WO2014191712A1 - A method of analyzing radio-frequencv signals using sub-nyquist sampling - Google Patents

Abstract

A method of analyzing a radio-frequency input signal using sub-Nyquist sampling comprises receiving the input signal, applying a relative time delay between a first channel and a second channel, sampling at a respective sampling rate each of the first channel and the second channel, wherein for each channel the sampling rate is less than the Nyquist rate, applying a respective time-frequency transform to each of the first channel and the second channel, identifying at least one time-frequency component in the first channel and the second channel, and, for the or each time- frequency component, obtaining a comparison of the time-frequency component from the first channel and the time-frequency component from the second channel, and distinguishing the time-frequency component from a plurality of aliases of the time- frequency component in dependence on the comparison.

Description

A method of analyzing radio-frequency signals using sub-Nyquist sampling Field of the invention

The present invention relates to the analysis of radio-frequency input signals, for example wideband radio-frequency signals, using sub-Nyquist sampling. The invention has particular application to, but is not limited to, electronic surveillance systems.

Background

The Nyquist-Shannon Sampling theorem states that a function x(t) that has a bandwidth of B Hertz may be completely determined by sampling the function at a sampling interval of T - 1/2B seconds. The frequency 2B is known as the Nyquist rate. The sampling interval T = 1/2B s is the sampling interval that is theoretically needed to digitize an analog signal while retaining all the information in the signal. In the case of a baseband signal, B will be equal to the highest signal frequency.

For applications that involve high frequencies and large quantities of data, sampling with a sampling interval T = \I2B s may be a challenging requirement. For example, to fully determine a function with a bandwidth of 2 GHz to 18 GHz would require a sampling rate of 32 Gsps (Giga-samples-per-second), and generate very large quantities of data (32 GB of data per second if 8 bits per sample are collected).

Therefore there has been significant interest in recent years in applications that use sub-Nyquist sampling (sampling at a lower rate than the Nyquist rate 2B ) to detect, identify and reconstruct relevant functions.

Electronic Surveillance (ES) is concerned generally with the gathering of information about electromagnetic signals through their passive interception. Radio-frequencies in general may be considered to cover the range from, for example, 3 kHz to 300 GHz. The signals of interest to an electronic surveillance system may be radar signals or communication signals, which may for example fall within the range of 100 MHz to 30 GHz, with 2 GHz to 18 GHz being of particular interest for radar applications. There is a need in Electronic Surveillance (ES) to monitor a wide frequency band (for example, 2 to 18 GHz) where radar and communication signals may occur. An ES system may be required to detect, analyse and, if necessary, reconstruct, signals anywhere in its frequency band with sufficient sensitivity and in real-time or near realtime. This can be particularly challenging where the signals involve, for example, rapid changes of frequency or other Low Probability of Intercept (LPI) characteristics.

Early solutions used instantaneous frequency measurements to detect and categorise radio-frequency (RF) signals. However, such systems have limited sensitivity and cannot sort multiple signals simultaneously. In more recent years, digital receivers have been preferred due to their ability to process a wider range of signals including Frequency Modulated Continuous Wave (FMCW) and other Low Probability of Intercept (LPI) radar signals. However, size, weight and power (SWAP) requirements impose limitations on the sampling rates and hence the bandwidths that are viable for such digital receivers.

In principle, a full band channelized receiver can be constructed using a bank of digital receivers to cover a wide band. Each of the receivers covers a respective narrower band within the wide band, and the combination of the narrower bands covers the full band. However, such a solution may not meet practical SWAP requirements. An alternative strategy that is used in practice is to employ a rapidly swept super-heterodyne receiver (RSSR). At any instant, an RSSR monitors only a single narrow frequency band. The receiver observes the full frequency range by sweeping a narrow band filter across the full band. While an RSSR has relatively high sensitivity due to its narrow bandwidth, it tends to have a poor Probability of Intercept (POI) and performs poorly in detecting frequency-agile sources (sources that rapidly change their operating frequency, for example to avoid detection). In order for a high probability of detection of an individual radar pulse, the dwell time of the RSSR must be considerably shorter than the radar pulse width, to make it likely that the RSSR will sweep through the pulse at some time within the pulse. Use of a short dwell time sacrifices the sensitivity of the receiver, which would be higher if it could integrate over the whole pulse.

