Exploitation of pilot signals for blind resilient detection and geo-observable estimation of navigation signals

Abstract

A method and apparatus detects and estimates geo-observables of navigation signals employing civil formats with repeating baseband signal components, i.e., “pilot signals,” including true GNSS signals generated by satellite vehicles (SV's) or ground beacons (pseudolites), and malicious GNSS signals, e.g., spoofers and repeaters. Multi-subband symbol-rate synchronous channelization can exploit the full substantive bandwidth of the GNSS signals with managed complexity in each subband. Spatial/polarization receivers can be provided to remove interference and geolocate non-GNSS jamming sources, as well as targeted GNSS spoofers that emulate GNSS signals. This can provide time-to-first-fix (TTFF) over much smaller time intervals than existing GNSS methods; can operate in the presence of signals with much wider disparity in received power than existing techniques; and can operate in the presence of arbitrary multipath.

Claims

1. A method, comprising: channelizing a received navigation signal into a plurality of subband signals, each subband signal comprising a plurality of frequency channels; for each subband signal, computing linear combiner weights for the plurality of frequency channels based on one or more exploitable symbol stream properties of the received navigation signal; using the linear combiner weights to combine the frequency channels and excise interference in the each subband signal, thereby increasing signal-to-noise-and-interference of the each subband signal; and combining the plurality of subband signals to produce at least one of a detection statistic and a geo-observable estimate of the received navigation signal.

2. The method of claim 1, wherein channelizing employs an analog-to-digital converter (ADC) configured to perform symbol-rate synchronous reception and channelization of the received navigation signal.

3. The method of claim 1, wherein channelizing employs an analog-to-digital converter sampling rate that is equal to an integer multiple of the navigation signal's baseband symbol rate.

4. The method of claim 1, wherein channelizing employs an analog-to-digital converter sampling rate that is less than the received navigation signal's bandwidth.

5. The method of claim 1, wherein channelizing employs a fast Fourier transform (FFT), and FFT output bins are employed as the plurality of frequency channels.

6. The method of claim 1, wherein an interference-excising linear algebraic combiner is configured to perform at least one of using the linear combiner weights to combine the frequency channels and combining the plurality of subband signals.

7. The method of claim 1, wherein channelizing comprises varying the plurality of frequency channels in order to vary a number of processing degrees of freedom in the each subband signal.

8. The method of claim 1, wherein the plurality of frequency channels is selected to be equal to or greater than a number of interferers in the each subband signal.

9. The method of claim 1, wherein the one or more exploitable symbol stream properties comprises at least one of a periodic signal, known content in the symbol stream, a known pilot, a known modulation property, and a constant-modulus structure.

10. The method of claim 1, wherein the plurality of subband signals are contiguous or separated in frequency, and the plurality of frequency channels in the each subband signal are contiguous or separated in frequency.

11. The method of claim 1, further comprising using known positions of transmitters that transmit the at least one navigation signal in order to determine at least one of clock rate offset and ephemeris of a receiver that performs the method.

12. The method of claim 1, further comprising estimating geo-observables from the interference, and geolocating one or more sources of the interference from the geo-observables.

13. The method of claim 1, further comprising communicatively coupling to a data service or at least one navigation signal receiver for receiving third-party baseband symbol data or navigation data.

14. The method of claim 1, further comprising computing positioning, navigation, and timing (PNT) analytics using the at least one geo-observable estimate, wherein the PNT analytics comprises at least one of blind Resilient PNT (RPNT) analytics, non-blind RPNT analytics, pilot-exploiting RPNT, rate synchronization, geolocation, navigation signal geo-observable estimates, navigation signal quality estimates, accuracy of the geo-observable estimates, and accuracy of the navigation signal quality estimates.

15. The method of claim 14, wherein the computing performs analytics over integration times that are longer than a single baseband symbol, navigation symbol, or pilot period.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) Block diagrams and flow diagrams depicted herein can represent computer software instructions or groups of instructions. One or more of the processing blocks or steps may be implemented with modules (e.g., procedures, functions, and so on) that perform the functions described herein. The software codes may be stored in non-transitory computer-readable memory. Alternatively, processing blocks or steps described herein may represent steps performed by hardware, including one or more functionally equivalent circuits, such as a digital signal processor, an application specific integrated circuit, a programmable logic device, a field programmable gate array, a general-purpose processor programmed to perform one or more of the disclosed steps, or other electronic units designed to perform the functions disclosed herein. Disclosed aspects can employ multiple processors communicatively coupled together, and can employ distributed computing, such as Cloud computing, cluster computing, distributed computing, etc.

(2) FIG. 1 depicts a system in which aspects of the disclosure can be implemented.

(3) FIG. 2 illustrates operating parameters and geo-observables that can be employed in aspects disclosed herein.

(4) FIG. 3 is a block diagram that depicts generation of civil navigation signals that aspects of the disclosure can be configured to exploit.

(5) FIG. 4 is a Table of civil GNSS modulation formats that aspects of the disclosure can be configured to exploit.

(6) FIG. 5A, FIG. 5B, FIG. 6A, and FIG. 6B are block diagrams of receiver front-ends that can be configured according to aspects of the disclosure.

(7) FIG. 7 is a block diagram that depicts a single-feed channelizer in accordance with an aspect of the disclosure.

(8) FIG. 8 is a block diagram that depicts a single-feed FFT-based channelizer in accordance with an aspect of the disclosure.

(9) FIG. 9 is a block diagram of a single-feed FFT-based symbol-rate synchronous multi-subband channelizer according to an aspect of the disclosure.

(10) FIG. 10 is a block diagram of a single-feed polyphase filter based symbol-rate synchronous multi-subband channelizer according to an aspect of the disclosure.

(11) FIG. 11 is a block diagram of single-feed symbol-rate synchronous multi-subband channelizer according to another aspect of the disclosure.

(12) FIG. 12A and FIG. 12B depict exemplary channelizer metrics for a rectangularly windowed polyphase filter based symbol-rate synchronous channelizer.

(13) FIG. 13 is a block diagram that depicts a multifeed receiver and channelizer in accordance with an aspect of the disclosure.

(14) FIG. 14 is a block diagram of a multifeed FFT-based symbol-rate synchronous multi-subband channelizer according to an aspect of the disclosure.

(15) FIG. 15 is a block diagram of a multifeed polyphase filter based symbol-rate synchronous multi-subband channelizer according to an aspect of the disclosure.

(16) FIG. 16 and FIG. 17 illustrate despreader parameters that can be employed in aspects disclosed herein.

(17) FIG. 18 is a flow diagram of a multi-subband signal detection, geo-observable/quality estimation, and symbol estimation algorithm that can be employed in aspects disclosed herein.

(18) FIG. 19 is a flow diagram of a multi-subband signal detection and geo-observable/quality estimation, algorithm that can be employed in aspects disclosed herein.

(19) FIG. 20 is a flow diagram of a multi-subband signal detection and geo-observable/quality estimation, algorithm that can be employed in aspects disclosed herein.

(20) FIG. 21 is a flow diagram of a local-maxima search procedure according to an aspect of the disclosure.

(21) FIGS. 22A, 22B, and 22C and FIGS. 23A, 23B, and 23C FIG. 22 and FIG. 23 are plots that depict DFOA search sensitivity.

DETAILED DESCRIPTION

(22) Various aspects of the disclosure are described below. It should be apparent that the teachings herein may be instantiated in a wide variety of forms and that any specific structure, function, or both being disclosed herein are merely representative. Based on the teachings herein one skilled in the art should appreciate that an aspect disclosed herein may be implemented independently of any other aspects and that two or more of these aspects may be combined in various ways. For example, an apparatus may be implemented or a method may be practiced using any number of the aspects set forth herein. In addition, such an apparatus may be implemented or such a method may be practiced using other structure, functionality, or structure and functionality in addition to or other than one or more of the aspects set forth herein.

(23) In the following description, for the purposes of explanation, numerous specific details are set forth in order to provide a thorough understanding of the invention. It should be understood, however, that the particular aspects shown and described herein are not intended to limit the invention to any particular form, but rather, the invention is to cover all modifications, equivalents, and alternatives falling within the scope of the invention as defined by the claims.

(24) FIG. 1 depicts a model of the presumed normal condition wherein a receiver (1) is receiving GPS signals from an SV (3).

(25) FIG. 2 Illustrates exemplary observable parameters of signals transmitted from a GNSS SV (3) to a GNSS receiver (1) that can be used to geo-locate those signals (“geo-observables”) (6), as a function of parameters local to the GNSS SV (4) and the GNSS receiver (2). These geo-observables (6) include the time-of-arrival (TOA) (after removal of delays induced in transmitter and receiver electronics), received incident power (RIP), line-of-bearing (LOB), direction-of-arrival (DOA) relative to the platform orientation, frequency-of-arrival (FOA) (after removal of known frequency offset between the transmitter and receiver center frequencies), and carrier phase (CCP) (e.g., after calibration and removal of transmitter and receiver phase offsets), which are measurable and differentiable at the receiver (1).

(26) The TOA, RIP, FOA, LOB, DOA, and CCP are referred to as geo-observable, as they are observable parameters of the GNSS received signals that can be used to geo-locate the GNSS receiver, given knowledge of the GNSS transmitter locations over time, which are transmitted within the navigation stream of those signals. They are also referred to here as positioning, navigation, and timing (PNT) analytics, as they provide metadata of the transmitted signals that can be used for purposes beyond geo-location of the receiver, e.g., to assess quality and reliability of the received GNSS signals, in order to aid other PNT systems on board the receiver platform.

(27) FIG. 3 depicts operations used to generate the civil navigation signals that are the focus of the invention. It should be noted that the processing steps shown in FIG. 3 is one conceptual means for generating civil GNSS signals, and ignore other SiS's that may be transmitted simultaneously with civil GNSS signals, for example, the GPS P, P(Y), M, and L1C SiS's. In practice, these other SiS's are largely outside the band occupied by the civil GNSS signals, and moreover are received well below the noise floor. Therefore, their effect on aspects described herein is small. An important exception is military GNSS signals that may be generated by terrestrial pseudolites, as the non-civil components of those signals may be above the noise floor within the civil GNSS passband in some use scenarios.

(28) The processing steps shown in FIG. 3 generate a particular class of spread spectrum signal, referred to herein as a modulation-on-symbol direct-sequence spread spectrum (MOS-DSSS) signal, in which a data symbol sequence d.sub.T(n.sub.sym;l) with symbol rate f.sub.sym is spread by a M.sub.chp×1 dimensioned ranging code

(29) $c_T\left(\ell \right)={\left[c_T\left(m_{\mathrm{chp}};\ell \right)\right]}_{m_{\mathrm{chp}}=0}^{M_{\mathrm{chp}}-1}$
that is repeated over every data symbol. The spread signal can be represented as the multiplication of data symbol sequence d.sub.T(n.sub.sym;l) and code vector c.sub.T(l) (115), resulting in M.sub.chp×1 c.sub.T (l)d.sub.T(n.sub.sym;l). The vector signal is then passed through a M.sub.chp:1 parallel-to-serial (P/S) converter (116), resulting in a scalar spread signal with rate M.sub.chpf.sub.sym, which can be expressed as c.sub.T(n.sub.chp mod M.sub.chp;l)d.sub.T(└n.sub.chp/M.sub.chp┘;l) where (⋅)mod M and └⋅┘ denote the modulo-M operation and integer truncation or “floor” operations, respectively. As shall be described below, this property induces inherent massive spectral redundancy to the civil GNSS format that is exploited by disclosed aspects.

(30) The chip-rate spread signal is then passed through digital-to-analog conversion (DAC) (118), and transmission/upconversion stages (120), controlled by an internal clock (clk) (117) and local-oscillator (LO) (119) stages, and transmitted from an antenna (121) to generate a signal-in-space (SiS) given by √{square root over (2P.sub.T(l))}Re{s.sub.T(t−τ.sub.T(l);l)e.sup.j(2πd.sup.T.sup.t+φ.sup.T.sup.(l))} for transmitter l (122), where f.sub.T(l) is the transmit frequency, typically common to all SV's (f.sub.T(l)≡f.sub.T), P.sub.T(l), φ.sub.T(l) and τ.sub.T(l) are the transmit power, phase, and electronics delay for SiS l, respectively, and where s.sub.T(t;l) is the complex baseband representation of the transmitted signal.

