APPARATUS AND METHODS FOR DC-OFFSET ESTIMATION

Abstract

A radio system comprises a radio transmitter apparatus and a radio receiver apparatus. The radio transmitter apparatus is configured to transmit a continuous-wave radio-frequency signal having a first frequency. The radio receiver apparatus comprises: an antenna for receiving the continuous-wave radio-frequency signal; a local oscillator for generating a periodic signal at a second frequency which differs from the first frequency by a frequency offset; a mixer for mixing the received continuous-wave radio-frequency signal with the periodic signal to generate a down-mixed signal; and a processor or other circuitry configured to generate frequency-offset data from the down-mixed signal, wherein the frequency-offset data is representative of an estimate of the frequency offset. The processor or other circuitry is configured to use the frequency-offset data to generate DC-offset data representative of an estimate of a DC offset component of the down-mixed signal.

Claims

1. A radio-frequency reception method comprising: receiving a continuous-wave radio-frequency signal having a first frequency; generating a periodic signal at a second frequency which differs from the first frequency by a frequency offset; mixing the received radio-frequency signal with the periodic signal to generate a down-mixed signal; processing the down-mixed signal to generate frequency-offset data representative of an estimate of the frequency offset; and processing the frequency-offset data to generate DC-offset data representative of an estimate of a DC offset component of the down-mixed signal, from the frequency-offset data.

2. The radio-frequency reception method of claim 1, further comprising using the DC-offset data to reduce or remove a DC offset component from the down-mixed signal.

3. The radio-frequency reception method of claim 1, wherein the frequency offset is in the range 5 kHz to 100 kHz.

4. The radio-frequency reception method of claim 1, wherein the frequency offset comprises a predetermined intermediate-frequency component and an error component, and wherein the predetermined intermediate-frequency component is around 10 kHz or is in the range 5 kHz to 100 kHz.

5. The radio-frequency reception method of claim 1, comprising processing at most one cycle period of the down-mixed signal to generate the frequency-offset data and the DC-offset data.

6. The radio-frequency reception method of claim 1, further comprising generating phase-difference data representative of a phase difference between the received continuous-wave radio-frequency signal and the generated periodic signal.

7. The radio-frequency reception method of claim 6, further comprising using the phase-difference data to determine a distance a transmitter of the continuous-wave radio-frequency signal and a receiver of the continuous-wave radio-frequency signal.

8. The radio-frequency reception method of claim 1, wherein generating the frequency-offset data comprises generating difference data representative of a difference signal comprising a sequence of differences between pairs of sample values of the down-mixed signal, wherein the sample values of each pair are separated by a common distance.

9. The radio-frequency reception method of claim 1, wherein the DC-offset data represents a maximum-likelihood estimate of the DC offset component of the down-mixed signal.

10. A radio receiver apparatus comprising: an electrical input for receiving a continuous-wave radio-frequency signal having a first frequency; a local oscillator for generating a periodic signal at a second frequency which differs from the first frequency by a frequency offset; a mixer for mixing the received continuous-wave radio-frequency signal with the periodic signal to generate a down-mixed signal; and a processor or other circuitry configured to generate frequency-offset data from the down-mixed signal, wherein the frequency-offset data is representative of an estimate of the frequency offset, and configured to process the frequency-offset data to generate DC-offset data representative of an estimate of a DC offset component of the down-mixed signal, from the frequency-offset data.

11. The radio receiver apparatus of claim 10, wherein the processor or other circuitry is further configured to use the DC-offset data to reduce or remove a DC offset component from the down-mixed signal.

12. The radio receiver apparatus of claim 10, wherein the processor or other circuitry is further configured to process at most one cycle period of the down-mixed signal to generate the frequency-offset data and the DC-offset data.

13. The radio receiver apparatus of claim 10, wherein the processor or other circuitry is further configured to generate phase-difference data representative of a phase difference between the received continuous-wave radio-frequency signal and the generated periodic signal.

14. The radio receiver apparatus of claim 13, wherein the processor or other circuitry is further configured to use the phase-difference data to determine a distance between the radio receiver apparatus and a transmitter of the continuous-wave radio-frequency signal.

15. The radio receiver apparatus of claim 10, wherein the processor or other circuitry is configured to generate the frequency-offset data by generating difference data representative of a difference signal comprising a sequence of differences between pairs of sample values of the down-mixed signal, wherein the sample values of each pair are separated by a common distance.