A radio-frequency ES receiver covering 2 GHz to 18 GHz can potentially see millions of pulses per second. However, at a given time instant the number of active frequencies may be relatively sparse (small compared to the number of possible frequencies). It can be concluded that capturing the entire bandwidth at the Nyquist rate would contain significant amounts of redundant information.

A new sampling theory has emerged in the last few years known as compressed sensing. Compressed sensing theory states that if an input signal is sparse (contains relatively few frequency components at a given time) then the input signal can be digitized with a sample rate much less than the Nyquist rate. Several techniques have been described in the literature.

The earliest compressed sensing system was proposed by Feng and Bresler (P Feng and Y Bresler, 'Spectrum-blind minimum-rate sampling and reconstruction of multiband signals', Proc. IEEE International Conf. on Acoustics, Speech and Signal Processing, vol. 3, pp. 1688-1691). The system of Feng and Bresler exploited a multi-coset sampling strategy using a parallel bank of sub-Nyquist sampling channels.

A multi-coset sampling strategy involves sampling an input signal (in this case, sampling the signal in time) using a periodic non-uniform sampling pattern. The sampling is carried out using a first sampling channel and one or more further sampling channels (collectively referred to as a set of multi-coset channels). Each channel samples the input signal at a common periodic sampling rate l/LT , where T - 1/2B is the Nyquist interval for the frequency band in which the input signal is acquired (II T is the Nyquist rate) and L is an integer value which is a system parameter. For example, if L - 32, then each channel undersamples the input signal by a factor of 32, the sampling interval being 32 times the Nyquist interval.

There is a relative time delay between sampling on the different channels, each channel having its own unique time delay with regard to a regular periodic l/LT sampling pattern which may be taken to start at t = 0.

Consider the case of q channels, i = \,...,q . Each channel has an associated time delay c_tT , where each c, is less than L and is not the same as any other c, , and T is the Nyquist interval. Therefore the samples from each channel may be described as a coset of the samples from any other of the channels with respect to the full Nyquist sampling sequence, because the samples for each channel comprise the same regular periodic sampling pattern at the same sampling interval LT , but shifted by a relative time delay that is defined by the respective channel's value of c_t .

Each channel individually undersamples the signal by a factor L and therefore there are L distinct frequency subbands covering the frequency ranges [/ / LT ',(/ + 1) / LT] that are aliased together, (/ = 0..J - 1) . A frequency subband that contains a component of the input signal x(t) is referred to as an active subband.

For sub-Nyquist sampling, the number of channels q must be less than L (q = L would be Nyquist rate sampling). On average, the input signal is undersampled by a factor of LI q . For example, if L = 32 and there are 4 channels, q = 4 , then the average sampling rate is ql LT - 1 /ST and so the input signal is undersampled by a factor of 8.

In the system of Feng and Bresler, signal reconstruction is possible as long as the number of active subbands in the signal is fewer than the number of multi-coset channels. Identification of the active subbands is achieved using a variant of the MUSIC algorithm (RO Schmidt, 'Multiple emitter location and signal parameter estimation', Proceedings of RADC Spectral Estimation Workshop, pp 243-258, 1979) which involves a computationally expensive eigenvalue decomposition and requires an accumulation of sufficient samples to calculate an accurate cross channel covariance matrix.