(31) Based on these operations, the complex-baseband signal can be modeled by

(32) $\begin{array}{cc}{s}_T\left(t;\ell \right)=\underset{n_{\mathrm{sym}}}{.Math.}{d}_T\left(n_{\mathrm{sym}};\ell \right)h_T\left(t-T_{\mathrm{sym}}{n}_{\mathrm{sym}};\ell \right),& \mathrm{Eqn}\left(1\right)\end{array}$
where d.sub.T(n.sub.sym;l) is the encoded symbol stream, T.sub.sym=1/f.sub.sym, is the symbol period, and h.sub.T(t;l) is a spread symbol given by

(33) $\begin{array}{cc}\begin{array}{cc}{h}_T\left(t;\ell \right)=\stackrel{M_{\mathrm{chp}}-1}{\underset{m_{\mathrm{chp}}=0}{.Math.}}{c}_T\left(m_{\mathrm{chp}};\ell \right)g_T\left(t-T_{\mathrm{chp}}{m}_{\mathrm{chp}};\ell \right),& T_{\mathrm{chp}}=\frac{T_{\mathrm{sym}}}{M_{\mathrm{chp}}},\end{array}& \mathrm{Eqn}\left(2\right)\end{array}$
and where g.sub.T (t;l) is the chip symbol modulated by each chip, referred to here as the chip shaping. The spread symbol can also be expressed by it's continuous Fourier transform H.sub.T(f;l)=∫h.sub.T(f;l)e.sup.−j2πft df,
H.sub.T(f;l)=C.sub.T(e.sup.j2πfT.sup.chp;l)G.sub.T(f;l),  Eqn (3)
where

(34) $C_T\left(e^{j2\pi f};\ell \right)=\underset{m_{\mathrm{chp}}}{.Math.}{c}_T\left(m_{\mathrm{chp}}\right)e^{-j2\pi {\mathrm{fm}}_{\mathrm{chp}}}$
is the discrete Fourier transform of C.sub.T(m.sub.chp;l), and where G.sub.T(f;l) is the continuous Fourier transform of g.sub.T(t;l). The spread signal can therefore be modeled as a pulse-amplitude modulated (PAM) waveform in which baseband symbol sequence modulates a symbol waveform that is itself direct-sequence spread by the ranging code, leading to the MOS-DSSS nomenclature employed here.

(35) In typical navigation systems, the chip shaping is identical for each transmitted signal (g.sub.T(t;l)≡g.sub.T(t)); however, in practice, the shaping can also vary between transmitters. For typical navigation systems, the chip shaping is also nominally rectangular, such that G.sub.T(f;l)≡T.sub.chpSa(πfT.sub.chp), where Sa(x)=sin(x)/x; however, different chip shapings can be employed without changing the basic structure of the MOS-DSSS modulation format.

(36) In typical navigation systems, the spreading code C.sub.T(l) employed at transmitter l is unique to that transmitter, and is in fact provisioned by the navigation system. In contrast, the transmit power and center frequency are typically (but not necessarily) identical for each transmitter, while the transmitter carrier phase, delay, and other impairments will vary between the transmitters. However, any of these typical assumptions can be varied in practice, or in different instantiations of a navigation system.

(37) As shown in FIG. 3, the full symbol sequence d.sub.T(n.sub.sym;l) is generated from an underlying navigation sequence b.sub.T(n.sub.NAV;l)|.sub.NAV with navigation rate f.sub.NAV, which contains information needed to generate a full positioning, navigation, and timing (PNT) solution at navigation receiver. The navigation sequence typically provides timing and ephemeris (time-stamped position, velocity, and acceleration) of the navigation transmitter and larger network, known impairments at the transmitter, and additional information needed to determine quality and timing of the transmitted signal. The navigation sequence also typically provides information about the specific spreading code(s) employed at the transmitter. This navigation sequence is then passed through processing operations to generate an encoded signal sequence b.sub.T(n.sub.sym;l)|.sub.sym, with symbol rate f.sub.sym and for some modulation formats combined with a known pilot sequence p.sub.T(n.sub.sym;l), with symbol rate f.sub.sym (112b) to generate the full symbol sequence signal d.sub.T(n.sub.sym;l). All of these additional processing steps add additional known structure to the transmitted MOS-DSSS signal, which can be exploited in subsequent processing stages. Additionally, the navigation sequence, the information contained within that sequence (e.g., the transmitter ephemeres) can be obtained after transmission from third-party sources, providing additional exploitable structure in applications some aspects of the invention.

(38) FIG. 4 is a tabular description of GNSS networks employing civil GNSS signals that can be modeled as MOS-DSSS signals. As this Figure shows, every GNSS network deployed to date possesses at least one signal type that can be modeled by FIG. 3. Moreover, all of these signals have a 1 millisecond (ms) ranging code period, and therefore a 1 ksps (kilosample per second) MOS-DSSS baseband symbol rate, albeit with widely varying chip rates and SiS bandwidths, allowing their demodulation using a common receiver structure.

(39) As the last column of FIG. 4 shows, the symbol streams of each of these signals possess properties that can be used to detect and differentiate them from other signals in the environment, including military GNSS signals, jammers, and civil GNSS signals with the same or similar properties. For example, the GPS coarse acquisition (C/A) or “legacy” signal listed in the first row of FIG. 4 is constructed from a 50 bit/second (bps) BPSK (real) navigation sequence b.sub.T(n.sub.NAV;l)|.sub.NAV, repeated 20 times per MOS-DSSS symbol to generate 1 ksps symbol sequence d.sub.T(n.sub.sym;l)=b.sub.T(n.sub.sym;l)|.sub.sym=b.sub.T(└n.sub.sym/M.sub.NAV┘;l)|.sub.NAV where M.sub.NAV=20. Moreover, b.sub.T(n.sub.NAV;l)|.sub.NAV possesses internal fields that can be further exploited to aid the demodulation process, e.g., the TLM Word Preamble transmitted within every 300-bit Navigation subframe.

(40) Similarly, the GPS L5, QZSS, Galileo E5B, E6B, and Beidou 1B transmitters all add a periodic pilot to the quadrature rail of their navigation signals, and the NAVIC RS BOC transmitter adds a periodic pilot to the in-phase rail of its navigation signal. Means for exploiting periodic pilots are a focus of some aspects of this invention.

(41) FIG. 5A and FIG. 5B each depict exemplary direct-conversion front-end reception processing for a single-feed receiver receiving GNSS signals. FIG. 5A shows a direct-conversion receiver front-end that is dedicated to a specific GNSS band, and FIG. 5B shows a direct-conversion receiver front-end that can be flexibly tasked to any GNSS band, consistent with a software defined radio (SDR) architecture. Each can be used as a part of, and in, some of the disclosed aspects.

(42) The direct-conversion receivers shown in FIG. 5A and FIG. 5B each comprise the following: An antenna (128) and low-noise amplifier (LNA) (131), which receives an SiS containing GNSS emissions within the FoV of the receiver. A local oscillator (LO) (133), which generates a complex sinusoid with frequency f.sub.R and phase φ.sub.R, comprising an in-phase (I) real sinusoid, and a quadrature (Q) sinusoid shifted by 90° in phase from the in-phase sinusoid; A complex mixer (134), which multiplies the LNA output signal by the conjugate of the complex sinusoid, thereby creating a complex signal with real and imaginary parts, referred to herein as the in-phase (I) and quadrature (Q) rails of that signal, which has been downconverted by frequency shift f.sub.R A dual lowpass filter (LPF) (135), which filters each signal rail to remove unwanted signal energy from the complex downconverted signal, yielding analog scalar complex-baseband signal x.sub.R(t), centered at 0 MHz if f.sub.R=f.sub.T. A dual analog-to-digital converter (ADC) (136), which samples each rail of x.sub.R(t) at rate f.sub.ADC=M.sub.ADCf.sub.sym, where f.sub.sym is the civil GNSS signal symbol rate shown in FIG. 3 and M.sub.ADC is a positive integer, thereby generating complex dual-ADC output signal x.sub.ADC(n.sub.ADC) with time index n.sub.ADC (137).
The receiver structure shown in FIG. 5A possesses an additional bandpass filter (BPF) (130), inserted between the antenna and LNA, and dedicated to a specific GNSS band, which removes unwanted adjacent-channel interference (ACI) from the received GNSS SiS, and depicts an LO that is locked to specific, preset value, consistent with a dedicated receiver that is optimized for a specific GNSS band. This structure is more typical of existing GNSS receivers, and provides enhanced signal quality at cost of receiver flexibility.

(43) In contrast, the receiver structure shown in FIG. 5B is consistent with SDR receivers that can be tasked to any GNSS band under user control. This receiver structure can also be used to receive a segment of a GNSS band, e.g., a 1 MHz segment of the GPS L5 band, if f.sub.R is significantly offset from the GNSS band center f.sub.T. The ability to blindly detect, demodulate, and estimate geo-observables of any portion of any GNSS band using the same reception, downconversion, and sampling hardware regardless of the actual bandwidth of the GNSS signals present in that band can be an advantageous feature of at least some of the disclosed aspects.

(44) FIG. 6A and FIG. 6B each depict exemplary front-end reception processing for a single-feed superheterodyne receiver receiving GNSS signals; with FIG. 6A showing a superheterodyne receiver front-end that is dedicated to a specific GNSS band, and FIG. 6B showing a superheterodyne receiver front-end that can be flexibly tasked to multiple GNSS bands. Each can be used as a part of, and in, at least some aspects disclosed herein.

(45) The heterodyne receivers shown in FIG. 6A and FIG. 6B shift the SiS received by an antenna (128) and amplified in an LNA (130) to IF frequency f.sub.IF, typically using a two-stage mixer 134, 141, yielding analog scalar real-IF signal x.sub.R(t) centered on IF frequency f.sub.IF if f.sub.R=f.sub.T, with receive phase-shift φ.sub.R=φ.sub.R.sup.(1)+φ.sub.R.sup.(2). The analog signal is then passed to an ADC (143) that samples x.sub.R(t) at rate M.sub.ADCf.sub.sym, where f.sub.sym is the MOS-DSSS signal symbol rate shown in FIG. 6A and FIG. 6B, and M.sub.ADC is a positive integer, yielding real ADC output signal X.sub.ADC(n.sub.ADC) with time index n.sub.ADC (137). The receiver structure shown in FIG. 6A can be optimized for a specific GNSS band, while the receiver structure shown in FIG. 6B can be flexibly tasked to different navigation signal bands, or to segments of such bands, under software control.

(46) In all four cases, the receiver can induce additional delays, frequency offsets, and sampling rate offsets due to internal electronics in the receiver systems, and due to clock rate and timing offset from universal time coordinates (UTC). These are subsumed into the overall channel response, and measured and removed as part of the geo-location solution.

(47) As will be discussed below, the ADC output signal provided by either the direct-conversion or the superheterodyne receivers can be used in disclosed aspects without any change to the design, except control parameters used in channelization operations immediately after that ADC. The disclosed aspects can also be practiced with many other receiver front-ends known to the art. In many cases, the disclosed aspects are blind to substantive differences between those receiver front-ends.