16. The radio receiver apparatus of claim 10, wherein the DC-offset data represents a maximum-likelihood estimate of the DC offset component of the down-mixed signal.

17. A radio system comprising a radio transmitter apparatus and a radio receiver apparatus, wherein the radio transmitter apparatus is configured to transmit a continuous-wave radio-frequency signal having a first frequency, and wherein the radio receiver apparatus comprises: an antenna for receiving the continuous-wave radio-frequency signal; a local oscillator for generating a periodic signal at a second frequency which differs from the first frequency by a frequency offset; a mixer for mixing the received continuous-wave radio-frequency signal with the periodic signal to generate a down-mixed signal; and a processor or other circuitry configured to generate frequency-offset data from the down-mixed signal, wherein the frequency-offset data is representative of an estimate of the frequency offset, and configured to process the frequency-offset data to generate DC-offset data representative of an estimate of a DC offset component of the down-mixed signal, from the frequency-offset data.

18. The radio system of claim 17, wherein the frequency offset is in the range 5 kHz to 100 kHz.

19. The radio system of claim 17, wherein: the radio transmitter apparatus comprises a transmitter crystal oscillator, having a nominal transmitter oscillator frequency, and transmitter local-oscillator circuitry configured to generate a transmitter local-oscillator signal from the transmitter crystal oscillator; the radio receiver apparatus comprises a receiver crystal oscillator, having a nominal receiver oscillator frequency, and receiver local-oscillator circuitry configured to generate a receiver local-oscillator signal from the receiver crystal oscillator; the transmitter local-oscillator circuitry and the receiver local-oscillator circuitry are configured so that the transmitter local-oscillator signal and the receiver local-oscillator signal are offset by a frequency offset in the range 5 kHz to 100 kHz when the transmitter crystal oscillator oscillates at the transmitter oscillator frequency and when the receiver crystal oscillator oscillates at the receiver oscillator frequency.

20. The radio system of any of claim 17, configured to generate phase-difference data representative of a phase difference between the received continuous-wave radio-frequency signal and the generated periodic signal, and to use the phase-difference data to determine a distance between the radio transmitter apparatus and the radio receiver apparatus.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Certain preferred embodiments of the invention will now be described, by way of example only, with reference to the accompanying drawings, in which:

[0059] FIG. 1 is a schematic diagram of a radio communication system embodying the invention;

[0060] FIG. 2 is a schematic diagram of a radio receiver embodying the invention;

[0061] FIG. 3 is a flow chart of a DC offset estimation process embodying the invention; and

[0062] FIG. 4 is an I-Q phase diagram showing outputs from a simulation of the DC offset estimation process.

DETAILED DESCRIPTION

[0063] FIG. 1 shows a first radio transceiver device 1 and a second radio transceiver device 2. The devices 1, 2 could be any radio-equipped devices, such as smartphones, wireless sensors, household appliances, land vehicles, satellites, etc. In one set of embodiments, the devices 1, 2 are consumer electronics products, and can communicate with each other over a Bluetooth™ Low Energy communication link.

[0064] The devices 1, 2 are separated by a distance, d. This distance could be of the order of centimetres, metres, kilometres, tens of kilometres, hundreds of kilometres or more.

[0065] The devices 1, 2 are configured to measure the distance d using multicarrier phase-based ranging. Note that phase-based ranging is quite different from radar ranging, in which a radar device receives reflections of radio signals off a target. In phase-based ranging, radio waves are transmitted from a radio transmitter device and are received by a radio receiver device that is located away from the radio transmitter device.

[0066] The devices 1, 2 are not synchronized, but perform multicarrier phase-based ranging to estimate d. An initiator—e.g., the first device 1—first sends a continuous-wave (CW) radio signal that is an amplified version of a sine wave, of frequency f.sub.i,LO, generated by its local oscillator (LO). This signal is then received by a responder—e.g., the second device 2—that has its LO set to a similar frequency, f.sub.r,LO, (as explained in more detail below). The responder 2 measures a first phase difference between its LO and the incoming signal.

[0067] The initiator 1 and responder 2 then switch roles, and the responder 2 transmits a CW signal which is an amplified version of its local oscillator (LO)—i.e. a sine wave of frequency f.sub.r,LO. This is then received by the initiator 1, which measures a second phase difference between the initiator's LO, set to the frequency f.sub.i,LO, and the incoming signal. The LOs of both devices 1, 2 are kept constant during the two phases of the procedure.