Multi-coset sampling was rediscovered by Mishali and Eldar (M Mishali and Y Eldar, 'Blind multiband signal reconstruction; Compressed sensing for analog signals', IEEE Trans. Signal Processing, vol. 57, no. 3, pp 993-1009, Mar. 2009) and variations of standard compressed sensing algorithms were proposed for signal reconstruction. Other sub-Nyquist sampling strategies have also been proposed based on the ideas of compressed sensing (D Donoho, 'Compressed Sensing', IEEE Trans. Info. Th., vol. 52, no. 4, pp 1289-1306, Apr. 2006). For example, random modulation sampling strategies using spread spectrum techniques similar to those used in telecommunications have been proposed (M Mishali and Y Eldar, 'From theory to practice: Sub-Nyquist sampling of sparse wideband analog signals', Selected Topics in Signal Processing, IEEE Journal of, vol. 4, no. 2, pp 375-391 , Apr 2010; J Tropp, J Laska, M Duarte, J Romberg and R Baraniuk, 'Beyond Nyquist: Efficient sampling of sparse bandlimited signals', IEEE Trans. Info. Th., vol. 56, no. 1, pp 520-544, Jan. 2010). However, the reconstruction techniques proposed for all these systems are based on standard compressed sensing algorithms.

Compressed sensing algorithms tend to fall into two categories:- greedy methods (for example, JA Tropp, 'Greed is good: algorithmic results for sparse approximation', IEEE Transactions on Information Theory, vol. 50, no. 10, pp 2231-2242, 2004) or optimization-based algorithms (for example, MAT Figueiredo, RD Nowak and SJ Wright, 'Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems', Selected Topics in Signal Processing, IEEE Journal of, vol. 1 , pp 586-597, 2007).

With regard to hardware, existing methods as described above usually require a complex digitizer structure (for some compressed sensing digitizers, pseudo-random sequences at the Nyquist rate are required to be generated). Furthermore, the reconstruction of the signal of interest may require complex iterative optimization algorithms. Although there have been significant advances in the speed of compressed sensing algorithms, iterative algorithms are unlikely to be suited to efficient low SWAP hardware implementations. When there is a requirement to simplify the architecture of a wideband ES receiver (for example, to reduce at least one of size, weight and power), a complex digitizer structure and the use of complex iterative algorithms are both undesirable traits.

Although analog to digital converter devices are available that are capable of producing sample rates up to 50 Gsps, they may suffer from poor dynamic range (around 40 dB, compared to a requirement of, for example, 70 dB) due to a low number of output bits (around 8 bits). Furthermore, such analog to digital converter devices require very large data bandwidths that place a heavy burden on the front end data interfaces and signal processing (such as the 32 GB/s data generation described for the 2 GHz to 18 GHz signal above).

Summary of the invention

In a first aspect of the invention there is provided a method of analyzing a radio- frequency input signal using sub-Nyquist sampling, comprising receiving the input signal, applying a relative time delay between a first channel and a second channel, sampling at a respective sampling rate each of the first channel and the second channel, wherein for each channel the sampling rate is less than the Nyquist rate, applying a respective time-frequency transform to each of the first channel and the second channel, identifying at least one time-frequency component of the input signal in the first channel and the second channel, and, for the or each time-frequency component, obtaining a comparison of the time-frequency component from the first channel and the time-frequency component from the second channel, and distinguishing the time-frequency component from a plurality of aliases of the time- frequency component in dependence on the comparison.

Sampling at a sub-Nyquist rate reduces the data bandwidth required by a system compared to sampling at the Nyquist rate. The method may be non-iterative and may be pipelined, and therefore may be suitable for a low SWAP implementation. It may avoid the need for complex digitizer systems and/or complex and iterative reconstruction algorithms.

The comparison may comprise a phase comparison of the first channel and the second channel.

Obtaining the comparison may comprise maximising an inner product between a phase vector and a cross-channel vector of time-frequency coefficients.

Distinguishing the or each time-frequency component may comprise allocating the or each time-frequency component to a subband.

The method may further comprise calculating a cross-channel vector of time- frequency coefficients. The method may further comprise calculating a magnitude of time-frequency coefficients on each channel.

The method may further comprise comparing time-frequency coefficients obtained from the time-frequency transforms to a threshold value. The threshold value may be representative of a threshold noise level.

