup.−jθ[n]=Ω[k]e.sup.j2πkn/K,  (1.8b)
where

(60) $W\left[k\right]=\frac{1}{K}{.Math.}_{n=0}^{K-1}\tilde{w}\left[n\right]e^{-j2\pi \mathrm{kn}/K}\mathrm{and}\Omega \left[k\right]=\frac{1}{K}{.Math.}_{n=0}^{K-1}{e}^{-j\theta \left[n\right]}{e}^{-j2\pi \mathrm{kn}/K}$
are the corresponding nDFT coefficients. Here nDFT is an unorthodox definition for Discrete Fourier Transform, where the normalization by K is performed while finding W[k],Ω[k] instead of in (1.8). These nDFT coefficients are periodic with period K and satisfy the following lemma:

(61) Lemma 3.1 The nDFT coefficients of e.sup.−jθ[n] for 0≤n<K satisfy:

(62) $\begin{array}{cc}{.Math.}_{k\in 𝒦}\Omega \left[k\right]{\Omega \left[k+k_1\right]}^{*}={\delta }_{0,k_1}^K,& \left(1.9a\right)\end{array}$$\begin{array}{cc}{\Delta }_{k_1,k_2}\stackrel{\Delta }{=}𝔼\left\{\Omega \left[k_1\right]{\Omega \left[k_2\right]}^{*}\right\}\approx \frac{{\delta }_{k_1,k_2}^K}{K}\left[\frac{1-e^{-\left(\frac{{\sigma }_{\theta }^2{T}_s-j4\pi k_1}{4}\right)}}{e^{\frac{{\sigma }_{\theta }^2{T}_s-j4\pi k_1}{2K}}-1}+\frac{1-e^{-\left(\frac{{\sigma }_{\theta }^2{T}_s+j4\pi k_1}{4}\right)}}{1-e^{\frac{{\sigma }_{\theta }^2{T}_s+j4\pi k_1}{2K}}}\right],& \left(1.9b\right)\end{array}$
for arbitrary integers k.sub.1, k.sub.2, where δ.sub.a,b.sup.K=1 if a=b (mod K) or δ.sub.a,b.sup.K=0 otherwise.

(63) Proof. See Appendix 1A.

(64) To test the accuracy of the approximation in Lemma 3.1, the Monte-Carlo simulations of Δ.sub.k,k, Δ.sub.k,k+1 and Δ.sub.k,k+100 for a typical phase-noise process (−93 dBc/Hz at 10 MHz offset) are compared to (1.9b) in FIG. 7. As is evident from the results, (1.9b) is accurate for k.sub.1=k.sub.2. Similarly, simulated Δ.sub.k,k+1, Δ.sub.k,k+100 values are ≈20 dB lower than Δ.sub.k,k for all k, i.e., ≈0 as in (1.9b). The analogous version of Lemma 3.1 for phase-noise modeled as an OU process is presented in Appendix 1C. In a similar way, for nDFT coefficients of the channel noise we have:

(65) Lemma 3.2 The nDFT coefficients of {tilde over (w)}[n], i.e., {W[k]|∀k}, are jointly Gaussian with:

(66) $\begin{array}{cc}𝔼\left\{W\left[k_1\right]{W\left[k_2\right]}^{†}\right\}={\delta }_{k_1,k_2}^K\frac{N_0}{T_s}{M_{\mathrm{rx}}}_{},\mathrm{and}𝔼\left\{W\left[k_1\right]{W\left[k_2\right]}^T\right\}={𝕆}_{M_{\mathrm{rx}},M_{\mathrm{rx}}},& \left(1.1\right)\end{array}$
for arbitrary integers k.sub.1, k.sub.2, where δ.sub.a,b.sup.K=1 if a=b (mod K) or δ.sub.a,b.sup.K=0 otherwise.

(67) Proof. See Appendix 1B.

(68) Note that using these nDFT coefficients, the low-pass filtered versions of {tilde over (w)}[n] and e.sup.−jθ[n] in (1.7) can be approximated as:
ŵ[n]≈W[k]e.sup.j2πkn/K,  (1.11a)
A[n]≈Ω[k]e.sup.j2πkn/K,  (1.11b)
where ={−ĝ, . . . , ĝ} and the approximations are obtained by replacing the linear convolution of {tilde over (s)}.sub.rx,BB(t) and the filter response LPF.sub.ĝ{ } with a circular convolution. This is accurate when the filter response has a narrow support, i.e., for ĝ»1. The remaining results in this paper are based on the approximations in (1.9)-(1.11) and on an additional approximation discussed later in Remark 3.1. While we still use the ≤, =, ≥ operators in the following results for convenience of notation, it is emphasized that these equations are true in the strict sense only if the approximations in (1.9)-(1.11) and Remark 3.1 are met with equality. However simulation results are also used in Section VI to test the validity of these approximations. Substituting (8) and (11) into (7), the k-th OFDM demodulation output can be expressed as:

(69) $\begin{array}{cc}\begin{array}{c}{Y}_k=\frac{T_s}{K}{.Math.}_{n=0}^{K-1}y\left[n\right]e^{-j2\pi \frac{\mathrm{kn}}{K}}\\ ={.Math.}_{\stackrel{.}{k}\in \widetilde{𝒢}}{t}^{†}{\mathscr{H}\left(0\right)}^{†}({.Math.}_{\stackrel{\_}{k}\in 𝒦\backslash 𝒢}\mathscr{H}\left(f_{\stackrel{\_}{k}}\right)t\sqrt{E^{\left(r\right)}}{x}_{\stackrel{\_}{k}}{\Omega }^{*}\left[\stackrel{.}{k}\right]\Omega \left[\stackrel{.}{k}+k-\stackrel{\_}{k}\right]+\\ \mathscr{H}\left(0\right){\mathrm{tE}}^{\left(r\right)}{\Omega }^{*}\left[\stackrel{.}{k}\right]\Omega \left[\stackrel{.}{k}+k\right])+{.Math.}_{\stackrel{.}{k}\in \widetilde{𝒢}}\sqrt{T_s}{W\left[\stackrel{.}{k}\right]}^{†}\\ (\mathscr{H}\left(0\right)t\sqrt{E^{\left(r\right)}}\Omega \left[k+\stackrel{.}{k}\right]+{.Math.}_{\stackrel{\_}{k}\in 𝒦\backslash 𝒢}\mathscr{H}\left(f_{\stackrel{\_}{k}}\right){\mathrm{tx}}_{\stackrel{\_}{k}}\Omega \\ \left[k+\stackrel{.}{k}-\stackrel{\_}{k}\right])+{.Math.}_{\stackrel{.}{k}\in \widetilde{𝒢}}\sqrt{T_s}{t}^{†}\mathscr{H}\left(0\right)W\left[k+\stackrel{.}{k}\right]\\ \sqrt{E^{\left(r\right)}}{\Omega }^{*}\left[\stackrel{.}{k}\right]+T_s{.Math.}_{\stackrel{.}{k}\in \widehat{𝒢}}{W}^{†}\left[\stackrel{.}{k}\right]W\left[k+\stackrel{.}{k}\right].\end{array}& \left(1.12\right)\end{array}$

(70) We shall split Y.sub.k as Y.sub.k=S.sub.k+I.sub.k+Z.sub.k where S.sub.k, referred to as the signal component, involves the terms in (1.12) containing x.sub.k and not containing the channel noise, I.sub.k, referred to as the interference component, involves the terms containing E.sup.(r), {x.sub.k|k∈\{k}} and not containing the channel noise, and Z.sub.k, referred to as the noise component, containing the remaining terms. These signal, interference and noise components are analyzed in the following subsections.

(71) A. Signal Component Analysis

(72) From (1.12), the signal component for k∈\ can be expressed as:
S.sub.k=M.sub.rxβ.sub.0,k√{square root over (E.sup.(r))}x.sub.k[|Ω[{dot over (k)}]|.sup.2],  (1.13)
where we define β.sub.k.sub.1.sub.,k.sub.2t.sup.†(k.sub.1).sup.†(k.sub.2)t/M.sub.rx. Since ⊂, note that |Ω[{dot over (k)}]|.sup.2<1 from Lemma 3.1, i.e., the phase-noise causes some loss in power of the signal component. However this loss is much smaller than in PACE (V. V. Ratnam and A. F. Molisch, “Periodic analog channel estimation aided beamforming for massive MIMO systems,” IEEE Transactions on Wireless Communications, 2019, accepted) or digital CE based beamforming, where only |Ω[0]| contributes to the signal component unless additional phase-noise compensation is used. As is evident from (1.13), CACE based beamforming utilizes the (filtered) received signal vector corresponding to the reference tone as weights to combine the received signal vector corresponding to the data sub-carriers, i.e., it emulates imperfect maximal ratio combining (MRC). The imperfection is because the reference tone and the k-th data stream pass through slightly different channels owing to the difference in their modulating frequencies. However the resulting loss in beamforming gain is expected to be small for sparse channels, i.e., for L«M.sub.tx, M.sub.rx, as shall be shown in Sections IV and VI. The second moment of the signal component, averaged over the phase-noise and data symbols, can be computed as:
{|S.sub.k|.sup.2}=M.sub.rx.sup.2|β.sub.0,k|.sup.2E.sup.(r)E.sup.(d){[|Ω[{dot over (k)}]|.sup.2].sup.2}≥M.sub.rx.sup.2|β.sub.0,k|.sup.2E.sup.(r)E.sup.(d)μ(0,ĝ).sup.2  (1.14)
where we define μ(a,ĝ)Δ.sub.a+{dot over (k)},a+{dot over (k)}, and (114) follows from Jensen's inequality and (1.9b).