(48) Assuming the reception scenario shown in FIG. 2, and the direct-conversion receiver structure shown in FIG. 5, the downconverted and filtered signal shown in x.sub.R(t) is given by

(49) $\begin{array}{cc}{x}_R\left(t\right)=i_R\left(t\right)+\stackrel{L}{\underset{\ell =1}{.Math.}}{s}_R\left(t;\ell \right),& \mathrm{Eqn}\left(4\right)\end{array}$
where i.sub.R(t) is the noise and interference received in the receiver passband and s.sub.R(t;l) is the complex-baseband representation of the signal received from transmitter l. Over time intervals on the order of 1-10 seconds from reception time t.sub.0, s.sub.R(t;l) can be modeled by

(50) $\begin{array}{cc}{s}_R\left(t_0+t;\ell \right)\approx g_{\mathrm{TR}}\left(\ell \right)e^{j2\pi \left({\alpha }_{\mathrm{TR}}\left(\ell \right)t+\frac{1}{2}\alpha \begin{array}{c}\left(1\right)\\ \mathrm{TR}\end{array}\left(\ell \right)t^2\right)}{s}_T\left(\left(1+\tau \begin{array}{c}\left(1\right)\\ \mathrm{TR}\end{array}\left(\ell \right)\right)t-{\tau }_{\mathrm{TR}}\left(\ell \right);\ell \right),& \mathrm{Eqn}\left(5\right)\end{array}$
at the input to the dual-ADC's shown in FIG. 5, where

(51) $\begin{array}{cc}{g}_{\mathrm{TR}}\left(\ell \right)\stackrel{△}{=}\sqrt{P_R\left(t_0;\ell \right)}\exp \left(j{\phi }_{\mathrm{TR}}\left(\ell \right)\right),\mathrm{end}‐\mathrm{to}‐\mathrm{end}\mathrm{complex}\mathrm{channel}\mathrm{gain}& \mathrm{Eqn}\left(6\right)\end{array}$$\begin{array}{cc}{\phi }_{\mathrm{TR}}\left(\ell \right)\stackrel{△}{=}{\phi }_T\left(\ell \right)-{\phi }_R-2\pi f_T\left(\ell \right){\tau }_{\mathrm{TR}}\left(t_0;\ell \right),\mathrm{end}‐\mathrm{to}‐\mathrm{end}\mathrm{channel}\mathrm{phase}& \mathrm{Eqn}\left(7\right)\end{array}$$\begin{array}{cc}\begin{array}{cc}{\tau }_{\mathrm{TR}}\left(\ell \right)\stackrel{△}{=}{\tau }_T\left(\ell \right)+{\tau }_R+{\tau }_{\mathrm{TR}}^{\left(0\right)}\left(t_0;\ell \right),& \mathrm{end}‐\mathrm{to}‐\mathrm{end}\mathrm{observed}TOA\end{array}& \mathrm{Eqn}\left(8\right)\end{array}$$\begin{array}{cc}\begin{array}{cc}{\alpha }_{\mathrm{TR}}\left(\ell \right)\stackrel{△}{=}{f}_T\left(\ell \right)+f_R-f_T\left(\ell \right){\tau }_{\mathrm{TR}}^{\left(1\right)}\left(t_0;\ell \right),& \mathrm{end}‐\mathrm{to}‐\mathrm{end}\mathrm{observed}FOA\end{array}& \mathrm{Eqn}\left(9\right)\end{array}$$\begin{array}{cc}\begin{array}{cc}{\tau }_{\mathrm{TR}}^{\left(1\right)}\left(\ell \right)\stackrel{△}{=}{\tau }_{\mathrm{TR}}^{\left(1\right)}\left(t_0;\ell \right),& \mathrm{observed}\mathrm{differential}TOA\left(DTOA\right)\end{array}& \mathrm{Eqn}\left(10\right)\end{array}$$\begin{array}{cc}\begin{array}{cc}{\alpha }_{\mathrm{TR}}^{\left(1\right)}\left(\ell \right)\stackrel{△}{=}-f_T\left(\ell \right){\tau }_{\mathrm{TR}}^{\left(2\right)}\left(t_0;\ell \right),& \mathrm{observed}\mathrm{differential}FOA\left(DFOA\right).\end{array}& \mathrm{Eqn}\left(11\right)\end{array}$
and where φ.sub.T(l), f.sub.T(l), and τ.sub.T(l) are the carrier phase, frequency, and timing offset induced at transmitter l; φ.sub.R, f.sub.R, and τ.sub.R are the carrier phase, frequency, and timing offset induced at the receiver; P.sub.R(t.sub.0;l) is the received incident power of SiS l at the receiver; and τ.sub.TR.sup.(q)(t.sub.0;l)=dτ.sub.TR(t;l)/dt.sup.q is the q.sup.th derivative of the observed TOA between transmitter l and the receiver at time t. Given observed position, velocity, and acceleration p.sub.TR(t;l)=p.sub.T(t;l)−p.sub.R(t;l), v.sub.TR(t;l)=v.sub.T(t;l)−v.sub.R(t;l), and a.sub.TR(t;l)=a.sub.T(t;l)−a.sub.R(t;l), where {p.sub.T(t;l),v.sub.T(t;l),a.sub.T(t;l)}.sub.l=1.sup.L and {p.sub.R(t),v.sub.R(t),a.sub.R(t)} are the transmitter and receiver ephemeres, respectively, then τ.sub.TR.sup.(q)(t;l) is given by

(52) $\begin{array}{cc}{\tau }_{\mathrm{TR}}^{\left(0\right)}\left(t;\ell \right)=\frac{1}{c}{.Math.p_{\mathrm{TR}}\left(t;\ell \right).Math.}_2,& \mathrm{Eqn}\left(12\right)\end{array}$$\begin{array}{cc}{\tau }_{\mathrm{TR}}^{\left(1\right)}\left(t;\ell \right)=\frac{1}{c}{u}_{\mathrm{TR}}^T\left(t;\ell \right)v_{\mathrm{TR}}\left(t;\ell \right),& \mathrm{Eqn}\left(13\right)\end{array}$$\begin{array}{cc}{\tau }_{\mathrm{TR}}^{\left(2\right)}\left(t;\ell \right)=\frac{1}{c}\left(u_{\mathrm{TR}}^T\left(t;\ell \right)a_{\mathrm{TR}}\left(t;\ell \right)+{.Math.P_{\perp }\left(p_{\mathrm{TR}}\left(t;\ell \right)\right)v_{\mathrm{TR}}\left(t;\ell \right).Math.}_2\right),& \mathrm{Eqn}\left(14\right)\end{array}$
where u.sub.TR(t;l) is the line-of-bearing (LOB) direction vector from the receiver to transmitter l,

(53) $\begin{array}{cc}{u}_{\mathrm{TR}}\left(t;\ell \right)=\frac{p_{\mathrm{TR}}\left(t;\ell \right)}{{.Math.p_{\mathrm{TR}}\left(t;\ell \right).Math.}_2},& \mathrm{Eqn}\left(15\right)\end{array}$
and P.sub.⊥(p.sub.TR(t;l)) is the projection matrix orthogonal to p.sub.TR(t;l), given by

(54) 0$\begin{array}{cc}{P}_{\perp }\left(p_{\mathrm{TR}}\left(t;\ell \right)\right)=I_3-\frac{p_{\mathrm{TR}}\left(t;\ell \right)p_{\mathrm{TR}}^T\left(t;\ell \right)}{{.Math.p_{\mathrm{TR}}\left(t;\ell \right).Math.}_2^2},& \mathrm{Eqn}\left(16\right)\end{array}$
and where ∥⋅∥.sub.2 and I.sub.3 denote the L2 Euclidean vector norm and 3×3 identity matrix, respectively. As shown in the '417 NPA, incorporated herein by reference, the RIP and LOB adheres closely to a zero-order model over time intervals on the order of 1-to-10 seconds, for transmitters in medium-Earth orbits and receivers with dynamics commensurate with high-velocity airborne vehicles.

(55) The local direction-of-arrival (DOA) of the signal received from transmitter l can then be represented by local DOA direction vector u.sub.R(t;l)=Ψ.sub.R(t)u.sub.TR(t;l), which can be converted to azimuth relative to the local receiver heading and elevation relative to the plane of receiver motion using well-known directional transformations. As shown in the '417 NPA, incorporated herein by reference, the DOA also adheres closely to a first-order model over time intervals on the order of 1-to-10 seconds, for transmitters in medium-Earth orbits and receivers with dynamics commensurate with high-velocity airborne vehicles. Moreover, the variation in local DOA is nearly identical for all of the received navigation signals under this scenario, as it primarily due to motion/orientation of the receiver, which equally affects all of the signal LOB's.

(56) FIG. 7 shows a single-feed receiver structure used in one aspect of the disclosure, describing the symbol-rate synchronous vector channelizer (150) integral to the disclosure. A SiS is received at an antenna (128), and downconverted to complex baseband using a direct-conversion receiver (100) implementing operations such as those shown in FIG. 5A and FIG. 5B, comprising reception, frequency downconversion by receive frequency f.sub.R to complex baseband representation, and sampling using a dual ADC (136) with sampling rate M.sub.ADCf.sub.sym, where M.sub.ADC is a positive integer. The sampled signal is then passed to a symbol-rate synchronous channelizer (150) comprising the following conceptual operations: A 1:M.sub.ADC serial-to-parallel conversion operation that transforms the rate M.sub.ADCf.sub.sym ADC output signal to an M.sub.ADC×1 vector signal x.sub.sym(n.sub.sym) with rate f.sub.sym, given by

(57) $\begin{array}{cc}{x}_{\mathrm{sym}}\left(n_{\mathrm{sym}}\right)=\left(\begin{array}{c}{x}_R\left(T_{\mathrm{sym}}{n}_{\mathrm{sym}}\right)\\ .Math.\\ x_R\left(T_{\mathrm{sym}}\left(n_{\mathrm{sym}}+\frac{m_{\mathrm{ADC}}^{-1}}{M_{\mathrm{ADC}}}\right)\right)\end{array}\right).& \mathrm{Eqn}\left(17\right)\end{array}$ An M.sub.ADC:K.sub.chn channelization operation (155), conceptually performed by operating on x.sub.sym(n.sub.sym) using K.sub.chn×M.sub.ADC channelization matrix operator T.sub.chn (153) to form K.sub.chn×1 channelizer output signal x.sub.chn(n.sub.sym)=T.sub.chn∘x.sub.sym(n.sub.sym) (159), where K.sub.chn is the symbol-rate synchronous output channels used in subsequent DSP operations and “∘” denotes a general filtering operation. In some aspects of the invention, T.sub.chn (153) is a memoryless matrix and the operation is a simple matrix multiplication. In other aspect of the invention, T.sub.chn (153) is a more linear-time-invariant (LTI) impulse response, and the operation is a matrix convolution.
The data rate of x.sub.chn(n.sub.sym) is nominally equal to the symbol rate of the navigation signal of interest to the system (less offset due to nonideal error in the ADC clock), e.g., 1 ksps for the signals listed in FIG. 4, and the dimensionality of x.sub.chn(n.sub.sym) is K.sub.chn×1, where K.sub.chn Is the number of frequency channels used by subsequent DSP processing stages. Hence this is referred to here as a symbol-rate synchronous vector channelizer.

(58) A variety of different channelization methods can be implemented using this general processing structure, including time and frequency domain channelization methods, or channelizations employing mixed time-frequency channelizations such as wavelet-based channelizers.