The method may further comprise processing the input signal on at least one further channel, wherein processing the input signal on the further channel comprises applying a relative time delay to the further channel with respect to the first channel and to the second channel, sampling the further channel at the sampling rate, applying the time-frequency transform to the further channel, and identifying the or each time-frequency component in the or each further channel, and wherein distinguishing the or each time-frequency component from a plurality of aliased time- frequency components is based on a comparison of the or each time-frequency component from the first channel, the or each time-frequency component from the second channel and the or each time-frequency component from the or each further channel.

Using more than two channels may improve the sensitivity of the system where that is required, as the channels may be averaged to improve the signal to noise ratio. Using more than two channels can also help to distinguish each time-frequency component from its aliases.

Distinguishing the or each time-frequency component from the or each plurality of aliased time-frequency components may in some cases be based only on a comparison of a time-frequency component from the first channel and a time- frequency component from the second channel, wherein no further channel is provided and/or used. The method may enable good approximate reconstruction of the input signal from as few as two sub-Nyquist channels that both sample at the same rate.

The number of active subbands in the input signal may be greater than the number of channels. Where the input signal comprises at least M active subbands, distinguishing the or each time-frequency component may be based only on a comparison of time-frequency components on N channels, where M is greater than N.

The method may further comprise reconstructing at least part of the input signal.

Reconstructing at least part of the input signal may comprise reconstructing a discrete-time representation of at least part of the input signal. Reconstructing at least part of the input signal may comprise reconstruction of the full band discrete representation of at least part of the signal.

The reconstruction of the signal may comprise reconstruction of an analog representation of the signal. The reconstruction of the signal may be performed non-iteratively. A non-iterative reconstruction scheme may require reduced processing power. Therefore, the use of a non-iterative method of reconstruction may enable the development of a compact and cost-effective ES receiver that may meet SWAP requirements.

The method may further comprise continuously monitoring a frequency band, wherein the received input signal is received within the frequency band. Thus, longer analysis windows may be used than in systems which use rapid frequency sweeping, providing associated processing gain.

The Nyquist rate may be taken to equal 2B, where B is the bandwidth of the signal in the case of a non-baseband signal, or the maximum frequency of the signal in the case of a baseband signal.

The monitored frequency band may be between 100 MHz and 20 GHz, optionally between 2 GHz and 18 GHz.

The monitored frequency band may be wider than 5 GHz, optionally wider than 10 GHz, further optionally wider than 15 GHz.

The relative time delay may be selected for reduction of sampling jitter.

A sampling interval of the sampling may comprise an integer multiple of the Nyquist interval.

A sampling interval of the sampling may be a periodic sampling interval. The sampling may have a sampling pattern that is neither random nor pseudorandom. The sampling may have a sampling pattern that comprises a periodic non-uniform sampling pattern.

The sampling rate for the first channel may be the same as the sampling rate for the second channel. The use of the same sampling rate may aid synchronization of the system. The use of the same sampling rate for two or more channels may also allow the use of a receiver architecture which may be similar to that of existing interleaved analog to digital converters. This may also, for example, allow the use of existing calibration techniques. Use of a periodic sampling pattern may require less processing power than use of a random or pseudorandom sampling pattern. Use of a periodic sampling pattern may have improved noise characteristics over a random or pseudorandom sampling pattern.

The method may further comprise applying a filter to the input signal, optionally before applying the relative time delay, performing the sampling, or applying the time- frequency transform. Filtering out frequencies that are not of interest may improve the sensitivity of the system and the ability to identify the correct subband for each signal component.

The filter may comprise at least one of a frequency selective filter, a lowpass filter, a highpass filter, a bandpass filter.

The method may further comprise performing a relative calibration of the first channel and the second channel. The performance of the sampling scheme may be sensitive to gain and timing mismatches between the channels. Calibration of the system by adjusting the digital part of the system may improve performance.