[0068] Assuming no multipath interference, noise or other impairments, the sum ψ.sub.ir of the two phase differences is related to the distance, d, between the devices 1, 2 by

[00001]$\begin{array}{cc}{\psi }_{\mathrm{ir}}\approx \frac{2\pi \left(f_{i,\mathrm{LO}}+f_{r,\mathrm{LO}}\right)d}{c}-2\mathrm{\pi \Delta }fT+C& \left(1\right)\end{array}$

[0069] where f.sub.i,LO and f.sub.r,LO are the LO frequencies of the initiator 1 and responder 2 respectively, T is the time between phase measurements, Δf is the offset between the two LO frequencies, and C is a constant.

[0070] The frequency offset Δf contains two components: a first component, f.sub.IF, is an intentional, predetermined offset (i.e. an intermediate mixing frequency), while a second component, f.sub.errOr, is an unknown offset due to oscillator inaccuracies between the two devices 1, 2. I.e.

[00002]$\Delta f=f_{\mathrm{IF}}+f_{\mathrm{error}}.$

[0071] The two LO frequencies are thus related by

[00003]$f_{r,\mathrm{LO}}=f_{i,\mathrm{LO}}-\Delta f=f_{i,\mathrm{LO}}-\left(f_{\mathrm{IF}}+f_{\mathrm{error}}\right).$

[0072] By taking phase measurements over at least two different f.sub.i,LO frequencies (e.g., over a set of two or more Bluetooth™ Low Energy carrier frequencies), and calculating the gradient of ψ.sub.ir as a function of f.sub.i,LO, the distance, d, can be estimated.

[0073] The responder 2 may transmit its phase measurements to the initiator 1, e.g., as data in one or more radio packets, and the initiator 1 may use these measurements to estimate the distance, d. Alternatively, the phase measurements could be sent to the responder 2, or to a third device, such as a network server, for performing the distance estimation.

[0074] Note that, if Δf=0, the time dependence would disappear. Correcting precisely for the cases where f.sub.IF≠0 and f.sub.error≠0 requires knowledge of T. This may be difficult to achieve accurately in the absence of modulated data to provide timing information, because the devices 1, 2 are not synchronous. Therefore it is desirable to minimize Lf so that the time dependence is sufficiently small that it can be disregarded while preserving a desired level of accuracy in the distance estimation.

[0075] However, the negative impact of DC offset on the ability to measure the phase increases as Δf reduces. If Δf=0, the DC offset can completely mask the received signal.

[0076] The present embodiments address this by setting an intentional non-zero frequency offset, f.sub.IF, between the nominal frequencies of the receiver and transmitter local oscillators, and by seeking to minimise f.sub.error, so as to provide a small, but non-zero, frequency offset Δf. This frequency offset may be of the order of 10 kHz—e.g., around kHz, 10 kHz, 20 kHz or so. The devices 1,2 between them estimate the actual frequency offset, Δf, including an error, and use this to estimate the DC offset instantaneously—i.e. while receiving the constant-wave signal—rather than requiring a separate DC-offset calibration phase. This approach therefore doesn't rely on the accuracy of an earlier-measured estimate of the DC offset measurement, and may therefore provide more accurate distance estimates.

[0077] FIG. 2 provides more detail of some of the internal components of the initiator device 1 in an exemplary implementation. The responder device 2 may be similar.

[0078] The device 1 contains, within a housing 3, a radio microcontroller chip 4 that supports Bluetooth™ Low Energy communications. It may additionally or instead support other radio protocols such as IEEE 802.11, 3GPP LTE Cat-M1, 3GPP LTE NB-IoT, IEEE 802.15.4, Zigbee™, Thread™, ANT™, etc.

[0079] The radio chip 4 contains a low-noise amplifier (LNA) 5, a local oscillator (LO) 6, a quadrature mixer 7 for mixing an incoming signal with a periodic LO signal, receive-path filtering 8, an analog-to-digital converter (ADC) 9, radio memory 10 and a radio processor 11. On a transmit path, it contains a power amplifier (PA) 12, which can also receive the periodic LO signal from the local oscillator 6. The radio memory 10 may include volatile memory (e.g., RAM) and non-volatile memory (e.g., flash). The radio processor 11 may be a general purpose processor such as an Arm™ Cortex-M™ processor; it may also include one or more DSPs.