(59) FIG. 8 depicts a specific instantiation of the symbol-rate synchronous vector channelizer (150), using Fast-Fourier Transform (FFT) operations. The channelizer first applies a (nominally zero-padded) discrete Fourier transform (DFT) with M.sub.ADC input samples, K.sub.FFT≥M.sub.ADC output bins (152), and a (nominally rectangular) window

(60) $\begin{array}{cc}{\left\{w_{\mathrm{chn}}\left(m_{\mathrm{ADC}}\right)\right\}}_{m_{\mathrm{ADC}}=0}^{M_{\mathrm{ADC}}-1}& \left(156\right)\end{array}$
to each S/P output vector, where K.sub.FFT can be conveniently chosen to minimize computational complexity and/or cost of the DFT using an FFT algorithm. The channelizer then selects K.sub.chn output bins

(61) ${\left\{k_{\mathrm{bin}}(\left(k_{\mathrm{chn}}\right)\right\}}_{k_{\mathrm{chn}}=0}^{K_{\mathrm{chn}}-1}$
for use in subsequent adaptive processing algorithms (154). The channelization operation can be expressed as a multiplication of x.sub.sym(n.sub.sym) by K.sub.chn×M.sub.ADC matrix

(62) $\begin{array}{cc}{T}_{\mathrm{chn}}=\left[\exp \left(-j2\pi k_{\mathrm{bin}}\left(k_{\mathrm{chn}}\right)\frac{m_{\mathrm{ADC}}}{K_{\mathrm{FFT}}}\right)w_{\mathrm{chn}}\left(m_{\mathrm{ADC}}\right)\right].& \left(153\right)\end{array}$

(63) In one aspect of the invention, the output channelizer bins are selected contiguously over the passband of the receiver, e.g., using formula k.sub.bin(k.sub.chn)=(k.sub.init+k.sub.chn)mod K.sub.FFT, where k.sub.init is the first FFT bin output from the channelizer (FFT bin associated with channelizer bin 0). As will be discussed herein, an advantage of this approach is minimization of channel signature blur due to movement of the transmitter and/or receiver platforms. In another aspect of the invention, the output channelizer bins are selected sparsely over the passband of the receiver, e.g., using formula k.sub.bin(k.sub.chn)=(k.sub.init+K.sub.sepk.sub.chn)mod K.sub.FFT, where K.sub.sep is an integer separation factor between channelizer bins. An advantage of this approach is ability to avoid known receiver impairments such as LO leakage (e.g., by setting k.sub.init to a non-multiple of K.sub.sep), or to reduce complexity of the channelization operation, e.g., using sparse FFT methods.

(64) In some aspects of the invention, the output channelizer bins are preselected, e.g., to avoid known receiver impairments such as LO leakage, or to reduce complexity of the channelization operation. In other aspects of the invention, the output channelizer bins are adaptively selected, e.g., to avoid dynamic narrowband co-channel interferers (NBCCI).

(65) In, the '417 NPA, which has been incorporated herein by reference, means for processing the vector channelizer output signal, in that case with K.sub.chn=M.sub.DoF. However, exploitation of channelization dimensionalities covering the full active bandwidth of the GPS L1 legacy signal, much less the much wider bandwidth GPS L5 and other civil GNSS signals, would substantively increase both the complexity of the linear-algebraic methods disclosed in the '417 NPA, and the reception time needed to implement those methods. For this reason, the '417 NPA focused on partial-band and subband methods that only exploited a part of that active bandwidth. Methods for overcoming this limitation without unduly affecting complexity or reception time is a primary focus of this invention.

(66) FIG. 9 depicts an instantiation of a single-feed FFT-based symbol-rate synchronous multi-subband channelizer, intended to overcome these limitations. This instantiation is identical to the FFT-based symbol-rate synchronous channelizer shown in FIG. 8, except that the K.sub.chn-channel selection operation (154) is replaced by a subband selection operation (157) that selects K.sub.sub subbands from the K.sub.FFT FFT output bins provided by the FFT operation (152). In the aspect of the invention shown in this Figure, the number of bins in each subband is equal to the same number, M.sub.DoF, referred to herein as the number of degrees of freedom of the subband. This has the benefit of regularizing linear-algebraic DSP operations performed on each subband in subsequent processing steps. However, other aspects of the invention can vary the number of bins, and therefore processing degrees of freedom, e.g., based on numbers of interferers present in each subband or other factors. The FFT output bins selected for the subbands are then organized into K.sub.sub M.sub.DoF×1 vector signals given by

(67) $\begin{array}{cc}{x}_{\mathrm{DoF}}\left(n_{\mathrm{sym}};k_{\mathrm{sub}}\right)=\left(\begin{array}{c}{x}_{\mathrm{FFT}}\left(k_{\mathrm{bin}}\left(0;k_{\mathrm{sub}}\right),n_{\mathrm{sym}}\right)\\ .Math.\\ x_{\mathrm{FFT}}\left(k_{\mathrm{bin}}\left(M_{\mathrm{DoF}}-1;k_{\mathrm{sub}}\right),n_{\mathrm{sym}}\right)\end{array}\right)k_{\mathrm{sub}}=0,.Math.,K_{\mathrm{sub}}-1,& \mathrm{Eqn}\left(18\right)\end{array}$
each with output rate f.sub.sym, where

(68) ${\left\{x_{\mathrm{FFT}}\left(k_{\mathrm{bin}},n_{\mathrm{sym}}\right)\right\}}_{k_{\mathrm{bin}}=0}^{K_{\mathrm{FFT}}-1}$
are the bins output from the FFT (152).

(69) In one aspect of the invention, the FFT bins selected for each subband are contiguous, e.g., set using formula k.sub.bin(m.sub.DoF;k.sub.sub)=(k.sub.init(k.sub.sub)+m.sub.DoF)mod K.sub.FFT, where k.sub.init(k.sub.sub) denotes the initial FFT bin selected for subband k.sub.sub. As will be discussed herein, an advantage of this approach is minimization of channel signature blur due to movement of the transmitter and/or receiver platforms. In another aspect of the invention, the FFT bins selected for each subband are more sparsely distributed, e.g., using formula k.sub.bin(m.sub.DoF;k.sub.sub)=(k.sub.init(k.sub.sub)+K.sub.sepm.sub.DoF)mod K.sub.FFT, where K.sub.sep is an integer separation between bins within the subband. An advantage of this approach is ability to avoid known receiver impairments such as LO leakage, or to reduce complexity of the channelization operation, e.g., using sparse FFT methods. The bin spacing K.sub.sep can be equal for each subband, or be different between subbands.

(70) In one aspect of the invention, the subband are themselves contiguous, e.g., such that k.sub.init(k.sub.sub+1)=(k.sub.init(k.sub.sub)+M.sub.DoF)mod K.sub.FFT for subbands with contiguous bins within each subband, and k.sub.init(k.sub.sub+1)=(k.sub.init(k.sub.sub)+K.sub.sepM.sub.DoF)mod K.sub.FFT for subbands with sparsely separated bins within each subband. In other aspects of the invention, subbands may be widely separated. In yet other aspects of the invention, subbands may be interleaved, e.g., by setting K.sub.sep=K.sub.sub and k.sub.init(k.sub.sub)=k.sub.init(0)+k.sub.sub, such that k.sub.bin(m.sub.DoF;k.sub.sub)=(k.sub.init(0)+(K.sub.subm.sub.DoF+k.sub.sub))mod K.sub.FFT.

(71) In some aspects of the invention, the output subband bins are preselected, e.g., to avoid known receiver impairments such as LO leakage, or to reduce complexity of the channelization operation. In other aspects of the invention, the output subband bins are adaptively selected, e.g., to avoid dynamic narrowband co-channel interferers (NBCCI).

(72) FIG. 10 is a block diagram of an alternate single-feed polyphase filter based symbol-rate synchronous multi-subband channelizer, according to an aspect of the disclosure. This block diagram also depicts a simple means for separating the channelizer output signal into contiguous subbands. The ADC output data x.sub.R(T.sub.ADCn.sub.ADC) is passed through a 1:M.sub.ADC serial-to-parallel converter (151) and a Q.sub.ADCM.sub.ADC-tap polyphase filter (160) with (typically real, potentially complex) channelizer weights

(73) $\begin{array}{cc}{\left\{w_{\mathrm{chn}}\left(m_{\mathrm{ADC}}\right)\right\}}_{m_{\mathrm{ADC}}=0}^{Q_{\mathrm{ADC}}{M}_{\mathrm{ADC}}},& 158\end{array}$
given by

(74) $\begin{array}{cc}{x}_{\mathrm{chn}}\left(k_{\mathrm{chn}},n_{\mathrm{sym}}\right)=\underset{m_{\mathrm{ADC}}=0}{\overset{Q_{\mathrm{ADC}}{M}_{\mathrm{ADC}}-1}{.Math.}}{w}_{\mathrm{chn}}^{*}\left(m_{\mathrm{ADC}}\right)x_R\left(T_{\mathrm{ADC}}\left(M_{\mathrm{ADC}}{n}_{\mathrm{sym}}+m_{\mathrm{ADC}}\right)\right)e^{-j2\pi f_{\mathrm{chn}}\left(k_{\mathrm{chn}}\right)T_{\mathrm{ADC}}{m}_{\mathrm{ADC}}},& \mathrm{Eqn}\left(19\right)\end{array}$
such that x.sub.chn(n.sub.sym) is given by

(75) $$\begin{gathered} \begin{array}{cc}{x}_{\mathrm{chn}}\left(n_{\mathrm{sym}}\right)=\underset{q_{\mathrm{ADC}}=0}{\stackrel{Q_{\mathrm{ADC}}-1}{.Math.}}{T}_{\mathrm{chn}}\left(q_{\mathrm{sym}}\right)x_{\mathrm{sym}}\left(n_{\mathrm{sym}}+q_{\mathrm{sym}}\right) \\ {T}_{\mathrm{chn}}\left(q_{\mathrm{sym}}\right)=\left[w_{\mathrm{chn}}^{*}\left(M_{\mathrm{ADC}}{q}_{\mathrm{sym}}+m_{\mathrm{ADC}}\right)e^{-j2\pi f_{\mathrm{chn}}\left(k_{\mathrm{chn}}\right)T_{\mathrm{ADC}}\left(M_{\mathrm{ADC}}{q}_{\mathrm{sym}}+m_{\mathrm{ADC}}\right)}\right].& \mathrm{Eqn}\left(20\right)\end{array} \end{gathered}$$
Relating to FIG. 7, this channelizer employs a transformation x.sub.chn(n.sub.sym)=T.sub.chn∘x.sub.sym(n.sub.sym), where T.sub.chn is a K.sub.chn×M.sub.ADC polyphase filter with response length Q.sub.ADC. In the aspect shown in FIG. 10, the channel frequencies

(76) 0${\left\{f_{\mathrm{chn}}\left(k_{\mathrm{chn}}\right)\right\}}_{k_{\mathrm{chn}}=0}^{K_{\mathrm{chn}}-1}$
are furthermore assumed to satisfy

(77) $\begin{array}{cc}{f}_{\mathrm{chn}}\left(k_{\mathrm{chn}}\right)=\left(k_{\mathrm{chn}}-\frac{K_{\mathrm{chn}}-1}{2}\right)f_{\mathrm{sym}}& \mathrm{Eqn}\left(21\right)\end{array}$
such that

(78) $\begin{array}{cc}{T}_{\mathrm{chn}}\left(q_{\mathrm{sym}}\right)={\left(-1\right)}^{\left(K-1\right)q_{\mathrm{sym}}}\left[w_{\mathrm{chn}}^{*}\left(M_{\mathrm{ADC}}{q}_{\mathrm{sym}}+m_{\mathrm{ADC}}\right)e^{-j2\pi k_{\mathrm{chn}}{m}_{\mathrm{ADC}}/M_{\mathrm{ADC}}}\right].& \mathrm{Eqn}\left(22\right)\end{array}$
This channelizer can be implemented using methods well-known to those of ordinary skill in the art.