(73) B. Interference Component Analysis

(74) From (1.12), the interference component for k∈\ can be expressed as:
I.sub.k=M.sub.rxβ.sub.0,k√{square root over (E.sup.(r))}x.sub.kΩ[{dot over (k)}]*Ω[{dot over (k)}+k−k]+Σ.sub.{dot over (k)}∈ĜM.sub.rxβ.sub.0,0E.sup.(r)Ω[{dot over (k)}]*Ω[{dot over (k)}+k]  (1.15)

(75) As is clear from above, the demodulation output Y.sub.k for k∈\ suffers inter-carrier interference (ICI) from other sub-carrier data streams due to the RX phase-noise. The first and second moments of I.sub.k, averaged over the other sub-carrier data {k∈\[∪{k}]} and the phase-noise, can be expressed as:

(76) $\begin{array}{cc}𝔼\left\{I_k\right\}\stackrel{\left(1\right)}{=}\underset{\stackrel{.}{k}\in \hat{G}}{.Math.}{M}_{\mathrm{rx}}{\beta }_{0,0}{E}^{\left(r\right)}{\Delta }_{\stackrel{.}{k}+k,\stackrel{.}{k}}=0& \left(1.16a\right)\\ 𝔼\left\{{.Math.I_k.Math.}^2\right\}\stackrel{\left(2\right)}{=}\underset{\stackrel{\_}{k}\in 𝒦\mathrm{\backslash [}𝒢.Math.\left\{k\right\}]}{.Math.}{M}_{\mathrm{rx}}^2{.Math.{\beta }_{0,\stackrel{\_}{k}}.Math.}^2{E}^{\left(r\right)}{E}^{\left(d\right)}𝔼\left\{{.Math.\underset{\stackrel{.}{k}\in \widehat{𝒢}}{.Math.}{\Omega \left[\stackrel{.}{k}\right]}^{*}\Omega \left[\stackrel{.}{k}+k-\stackrel{\_}{k}\right].Math.}^2\right\}+M_{\mathrm{rx}}^2{{.Math.{\beta }_{0,0}.Math.}^2\left[E^{\left(r\right)}\right]}^2𝔼\left\{{.Math.\underset{\stackrel{.}{k}\in \widehat{𝒢}}{.Math.}{\Omega \left[\stackrel{.}{k}\right]}^{*}\Omega \left[\stackrel{.}{k}+k\right].Math.}^2\right\}\stackrel{\left(3\right)}{\le }\underset{\stackrel{\_}{k}\in 𝒦\backslash \left\{k\right\}}{.Math.}{M}_{\mathrm{rx}}^2{.Math.{\beta }_{\max }.Math.}^2{E}^{\left(r\right)}{E}^{\left(d\right)}𝔼\left\{{.Math.\underset{\stackrel{.}{k}\in \widehat{𝒢}}{.Math.}{\Omega \left[\stackrel{.}{k}\right]}^{*}\Omega \left[\stackrel{.}{k}+k-\stackrel{\_}{k}\right].Math.}^2\right\}+M_{\mathrm{rx}}^2{{.Math.{\beta }_{\max }.Math.}^2\left[E^{\left(r\right)}\right]}^2𝔼\left\{\left[\underset{\stackrel{.}{k}\in \widehat{𝒢}}{.Math.}{.Math.\Omega \left[\stackrel{.}{k}\right].Math.}^2\right]\left[\underset{\stackrel{.}{k}\in \widehat{𝒢}}{.Math.}{.Math.\Omega \left[\stackrel{.}{k}+k\right].Math.}^2\right]\right\}\stackrel{\left(4\right)}{\le }{M}_{\mathrm{rx}}^2{.Math.{\beta }_{\max }.Math.}^2{E}^{\left(r\right)}{E}^{\left(d\right)}𝔼\left\{\underset{\stackrel{.}{k},\stackrel{\mathrm{.Math.}}{k}\in \widehat{𝒢}}{.Math.}{\Omega \left[\stackrel{.}{k}\right]}^{*}\left[\underset{\stackrel{\_}{k}\in 𝒦\backslash \left\{k\right\}}{.Math.}\Omega \left[\stackrel{.}{k}+k-\stackrel{\_}{k}\right]{\Omega \left[\stackrel{\mathrm{.Math.}}{k}+k-\stackrel{\_}{k}\right]}^{*}\right]\Omega \left[\stackrel{\mathrm{.Math.}}{k}\right]\right\}+M_{\mathrm{rx}}^2\underset{\stackrel{.}{k}\in \widehat{𝒢}}{.Math.}{{.Math.{\beta }_{\max }.Math.}^2\left[E^{\left(r\right)}\right]}^2𝔼\left\{{.Math.\Omega \left[\stackrel{.}{k}+k\right].Math.}^2\right\}\stackrel{\left(5\right)}{=}{M}_{\mathrm{rx}}^2{.Math.{\beta }_{\max }.Math.}^2{E}^{\left(r\right)}\left[E^{\left(d\right)}𝔼\left\{\underset{\stackrel{.}{k},\stackrel{\mathrm{.Math.}}{k}\in \widehat{𝒢}}{.Math.}{\Omega \left[\stackrel{.}{k}\right]}^{*}\Omega \left[\stackrel{\mathrm{.Math.}}{k}\right]-{\left[\underset{\stackrel{.}{k}\in \widehat{𝒢}}{.Math.}{.Math.\Omega \left[\stackrel{.}{k}\right].Math.}^2\right]}^2\right\}+\underset{\stackrel{.}{k}\in \widehat{𝒢}}{.Math.}{E}^{\left(r\right)}𝔼{.Math.\Omega \left[\stackrel{.}{k}+k\right].Math.}^2\right]\stackrel{\left(6\right)}{\le }{M}_{\mathrm{rx}}^2{.Math.{\beta }_{\max }.Math.}^2{E}^{\left(r\right)}\left[E^{\left(d\right)}\mu \left(0,\hat{g}\right)\left[1-\mu \left(0,\hat{g}\right)\right]+E^{\left(r\right)}\mu \left(k,\hat{g}\right)\right]& \left(1.16b\right)\end{array}$
where , are obtained using the fact that {x.sub.k|k∈} have a zero-mean and are independently distributed; is obtained by defining β.sub.maxmax|β.sub.k,k|, observing |β.sub.0,k|≤β.sub.max, and using \[∪{k}].Math.\{k} in first term and using the Cauchy-Schwarz inequality for the second term; follows by changing the summation order in the first term and by using (11) for the second term; follows by using Ω[k]=Ω[k+K] and (11) for the first term and follows by using (12) and the Jensen's inequality. As shall be shown in Section VI, (21) may be a loose bound on ICI for lower subcarriers, i.e., |k|«K.

(77) Remark 3.1 A tighter approximation for {|I.sub.k|.sup.2} can be obtained by replacing μ(k,ĝ) in (21) with {tilde over (μ)}(k,ĝ)Δ.sub.{dot over (k)},{dot over (k)}Δ.sub.{dot over (k)}+k,{dot over (k)}+k.

(78) This heuristic is obtained by assuming Ω[{dot over (k)}] and Ω[{umlaut over (k)}+k] to be independently distributed for {dot over (k)}, {umlaut over (k)}∈ and k∈\ in step of (21), but we skip the proof for brevity. As shall be verified in Section VI, Remark 3.1 offers a much better ICI approximation ∀k and hence we shall use {tilde over (μ)}(k,ĝ) instead of μ(k,ĝ) in the forthcoming derivations in Section VI.