Performing the relative calibration may comprise injecting a pilot signal within a known subband.

The method may further comprise receiving the pilot signal on each of the first channel and the second channel, and measuring at least one of the amplitude, frequency and phase of the injected signal on each of the first channel and the second channel.

The input signal may comprise an analog signal.

The input signal may be approximately sparse in at least one time-frequency representation. The input signal may comprise a plurality of signal components that are sparsely distributed.

The time-frequency representation in question may be any appropriate time- frequency representation for the system. This may enable sampling of a broad range of signals that are approximately sparse in some time-frequency representation, including chirped pulses which are of particular interest to radar. The input signal may comprise at least one of a radar signal and a communication signal.

The time frequency transform may comprise any suitable transform that transforms a signal into at least one time-frequency component representing the signal in the time and frequency domains.

The time-frequency transform may comprise at least one of a Gabor transform, a Gabor-type transform, a short-time Fourier transform, a short-time fractional Fourier transform, a wavelet transform, a chirplet transform or a polyphase filterbank.

Applying a respective time-frequency transform to each of the first channel and the second channel may comprise modifying each time-frequency transform to implement a digital fractional delay, the digital fractional delay corresponding to the relative time delay between the first channel and the second channel.

In another independent aspect of the invention there is provided a radio-frequency receiver apparatus comprising receiving means for receiving an input signal, time delay means for applying a relative time delay between a first channel and a second channel, sampling means for sampling at a respective sampling rate each of the first channel and the second channel, wherein for each channel the sampling rate is less than the Nyquist rate, transform means for applying a respective time-frequency transform to each of the first channel and the second channel and processing means for identifying at least one time-frequency component in the first channel and the second channel, and, for the or each time-frequency component, obtaining a comparison of the time-frequency component from the first channel and the time- frequency component from the second channel and distinguishing the time-frequency component from a plurality of aliases of the time-frequency component in dependence on the comparison.

In a further aspect of the invention, which may be provided independently, there is provided a radio-frequency receiving apparatus comprising a receiver for receiving an input signal, a time delay component for applying a relative time delay between a first channel and a second channel, a sampler for sampling at a respective sampling rate each of the first channel and the second channel, wherein for each channel the sampling rate is less than the Nyquist rate, a transform component for applying a respective time-frequency transform to each of the first channel and the second channel and a processor for identifying at least one time-frequency components in the first channel and the second channel, and, for the or each time-frequency component, obtaining a comparison of the time-frequency component from the first channel and the time-frequency component from the second channel and distinguishing the time-frequency component from a plurality of aliases of the time- frequency component in dependence on the comparison.

There may be provided a radio-frequency receiver apparatus or a method of analyzing a radio-frequency input signal substantially as described herein with reference to the accompanying drawings.

Any feature in one aspect of the invention may be applied to other aspects of the invention, in any appropriate combination. For example, apparatus features may be applied to method features and vice versa.

Detailed description of embodiments

Embodiments of the invention are now described, by way of non-limiting example, and are illustrated in the following figures, in which:-

Figure 1 is a schematic diagram of a sub-Nyquist sampling system according to one embodiment.

Figure 2 is a flow chart of a process according to an embodiment;

Figure 3 is a diagram demonstrating a process of combining a digital fractional delay filter and time-frequency transform filter;

Figures 4a, 4b and 4c are plots of a chirped Hanning filter and its discrete versions with and without delay;

Figure 5a is a spectogram of a noisy input signal;

Figure 5b is a plot of LoCoMC reconstruction with Fourier domain phase shift;

Figure 5c is a plot of LoCoMC reconstruction with a modified TF transform;

Figures 6a to 6f are plots showing a set of radar ES signals and their reconstructions; Figures 7a to 7f are zoomed versions of Figures 6d to 6f.

In a first embodiment, an input signal x(t) is received and processed by a receiver apparatus 2. Receiver apparatus 2 is schematically illustrated in Figure 1. The input signal x(t) is wideband (for example, 2 GHz to 18 GHz) and approximately sparse, and may comprise radar and/or communications signals from multiple radio- frequency sources.