[0080] The device 1 may also contain a further system processor 13, system memory 14, peripherals 15 such as a temperature sensor or I/O modules, and a battery 16. A radio antenna 17 may be within or external to the housing 3 and is connected to the radio chip 4 by appropriate components. Of course, it will be appreciated that the device 1 may contain other elements, such as buses, crystals, digital logic, analog circuitry, discrete active components, discrete passive components, further processors, user interface components, etc. which are not shown here for the sake of simplicity. Further details of the transmit path, e.g. as used when transmitting encoded data, are also omitted here for simplicity. The device 1 may be a component of a larger device, such as a car, or it may be a standalone device.

[0081] In use, software stored in the radio memory 10 is executed by the radio processor 11 to perform a distance estimation process as disclosed herein. The processor 11 is able to set the frequency of the local oscillator 6 to different frequencies as required. The second device 2 contains complementary software for carrying out its part of the process. In other embodiments, some or all of the process may be carried out by hard-wired logic in the radio devices 1, 2. The resulting distance estimate, d, may be output to the system processor 13 for further use, or may be stored for use by the radio chip 4 itself.

[0082] FIG. 3 is a flowchart showing the main steps of the distance estimation process in one embodiment.

[0083] In a first phase, the initiator device 1 sets its local oscillator 6 to the frequency f.sub.i,LO1 and transmits 30 a sinusoidal continuous-wave radio signal of frequency f.sub.i,LO1. The responder device 2 mixes 31 the incoming signal with a signal from its local oscillator having a frequency f.sub.r,LO1. The devices 1, 2 may be configured for an intended frequency offset Δf=f.sub.i,LO1−f.sub.r,LO1 of around 10 kHz. The responder device 2 estimates 32 the actual frequency offset, Δf, which may include an unknown error term, as explained in more detail below. It then estimates 33 a DC offset, as explained in more detail below. It then compensates 34 the sampled received signal to remove the estimated DC offset. Next, the responder device 2 measures 35 the phase difference ψ.sub.r,LO1 between the compensated received signal and the locally-generated signal.

[0084] In a second phase, the respond device 2 keeps its local oscillator at f.sub.r,LO1 and transmits 36 a sinusoidal continuous-wave radio signal of frequency f.sub.r,LO1. The initiator device 1 mixes 37 the incoming signal with a signal from its local oscillator 6 which remains set at f.sub.i,LO1. The initiator device 1 estimates 38 the frequency offset Δf=f.sub.i,LO1−f.sub.r,LO1 as explained in more detail below. It then estimates 39 a DC offset, as explained in more detail below. It then compensates 40 the sampled received signal to remove the estimated DC offset. Next, the initiator device 1 measures 41 the phase difference ψ.sub.i,LO1 between the compensated received signal and the locally-generated signal.

[0085] These steps 30 to 41 are then repeated, in steps 30′ to 41′. Both phases are the same, except that a different pair of local oscillator values, f.sub.i,LO2, f.sub.r,LO2, is used. The same intended frequency offset, Δf, may be used.

[0086] Although only two instances are shown in FIG. 3, the steps 30 to 41 may be performed any number of times, n, using different f.sub.i,LO values—e.g. n=2, 3, 5, 10 or more—depending on requirements.

[0087] Once sufficient phase differences have been collected, the responder device 2 transmits 42 all the phase differences it has measured, ψ.sub.r,LO1, ψ.sub.r,LO2, . . . , ψ.sub.r,LOn, to the initiator device 1, as data encoded in a data packet.

[0088] The initiator device 1 then sums 43 the two ψ.sub.i,Lo, ψ.sub.r,LO values for each frequency pair to get ψ.sub.r,LO1, ψ.sub.r,LO2, . . . , ψ.sub.i,r,LOn. It then estimates 44 the separation distance, d, from the gradient of ψ.sub.i,r over frequency, according to equation (1) above.

[0089] Of course, many variations are possible. For example, the responder device 2 could transmit its frequency offset estimates to the initiator device 1, instead of the initiator device 1 calculating these independently (although, if there is a significant time delay between the sine wave transmissions 30, 31, it may be desirable for both devices 1, 2 to perform independent estimations).

[0090] The continuous-wave signals may be transmitted as elements within respective data packets, or they may be standalone transmissions.

[0091] Exemplary methods for estimating the frequency offset and the DC offset, according to some embodiments, will now be described.

System Model

[0092] Assume that a continuous wave is transmitted by the initiator 1 for a period of Tμs. Define c as the DC offset. The initial phase and amplitude of the incoming signal as seen at the output of an ADC of the responder 2 is a=b exp(iψ) where b>0 and ψ∈[0,2π). The frequency offset is Δf. N samples are collected at a sample rate f.sub.s with sampling period T.sub.s=1/f.sub.s.