(79) If K.sub.chn=K.sub.subM.sub.DoF, then x.sub.chn(n.sub.sym) can be directly transformed to K.sub.sub contiguous M.sub.DoF×1 vector subband signals

(80) ${\left\{x_{\mathrm{DoF}}\left(n_{\mathrm{sym}};k_{\mathrm{sub}}\right)\right\}}_{k_{\mathrm{sub}}=0}^{K_{\mathrm{sub}}-1}$
with contiguous in-subband channels using a M.sub.DoF×K.sub.sub reshaping operation (161), such that

(81) $x_{\mathrm{DoF}}\left(n_{\mathrm{sym}};k_{\mathrm{sub}}\right)={\left[x_{\mathrm{chn}}\left(M_{\mathrm{DoF}}{k}_{\mathrm{sub}}+m_{\mathrm{DoF}},n_{\mathrm{sym}}\right)\right]}_{m_{\mathrm{DoF}}=0}^{M_{\mathrm{DoF}}-1}.$
The subband center frequencies

(82) ${\left\{f_{\mathrm{sub}}\left(k_{\mathrm{sub}}\right)\right\}}_{k_{\mathrm{sub}}=0}^{K_{\mathrm{sub}}-1}$
are then given by

(83) $\begin{array}{cc}{f}_{\mathrm{sub}}\left(k_{\mathrm{sub}}\right)=\left(k_{\mathrm{sub}}-\frac{K_{\mathrm{sub}}-1}{2}\right)M_{\mathrm{DoF}}{f}_{\mathrm{sym}},k_{\mathrm{sub}}=0,.Math.,K_{\mathrm{sub}}-1& \mathrm{Eqn}\left(23\right)\end{array}$
while the frequency channels within each subband are locally given by

(84) $\begin{array}{cc}{f}_{\mathrm{chn}}\left(m_{\mathrm{DoF}}\right)=\left(m_{\mathrm{DoF}}-\frac{M_{\mathrm{DoF}}-1}{2}\right)f_{\mathrm{sym}},m_{\mathrm{DoF}}=0,.Math.,M_{\mathrm{DoF}}-1& \mathrm{Eqn}\left(24\right)\end{array}$
such that f.sub.chn(M.sub.DoFk.sub.sub+m.sub.DoF)=f.sub.sub(k.sub.sub)+f.sub.chn(m.sub.DoF).

(85) FIG. 11 is a block diagram of a single-feed symbol-rate synchronous multi-subband channelizer according to another aspect of the disclosure. In this aspect, the ADC output signal is passed through a set of K.sub.sub frequency converters (162a)-(162b) and decimators (163a)-(163b), each of which frequency downconverts the ADC output signal by frequency f.sub.sub(k.sub.sub), and decimates that signal to intermediate rate f.sub.dec=M.sub.decf.sub.sym, where M.sub.dec is a positive integer. Each decimated signal x.sub.dec(n.sub.dec;k.sub.sub) is then passed through a symbol-rate synchronous channelization operation (164a)-(164b) comprising a 1:M.sub.dec serial-to-parallel converter and a M.sub.DoF×M.sub.dec channelization operator T.sub.DoF (165), to form channelized subband signal x.sub.DoF(n.sub.sym;k.sub.sub)=T.sub.DoF∘x.sub.sym(n.sub.sym;k.sub.sub) where

(86) $x_{\mathrm{sym}}\left(n_{\mathrm{sym}};k_{\mathrm{sub}}\right)={\left[x_{\mathrm{dec}}\left(M_{\mathrm{dec}}{n}_{\mathrm{sym}}+m_{\mathrm{dec}};k_{\mathrm{sub}}\right)\right]}_{m_{\mathrm{dec}}=0}^{M_{\mathrm{dec}}-1}.$
The subband frequencies

(87) ${\left\{f_{\mathrm{sub}}\left(k_{\mathrm{sub}}\right)\right\}}_{k_{\mathrm{sub}}=0}^{K_{\mathrm{sub}}-1}$
can be organized in accordance with Eqn (23); set to a preselected set of arbitrary frequencies; or adaptively determined based on channel dynamics or co-channel interference considerations.

(88) Given the reception scenario described in FIG. 1 and FIG. 2, and the received signal model given in Eqn (4)-Eqn (14), then the single-sample channelizer output signal shown in FIG. 7 can be modeled by

(89) 0$\begin{array}{cc}{x}_{\mathrm{chn}}\left(n_{\mathrm{sym}}\right)=i_{\mathrm{chn}}\left(n_{\mathrm{sym}}\right)+\underset{\ell =1}{\overset{L_T}{.Math.}}{s}_{\mathrm{chn}}\left(n_{\mathrm{sym}};\ell \right),& \mathrm{Eqn}\left(25\right)\end{array}$
where i.sub.chn(n.sub.sym)=T.sub.chn∘i.sub.sym(n.sub.sym) and s.sub.chn(n.sub.sym;l)=T.sub.chn∘s.sub.sym(n.sub.sym;l) are the K.sub.chn×1 interference and navigation signal l channelizer output signals, respectively, and where

(90) $i_{\mathrm{sym}}\left(n_{\mathrm{sym}}\right)=\left[i_R\left(T_{\mathrm{sym}}\left(n_{\mathrm{sym}}+\frac{m_{\mathrm{ADC}}}{M_{\mathrm{ADC}}}\right)\right)\right]\mathrm{and}{s}_{\mathrm{sym}}\left(n_{\mathrm{sym}};\ell \right)=\left[s_R\left(T_{\mathrm{sym}}\left(n_{\mathrm{sym}}+\frac{m_{\mathrm{ADC}}}{M_{\mathrm{ADC}}}\right);\ell \right)\right]$
are the M.sub.ADC×1 interference and GNSS signal l S/P output vectors, respectively.

(91) The '417 NPA, incorporated herein by reference, provides a detailed description of the signals obtaining at the output of a symbol-rate-synchronous channelizer with K.sub.chn=M.sub.DoF. A simpler channel model can be obtained using the polyphase filter aspect shown in FIG. 10. In this case, under the simplifying assumption

(92) $\begin{array}{cc}{s}_R\left(t;\ell \right)\approx g_{\mathrm{TR}}\left(\ell \right)e^{j2\pi {\alpha }_{\mathrm{TR}}\left(\ell \right)t}{s}_T\left(t-{\tau }_{\mathrm{TR}}\left(\ell \right);\ell \right),& \mathrm{Eqn}\left(26\right)\end{array}$
i.e., ignoring affects of platform dynamics other than Doppler shift caused by the transmitter velocity observed at the receiver, then the channelizer output signal s.sub.chn(k.sub.chn,n.sub.sym;l) is given by

(93) $\begin{array}{cc}{s}_{\mathrm{chn}}\left(k_{\mathrm{chn}},n_{\mathrm{sym}};\ell \right)=e^{j2\pi {\alpha }_{\mathrm{TR}}\left(\ell \right)T_{\mathrm{sym}}{n}_{\mathrm{sym}}}\underset{q_{\mathrm{sym}}}{.Math.}{a}_{\mathrm{chn}}\left(k_{\mathrm{chn}},q_{\mathrm{sym}};\ell \right)d_T\left(n_{\mathrm{sym}}-q_{\mathrm{sym}};\ell \right),& \mathrm{Eqn}\left(27\right)\end{array}$
where a.sub.chn(k.sub.chn,q.sub.sym;l) is given by

(94) $\begin{array}{cc}{a}_{\mathrm{chn}}\left(k_{\mathrm{chn}},q_{\mathrm{sym}};\ell \right)=e^{j2\pi \left(q_{\mathrm{sym}}{T}_{\mathrm{sym}}-{\tau }_{\mathrm{TR}}\left(\ell \right)\right)f_{\mathrm{chn}}\left(k_{\mathrm{chn}}\right)}{e}^{-j2\pi q_{\mathrm{sym}}{\alpha }_{\mathrm{TR}}\left(\ell \right)}\times \underset{-\infty }{\overset{\infty }{\int }}{W}_{\mathrm{chn}}^{*}\left(e^{j2\pi {\mathrm{fT}}_{\mathrm{ADC}}}\right)H_{\mathrm{TR}}\left(f+f_{\mathrm{chn}}\left(k_{\mathrm{chn}}\right)\right)e^{j2\pi \left(q_{\mathrm{sym}}{T}_{\mathrm{sym}}-{\tau }_{\mathrm{TR}}\left(\ell \right)\right)f}\mathrm{df},& \mathrm{Eqn}\left(28\right)\end{array}$
in which W(e.sup.j2πf) is the discrete Fourier transform of w.sub.chn(m.sub.ADC) and H.sub.TR(f;l) is the frequency response of the end-to-end symbol shaping, given by
H.sub.TR(f;l)=g.sub.TR(l).sup.j2πα.sup.TR.sup.(l)H.sub.R(f)H.sub.T(f−α.sub.TR(l);l)√{square root over (P.sub.T(l))},  Eqn (29)
where H.sub.T(f;l) is the Fourier transform of h.sub.T(t;l) given in Eqn (2) and H.sub.R(f) is the frequency response of the receiver filtering operations.

(95) If H.sub.TR(f;l) is bandlimited below f.sub.ADC/2 e.g., using typical pre-ADC antialiasing filters, the bandwidth of W(e.sup.j2πf) is much less than the frequency variation in H.sub.TR(f;l) and f.sub.chn(k.sub.chn) is given by Eqn (21), then Eqn (28) can be approximated by
a.sub.chn(k.sub.chn,q.sub.sym;l)≈a.sub.chn(k.sub.chn;l)a.sub.sym(q.sub.sym;l)  Eqn (30)
where a.sub.chn(k.sub.chn;l) and a.sub.sym(q.sub.sym;l) are the frequency-varying and time-varying components of the channel signature,

(96) $\begin{array}{cc}{a}_{\mathrm{chn}}\left(k_{\mathrm{chn}};\ell \right)=f_{\mathrm{ADC}}{H}_{\mathrm{TR}}\left(f_{\mathrm{chn}}\left(k_{\mathrm{chn}}\right);\ell \right)e^{-j2\pi f_{\mathrm{chn}}\left(k_{\mathrm{chn}}\right){\tau }_{\mathrm{TR}}\left(\ell \right)},& \mathrm{Eqn}\left(31\right)\end{array}$$\begin{array}{cc}{a}_{\mathrm{sym}}\left(q_{\mathrm{sym}};\ell \right)={\left(-1\right)}^{\left(K_{\mathrm{chn}}-1\right)q_{\mathrm{sym}}}{w}_{\mathrm{sym}}^{*}\left(q_{\mathrm{sym}}-{\tau }_{\mathrm{TR}}\left(\ell \right)f_{\mathrm{sym}}\right)e^{-j2\pi q_{\mathrm{sym}}{\alpha }_{\mathrm{TR}}\left(\ell \right)},& \mathrm{Eqn}\left(32\right)\end{array}$
respectively, referred to herein as the channel frequency signature and channel time signature, and where w.sub.sym(τ) is the interpolated channelizer window, given by

(97) $\begin{array}{cc}{w}_{\mathrm{sym}}\left(\tau \right)=\underset{m_{\mathrm{ADC}}}{.Math.}{w}_{\mathrm{chn}}\left(m_{\mathrm{ADC}}\right)\mathrm{Sa}\left(\pi \left(\tau M_{\mathrm{ADC}}-m_{\mathrm{ADC}}\right)\right).& \mathrm{Eqn}\left(33\right)\end{array}$
The channelized received signal s.sub.chn(k.sub.chn,n.sub.sym;l) is then approximated by
s.sub.chn(k.sub.chn,n.sub.sym;l)≈a.sub.chn(k.sub.chn;l)d.sub.R(n.sub.sym;l),  Eqn (34)
where d.sub.R(n.sub.sym;l) is the signal l symbol sequence observed at the receiver, given by
d.sub.R(n.sub.sym;l)=(a.sub.sym(n.sub.sym;l)∘d.sub.T(n.sub.sym;l))e.sup.j2πα.sup.TR.sup.(l)T.sup.sym.sup.n.sup.sym,  Eqn (35)
and where “∘” here denotes the convolution operation. Further decomposing τ.sub.TR(l) and α.sub.TR(l) into symbol-normalized TOA and FOA components