Receiver apparatus 2 comprises an analog section 4 (providing a multi-coset sampler), an Analog to Digital Converter (ADC) 6, and a digital section 8.

Receiver apparatus 2 comprises a plurality q of sampling channels. In this embodiment, two sampling channels are used, which are represented in the schematic diagram of Figure 1 (with q = 2 ). In other embodiments, other numbers of channels are used, as indicated by the dotted arrows in Figure 1.

Analog section 4 comprises an input receiver 10 for receiving the input signal x(t) as an analog quantity, a bank of time delay filters 12 (one time delay filter 12 per channel) and a corresponding bank of sub-Nyquist sample and hold devices 14 (one sample and hold device 14 per channel). Each sample and hold device 14 is required to have an input bandwidth of at least the bandwidth of the signal to be received.

The analog section (or multi-coset sampling component) 4 may have a similar form to existing interleaved ADC designs and may therefore be implemented using any suitable existing technology.

In the present embodiment, analog section 4 further comprises a frequency selective filter 16 between the input receiver 10 and the time delay filters 12. In other embodiments, an alternative filter is used, or no filter is used. Any suitable frequency selective filter may be used.

The digital section 8 of receiver apparatus 2 comprises, for each channel, a digital fractional delay filter 18 and a time-frequency transform component 20. The digital section 8 also comprises a single subband classifier 22 that receives input from both channels. In embodiments that have a greater number of sampling channels, the subband classifier 22 receives input from all channels.

In the current embodiment, digital fractional delay filter 18, time-frequency transform component 20 and subband classifier 22 are implemented in a pipelined implementation on an FPGA (Field-Programmable Gate Array). In another embodiment, the digital fractional delay filter 18, time-frequency transform component 20 and subband classifier 22 are implemented using one or more DSPs (Digital Signal Processors). In further embodiments, the components are implemented in software on a processor. In other embodiments, the components may be implemented as a combination of hardware and software. More than one component may be included on a single hardware element, or a single component may be implemented across more than one hardware element.

If the time-frequency representation used in the implementation has narrowband time-frequency cells then the implementation of fractional delay filter 18 can be absorbed into the time-frequency transform 20 and approximated using a frequency dependent phase shift as used in the DUET algorithm (A Jourjine, S Rickard and O Yilmaz, 'Blind separation of disjoint orthogonal signals: Demixing n sources from 2 mixtures', Proc IEEE Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP), vol. V, 2000, pp 2985-2988).

When the time-frequency representation is of a Gabor type with a bandlimited kernel then the fractional delay filter can be absorbed into the time-frequency transform 20 and approximated using a modified time-frequency representation that comprises a combination of a frequency dependent phase shift and a resampled kernel function.

Figure 2 illustrates a flow chart for the process of acquiring, detecting, analyzing and reconstructing a signal using the apparatus of Figure 1. In the embodiment of Figure 2, reconstruction is of the discrete (sampled) signal only. In principle the process may also be extended to the full analog reconstruction if a suitably high bandwidth digital to analog convertor is available.

The process is first discussed in overview with reference to the flow chart of Figure 2, followed by a more mathematical analysis of the processing of the sampled signal.

At stage 30, an input signal x(t) is received by the input receiver 10. It is known that the input signal x(t) is approximately sparse, that is to say, it comprises relatively few frequencies at any given time. This means that when the input signal x(t) is represented in a suitable representation, such as by using a time-frequency transform, there will be relatively few non-negligible coefficients in the resulting matrix.

The input signal x(t) may occupy all of the frequency subbands within the frequency band, but only sparsely within the time frequency transform. Alternatively, the input signal x(t) may occupy a subset of the frequency subbands.

An assumption is made that the input signal has an approximate disjoint aliased support. That is to say that, to a good approximation, the input signal x(t) at any given time is comprised of a plurality of signal components, where the aliases of a given signal component do not coincide or overlap with any further signal components or any of the further signal component's aliases within the time frequency representation.