(79) C. Noise Component Analysis

(80) From (16), the noise component of Y.sub.k for k∈\ can be expressed as:
Z.sub.k=Z.sub.k.sup.(1)+Z.sub.k.sup.(2)+Z.sub.k.sup.(3), where:  (1.17)
Z.sub.k.sup.(1)=√{square root over (T.sub.s)}W[{dot over (k)}].sup.†((0)t√{square root over (E.sup.(r))}Ω[k+{dot over (k)}]+(f.sub.k)tx.sub.kΩ[k+{dot over (k)}−k])
Z.sub.k.sup.(2)=√{square root over (T.sub.s)}t.sup.†(0).sup.†W[k+{dot over (k)}]√{square root over (E.sup.(r))}Ω*[{dot over (k)}]
Z.sub.k.sup.(3)=T.sub.sW.sup.†[{dot over (k)}]w[k+{dot over (k)}]

(81) Note that the noise consists of both signal-noise and noise-noise cross product terms. From Lemma 3.2, it can readily be verified that {Z.sub.k}=0 and {Z.sub.k.sup.(i)[Z.sub.k.sup.(j)]*}=0 for i≠j, where the expectation is taken over the noise realizations. Thus the second moment of Z.sub.k, averaged over the TX data, phase-noise and channel noise, can be expressed as {|Z.sub.k|.sup.2}={|Z.sub.k.sup.(1)|.sup.2}+{|Z.sub.k.sup.(2)|.sup.2}+{|Z.sub.k.sup.(3)|.sup.2}, where:
{|Z.sub.k.sup.(1)|.sup.2}N.sub.0|(0)t√{square root over (E.sup.(r))}Ω[{dot over (k)}+k]+(k)tx.sub.kΩ[k+{dot over (k)}−k]|.sup.2M.sub.rxN.sub.0[β.sub.0,0E.sup.(r)Δ.sub.{dot over (k)}+k,{dot over (k)}+k+β.sub.k,kE.sub.k.sup.(d)Δ.sub.k+{dot over (k)}−k,k+{dot over (k)}−k]  (1.18a)
{|Z.sub.k.sup.(2)|.sup.2}=.sup.(3)N.sub.0|(0)t|.sup.2E.sup.(r){|Ω[{dot over (k)}]|.sup.2}=.sup.(4)M.sub.rxβ.sub.0,0N.sub.0E.sup.(r)Δ.sub.k,k  (1.18b)
{|Z.sub.k.sup.(3)|.sup.2}=T.sub.s.sup.2{W[{dot over (k)}].sup.†W[k+{dot over (k)}]W[k+{umlaut over (k)}].sup.†W[{umlaut over (k)}]}=.sup.(5)M.sub.rx||N.sub.0.sup.2  (1.18c)
where ||=2ĝ+1, , follow from Lemma 3.2; , follow from (12), and follows from Lemma 3.2, (1.9b) and the result on the expectation of the product of four Gaussian random variables 9W. Bar and F. Dittrich, “Useful formula for moment computation of normal random variables with nonzero means,” IEEE Transactions on Automatic Control, vol. 16, pp. 263-265, June 1971). From (1.18), we can then upper-bound the noise power as:
{|Z.sub.k|.sup.2}≤M.sub.rxβ.sub.maxN.sub.0[E.sup.(r)+||E.sup.(d)]+M.sub.rx||N.sub.0.sup.2(1.19)
where we use the fact that |β.sub.k,k|≤β.sub.max, [Δ.sub.k+k,{dot over (k)}+k+Δ.sub.k,{dot over (k)}]≤1 for k∈\ (as ĝ≤g/2) and Δ.sub.k+{dot over (k)}−k,k+{dot over (k)}−k≤1, from (0.9a).

(82) IV. Performance Analysis

(83) From (1.12)-(1.17), the effective single-input-single-output (SISO) channel between the k-th sub-carrier input and corresponding output can be expressed as:
Y.sub.k=M.sub.rxβ.sub.0,k√{square root over (E.sup.(r))}[|Ω[{dot over (k)}]|.sup.2]x.sub.k+I.sub.k+Z.sub.k, for k∈\  (1.20)
where I.sub.k and Z.sub.k are analyzed in Sections III-B and III-C, respectively. As is evident from (1.20), the signal component suffers from two kinds of fading: (i) a frequency-selective and channel dependent slow fading component represented by β.sub.0,k and (ii) a frequency-flat and phase-noise dependent fast fading component, represented by |Ω[{dot over (k)}]|.sup.2. The estimation of these fading coefficients is discussed later in this section. In this paper, we consider the simple demodulation approach where x.sub.k is estimated only from Y.sub.k, and the I.sub.k,Z.sub.k are treated as noise. For this demodulation approach, a lower bound to the signal-to-interference-plus-noise ratio (SINR) can be obtained from (1.14), (1.16b), Remark 3.1 and (1.19), as:

(84) $\begin{array}{cc}{\gamma }_k^{\mathrm{LB}}\left(\beta \right)\stackrel{\Delta }{=}\frac{M_{\mathrm{rx}}{.Math.{\beta }_{0,k}.Math.}^2{E}^{\left(r\right)}{E}^{\left(d\right)}{\mu \left(0,\hat{g}\right)}^2}{\begin{array}{c}{M}_{\mathrm{rx}}{.Math.{\beta }_{\max }.Math.}^2{E}^{\left(r\right)}\left[E^{\left(d\right)}\mu \left(0,\hat{g}\right)\left(1-\mu \left(0,\hat{g}\right)\right)+E^{\left(r\right)}\tilde{\mu }\left(k,\hat{g}\right)\right]+\\ {\beta }_{\max }{N}_0\left[E^{\left(r\right)}+.Math.\widehat{𝒢}.Math.E^{\left(d\right)}\right]+.Math.\widehat{𝒢}.Math.N_0^2\end{array}}& \left(1.21\right)\end{array}$

(85) where β{β.sub.0,k|} and we use the fact that {|I.sub.k+Z.sub.k|.sup.2}={|I.sub.k|.sup.2}+{|Z.sub.k|.sup.2}.

(86) Remark 4.1 If the RX array response vectors or the MPCs are mutually orthogonal i.e. a.sub.rx(.sub.1).sup.†a.sub.rx(.sub.2)=M.sub.rx then β.sub.{dot over (k)},{umlaut over (k)}=||.sup.2|a.sub.tx().sup.†t|.sup.2 and β.sub.max=||.sup.2.

(87) The orthogonality of array response vectors is approximately satisfied if the MPCs are well separated and M.sub.rx»L (O. El Ayach, R. Heath, S. Abu-Surra, S. Rajagopal, and Z. Pi, “The capacity optimality of beam steering in large millimeter wave MIMO systems,” in IEEE International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), pp. 100-104, June 2012). From Remark 4.1, note that even without explicit CE at the RX γ.sub.k.sup.LB(β) scales with M.sub.rx in the low SNR regime, which is a desired characteristic. While the ICI term also scales with M.sub.rx, its contribution can be kept small in the desired SNR range by picking ĝ such that μ(0,ĝ)≈1. In a similar way, with perfect knowledge of the fading coefficients at the RX, an approximate lower bound to the ergodic capacity can be obtained as:

(88) $\begin{array}{cc}C\left(\beta \right)\stackrel{\left(1\right)}{\ge }\frac{1}{K}\underset{k\in 𝒦\backslash 𝒢}{.Math.}{𝔼}_{{.Math.}_{k\in 𝒢}{.Math.\Omega \left[\stackrel{.}{k}\right].Math.}^2}\left\{\log \left[1+\frac{M_{\mathrm{rx}}^2{.Math.{\beta }_{0,k}.Math.}^2{E}^{\left(r\right)}{E^{\left(d\right)}\left[\underset{\stackrel{.}{k}\in \widehat{𝒢}}{.Math.}{.Math.\Omega \left[\stackrel{.}{k}\right].Math.}^2\right]}^2}{𝔼\left\{{.Math.I_k.Math.}^2+{.Math.Z_k.Math.}^2|\underset{\stackrel{.}{k}\in \widehat{𝒢}}{.Math.}{.Math.\Omega \left[\stackrel{.}{k}\right].Math.}^2\right\}}\right]\right\}\stackrel{\left(2\right)}{\approx }\frac{1}{K}\underset{k\in 𝒦\backslash 𝒢}{.Math.}\left(\log \left[𝔼\left\{{.Math.I_k.Math.}^2+{.Math.Z_k.Math.}^2\right\}+M_{\mathrm{rx}}^2{.Math.{\beta }_{0,k}.Math.}^2{E}^{\left(r\right)}{E}^{\left(d\right)}{\mu \left(\hat{g}\right)}^2\right]-\log \left[𝔼\left\{{.Math.I_k.Math.}^2+{.Math.Z_k.Math.}^2\right\}\right]\right)\stackrel{\left(3\right)}{\ge }\frac{1}{K}\underset{k\in 𝒦\backslash 𝒢}{.Math.}\log \left[1+{\gamma }_k^{\mathrm{LB}}\left(\beta \right)\right]\stackrel{\Delta }{=}{C}_{\mathrm{approx}}\left(\beta \right)& \left(1.22\right)\end{array}$
where is obtained by assuming I.sub.k, Z.sub.k to be Gaussian distributed and using the expression for ergodic capacity (A. Goldsmith and P. Varaiya, “Capacity of fading channels with channel side information,” IEEE Transactions on Information Theory, vol. 43, no. 6, pp. 1986-1992, 1997), follows by sending the outer expectation into the log(.Math.) functions and follows from (1.14), (1.16b) and (1.19). While is an approximation, it typically yields a lower bound since Variance{|Ω[{dot over (k)}]|.sup.2}≤μ(0,ĝ)[1−μ(0,ĝ)]«μ(0,ĝ).sup.2 (from (1.9a) and (R. Bhatia and C. Davis, “A better bound on the variance,” The American Mathematical Monthly, vol. 107, p. 353, April 2000).