(98) $\begin{array}{cc}{\tau }_{\mathrm{TR}}\left(\ell \right)=\left({\tilde{\tau }}_{\mathrm{TR}}\left(\ell \right)+n_{\mathrm{TR}}\left(\ell \right)\right)T_{\mathrm{sym}}\{\begin{array}{cc}{\tilde{\tau }}_{\mathrm{TR}}\left(\ell \right)\in [\begin{array}{cc}0& 1\end{array}),& \mathrm{fractional}\mathrm{fine}\mathrm{TOA}\\ n_{\mathrm{TR}}\left(\ell \right)\in \mathbb{Z},& \mathrm{TOA}\mathrm{symbol}\mathrm{offset}\end{array},& \mathrm{Eqn}\left(36\right)\end{array}$$\begin{array}{cc}{\alpha }_{\mathrm{TR}}\left(\ell \right)=\left({\tilde{\alpha }}_{\mathrm{TR}}\left(\ell \right)+k_{\mathrm{TR}}\left(\ell \right)\right)f_{\mathrm{sym}}\{\begin{array}{cc}{\alpha }_{\mathrm{TR}}\left(\ell \right)\in [\begin{array}{cc}-\frac{1}{2}& \frac{1}{2}\end{array}),& \mathrm{fractional}\mathrm{fine}\mathrm{FOA}\\ k_{\mathrm{TR}}\left(\ell \right)\in \mathbb{Z},& \mathrm{FOA}\mathrm{Nyquist}\mathrm{zone}\end{array},& \mathrm{Eqn}\left(37\right)\end{array}$
respectively, then Eqn (34) can be replaced by

(99) $\begin{array}{cc}{d}_R\left(n_{\mathrm{sym}};\ell \right)=\left(a_{\mathrm{sym}}\left(n_{\mathrm{sym}};\ell \right)\circ d_T\left(n_{\mathrm{sym}}-n_{\mathrm{TR}}\left(\ell \right);\ell \right)\right)e^{j2\pi {\tilde{\alpha }}_{\mathrm{TR}}\left(\ell \right)n_{\mathrm{sym}}},& \mathrm{Eqn}\left(38\right)\end{array}$$\begin{array}{cc}{a}_{\mathrm{sym}}\left(q_{\mathrm{sym}};\ell \right)={\left(-1\right)}^{\left(K_{\mathrm{chn}}-1\right)q_{\mathrm{sym}}}{w}_{\mathrm{sym}}^{*}\left(q_{\mathrm{sym}}-{\tilde{\tau }}_{\mathrm{TR}}\left(\ell \right)\right)e^{-j2\pi q_{\mathrm{sym}}{\alpha }_{\mathrm{TR}}\left(\ell \right)},& \mathrm{Eqn}\left(39\right)\end{array}$
except for phase shift φ(l)=2π(f.sub.chn(0)−{tilde over (α)}.sub.TR(l))n.sub.TR(l), which is subsumed into end-to-end link gain in g.sub.TR(l) in Eqn (5). Then d.sub.R(n.sub.sym;l) is completely characterized by the channel time signature defined (over fine TOA ranging between 0 and 1; the received TOA measured to integer number of symbols; and the fractional fine FOA ranging between −½ and ½.

(100) For the GNSS signals listed in FIG. 4, and assuming |α.sub.TR(l)|<<f.sub.chp, then a.sub.chn(k.sub.chn;l) can be further approximated by

(101) $\begin{array}{cc}{a}_{\mathrm{chn}}\left(k_{\mathrm{chn}};\ell \right)\approx \frac{M_{\mathrm{ADC}}}{\sqrt{M_{\mathrm{chp}}}}\mathrm{Sa}\left(\pi f_{\mathrm{chn}}\left(k_{\mathrm{chn}}\right)T_{\mathrm{chp}}\right)H_R\left(f_{\mathrm{chn}}\left(k_{\mathrm{chn}}\right)\right)g_{\mathrm{TR}}\left(\ell \right)\sqrt{P_T\left(\ell \right)}{\epsilon }_R\left(k_{\mathrm{chn}};\ell \right),& \mathrm{Eqn}\left(40\right)\end{array}$$\begin{array}{cc}{\epsilon }_R\left(k_{\mathrm{chn}};\ell \right)=\frac{1}{\sqrt{M_{\mathrm{chp}}}}{C}_T\left(e^{-j2\pi \left(f_{\mathrm{chn}}\left(k_{\mathrm{chn}}\right)-{\alpha }_{\mathrm{TR}}\left(\ell \right)\right)T_{\mathrm{chp}}};\ell \right)e^{-j2\pi \left(f_{\mathrm{chn}}\left(k_{\mathrm{chn}}\right)-{\alpha }_{\mathrm{TR}}\left(\ell \right);\ell \right){\tau }_{\mathrm{TR}}\left(\ell \right)},& \mathrm{Eqn}\left(41\right)\end{array}$
where normalized observed ranging code frequency signature ε.sub.R(k.sub.chn;l) captures the cross-channel frequency variability of each channel frequency signature due to the separate ranging code used at each transmitter, and the differing TOA on each transmission path. For well-designed pseudo-random code sequences and M.sub.chp>>1, ε.sub.R(k.sub.chn;l) can be modeled as a zero-mean, unit-variance, independent and identically-distributed (i.i.d.) complex-Gaussian random process. The channelized output signal can then be approximated by

(102) 0$\begin{array}{cc}{x}_{\mathrm{chn}}\left(n_{\mathrm{sym}}\right)\approx i_{\mathrm{chn}}\left(n_{\mathrm{sym}}\right)+\underset{\ell =1}{\overset{L_T}{.Math.}}{a}_{\mathrm{chn}}\left(\ell \right)d_R\left(n_{\mathrm{sym}};\ell \right),& \mathrm{Eqn}\left(42\right)\end{array}$
where

(103) $a_{\mathrm{chn}}\left(\ell \right)={\left[a_{\mathrm{chn}}\left(k_{\mathrm{chn}};\ell \right)\right]}_{k_{\mathrm{chn}}=0}^{K_{\mathrm{chn}}-1}$
is the K.sub.chn×1 signal l channel frequency signature and a.sub.chn(k.sub.chn;l) is given by Eqn (40)-Eqn (41) and d.sub.R(n.sub.sym;l) is given by Eqn (38).

(104) Assuming the background interference i.sub.R(t) given in Eqn (4) is complex-Gaussian and stationary with power spectral density (PSD) S.sub.i.sub.R.sub.i.sub.R (f), then the interference power level in each channel is given by

(105) $\begin{array}{cc}.Math.{.Math.i_{\mathrm{chn}}\left(k_{\mathrm{chn}},n_{\mathrm{sym}}\right).Math.}^2.Math.=\underset{-1/2}{\overset{1/2}{\int }}{.Math.W_{\mathrm{chn}}\left(e^{j2\pi f}\right).Math.}^2{\tilde{S}}_{i_R{i}_R}\left(f+f_{\mathrm{chn}}\left(k_{\mathrm{chn}}\right)T_{\mathrm{ADC}}\right)\mathrm{df},& \mathrm{Eqn}\left(43\right)\end{array}$
where

(106) ${\tilde{S}}_{i_R{i}_R}\left(f\right)=f_{\mathrm{ADC}}\underset{\ell }{.Math.}{.Math.H_R\left(\left(f+\ell \right)f_{\mathrm{ADC}}\right).Math.}^2{S}_{i_R{i}_R}\left(\left(f+\ell \right)f_{\mathrm{ADC}}\right)$
is the PSD of sampled interference signal i.sub.R(T.sub.ADCn.sub.ADC). Assuming the ADC input signal is bandlimited to less than f.sub.ADC/2 ahead of the sampling operation, and that the PSD frequency variation is low over the passband of the polyphase filter frequency response W.sub.chn(e.sup.j2πf), then Eqn (43) can be approximated by

(107) $\begin{array}{cc}\begin{array}{c}.Math.{.Math.i_{\mathrm{chn}}\left(k_{\mathrm{chn}},n_{\mathrm{sym}}\right).Math.}^2.Math.\approx f_{\mathrm{ADC}}{.Math.H_R\left(f_{\mathrm{chn}}\left(k_{\mathrm{chn}}\right)\right).Math.}^2{S}_{i_R{i}_R}\left(f_{\mathrm{chn}}\left(k_{\mathrm{chn}}\right)\right)\underset{-1/2}{\overset{1/2}{\int }}{.Math.W_{\mathrm{chn}}\left(e^{j2\pi f}\right).Math.}^2\mathrm{df}\\ =f_{\mathrm{ADC}}{.Math.H_R\left(f_{\mathrm{chn}}\left(k_{\mathrm{chn}}\right)\right).Math.}^2{S}_{i_R{i}_R}\left(f_{\mathrm{chn}}\left(k_{\mathrm{chn}}\right)\right){.Math.w_{\mathrm{chn}}.Math.}_2^2\\ =\frac{M_{\mathrm{ADC}}^2}{M_{\mathrm{chp}}}{f}_{\mathrm{chp}}{.Math.H_R\left(f_{\mathrm{chn}}\left(k_{\mathrm{chn}}\right)\right).Math.}^2{S}_{i_R{i}_R}\left(f_{\mathrm{chn}}\left(k_{\mathrm{chn}}\right)\right)\end{array},& \mathrm{Eqn}\left(44\right)\end{array}$
where

(108) ${.Math.w_{\mathrm{chn}}.Math.}_2^2=\underset{m_{\mathrm{ADC}}}{.Math.}{.Math.w_{\mathrm{chn}}\left(m_{\mathrm{ADC}}\right).Math.}^2$
denotes the squared L2 Euclidean norm of the channelizer weights {w.sub.chn(m.sub.ADC)} (158), and further assuming normalization constraint ∥w.sub.chn∥.sub.2.sup.2=M.sub.ADC. More generally, the interference cross-correlation across symbol lag and frequency channel offset can be approximated by

(109) $\begin{array}{cc}\begin{array}{c}{R}_{i_{\mathrm{chn}}{i}_{\mathrm{chn}}}\left(k_{\mathrm{chn}},{\ell }_{\mathrm{chn}};m_{\mathrm{sym}}\right)\stackrel{△}{=}.Math.i_{\mathrm{chn}}\left(k_{\mathrm{chn}},n_{\mathrm{sym}}\right)i_{\mathrm{chn}}^{*}\left({\ell }_{\mathrm{chn}},n_{\mathrm{sym}}-m_{\mathrm{sym}}\right).Math.\\ \approx \frac{M_{\mathrm{ADC}}^2}{M_{\mathrm{chp}}}{f}_{\mathrm{chp}}{S}_{i_R{i}_R}\left(\frac{f_{\mathrm{chn}}\left(k_{\mathrm{chn}}\right)+f_{\mathrm{chn}}\left({\ell }_{\mathrm{chn}}\right)}{2}\right)\\ \times {\left(-1\right)}^{\left(K_{\mathrm{chn}}-1\right)m_{\mathrm{sym}}}{\rho }_{\mathrm{chn}}\left(M_{\mathrm{ADC}}{m}_{\mathrm{sym}},\frac{k_{\mathrm{chn}}-{\ell }_{\mathrm{chn}}}{M_{\mathrm{ADC}}}\right)\end{array},& \mathrm{Eqn}\left(45\right)\end{array}$
where

(110) ${\rho }_{\mathrm{chn}}\left(m,\alpha \right)=\frac{1}{M_{\mathrm{ADC}}}\underset{n}{.Math.}{w}_{\mathrm{chn}}\left(n\right)w_{\mathrm{chn}}^{*}\left(n-m\right)e^{-j2\pi \alpha n}$
is the discrete-time ambiguity function for channelizer window {w.sub.chn(m.sub.ADC)} (158), normalized M.sub.ADC so that |ρ.sub.chn(m,α)|≤1.