The underlying assumption in the approach of the present embodiment is that the observed radio-frequency signals will only sparsely, or approximately sparsely, occupy the full time-frequency domain. This is required for retrieval of signals at a sub-Nyquist rate. Even with the current increase in pulse density observed by ES systems, this assumption is likely to hold.

In the embodiment of Figure 2, at stage 32 the input signal x(t) is filtered using the frequency selective filter 16. The frequency selective filter 16 is used to restrict attention to frequency bands of interest to maximize the sensitivity of the receiver apparatus 2.

In other embodiments, different filtering is carried out at stage 32. For example, in some embodiments only a lowpass filter is used to filter out frequencies above the maximum frequency of interest, or a bandpass filter is used to also filter out frequencies below the minimum frequency of interest. Any suitable combination of filters may be used. A frequency selective filter may filter out particular subbands within the overall wide frequency band which are known not to be of interest, resulting in a multiband signal.

As the sampling scheme is a multi-coset one, it is necessary to sample the signal at the same rate but with different time offsets on each channel. The sampling is implemented in practice by delaying the sample and hold within each channel by an appropriate time delay and then reversing the time delay on the digital signal so that the channels are once again aligned in time (the reversal of the time delay in the digital part of the system is described in stage 40).

At stage 34, for each channel , the filtered input signal is delayed by a time delay of c, seconds using a respective time delay filter 12, where each c_j is less than L and is different from every other c, . T is the Nyquist interval and l/LT is the common sampling rate at which the input signal x t) will be sampled by each channel.

In the present embodiment, having two channels, the first channel has an associated time delay of c_xT and the second channel has a time delay of c₂T . In this particular embodiment, c, = 0 and c₂ is a value less than L . The value c₂ may be chosen to minimize the probability of incorrect subband classification, as will be discussed further below.

In further embodiments, more than two channels are used. In the case of q channels, i = \,...,q , the i th channel has a unique time delay of c, seconds, where c_j is less than L .

At stage 36, each channel is sampled using a sample and hold device 14 at sampling rate l/LT . Both (or, in other embodiments, all) channels use the same sampling rate.

Operation of the sample and hold device 14 as described results in multi-coset sampling of the (filtered) input signal. The first channel samples at a regular periodic interval LT . The sampling pattern of the other channel (or each other channel, in embodiments with greater than two channels) is a coset of the sampling pattern of the first channel with a relative time shift.

At stage 38, Analog to Digital Converter 6 digitizes the signal from each sample and hold device 14 for subsequent digital processing and analysis. In the present embodiment, the signal from each sample and hold device 14 is digitized using Analog to Digital Converter 6, and the signal is digitized using 12 bits. In other embodiments, a different number of bits is used. The digitization produces the digital sampled equivalent >>,.[«] of the analog input signal x(t) on each channel z , /^' = \,...,q . (In the present embodiment, q = 2.)

The sampling and digitization process described above may be achieved using a receiver architecture that is similar to that of existing interleaved Analog to Digital Converters. An interleaved Analog to Digital Converter also samples an input signal using a sampling rate l/LT that is L times lower than the Nyquist rate, but as it uses L channels, each with an appropriate delay, it achieves an effective full Nyquist sampling rate from its multiple channels. This may be contrasted with the sub- Nyquist implementation which uses substantially fewer than L channels, in this embodiment as few as two channels. The effective or average sampling rate of the present system is always sub-Nyquist.

Turning to the digital part of the process of Figure 2, at stage 40 a respective digital fractional delay filter 18 applies a digital fractional delay -c_tT to each digital signal y_t[ri to align the different channels in time. The resulting digital signal on each channel is z,.[«] (in the present embodiment, z, [«] and z₂[ri\ ).

At stage 42, for each channel, the time-frequency transform component 20 transforms

using a suitable time-frequency transform at the sub-Nyquist rate, resulting in transformed signal r^_k where m is an index in (discrete) time and k is an index in (discrete) frequency.