(89) Note that for demodulating x.sub.k's and achieving the above SINR and capacity, the RX requires estimates of N.sub.0 and the SISO channel fading coefficients β and |Ω[{dot over (k)}]|.sup.2. Since the RX has a good beamforming gain (1.21), the channel parameters β, N.sub.0 can be tracked accurately at the RX with a low estimation overhead using pilot symbols and blanked symbols. These values, along with phase-noise parameter σ.sub.θ, can further be fed back to the TX for rate and power allocation. Note that since these pilots are only used to estimate the SISO channel parameters and not the actual MIMO channel, the advantages of simplified CE are still applicable for a CACE based RX. On the other hand, the low variance albeit fast varying component |Ω[{dot over (k)}]|.sup.2 can be estimated for every symbol using the 0-th sub-carrier output Y.sub.0. It can be shown from (1.12) that Y.sub.0=M.sub.rxβ.sub.0,0E.sup.(r)|Ω[{dot over (k)}]|.sup.2+I.sub.0+M.sub.rx|N.sub.0+Z.sub.0, where we have {|I.sub.0|.sup.2}≤{|I.sub.k|.sup.2} and {|Z.sub.0|.sup.2}≤2{|Z.sub.k|.sup.2} for any k∈\. Thus |Ω[{dot over (k)}]|.sup.2 can be estimated from Y.sub.0 with an

(90) $\mathrm{SINR}\ge \frac{E^r{\gamma }_k^{\mathrm{LB}}\left(\beta \right)}{2E_{\left(d\right)}},$
which is usually a large value.

(91) A. Optimizing System Parameters

(92) From (1.22), note that the approximate ergodic capacity C.sub.approx(β) is a decreasing function of g for g≥2ĝ. Thus a C.sub.approx(β) maximizing choice of g should satisfy g=2ĝ. From (1.22) and (1.21), we can also lower bound C.sub.approx(β) as:

(93) $\begin{array}{cc}{C}_{\mathrm{approx}}\left(\beta \right)\stackrel{\left(1\right)}{\ge }\frac{1}{K}\underset{k\in 𝒦\backslash 𝒢}{.Math.}\log \left({.Math.{\beta }_{0,k}.Math.}^2\right)+\frac{K-.Math.𝒢.Math.}{K}\log \left[\Xi \left(\beta \right)\right],\mathrm{where}\text{:}& \left(1.23a\right)\\ \Xi \left(\beta \right)\stackrel{\left(2\right)}{=}\frac{M_{\mathrm{rx}}{E}^{\left(r\right)}{E}^{\left(d\right)}{\mu \left(0,\hat{g}\right)}^2}{\begin{array}{c}{M}_{\mathrm{rx}}{.Math.{\beta }_{\max }.Math.}^2{E}^{\left(r\right)}\left(\frac{E_s}{K-.Math.𝒢.Math.}\right)\mu \left(0,\hat{g}\right)\left[1-\mu \left(0,\hat{g}\right)\right]+\\ {\beta }_{\max }{N}_0\left(E^{\left(r\right)}+.Math.\widehat{𝒢}.Math.E^{\left(d\right)}\right)+.Math.\widehat{𝒢}.Math.N_0^2\end{array}},& (1.23b\end{array}$
follows from the fact that log(1+γ.sub.k.sup.LB(β))≥log(γ.sub.k.sup.LB(β)) and by taking the summation over k in (1.22) into the denominator of the logarithm; and follows from the fact that {tilde over (μ)}(k,ĝ)≤μ(0,ĝ)Δ.sub.k,k and E.sup.(d)(K−||)+E.sup.(r)=E.sub.s. It can be verified that the numerator of Θ(β) is a differentiable, strictly concave function of E.sup.(r), while the denominator is a positive, affine function of E.sup.(r). Thus Θ(β) is a strictly pseudo-concave function of E.sup.(r) (S. Schaible, “Fractional programming,” Zeitschrift für Operations Research, vol. 27, pp. 39-54, dec 1983), and the C.sub.approx(β) maximizing power allocation can be obtained by setting

(94) 0$\frac{d\Xi \left(\beta \right)}{{\mathrm{dE}}^{\left(r\right)}}=0$
as:

(95) $\begin{array}{cc}{E}_{\mathrm{opt}}^{\left(r\right)}=E_s\frac{\sqrt{R^2+\mathrm{QR}}-R}{Q}& \left(1.24\right)\end{array}$
where Q=M.sub.rx|β.sub.max|.sup.2[1−μ(0,ĝ)]μ(0,ĝ)E.sub.s+β.sub.maxN.sub.0(K−||−||) and R=N.sub.0||[β.sub.max+N.sub.0(K−||)/E.sub.s]. As evident from (1.23b), ĝ offers a trade-off between the phase-noise induced ICI and channel noise accumulation. While finding a closed form expression for (1.23b) maximizing ĝ is intractable, it can be computed numerically by performing a simple line search over 1≤ĝ≤min{K.sub.1, K.sub.2}/2, with g=2ĝ and E.sup.(r) as given by (1.24).

(96) V. Initial Access, TX Beamforming and Uplink Beamforming

(97) In this section we briefly discuss stages (i) and (ii) of downlink transmission (see Section 2), and uplink TX beamforming for CACE aided UEs. In the suggested IA protocol for stage (i), the BS performs beam sweeping along different angular directions, possibly with different beam widths, similar to the approach of 3GPP New Radio (NR). For each TX beam, the BS transmits primary (PSS) and secondary synchronization sequences (SSS) with the reference signal, in a form similar to (1.1). The UEs use CACE aided RX beamforming, and initiate uplink random access to the BS upon successfully detecting a PSS/SSS. As shall be shown in Section VI, the SINR expression (1.21) is resilient to frequency mismatches between TX and RX oscillators, and thus is also applicable for the PSS/SSSs where frequency synchronization may not exist. Since angular beam-sweeping is only performed at the BS, the IA latency does not scale with M.sub.rx and yet the PSS/SSS symbols can exploit the RX beamforming gain, thus improving cell discovery radius and/or reducing IA overhead. This is in contrast to digital CE at the UE, which would require sweeping through many RX beam directions for each TX direction, necessitating several repetitions of the PSS/SSS for each TX beam. During downlink stage (ii), note that scheduling of UEs, designing TX beamformer and allocation of power requires knowledge of {|,a.sub.tx()} for all the UEs. Such rCSI can be acquired at the BS either by downlink CE with CSI feedback from the UEs or by uplink CE. The protocol for downlink CE with feedback is similar to the IA protocol, with the BS transmitting pilot symbols instead of PSS and SSS. Uplink CE can be performed by transmitting orthogonal pilots from the UEs omni-directionally, and using any of the digital CE algorithms from Section I at the BS. Note that CACE cannot be used at the BS since the pilots from multiple UEs need to be separated via digital processing.

(98) Note that the phase shifts used for RX beamforming at a CACE aided UE in downlink, can also be used for transmit beamforming in the uplink. However since the reference signal is not available at the UE during uplink transmission in time division duplexing systems, a mechanism for locking these phase shift values from a previous downlink transmission stage is required (similar to (V. V. Ratnam and A. F. Molisch, “Periodic analog channel estimation aided beamforming for massive MIMO systems,” IEEE Transactions on Wireless Communications, 2019, accepted to)). In contrast, frequency division duplexing can avoid such a mechanism due to continuous availability of the downlink reference, and consequently ŝ.sub.rx,BB(t).

(99) VI. Simulation Results

(100) For the simulations, we consider a single cell scenario with a λ/2-spaced 32×8 (M.sub.tx=256) antenna BS and one representative UE with a λ/2-spaced 16×4 (M.sub.rx=64) antenna army, having perfect timing synchronization to the BS, one down-conversion chain, and using CACE aided beamforming. The BS has apriori rCSI and transmits one spatial OFDM data stream with T.sub.s=1 μs, K.sub.1=K.sub.2+1=512 and f.sub.c=30 GHz along the strongest MPC, i.e., t=a.sub.tx(l) for l=argmax{||}. The UE oscillator has Wiener phase-noise with variance σ.sub.θ.sup.2 known both to the BS and UE. The UE also has perfect knowledge of β, N.sub.0 and |Ω[{dot over (k)}]|.sup.2.

(101) For testing the validity of the analytical results, we first consider a sample sparse channel matrix H(t) with L=3, ={0,20,40}ns, angles of arrival ψ.sub.azi.sup.rx={0, π/6, −π/6}, ψ.sub.ele.sup.rx={0.45π, π/2, π/2} and effective amplitudes

(102) $\frac{{\alpha }_{\ell }{a_{\mathrm{tx}}\left(\ell \right)}^{†}t}{\sqrt{{\beta }_{\max }}}=\left\{\sqrt{0.6},-\sqrt{0.3},\sqrt{0.1}\right\}.$
The UE uses ĝ=g/2=10, σ.sub.θ.sup.2=1/T.sub.s and E.sup.(r), E.sup.(d) from (1.24). For this model, the symbol error rates (SERs) for the sub-carriers, obtained by Monte-Carlo simulations, are compared to the analytical SERs for a Gaussian channel with SINR given by (1.21) (with/without Remark 3.1) in FIG. 8. For the Monte-Carlo results, we use truncated sinc filters: LPF.sub.ĝ(t)=sin(2πĝt/T.sub.s)/(πt) for |t|≤2T.sub.s/ĝ. As observed from the results and also mentioned in Section III-B, the use of Remark 3.1 in (1.21) provides a tight SINR bound even for small |k|, We also observe that the SER for k=22 (≈ĝ) is high due to the interference caused from the high power reference signal. However this interference diminishes very quickly with k, as evident from the SER for k=−40. While the mean RX oscillator frequency was assumed to be perfectly matched to the TX oscillator for the analytical results (see Section II), we also plot in FIG. 8 the case with a 5 MHz frequency mismatch. Results show a negligible degradation in performance, suggesting that the CACE design is resilient to frequency mismatches smaller than the filter bandwidth of LPF.sub.ĝ. Due to the accuracy of the bounds in FIG. 8 and for computational tractability, we shall henceforth use (1.21) and (1.22) to quantify performance of CACE in the rest of the section. We next plot the C.sub.approx(β) from (1.24) as a function of ĝ in FIG. 9, with (a) C.sub.approx(β) maximizing E.sup.(r) (obtained by exhaustive search over 0≤E.sup.r≤E.sub.s) and (b) E.sup.r chosen from (1.24), respectively. As observed from the results, the curves are very close, suggesting the accuracy of the power allocation in (1.24). While the poor system performance at lower ĝ is due to phase-noise induced IC1, the poor performance at high ĝ is due to noise accumulation and spectral efficiency reduction. We also note that the optimal ĝ increases with SNR.