(111) FIG. 12 depicts the channel time signature (FIG. 12A) and normalized ambiguity function (FIG. 12B) induced by the polyphase-filter based symbol-rate synchronous channelizer (160) shown in FIG. 10, using single-symbol rectangularly-windowed channelizer weights (158). This Figure also obtains for the FFT-based symbol-rate synchronous channelized shown in FIG. 8 and FIG. 9. As shown in FIG. 12A, the channel time signature is only substantive over a single symbol for fine TOA offsets between 0 and 1, and is nearly equal to unity within this region. The received symbol sequence d.sub.R(n.sub.sym;l) given in Eqn (38) then simplifies to

(112) $\begin{array}{cc}{d}_R\left(n_{\mathrm{sym}};\ell \right)\approx d_T\left(n_{\mathrm{sym}}-1-n_{\mathrm{TR}}\left(\ell \right);\ell \right)e^{j2\pi {\tilde{\alpha }}_{\mathrm{TR}}\left(\ell \right)n_{\mathrm{sym}}},& \mathrm{Eqn}\left(46\right)\end{array}$
where the phase term −2π{tilde over (α)}.sub.TR(l) is subsumed into g.sub.TR(l). Similarly, the normalized ambiguity function shown in FIG. 12B is equal to unity at zero channel and symbol offset. Hence the background interference for this channelizer window can be treated as independent between channels and symbols, yielding K.sub.chn×K.sub.chn interference autocorrelation matrix (ACM)

(113) $\begin{array}{cc}{R}_{i_{\mathrm{chn}}{i}_{\mathrm{chn}}}=\frac{M_{\mathrm{ADC}}}{M_{\mathrm{chp}}}{f}_{\mathrm{chp}}\mathrm{diag}{\left\{{.Math.H_R\left(f_{\mathrm{chn}}\left(k_{\mathrm{chn}}\right)\right).Math.}^2{S}_{i_R{i}_R}\left(f_{\mathrm{chn}}\left(k_{\mathrm{chn}}\right)\right)\right\}}_{k_{\mathrm{chn}}=0}^{K_{\mathrm{chn}}-1}.& \mathrm{Eqn}\left(47\right)\end{array}$
However, the channel model established above can be extended to any channelizer window.

(114) Multiplying x.sub.chn(n.sub.sym) by the inverse square-root of R.sub.i.sub.chn.sub.i.sub.chn yields normalized signal

(115) 0$\begin{array}{cc}\begin{array}{c}{\tilde{x}}_{\mathrm{chn}}\left(n_{\mathrm{sym}}\right)\approx R_{i_{\mathrm{chn}}{i}_{\mathrm{chn}}}^{-1/2}{x}_{\mathrm{chn}}\left(n_{\mathrm{sym}}\right),\\ \approx {\epsilon }_{\mathrm{chn}}\left(n_{\mathrm{sym}}\right)+\underset{\ell =1}{\overset{L_T}{.Math.}}{u}_{\mathrm{chn}}\left(\ell \right)d_R\left(n_{\mathrm{sym}};\ell \right),\end{array}& \mathrm{Eqn}\left(48\right)\end{array}$
where d.sub.R(n.sub.sym;l) is given by Eqn (46), ε.sub.chn(n.sub.sym) is a K.sub.chn×1 i.i.d. complex-Gaussian random process with zero mean and ACM I.sub.K.sub.chn, and U.sub.chn(l) is the K.sub.chn×1 signal l normalized channel frequency signature given by

(116) $\begin{array}{cc}{u}_{\mathrm{chn}}\left(\ell \right)=\left[\frac{\mathrm{Sa}\left(\pi f_{\mathrm{chn}}\left(k_{\mathrm{chn}}\right)T_{\mathrm{chp}}\right)}{\sqrt{f_{\mathrm{chp}}{S}_{i_R{i}_R}\left(f_{\mathrm{chn}}\left(k_{\mathrm{chn}}\right)\right)}}{\epsilon }_R\left(k_{\mathrm{chn}};\ell \right)\right]g_{\mathrm{TR}}\left(\ell \right)\sqrt{P_T\left(\ell \right)}.& \mathrm{Eqn}\left(49\right)\end{array}$
In theory, a set of K.sub.chn×1 combining weights {tilde over (w)}.sub.max-SINR(l) can then be developed that extracts d.sub.R(n.sub.sym;l) from {tilde over (x)}.sub.chn(n.sub.sym) with maximum-attainable signal-to-interference-and-noise ratio (max-SINR). For the channel model given in Eqn (48)-Eqn (49), and assuming the navigation symbol sequences are independent for each transmitter, {tilde over (w)}.sub.max-SINR(l) is given by

(117) $\begin{array}{cc}\begin{array}{c}{w}_{max-\mathrm{SINR}}\left(\ell \right)\propto {\left(I_{K_{\mathrm{chn}}}+U_{\mathrm{chn}}\left(~\ell \right)U_{\mathrm{chn}}^H\left(~\ell \right)\right)}^{-1}{u}_{\mathrm{chn}}\left(\ell \right)\\ =\left(I_{K_{\mathrm{chn}}}-U_{\mathrm{chn}}\left(~\ell \right){\left(I_{L_T-1}+U_{\mathrm{chn}}^H\left(~\ell \right)U_{\mathrm{chn}}\left(~\ell \right)\right)}^{-1}{U}_{\mathrm{chn}}^H\right)u_{\mathrm{chn}}\left(\ell \right)\end{array},& \mathrm{Eqn}\left(50\right)\end{array}$
where U.sub.chn(˜l)=[u.sub.chn(l′)].sub.l′≠l is the K.sub.chn×(L.sub.T−1) matrix of normalized frequency signatures interfering with signal l, and the max-SINR for symbol sequence l is given by

(118) $\begin{array}{cc}\begin{array}{c}{\gamma }_{max-\mathrm{SINR}}\left(\ell \right)\approx {.Math.u_{\mathrm{chn}}\left(\ell \right).Math.}_2^2\\ -u_{\mathrm{chn}}^H\left(\ell \right)U_{\mathrm{chn}}\left(~\ell \right){\left(I_{L_T-1}+U_{\mathrm{chn}}^H\left(~\ell \right)U_{\mathrm{chn}}\left(~\ell \right)\right)}^{-1}{U}_{\mathrm{chn}}^H\left(~\ell \right)u_{\mathrm{chn}}\left(\ell \right)\end{array}.& \mathrm{Eqn}\left(51\right)\end{array}$
Modeling ε.sub.R(k.sub.sub;l) as i.i.d. complex Gaussian with zero mean and unity variance, appropriate for unknown, well-modeled long ranging codes, then Eqn (51) has mean lower bound

(119) $\begin{array}{cc}\begin{array}{c}{\stackrel{\_}{\gamma }}_{max-\mathrm{SINR}}\left(\ell \right)=E\left\{{\gamma }_{max-\mathrm{SINR}}\left(\ell \right)\right\}\\ \ge ({\stackrel{\_}{\gamma }}_{max-\mathrm{SNR}}\left(\ell \right)-\left(L_T-1\right)\left(\frac{{\stackrel{\_}{\gamma }}_{max-\mathrm{INR}}\left(\ell \right)}{1+{\stackrel{\_}{\gamma }}_{max-\mathrm{INR}}\left(\ell \right)}\right)\underset{k_{\mathrm{chn}}}{\max }{\stackrel{\_}{\gamma }}_{\mathrm{Rx}-\mathrm{SNR}}\left(k_{\mathrm{chn}};\ell \right)\end{array},& \mathrm{Eqn}\left(52\right)\end{array}$$\begin{array}{cc}.fwdarw.\{\begin{array}{cc}{\stackrel{\_}{\gamma }}_{max-\mathrm{SNR}}\left(\ell \right),& {\stackrel{\_}{\gamma }}_{max-\mathrm{INR}}\left(\ell \right)1\\ \left({\stackrel{\_}{\gamma }}_{max-\mathrm{SNR}}\left(\ell \right)-\left(L_T-1\right)\underset{k_{\mathrm{chn}}}{\max }{\stackrel{\_}{\gamma }}_{max-\mathrm{SNR}}\left(k_{\mathrm{chn}};\ell \right)\right),& {\stackrel{\_}{\gamma }}_{max-\mathrm{INR}}\left(\ell \right)1\end{array},& \mathrm{Eqn}\left(53\right)\end{array}$
where γ.sub.Rx-SNR(k.sub.chn.sup.;l) and γ.sub.max-SNR(l) are the mean receive SNR (Rx-SNR) and maximum-attainable SNR (max-SNR) of signal l in the absence of interference navigation signals, respectively,

(120) $\begin{array}{cc}\begin{array}{c}{\stackrel{\_}{\gamma }}_{\mathrm{Rx}-\mathrm{SNR}}\left(k_{\mathrm{chn}};\ell \right)=E\left\{.Math.u_{\mathrm{chn}}\left(k_{\mathrm{chn}};\ell \right).Math.\right\}\\ =\frac{{\mathrm{Sa}}^2\left(\pi f_{\mathrm{chn}}\left(k_{\mathrm{chn}}\right)T_{\mathrm{chp}}\right)}{f_{\mathrm{chp}}{S}_{i_R{i}_R}\left(f_{\mathrm{chn}}\left(k_{\mathrm{chn}}\right)\right)}{.Math.g_{\mathrm{TR}}\left(\ell \right).Math.}^2{P}_T\left(\ell \right)\end{array},& \mathrm{Eqn}\left(54\right)\end{array}$$\begin{array}{cc}\begin{array}{c}{\stackrel{\_}{\gamma }}_{max-\mathrm{SNR}}\left(\ell \right)=E\left\{{.Math.u_{\mathrm{chn}}\left(\ell \right).Math.}_2^2\right\}\\ =\underset{k_{\mathrm{chn}}=0}{\overset{K_{\mathrm{chn}}-1}{.Math.}}{\stackrel{\_}{\gamma }}_{\mathrm{Rx}-\mathrm{SNR}}\left(k_{\mathrm{chn}};\ell \right)\end{array},& \mathrm{Eqn}\left(55\right)\end{array}$
and where γ.sub.max-INR(l) is the mean max-attainable SNR of navigation signals interfering with signal l,

(121) $\begin{array}{cc}{\stackrel{\_}{\gamma }}_{max-\mathrm{INR}}\left(\ell \right)=\frac{1}{L_T-1}\underset{{\ell }^{\prime }\ne \ell }{.Math.}{\stackrel{\_}{\gamma }}_{max-\mathrm{SNR}}\left({\ell }^{\prime }\right).& \mathrm{Eqn}\left(56\right)\end{array}$
As Eqn (53) shows, at high receive SNR, the max-SINR combiner uses L.sub.T−1 combiner degrees of freedom (DoF's) to excise the navigation signals impinging on the receiver, and employs the remaining K.sub.chn−L.sub.T+1 combiner DoF's to suppress the background interference i.sub.chn(n.sub.sym).

(122) As disclosed in the '417 NPA, incorporated herein by reference, linear algebraic methods to determine max-SINR weights can be devised given either knowledge of the content of {d.sub.T(n.sub.sym;l)}.sub.l=1.sup.L.sup.T, e.g., provided by third-party sources; knowledge of a component of {d.sub.T(n.sub.sym;l)}.sub.l=1.sup.L.sup.T, e.g., a known/periodic pilot signal; or knowledge of some other exploitable structure of {d.sub.T(n.sub.sym;l)}.sub.l=1.sup.L.sup.T, as listed in FIG. 4. These methods can be used to detect the navigation signals in the environment, estimate the max-SINR of the combiner output symbol sequences for those signals, and determine at least the geo-observables {n.sub.TR(l),{tilde over (α)}.sub.TR(l)}.sub.l=1.sup.L.sup.T for those signals, and if needed estimate information-bearing components of the transmitted symbol streams {d.sub.T(n.sub.sym;l)}.sub.l=1.sup.L.sup.T. Moreover, these techniques can perform these operations without prior knowledge of the signal ranging code or receive frequency signature for the signals, or a search over TOA and FOA is the ranging code is known.

(123) However, as also disclosed in the '417 NPA, linear-algebraic methods that can develop combiner weights covering the entire navigation signal passband, e.g., K.sub.chn ˜1,000 for the GPS L1 legacy signal, and K.sub.chn ˜10,000 for the GPS L5 civil signal, require O(K.sub.chn.sup.2) operations/symbol to be implemented. Moreover, they require O(K.sub.chn) symbols to converge to a useful solution, e.g., 2-to-4 seconds for the GPS L1 legacy signal, and 20-40 seconds for the GPS L5 civil signal. This convergence time introduces additional complexity in GNSS reception scenarios, due to channel dynamics caused by movement of the GNSS satellite vehicles.