[0093] The sampled down-mixed signal at the output of the ADC, y, is given by

[00004]$y=c1+\mathrm{an}+v$

where 1 is the N×1 vector containing only 1's, v is an N×1 vector of random complex noise, and n is the frequency rotation vector with elements n.sub.k=exp(i2πΔfT.sub.sk).

Algorithm

[0094] The algorithm proceeds by first obtaining an estimate of the frequency offset, and then solving for the DC offset.

Frequency Offset Estimation A frequency estimate may be obtained in the following fashion.

[0095] First, a difference signal, z, is generated as

[00005]$z_k=y_k-y_{k-D_1}$

where D.sub.1 is a positive integer.

[0096] This gives

[00006]$\begin{array}{c}{z}_k={\mathrm{an}}_k-{\mathrm{an}}_{k-D1}+v_k-v_{k-D1}\\ =a\exp \left(i2\mathrm{\pi \Delta }{\mathrm{fT}}_{s}k\right)\left[1-\exp \left(-i2\mathrm{\pi \Delta }{\mathrm{fT}}_s{D}_1\right)\right]+v_k-v_{k-D1}\end{array}$

[0097] Note that the DC offset from the signal is removed in the difference signal, z, but the frequency offset is still present.

[0098] The frequency of this difference signal is measured by an appropriate method—for example, with a D-spaced Kay's frequency estimator. However, other options are possible, such as methods disclosed in “Frequency estimation by phase unwrapping”, R. G. McKilliam, B. G. Quinn, B. Moran et al., IEEE Transactions on Signal Processing, vol. 58, no. 6, pp. 2953-2963, 2010.

[0099] Using a spacing of D.sub.2, a series, w, is calculated as

[00007]$w_k=z_k{z}_{k-D_2}^{*}$

[0100] Disregarding the noise term, W.sub.k is equal to

[00008]$\begin{array}{cc}{w}_k=& {.Math.a.Math.}^2\exp \left(i2\mathrm{\pi \Delta }{\mathrm{fT}}_{s}k\right)\left[1-\exp \left(-i2\mathrm{\pi \Delta }{\mathrm{fT}}_s{D}_1\right)\right]\exp \left(-i2\mathrm{\pi \Delta }{\mathrm{fT}}_s\left(k-D_2\right)\right)\\ & {\left[1-\exp \left(-i2\mathrm{\pi \Delta }{\mathrm{fT}}_s{D}_1\right)\right]}^{*}\\ =& {.Math.a.Math.}^2.Math.1-\exp \left(-i2\mathrm{\pi \Delta }{\mathrm{fT}}_s{D}_1\right).Math.\exp \left(i2\mathrm{\pi \Delta }{\mathrm{fT}}_{s}k\right)\exp \left(-i2\mathrm{\pi \Delta }{\mathrm{fT}}_s\left(k-D_2\right)\right)\\ =& b\exp \left(i2\mathrm{\pi \Delta }{\mathrm{fT}}_s{D}_2\right)\end{array}$

where b is a constant.

[0101] Therefore, the frequency offset is estimated as

[00009]$\Delta \hat{f}=\angle \left(\Sigma w_k\right)\text{/}\left(2\pi T_s{D}_2\right)$

DC Offset Estimation

[0102] Assuming the noise v.sub.k is circularly symmetric complex Gaussian white noise, the maximum likelihood estimate of the DC offset, c, can be obtained using the frequency offset estimate Δ{circumflex over (f)}.

[0103] Define {circumflex over (n)}.sub.k=exp(i2πΔfT.sub.sk). The log-likelihood function is given by

[00010]$L={.Math.y-c1-a\hat{n}.Math.}^2$

[0104] After some manipulation it can be shown that

[00011]$\frac{\partial L}{\partial a^{*}}=-{\hat{n}}^{*}\left(y-c1\right)+a{.Math.\hat{n}.Math.}^2$

[0105] Solving for

[00012]$\frac{\partial L}{\partial a^{*}}=0,$

gives me maximum-likelihood estimate of the initial phase and amplitude, a, as

[00013]$\hat{a}=\frac{{\hat{n}}^{*}\left(y-c1\right)}{{.Math.\hat{n}.Math.}^2}=\frac{{\hat{n}}^{*}\left(y-c1\right)}{N}$

[0106] Substituting â into L gives

[00014]$L={.Math.y-c1-\frac{{\hat{n}}^{*}\left(y-c1\right)}{N}\hat{n}.Math.}^2={.Math.\left(I-\frac{\hat{n}{\hat{n}}^{*}}{N}\right)\left(y-c1\right).Math.}^2={.Math.P\left(y-c1\right).Math.}^2$

where I is the N×N identity matrix and

[00015]$P=I-\frac{\hat{n}{\hat{n}}^{*}}{N}$

is a projection matrix.
Define {tilde over (y)}=Py and {tilde over (1)}=P1.