(103) FIG. 10A compares the achievable throughput for beamforming with digital CE and different ACE schemes: CACE, PACE (V. V. Ratnam and A. F. Molisch, “Periodic analog channel estimation aided beamforming for massive MIMO systems,” IEEE Transactions on Wireless Communications, 2019, accepted), MA-FSR (V. V. Ratnam and A. Molisch, “Multi-antenna FSR receivers: Low complexity, non-coherent, massive antenna receivers,” in IEEE Global Communications Conference (GLOBECOM), December 2018), respectively for the sparse channel defined above. For digital CE, the RX beamformer is aligned with the largest eigenvector of the effective RX correlation matrix

(104) $R_{\mathrm{rx}}\left(t\right)=\frac{1}{K}\underset{k\in 𝒦}{.Math.}\mathscr{H}\left(f_k\right){\mathrm{tt}}^{{}^{†}}{\mathscr{H}\left(f_k\right)}^{†}$
(P. Sudarshan, N. Mehta, A. Molisch, and J. Zhang, “Channel statistics-based RF pre-processing with antenna selection,” IEEE Transactions on Wireless Communications, vol. 5, pp. 3501-3511, December 2006), which in turn is either (a) known apriori at BS or (b) is estimated by nested array based sampling (P. Pal and P. P. Vaidyanathan, “Nested arrays: A novel approach to array processing with enhanced degrees of freedom,” IEEE Transactions on Signal Processing, vol. 58, pp. 4167-4181, August 2010). To decouple the loss in beamforming gain due to CE errors from loss due to phase-noise, we assume σ.sub.θ≈0. As is evident from FIG. 10A, PACE and CACE suffer only a ≤2 dB beamforming loss in compared to digital CE in sparse channels and above a threshold SNR. While CACE performs marginally worse than PACE at high SNR due to continuous transmission of the reference, unlike PACE it does not suffer from carrier recovery losses at low SNR. While MA-FSR performs poorly due to low bandwidth efficiency, it requires much simpler hardware then all other schemes. To demonstrate the phase-noise suppressing capability of CACE (and MA-FSR), we also plot the throughput of CACE (with optimal ĝ) and digital CE, with σ.sub.θ.sup.2=1/T.sub.s and without any additional phase-noise mitigation. As is evident from the results, both CACE and MA-FSR aid in mitigating oscillator phase-noise in addition to enabling RX beamforming. To study the impact of more realistic channels and number of MPCs, we also consider a rich scattering stochastic channel in FIG. 10B, having L/10 resolvable MPCs and 10 sub-paths per resolvable MPC. All channel parameters are generated according to the 3GPP TR38.900 Rel 14 channel model (UMi NLoS scenario)(TR38.900, “Study on channel model for frequency spectrum above 6 GHz (release 14),” Tech. Rep. V14.3.1, 3GPP, 2017), with the resolvable MPCs and sub-paths modeled as clusters and rays, respectively. However to model the sub-paths of each MPC as unresolvable, we use an intra-cluster delay spread of 1 ns and an intra-cluster angle spread of π/50 (for all elevation, azimuth, arrival and departure). As observed, the loss in beamforming gain with L is only slightly higher for ACE schemes than digital CE.

(105) Note that the throughputs in FIG. 10 do not include the CE overhead for PACE and digital CE. While nested array digital CE requires 21 dedicated pilot symbols (≈2√{square root over (M.sub.rx)}) for updating RX beamformer, PACE requires 6 symbols (O(1)) and CACE, MA-FSR only require a continuous reference tone. The corresponding overhead reduction can be significant when downlink CE with CSI feedback is used for rCSI acquisition at the BS (see Section V). For example, with exhaustive beam-scanning (C. Jeong, J. Park, and H. Yu, “Random access in millimeter-wave beamforming cellular networks: issues and approaches,” IEEE Communications Magazine, vol. 53, pp. 180-185, January 2015) at the BS and an rCSI coherence time of 10 ms, the BS rCSI acquisition overhead reduces from 40% for nested array digital CE to 11% for PACE and ≈||/K<5% for CACE.

(106) VII. Conclusions

(107) This paper proposes the use of a novel CE technique called CACE for designing the RX beamformer in massive MIMO systems. In CACE, a reference tone is transmitted along with the data signals. At each RX antenna, the received signal is converted to baseband, the reference component is isolated, and is used to control the analog phase-shifter through which the data component is processed. The resulting baseband phase-shifted signals from all the antennas are then added, and fed to the down-conversion chain. This emulates using the received signal for reference as a matched filter for data, and enables both RX beamforming and phase-noise cancellation. The performance analysis suggests that in sparse channels and for ĝ»1, the SINR with CACE scales linearly with M.sub.rx. The analysis and simulations also show that ĝ yields a trade-off between phase-noise induced ICI and noise accumulation. Simulations suggest that CACE suffers only a small degradation in beamforming gain in comparison to digital CE based beamforming in sparse channels, and is resilient to TX-RX oscillator frequency mismatch. In comparison to other ACE schemes, CACE performs marginally worse than PACE at high SNR but performs much better at lower SNR. It also performs much better than MA-FSR, albeit at a higher RX hardware complexity. Finally, CACE also provides phase-noise suppression unlike most other CE schemes. The CE overhead reduction with CACE is significant, especially when downlink CE with feedback is required. The IA latency reduction with CACE aided beamforming is also discussed. While baseband phase shifters are sufficient for a CACE based RX unlike in conventional analog beamforming, 2M.sub.rx mixers may be required for the baseband conversion at the RX; thus adding to the hardware cost.

APPENDIX 1.A

(108) Proof of Lemma \refLemma_PN_properties. Note that from the definition of Ω[k], we have

(109) $e^{-j\theta \left[n\right]}\stackrel{ℱ}{.fwdarw.}\Omega \left[K\right]\mathrm{and}{e}^{j\theta \left[n\right]}\stackrel{ℱ}{.fwdarw.}{\Omega }^{*}\left[-k\right],$
where represents the nDFT Operation. Then using convolution property of the nDFT, we have:

(110) $e^{-j\theta \left[n\right]}{e}^{j\theta \left[n\right]}\stackrel{ℱ}{.fwdarw.}\underset{a\in 𝒦}{.Math.}\Omega \left[a\right]{\Omega }^{*}\left[a+k\right].Math.1\stackrel{ℱ}{.fwdarw.}\underset{a\in 𝒦}{.Math.}\Omega \left[a\right]{\Omega }^{*}\left[a+k\right].Math.{\delta }_{0,k}^K=\underset{a\in 𝒦}{.Math.}\Omega \left[a\right]{\Omega }^{*}\left[a+k\right]$
which proves property (1.9a). Property (1.9b) can be obtained as follows:

(111) $\begin{array}{cc}{\Delta }_{k_1,k_2}\stackrel{\Delta }{=}𝔼\left\{\Omega \left[k_1\right]{\Omega \left[k_2\right]}^{*}\right\}=\frac{1}{K^2}\underset{\stackrel{.}{n},\stackrel{\mathrm{.Math.}}{n}=0}{\overset{K-1}{.Math.}}𝔼\left\{e^{-j[\theta \left[\stackrel{.}{n}\right]-\theta \left[\stackrel{\mathrm{.Math.}}{n}\right]}\right\}{e}^{-j2\pi \frac{\left[k_1\stackrel{.}{n}-k_2\stackrel{\mathrm{.Math.}}{n}\right]}{K}}\stackrel{\left(1\right)}{=}\frac{1}{K^2}\underset{\stackrel{.}{n},\stackrel{\mathrm{.Math.}}{n}=0}{\overset{K-1}{.Math.}}{e}^{-\frac{{\sigma }_{\theta }^2.Math.\stackrel{.}{n}-\stackrel{\mathrm{.Math.}}{n}.Math.T_s}{2K}}{e}^{-j2\pi \frac{\left[k_1\stackrel{.}{n}-k_2\stackrel{\mathrm{.Math.}}{n}\right]}{K}}\stackrel{\left(2\right)}{=}\frac{1}{K^2}\underset{\stackrel{\mathrm{.Math.}}{n}=0}{\overset{K-1}{.Math.}}\underset{u=-\stackrel{\mathrm{.Math.}}{n}}{\overset{K-1-\stackrel{\mathrm{.Math.}}{n}}{.Math.}}{e}^{-\frac{{\sigma }_{\theta }^2.Math.u.Math.T_s}{2K}}{e}^{-j2\pi \frac{\left[k_{1}u+\left(k_1-k_2\right)\stackrel{\mathrm{.Math.}}{n}\right]}{K}}\stackrel{\left(3\right)}{\approx }\frac{1}{K^2}\underset{\stackrel{\mathrm{.Math.}}{n}=0}{\overset{K-1}{.Math.}}\underset{u=-K/2}{\overset{K/2-1}{.Math.}}{e}^{-\frac{{\sigma }_{\theta }^2.Math.u.Math.T_s}{2K}}{e}^{-j2\pi \frac{\left[k_{1}u+\left(k_1-k_2\right)\stackrel{\mathrm{.Math.}}{n}\right]}{K}}\stackrel{\left(4\right)}{=}\frac{{\delta }_{k_1,k_2}^K}{K}\left[\frac{1-e^{-\left(\frac{{\sigma }_{\theta }^2{T}_s-j4\pi k_1}{4}\right)}}{e^{\frac{{\sigma }_{\theta }^2{T}_s-j4\pi k_1}{2K}}-1}+\frac{1-e^{-\left(\frac{{\sigma }_{\theta }^2{T}_s+j4\pi k_1}{4}\right)}}{1-e^{-\frac{{\sigma }_{\theta }^2{T}_s+j4\pi k_1}{2K}}}\right]& \left(1.25\right)\end{array}$
where follows by using the expression for the characteristic function of the Gaussian random variable θ[{dot over (n)}]−θ[{umlaut over (n)}]; follows by defining u={dot over (n)}−{umlaut over (n)} and follows by changing the inner summation limits which is accurate for σ.sub.θ.sup.2T.sub.s»1 and follows from the expression for the sum of a geometric series.