(124) In the '417 NPA, partial subband methods that reduce K.sub.chn to a manageable value are disclosed to overcome these issues. These methods can provide strong advantage in the presence of strong MOS-DSSS spoofing and jamming signals. However, they sacrifice substantive processing gain to achieve this capability at reasonable complexity and over short reception intervals. This issue can be especially significant for modern wideband navigation signals such as the GPS L5 civil signal. The multi-subband approach provides a path to overcome this limitation.

(125) Returning to FIG. 10, reshape operation (161) shown in FIG. 10 separates x.sub.chn(n.sub.sym) into K.sub.sub subbands, each comprising M.sub.DoF frequency channels such that K.sub.chn=K.sub.subM.sub.DoF, and the signal in each subband can be represented by

(126) $\begin{array}{cc}\begin{array}{c}{x}_{\mathrm{DoF}}\left(n_{\mathrm{sym}};k_{\mathrm{sub}}\right)=\left(\begin{array}{c}{x}_{\mathrm{chn}}\left(M_{\mathrm{DoF}}{k}_{\mathrm{sub}},n_{\mathrm{sym}}\right)\\ .Math.\\ x_{\mathrm{chn}}\left(M_{\mathrm{DoF}}{k}_{\mathrm{sub}}+M_{\mathrm{DoF}}-1,n_{\mathrm{sym}}\right)\end{array}\right)\\ \approx i_{\mathrm{DoF}}\left(n_{\mathrm{sym}};k_{\mathrm{sub}}\right)+\underset{\ell =1}{\overset{L_T}{.Math.}}{a}_{\mathrm{DoF}}\left(k_{\mathrm{sub}};\ell \right)d_R\left(n_{\mathrm{sym}};\ell \right)\end{array},& \mathrm{Eqn}\left(57\right)\end{array}$
where d.sub.R(n.sub.sym;l), given by Eqn (46) for the rectangularly-windowed channelizer, is shared for all subbands, and a.sub.DoF(k.sub.sub;l) is the M.sub.DoF×1 signal l frequency signature for subband k.sub.sub,

(127) $\begin{array}{cc}{a}_{\mathrm{DoF}}\left(k_{\mathrm{sub}};\ell \right)=\left(\begin{array}{c}{a}_{\mathrm{chn}}\left(M_{\mathrm{DoF}}{k}_{\mathrm{sub}},\ell \right)\\ .Math.\\ a_{\mathrm{chn}}\left(M_{\mathrm{DoF}}{k}_{\mathrm{sub}}+M_{\mathrm{DoF}}-1,\ell \right)\end{array}\right),& \mathrm{Eqn}\left(58\right)\end{array}$
and where i.sub.DoF(n.sub.sym;k.sub.sub) is the background noise and interference received in subband k.sub.sub. If i.sub.R(t) is stationary with PSD S.sub.i.sub.R.sub.i.sub.R(f) at the LPF output, then the zero-lag ACM of i.sub.DoF(n.sub.sym;k.sub.sub) is further approximated by

(128) $\begin{array}{cc}{R}_{i_{\mathrm{DoF}}{i}_{\mathrm{DoF}}}\left(k_{\mathrm{sub}}\right)\approx \frac{M_{\mathrm{ADC}}^2}{M_{\mathrm{chp}}}{f}_{\mathrm{chp}}{.Math.H_R\left(f_{\mathrm{sub}}\left(k_{\mathrm{sub}}\right)\right).Math.}^2{S}_{i_R{i}_R}\left(f_{\mathrm{sub}}\left(k_{\mathrm{sub}}\right)\right)P_{\mathrm{sub}},& \mathrm{Eqn}\left(59\right)\end{array}$
where

(129) 0$f_{\mathrm{sub}}\left(k_{\mathrm{sub}}\right)=\left(k_{\mathrm{sub}}-\frac{K_{\mathrm{sub}}-1}{2}\right)M_{\mathrm{DoF}}{f}_{\mathrm{sym}}i$
is the center frequency of subband k.sub.sub, and where

(130) $P_{\mathrm{sub}}={\left[\frac{1}{M_{\mathrm{ADC}}}\underset{m}{.Math.}{.Math.w_{\mathrm{chn}}\left(m\right).Math.}^2{e}^{-j2\pi \left(k-\ell \right)m/M_{\mathrm{ADC}}}\right]}_{k,\ell =0}^{M_{\mathrm{DoF}}-1}$
is the normalized subband interference ACM, which is equal to I.sub.M.sub.DoF for the single-symbol rectangular channelizer window.

(131) If M.sub.DoF≥L.sub.T, then navigation symbol sequences l can be received in each subband using an appropriately designed set of max-SINR linear in-subband linear combining weights w.sub.max-SINR(k.sub.sub;l), such that combiner output symbol sequence {circumflex over (d)}.sub.R(n.sub.sym;k.sub.sub;l)=w.sub.max-SINR.sup.H(k.sub.sub;l)x.sub.sub(n.sub.sym;k.sub.sub) excises the interfering navigation signals {d.sub.R(n.sub.sym;l′)}.sub.l′≠l also received in the subband using L.sub.T−1 degrees of freedom, and suppresses the background interference i.sub.sub(n.sub.sym;k.sub.chn) in the subband using the remaining M.sub.DoF−L.sub.T+1 degrees of freedom, and where (⋅).sup.H denotes the Hermitian transpose operation. Assuming a nearly-flat signature and interference response over each subband, and rectangular channelizer windows, then the SINR achievable using this linear combiner satisfies mean lower bound

(132) $\begin{array}{cc}{\stackrel{\_}{\gamma }}_{max-\mathrm{SINR}}\left(k_{\mathrm{sub}};\ell \right)\ge \left(M_{\mathrm{DoF}}-\left(L_T-1\right)\left(\frac{{\stackrel{\_}{\gamma }}_{max-\mathrm{INR}}\left(k_{\mathrm{sub}};\ell \right)}{1+{\stackrel{\_}{\gamma }}_{max-\mathrm{INR}}\left(k_{\mathrm{sub}};\ell \right)}\right)\right){\stackrel{\_}{\gamma }}_{\mathrm{Rx}-\mathrm{SNR}}\left(k_{\mathrm{sub}};\ell \right),& \mathrm{Eqn}\left(60\right)\end{array}$$\begin{array}{cc}.fwdarw.\{\begin{array}{cc}{M}_{\mathrm{DoF}}{\stackrel{\_}{\gamma }}_{\mathrm{Rx}-\mathrm{SNR}}\left(k_{\mathrm{sub}};\ell \right),& {\stackrel{\_}{\gamma }}_{max-\mathrm{INR}}\left(k_{\mathrm{sub}};\ell \right)1\\ \left(M_{\mathrm{DoF}}-L_T+1\right){\stackrel{\_}{\gamma }}_{\mathrm{Rx}-\mathrm{SNR}}\left(k_{\mathrm{sub}};\ell \right),& {\stackrel{\_}{\gamma }}_{max-\mathrm{INR}}\left(k_{\mathrm{sub}};\ell \right)1\end{array}& \mathrm{Eqn}\left(61\right)\end{array}$
in each subband, where γ.sub.Rx-SNR(k.sub.sub;l) and γ.sub.max-INR(k.sub.sub;l) are the mean receive signal l SNR and the max-attainable navigation signal interference received along with that signal in subband k.sub.sub, respectively

(133) $\begin{array}{cc}{\stackrel{\_}{\gamma }}_{\mathrm{Rx}-\mathrm{SNR}}\left(k_{\mathrm{sub}};\ell \right)\approx \frac{{\mathrm{Sa}}^2\left(\pi f_{\mathrm{sub}}\left(k_{\mathrm{sub}}\right)T_{\mathrm{chp}}\right)}{f_{\mathrm{chp}}{S}_{i_R{i}_R}\left(f_{\mathrm{sub}}\left(k_{\mathrm{sub}}\right)\right)}{.Math.g_{\mathrm{TR}}\left(\ell \right).Math.}^2{P}_T\left(\ell \right),& \mathrm{Eqn}\left(62\right)\end{array}$$\begin{array}{cc}{\stackrel{\_}{\gamma }}_{max-\mathrm{INR}}\left(k_{\mathrm{sub}};\ell \right)\approx \frac{M_{\mathrm{DoF}}}{L_T-1}\underset{{\ell }^{\prime }\ne \ell }{.Math.}{\stackrel{\_}{\gamma }}_{\mathrm{Rx}-\mathrm{SNR}}\left(k_{\mathrm{sub}};{\ell }^{\prime }\right).& \mathrm{Eqn}\left(63\right)\end{array}$
As Eqn (61) shows, the max-SINR weights can excise up to L.sub.T−1 strong MOS-DSSS signals impinging on the receiver in each subband, and can provide an additional factor of M.sub.DoF−L.sub.T+1 SNR gain to suppress the remaining background noise and interference in each subband. In low-SNR environments where all of the navigation signals are received well below the noise floor (γ.sub.RX-SNR(k.sub.sub;l)<<1/M.sub.DoF), the max-SINR weights can provide the full factor-of-M.sub.DoF SNR gain available to the receiver.

(134) In aspects of the invention, the symbol sequence estimates from each subband are then further combined using cross-band linear combining weights

(135) ${\left\{g_{\mathrm{sub}}\left(k_{\mathrm{sub}};\ell \right)\right\}}_{k_{\mathrm{sub}}=0}^{K_{\mathrm{sub}}-1},$
yielding multi-subband symbol sequence estimate

(136) $\begin{array}{cc}\begin{array}{c}{\hat{d}}_R\left(n_{\mathrm{sym}};\ell \right)\approx \underset{k_{\mathrm{sub}}=0}{\overset{K_{\mathrm{sub}}-1}{.Math.}}{g}_{\mathrm{sub}}^{*}\left(k_{\mathrm{sub}};\ell \right){\hat{d}}_R\left(n_{\mathrm{sym}};k_{\mathrm{sub}};\ell \right)\\ \approx g_{\mathrm{sub}}^H\left(\ell \right){\hat{d}}_{\mathrm{sub}}\left(n_{\mathrm{sym}};\ell \right)\end{array}.& \mathrm{Eqn}\left(64\right)\end{array}$
Assuming each subband has substantively excised the MOS-DSSS signals impinging on the receiver, and has provided a unit-power output signal, then {circumflex over (d)}.sub.sub(n.sub.sym;l) can be approximated by

(137) $\begin{array}{c}{\hat{d}}_R\left(n_{\mathrm{sym}};\ell \right)\approx {.Math.}_{max-\mathrm{SINR}}^{1/2}\left(\ell \right)\epsilon \left(n_{\mathrm{sym}};\ell \right)+a_{max-\mathrm{SINR}}\left(\ell \right){\hat{d}}_R\left(n_{\mathrm{sym}};\ell \right)\\ {.Math.}_{max-\mathrm{SINR}}\left(\ell \right)=\mathrm{diag}\left\{\frac{{\gamma }_{max-\mathrm{SINR}}\left(k_{\mathrm{sub}};\ell \right)}{{\left(1+{\gamma }_{max-\mathrm{SINR}}\left(k_{\mathrm{sub}};\ell \right)\right)}^2}\right\}\\ a_{max-\mathrm{SINR}}\left(\ell \right)=\left[\frac{{\gamma }_{max-\mathrm{SINR}}\left(k_{\mathrm{sub}};\ell \right)}{1+{\gamma }_{max-\mathrm{SINR}}\left(k_{\mathrm{sub}};\ell \right)}\right]\end{array}.$
The SINR of this signal is maximized by setting g.sub.sub(l)=[1+γ.sub.Max-SINR(k.sub.sub;l)], yielding output SINR

(138) $\begin{array}{cc}{\gamma }_{max-\mathrm{SINR}}\left(n_{\mathrm{sym}};\ell \right)=\underset{k_{\mathrm{sub}}=0}{\overset{K_{\mathrm{sub}}-1}{.Math.}}{\gamma }_{max-\mathrm{SINR}}\left(k_{\mathrm{sub}};\ell \right),& \mathrm{Eqn}\left(65\right)\end{array}$
which has mean lower bound