[0107] Then

[00016]$\frac{\partial L}{\partial c^{*}}=-{\tilde{1}}^{*}\tilde{y}+c{.Math.\tilde{1}.Math.}^2$

[0108] Solving for

[00017]$\frac{\partial L}{\partial c^{*}}=0,$

gives the maximum-likelihood estimate of the DC offset as

[00018]$\begin{array}{cc}\hat{c}=\frac{{\tilde{1}}^{*}\tilde{y}}{{.Math.\tilde{1}.Math.}^2}& \left(2\right)\end{array}$

[0109] Note that this does not require the devices 1, 2 to perform actual matrix computations involving P.

[0110] In practice the devices 1, 2 calculate {tilde over (y)} and {tilde over (1)} as

[00019]$\tilde{y}=y-\frac{{\hat{n}}^{*}\hat{n}}{N}y,\mathrm{and}$$\tilde{1}=1-\frac{{\hat{n}}^{*}\hat{n}}{N}1$

which they input to equation (2), above, to determine an estimate, ĉ, of the DC offset.

Phase Estimation

[0111] Once the DC offset has been estimated, the device 1, 2 can simply subtract the contribution to the sampled signal, y.sub.k, arising from the device's own local oscillator, and can then determine the phase difference of the incoming signal.

[0112] This can be done by calculating ŷ using the formula

[00020]$\hat{y}=y-\hat{c}1=\left(c-\hat{c}\right)1+\mathrm{an}+v$

and obtaining a phase estimate, {circumflex over (ψ)}, by taking the complex argument of the average of the values of ŷ, i.e.

[00021]$\hat{\psi }=\arg \left\{1^{*}\hat{y}\right\}$

[0113] Simulations

[0114] FIG. 4 shows results of a simulation using the methodology for joint estimation of the DC offset and frequency offset as disclosed above.

[0115] In this simulation, a measurement of a received continuous-wave was made over a period of 32 μs. A sample rate of 8 MHz was employed (i.e., ˜256 samples in total). A value of D.sub.1=D.sub.2=64 was used, so that D.sub.1T=D.sub.2 T=8 μs, where T is the sample period of 0.125 μs. The ADC range was −1023 to 1023. The received amplitude |a| was 511.5 (representing automatic gain control). The phase of a and the DC offset c were both randomly chosen from the range [0, 2π). The amplitude of the DC offset was chosen from a uniform distribution with amplitude [0, 511.5). Complex Gaussian noise was added with a real and imaginary standard deviation of ˜20.

[0116] FIG. 4 shows an example output of the algorithm under a simulated scenario in which the local oscillator is offset from the frequency of the received signal by 7.80 kHz. In this simulation run, the actual DC offset was randomly selected to equal −199-25i. This is represented in FIG. 4 by a point 50.

[0117] The graph also shows an arced cluster of points 51 which represent the received down-mixed signal, at the output of the ADC, at different sample times, under the presence of simulated Gaussian random noise. Over the 32 μs time frame, these samples cover a quarter rotation of the unit circle.

[0118] A further cluster of points 52 represents the difference signal, z.sub.k, calculted as described above, in which the DC offset has been removed but the frequency offset is still present.

[0119] The frequency offset estimate obtained from the difference signal in this simulation, using the algorithm described above, was 7.96 kHz. This represents an accuracy of approximately 160 Hz.

[0120] The graph also includes a point 53 representing the estimated DC offset, calculated as described above. This was estimated as having the value −211-22i using the algorithm disclosed above. This represents an accuracy of approximately 6% in the DC offset estimation.

[0121] Thus, with only a quarter cycle of the down-mixed signal, the DC offset can be estimated with an accuracy of ˜6%. This demonstrates that this method can enable rapid and accurate estimation of DC offset.

[0122] It will be appreciated by those skilled in the art that the invention has been illustrated by describing one or more specific embodiments thereof, but is not limited to these embodiments; many variations and modifications are possible, within the scope of the accompanying claims.