APPENDIX 1B

(112) Proof of Lemma \refLemma_N_properties. Note that each component of {tilde over (w)}(t) is independent and identically distributed as a circularly symmetric Gaussian random process. Hence its nDFT coefficients, obtained as

(113) $W\left[k\right]=\frac{1}{K}\underset{n=0}{\overset{K-1}{.Math.}}\tilde{w}\left({\mathrm{nT}}_s/K\right)e^{-j2\pi \frac{\mathrm{kn}}{K}}$
are also jointly Gaussian and circularly symmetric. For these coefficients at RX antennas a, b we obtain:

(114) $\begin{array}{cc}𝔼\left\{W_a\left[k_1\right]W_b\left[k_2\right]\right\}=\frac{1}{K^2}\underset{n_1,n_2=1}{\overset{K}{.Math.}}𝔼\left\{{\tilde{w}}_a\left(n_1{T}_s/K\right){\tilde{w}}_b\left(n_2{T}_s/K\right)\right\}{e}^{-j2\pi \frac{k_1{n}_1+k_2{n}_2}{K}}=0𝔼\left\{W_a\left[k_1\right]W_b^{*}\left[k_2\right]\right\}=\frac{1}{K^2}\underset{n_1,n_2=1}{\overset{K}{.Math.}}𝔼\left\{{\tilde{w}}_a\left(n_1{T}_s/K\right){{\tilde{w}}_b\left(n_2{T}_s/K\right)}^{*}\right\}{e}^{-j2\pi \frac{k_1{n}_1-k_2{n}_2}{K}}=\frac{{\delta }_{a,b}^{\infty }}{K^2}\underset{n_1,n_2=1}{\overset{K}{.Math.}}{R}_{\tilde{w}}\left(\left[n_1-n_2\right]T_s/K\right)e^{-j2\pi \frac{k_1{n}_1-k_2{n}_2}{K}}={\delta }_{a,b}^{\infty }{\delta }_{k_1,k_2}^K{N}_0/T_s& \left(1.26\right)\end{array}$
where we use the auto-correlation function of the channel noise at any RX antenna as: R.sub.{tilde over (w)}(t)=N.sub.0 sin(πKt/T.sub.s)exp{−jπ(K.sub.1−K.sub.2)t/T.sub.s}/πt.

APPENDIX 1C

(115) Here we model the RX phase-noise θ(t) as a zero mean Omstein-Ulhenbeck (OU) process (J. L. Doob, “The brownian movement and stochastic equations,” The Annals of Mathematics, vol. 43, p. 351, April 1942), which is representative of the output of a type-1 phase-locked loop with a linear phase detector (A. Viterbi, Principles of coherent communication. McGraw-Hill series in systems science, McGraw-Hill, 1966; D. Petrovic, W. Rave, and G. Fettweis, “Effects of phase noise on OFDM systems with and without PLL: Characterization and compensation,” IEEE Transactions on Communications, vol. 55, pp. 1607-1616, August 2007; A. Mehrotra, “Noise analysis of phase-locked loops,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 49, pp. 1309-1316, September 2002). For such a model, θ(t) satisfies:

(116) $\begin{array}{cc}\frac{d\theta \left(t\right)}{\mathrm{dt}}=-{\eta }_{\theta }\theta \left(t\right)+{\sigma }_{\theta }{w}_{\theta }\left(t\right)& \left(1.27\right)\end{array}$
where, w.sub.θ(t) is a standard real white Gaussian process, and η.sub.θ, σ.sub.θ are system parameters. From (6) it can be shown that θ(t) is a stationary Gaussian process (in steady state), with an auto-correlation function given by:

(117) 0$R_{\theta }\left(\tau \right)=𝔼\left\{\theta \left(t\right)\theta \left(t+\tau \right)\right\}=\frac{{\sigma }_{\theta }^2}{2{\eta }_{\theta }}{e}^{-{\eta }_{\theta }.Math.\tau .Math.}$
(D. Petrovic, W. Rave, and G. Fettweis, “Effects of phase noise on OFDM systems with and without PLL: Characterization and compensation,” IEEE Transactions on Communications, vol. 55, pp. 1607-1616, August 2007).

(118) Lemma 10.1 For phase-noise modeled as an OU process we have:

(119) $\begin{array}{cc}{.Math.}_{k\in 𝒦}\Omega \left[k\right]{\Omega \left[k+k_1\right]}^{*}={\delta }_{0,k_1}^K,& \left(1.28b\right)\end{array}$$\begin{array}{cc}{\Delta }_{k_1,k_2}\stackrel{△}{=}𝔼\left\{\Omega \left[k_1\right]{\Omega \left[k_2\right]}^{*}\right\}\approx \frac{{\delta }_{k_1,k_2}^K{e}^{-R_{\theta }\left[0\right]}}{K}{.Math.}_{u=-.Math.K/2.Math.}^{.Math.\left(K-1\right)/2.Math.}{e}^{R_{\theta }\left[u\right]}{e}^{-j2\pi \frac{k_{1}u}{K}}& \left(1.28b\right)\end{array}$
for arbitrary integers k.sub.1, k.sub.2, where δ.sub.a,b.sup.K=1 if a=b (mod K) or δ.sub.a,b.sup.K=0 otherwise, and

(120) $R_{\theta }\left[n\right]\stackrel{△}{=}𝔼\left\{\theta \left[\stackrel{.}{n}\right]\theta \left[\stackrel{.}{n}+n\right]\right\}=\frac{{\sigma }_{\theta }^2}{2{\eta }_{\theta }}{e}^{-{\eta }_{\theta }.Math.{\mathrm{nT}}_s/K.Math.}.$

(121) Proof of Lemma \refLemma_OU_PN_properties. Note that from the definition of Ω[k], we have

(122) $e^{-j\theta \left[n\right]}\stackrel{ℱ}{.fwdarw.}\Omega \left[k\right]\mathrm{and}{e}^{j\theta \left[n\right]}\stackrel{ℱ}{.fwdarw.}{\Omega }^{*}\left[-k\right],$
where represents the nDFT Operation. Then using convolution property of the nDFT, we have:

(123) $e^{-j\theta \left[n\right]}{e}^{j\theta \left[n\right]}\stackrel{ℱ}{.fwdarw.}{.Math.}_{a\in 𝒦}\Omega \left[a\right]{\Omega }^{*}\left[a+k\right].Math.1\stackrel{ℱ}{.fwdarw.}{.Math.}_{a\in 𝒦}\Omega \left[a\right]{\Omega }^{*}\left[a+k\right].Math.{\delta }_{0,k}^K={.Math.}_{a\in 𝒦}\Omega \left[a\right]{\Omega }^{*}\left[a+k\right]$

(124) which proves property (1.9a). Property (1.9b) can be obtained as follows:

(125) $\begin{array}{cc}{\Delta }_{k_1,k_2}\stackrel{△}{=}𝔼\left\{\Omega \left[k_1\right]{\Omega \left[k_2\right]}^{*}\right\}=\frac{1}{K^2}{.Math.}_{\stackrel{.}{n},\stackrel{.Math.}{n}=0}^{K-1}𝔼\left\{e^{-j[\theta \left[\stackrel{.}{n}\right]-\theta \left[\stackrel{.Math.}{n}\right]}\right\}{e}^{-j2\pi \frac{\left[k_1\stackrel{.}{n}-k_2\stackrel{.Math.}{n}\right]}{K}}\stackrel{\left(1\right)}{=}\frac{1}{K^2}{.Math.}_{\stackrel{.}{n},\stackrel{.Math.}{n}=0}^{K-1}{e}^{-R_{\theta }\left[0\right]+R_{\theta \left[\stackrel{.}{n}-\stackrel{.Math.}{n}\right]}}{e}^{-\mathrm{j2}\pi \frac{\left[k_1\stackrel{.}{n}-k_2\stackrel{.Math.}{n}\right]}{K}}\stackrel{\left(2\right)}{=}\frac{1}{K^2}{.Math.}_{\stackrel{.Math.}{n}=0}^{K-1}{.Math.}_{u=-\stackrel{.Math.}{n}}^{K-1-\stackrel{.Math.}{n}}{e}^{-R_{\theta }\left[0\right]+R_{\theta }\left[u\right]}{e}^{-j2\pi \frac{\left[k_{1}u+\left(k_1-k_2\right)\stackrel{.Math.}{n}\right]}{K}}\stackrel{\left(3\right)}{\approx }\frac{1}{K^2}{.Math.}_{\stackrel{.Math.}{n}=0}^{K-1}{.Math.}_{u=-.Math.K/2.Math.}^{.Math.\left(K-1\right)/2.Math.}{e}^{-R_{\theta }\left[0\right]+R_{\theta }\left[u\right]}{e}^{-j2\pi \frac{\left[k_{1}u+\left(k_1-k_2\right)\stackrel{.Math.}{n}\right]}{K}}=\frac{{\delta }_{k_1,k_2}{e}^{-R_{\theta }\left[0\right]}}{K}{.Math.}_{u=-.Math.K/2.Math.}^{.Math.\left(K-1\right)/2.Math.}{e}^{-R_{\theta }\left[0\right]}{e}^{-j2\pi \frac{k_{1}u}{K}}& \left(1.29\right)\end{array}$

(126) where follows by using the expression for the characteristic function of the Gaussian random variable θ[{dot over (n)}]−θ[{umlaut over (n)}]; follows by defining u={dot over (n)}−{umlaut over (n)} and follows by using the fact that R.sub.θ[u] has a limited support around u=0 and hence R.sub.θ[u]≈R.sub.θ[u−K]≈0 for u>(K−1)/2. Note that since e.sup.−R.sup.θ.sup.[0]+R.sup.θ.sup.[u] is an auto-correlation function, its nDFT is non-negative, thus ensuring that Δ.sub.k.sub.1.sub.,k.sub.1≥0 in (1.29).

(127) 2. Periodic Analog Channel Estimation Aided Beamforming for Massive MIMO Systems

(128) I. Introduction

(129) In the present embodiment, a novel ACE scheme, referred to as periodic ACE (PACE) is provided. In this embodiment, the reference is transmitted judiciously, and its amplitude and phase are explicitly estimated to drive an RX phase shifter array. In contrast to CACE, PACE requires one carrier recovery circuit and M.sub.rx phase shifters (see FIG. 3) and can support both homo/heter-dyne reception. In PACE, the TX transmits a reference tone at a known frequency during each periodic RX beamformer update phase. One carrier recovery circuit, involving phase-locked loops (PLLs), is used to recover the reference tone from one or more antennas, as shown in FIG. 3. This recovered reference tone, and its quadrature component, are then used to estimate the phase off-set and amplitude of the received reference tone at each RX antenna, via a bank of ‘filter, sample and hold’ circuits (represented as integrators in FIG. 1). As shall be shown, these estimates are proportional to the channel response at the reference frequency. These estimates are used to control an array of variable gain phase-shifters, which generate the RX analog beam. During the data transmission phase, the wide-band received data signals pass through these phase-shifters, are summed and processed similar to conventional analog beamforming. As the phase and amplitude estimation is done in the analog domain, O(1) pilots are sufficient to update the RX beamformer. Additionally, the power from multiple channel MPCs is accumulated by this approach, increasing the system diversity against MPC blocking. Furthermore, the same variable gain phase-shifts can also be used for transmit beamforming on the reverse link. Finally, by providing an option for digitally controlling the inputs to the phase-shifters, the proposed architecture can also support conventional beamforming approaches.

(130) On the flip side, PACE requires some additional analog hardware components, such as mixers and filters, in comparison to conventional digital CE. Additionally, the accumulation of power from multiple MPCs may cause frequency selective fading in a wide-band scenario, which can degrade performance. Finally, the proposed approach in its current suggested form does not support reception of multiple spatial data streams and can only be used for beamforming at one end of a communication link. This architecture is therefore more suitable for use at the user equipment (UEs). The possible extensions to multiple spatial stream reception shall be explored in future work. While the proposed architecture is also applicable in narrow-band scenarios, in this paper we shall focus on the analysis of a wide-band scenario where the repetition interval of PACE and beamformer update is of the order of aCSI coherence time, i.e. time over which the aCSI stays approximately constant (also called stationarity time in some literature). The contributions of the present embodiment include:

(131) 1. The development of a novel transmission technique, namely PACE, and a corresponding RX architecture that enable RX analog beamforming with low CE overhead.

(132) 2. To enable the RX operation, two novel reference recovery circuits are explored. These circuits are non-linear, making their analysis non-trivial. We provide an approximate analysis of their phase-noise and the resulting performance that is tight in the high SNR regime.

(133) 3. The achievable system throughput with PACE aided beamforming in a wide-band channel is analytically characterized.

(134) 4. Simulations with practically relevant channel models are used to support the analytical results and compare performance to existing schemes.

(135) Notation: scalars are represented by light-case letters; vectors by bold-case letters; and sets by calligraphic letters. Additionally, j=√{square root over (−1)}, a* is the complex conjugate of a complex scalar a, |a| represents the .sub.2-norm of a vector a and A.sup.† is the conjugate transpose of a complex matrix A. Finally, { } represents the expectation operator, .Math. represents the Kronecker product, represents equality in distribution, Re{.Math.}/Im{.Math.} refer to the real/imaginary component, respectively, (a,B) represents a circularly symmetric complex Gaussian vector with mean a and covariance matrix B, Exp{a} represents an exponential distribution with mean a and Uni{a, b} represents a uniform distribution in range [a, b].

(136) II. PACE General Assumptions and System Model

(137) We consider the downlink of a single-cell MIMO system, wherein one base station (BS) with M.sub.tx antennas transmits to several UEs with M.sub.rx antennas each. Since focus is on the downlink, we shall use abbreviations BS & TX and UE & RX interchangeably. Each UE is assumed to have one up/down-conversion chain, while no assumptions are made regarding the BS architecture.

(138) Here we assume the communication between the BS and UEs to involve three important phases: (i) initial access (IA)—where the BS and UEs find each other, timing/frequency synchronization is attained and spectral resources are allocated; (ii) analog beamformer design—where the BS and UEs obtain the required aCSI to update the analog precoding/combining beams; and (iii) data transmission. The relative time scale of these phases are illustrated in FIG. 11. Through most of this paper (Sections 2-4), we assume that the IA and beamformer design at the BS are already achieved, and we mainly focus on the beamformer design phase at the UE and the data transmission phase. Therefore we assume perfect timing and frequency synchronization between the BS and UE, and assume that the TX beamforming has been pre-designed based on aCSI at the BS. Later in Section V, we also briefly discuss how aCSI can be acquired at the BS, how IA can be performed and how the use of PACE can be advantageous in those phases.

(139) The BS transmits one spatial data-stream to each scheduled UE, and all such scheduled UEs are served simultaneously via spatial multiplexing. Furthermore, the data to the UEs is assumed to be transmitted via orthogonal precoding beams, such that, there is no inter-user interference. Under these assumptions and given transmit precoding beams and power allocation, we shall restrict the analysis to one representative UE without loss of generality. For convenience, we shall also assume the use of noise-less and perfectly linear antennas, filters, amplifiers and mixers at both the BS and UE. An analysis including the non-linear effects of these components is beyond the scope of this paper. The BS transmits orthogonal frequency division multiplexing (OFDM) symbols with K sub-carriers, indexed as ={−K.sub.1, . . . , K.sub.2−1, K.sub.2} with K.sub.1+K.sub.2+1=K, to this representative UE. The BS transmits two kinds of symbols: reference symbols and data symbols. In a reference symbol, only a reference tone, i.e., a sinusoidal signal with a pre-determined frequency known both to the BS and UE, is transmitted on the 0-th subcarrier, and the remaining sub-carriers are all empty. On the other hand, in a data symbol all the K sub-carriers are used for data transmission. The purpose of the reference symbols is to aid PACE and beamformer design at the RX, as shall be explained later. Since the BS can afford an accurate oscillator, we shall assume that the BS suffers negligible phase noise. The M.sub.tx×1 complex equivalent transmit signal for the 0-th symbol, if it is a reference or data symbol, respectively, can then be expressed as:

(140) $\begin{array}{cc}{\tilde{s}}_{\mathrm{tx}}^{\left(r\right)}\left(t\right)=\sqrt{\frac{2}{T_{\mathrm{cs}}}}t\sqrt{E^{\left(r\right)}}{e}^{j2\pi f_{c}t}& \left(2.1a\right)\end{array}$$\begin{array}{cc}{\tilde{s}}_{\mathrm{tx}}^{\left(d\right)}\left(t\right)=\sqrt{\frac{2}{T_{\mathrm{cs}}}}t\left[{.Math.}_{k\in 𝒦}{x}_k^{\left(d\right)}{e}^{j2\pi f_{k}t}\right]e^{j2\pi f_{c}t},& \left(2.1b\right)\end{array}$
for −T.sub.cp≤t≤T.sub.s, where t is the M.sub.tx×1 unit-norm TX beamforming vector for this UE with |t|=1, x.sub.k.sup.(d) is the data signal at the k-th OFDM sub-carrier, j=√{square root over (−1)}, f.sub.c is the carrier/reference frequency, f.sub.k=k/T.sub.s represents the frequency offset of the k-th sub-carrier, T.sub.cs=T.sub.cp+T.sub.s and T.sub.s, T.sub.cp are the symbol duration and the cyclic prefix duration, respectively. Here we define the complex equivalent signal such that the actual (real) transmit signal is given by s.sub.tx.sup.(.Math.)(t)=Re{{tilde over (s)}.sub.tx(t)}. For the data symbols, we assume the use of Gaussian signaling with E.sub.k.sup.(d)={|x.sub.k|.sup.2}, for each k∈. The total average transmit OFDM symbol energy (including cyclic prefix) allocated to the UE is defined as E.sub.cs, where E.sub.cs≥E.sup.(r) and E.sub.cs≥E.sub.k.sup.(d). For convenience we also assume that f.sub.c is a multiple of 1/T.sub.cs, which ensures that the reference tone has the same initial phase in consecutive reference symbols.
The channel to the representative UE is assumed to be sparse with L resolvable MPCs (L«M.sub.tx, M.sub.rx), and the corresponding M.sub.rx×M.sub.tx channel impulse response matrix is given as (M. Akdeniz, Y. Liu, M. Samimi, S. Sun, S. Rangan, T. Rappaport, and E. Erkip, “Millimeter wave channel modeling and cellular capacity evaluation,” IEEE Journal on Selected Areas in Communications, vol. 32, pp. 1164-1179, June 2014):
H(t)=a.sub.rx()a.sub.tx().sup.†δ(t−),  (2.2)
where is the complex amplitude and is the delay and a.sub.tx(), a.sub.rx() are the TX and RX array response vectors, respectively, of the -th MPC. As an illustration, the -th RX array response vector for a uniform planar array with M.sub.H horizontal and M.sub.V vertical elements (M.sub.rx=M.sub.HM.sub.V) is given by a.sub.rx()=ā.sub.rx(ω.sub.azi.sup.rx(), ψ.sub.ele.sup.rx()), where we define:

(141) $\begin{array}{cc}{\stackrel{\_}{a}}_{\mathrm{rx}}\left({\psi }_{\mathrm{azi}}^{\mathrm{rx}},{\psi }_{\mathrm{ele}}^{\mathrm{rx}}\right)\stackrel{\Delta }{=}\left[\begin{array}{c}1\\ e^{j2\pi \frac{{\Delta }_H\sin \left({\psi }_{\mathrm{azi}}^{\mathrm{rx}}\right)\sin \left({\psi }_{\mathrm{ele}}^{\mathrm{rx}}\right)}{\lambda }}\\ e^{j2\pi \frac{{\Delta }_H\left(M_H-1\right)\sin \left({\psi }_{\mathrm{azi}}^{\mathrm{rx}}\right)\sin \left({\psi }_{\mathrm{ele}}^{\mathrm{rx}}\right)}{\lambda }}\end{array}\right].Math.\left[\begin{array}{c}1\\ e^{j2\pi \frac{{\Delta }_V\cos \left({\psi }_{\mathrm{ele}}^{\mathrm{rx}}\right)}{\lambda }}\\ e^{j2\pi \frac{{\Delta }_V\left(M_V-1\right)\cos \left({\psi }_{\mathrm{ele}}^{\mathrm{rx}}\right)}{\lambda }}\end{array}\right],& \left(2.3\right)\end{array}$
ψ.sub.azi.sup.rx(), ψ.sub.ele.sup.rx() are azimuth and elevation angles of arrival for the -th MPC, Δ.sub.H, Δ.sub.V are the horizontal and vertical antenna spacings and A is the wavelength of the carrier signal. Expressions for a.sub.tx() can be obtained similarly. Note that in (2.2) we implicitly assume frequency-flat MPC amplitudes {α.sub.0, . . . , α.sub.L−1} and ignore beam squinting effects (S. K. Garakoui, E. A. M. Klumperink, B. Nauta, and F. E. van Vliet, “Phased-army antenna beam squinting related to frequency dependency of delay circuits,” in European Microwave Conference, pp. 1304-1307, October 2011), which are reasonable assumptions for moderate system bandwidths. To prevent inter symbol interference, we also let the cyclic prefix be longer than the maximum channel delay: T.sub.cp>τ.sub.L−1. To model a time varying channel, we treat {,a.sub.tx(),a.sub.rx()} as aCSI parameters, that remain constant within an aCSI coherence time and may change arbitrarily afterwards. However since the channel is more sensitive to delay variations, the MPC delays {τ.sub.0, . . . , τ.sub.L−1} are modeled as iCSI parameters that only remain constant within a shorter interval called the iCSI coherence time. Note that this time variation of delays is an equivalent representation of the Doppler spread experienced by the RX. Finally, we do not assume any distribution prior or side information on {,a.sub.tx(),a.sub.rx(),}.

(142) The RX front-end is assumed to have a low noise amplifier followed by a band-pass filter at each antenna element that leaves the desired signal un-distorted but suppresses the out-of-band noise. The M.sub.rx×1 filtered complex equivalent received waveform for the 0-th symbol can then be expressed as:
{tilde over (s)}.sub.rx.sup.(.Math.)(t)=a.sub.rx()a.sub.tx().sup.†{tilde over (s)}.sub.tx.sup.(.Math.)(t−)+√{square root over (2)}{tilde over (w)}.sup.(.Math.)(t)e.sup.j2πf.sup.c.sup.t  (2.4)
for 0≤t≤T.sub.s, where (.Math.)=(r)/(d), {tilde over (w)}.sup.(.Math.)(t) is the M.sub.rx×1 complex equivalent, baseband, stationary, additive, vector Gaussian noise process, with individual entries being circularly symmetric, independent and identically distributed (i.i.d.), and having a power spectral density: .sub.w(f)=N.sub.0 for −f.sub.K.sub.1≤f≤f.sub.K.sub.2. During the data transmission phase, the M.sub.rx×1 received data waveform {tilde over (s)}.sub.rx.sup.(d)(t) is phase shifted by a bank of phase-shifters, whose outputs are summed and fed to a down-conversion chain for data demodulation, as in conventional analog beamforming. However unlike conventional CE based analog beamforming, the control signals to the phase-shifters are obtained using the reference symbols {tilde over (s)}.sub.rx.sup.(r)(t) and using PACE, as shall be discussed in the next section.

(143) III. Analog Beamformer Design at the Receiver

(144) During each beamformer design phase, the BS transmits D consecutive reference symbols to facilitate PACE at the RX. This process involves two steps: locking a local RX oscillator to the received reference tone and using this locked oscillator to estimate the amplitude and phase-offsets at each antenna. Here locking refers to ensuring that the phase difference between the oscillator and the received reference tone is approximately constant. The first D.sub.1 reference symbols are used for the former step and the remaining D.sub.2=D−D.sub.1 symbols are used for the latter step. Therefore D is independent of M.sub.rx and is mainly determined by the time required for oscillator locking (see Remark 3.1). The first step shall be referred to as recovery of the reference tone and is analyzed in Section 3.1 and while the latter step is discussed in Section 3.2. As shall be shown both steps are significantly impaired by channel noise. Therefore in Section 3.3, we propose an improved architecture for reference tone recovery that provides better noise performance, albeit with a slightly higher hardware complexity. For convenience, we shall assume that the MPC delays do not change within the beamformer design phase, and are represented as {{circumflex over (τ)}.sub.0, . . . , {circumflex over (τ)}.sub.L−1} (see also Remark 3.2). However the delays may be different during the data transmission phase, as shall be considered in Section 4. Without loss of generality, assuming the first reference symbol to be the 0-th OFDM symbol, the complex equivalent RX signal for the D reference symbols at antenna m can be expressed as:

(145) $\begin{array}{cc}{\tilde{s}}_{\mathrm{rx},m}^{\left(r\right)}\left(t\right)=\sqrt{2}{A}_m^{\left(r\right)}{e}^{j2\pi f_{c}t}+\sqrt{2}{\tilde{w}}_m^{\left(r\right)}\left(t\right)e^{j2\pi f_{c}t}\mathrm{for}0\le t\le {\mathrm{DT}}_{\mathrm{cs}}-T_{\mathrm{cp}},\mathrm{where}{A}_m^{\left(r\right)}\stackrel{\Delta }{=}{.Math.}_{\ell =0}^{L-1}\sqrt{\frac{1}{T_{\mathrm{cs}}}}{{\alpha }_{\ell }\left[a_{\mathrm{rx}}\left(\ell \right)\right]}_m{a_{\mathrm{tx}}\left(\ell \right)}^{†}t\sqrt{E^{\left(r\right)}}{e}^{-j2\pi f_c{\tau }_{\ell }}& \left(2.5\right)\end{array}$
is the amplitude of the reference tone at antenna m.