In the present embodiment, the time-frequency transform is a Short Time Fourier Transform (STFT). In further embodiments, the time-frequency transform may be any Gabor or Gabor-like time-frequency transform. The time-frequency transform may be a short-time fractional Fourier transform, wavelet transform, chirplet transform or polyphase filterbank, or any other suitable time-frequency transform. A chirplet transform may, for example, be appropriate when the input signal comprises chirped radar pulses.

It is expected that the transform results in a time-frequency representation with relatively few non-zero and non-negligible coefficients. Since the sampling of each channel is sub-Nyquist, it will not be possible straight away to uniquely identify each time-frequency component from the transformed signal. Due to the undersampling, when the sampled input signal is transformed, each resulting time-frequency component appears as a set of aliased time-frequency components, one in each of L aliased frequency subbands. At this stage, it is only known that the correct time-frequency component is one of the L aliased time- frequency components.

It is necessary to distinguish between the L possibilities and establish to which of the L aliased subbands the time-frequency component should correctly be allocated. This is achieved by combining information from the different channels.

At stage 44, the transformed signals r^_k for both or all of the channels are compared in the subband classifier 22 to distinguish each individual subband.

The assumption of approximate disjoint aliased support means that by such a comparison each time-frequency component can be distinguished from its aliases.

First it is necessary to identify the significant time-frequency coefficients, by determining which of the time-frequency coefficients have a value that is greater than noise. This may be achieved by thresholding of the time-frequency coefficients r^_k , where a time-frequency coefficient is said to be significant if its magnitude over all channels is greater than a threshold value. It is assumed that the noise in each channel is independent.

Each significant time-frequency coefficient having been recognised, it is then required to classify each of the respective time-frequency components (allocate the time-frequency component to a single one of the L aliased subbands). This is achieved in the subband classifier 22 by comparing the phases on the two channels. Although each alias of a given time-frequency component has the same magnitude, it has a different phase term. A comparison of the relative phases between the first and second channel (or all the channels, in embodiments with more than two channels) is used to determine which of the aliased subbands each time-frequency component should be allocated to. Therefore each individual time-frequency component is uniquely classified and the subband classifier 22 outputs the resulting coefficients s_{m k} . It is then possible to reconstruct the discrete time representation of the input signal x(t) . In principle, it is also possible to reconstruct the analog signal by passing the digital signal through a digital to analog converter.

The embodiment described above with reference to Figure 1 and Figure 2 is described with reference to only two channels. However, the method may be implemented with any number of channels, as long as the number of channels q is less than L , where l/LT is the sampling interval on each channel. Averaging the output from the different channels reduces the noise and therefore increases the overall Signal-to-Noise ratio (SNR).

The whole process of Figure 2 can be pipelined and is non-iterative. It may therefore be suited for a low SWAP implementation. It avoids the need for complex digitizer systems and/or complex and iterative reconstruction algorithms.

The embodiment of Figure 1 can provide a practical acquisition, detection and analysis system for the observation of multiple radio-frequency signals in a wideband spectral sensing scenario. An efficient compressed sensing style receiver allows continuous monitoring of bandwidths many times greater than the sampling rate used, while requiring minimal computation to process the digital samples.

The resulting system requires as few as 2 sub-Nyquist digitizers (2 channels) running with the same sample rate, with signal reconstruction and/or analysis performed non- iteratively. All that is required is that the radio-frequency input to one of the digitizers is delayed with respect to the radio-frequency input to the other digitizer and that each sample and hold circuit 14 has an input bandwidth at least twice the highest frequency to be received or, for non-baseband signals, that each sample and hold circuit 14 has an input bandwidth of at least the bandwidth of the input signal.

The system may match or exceed the performance of current in-service ES receivers (both analog and digital) along with a significant reduction in hardware complexity. Digitization of large bandwidths (for example 2 GHz to 18 GHz) avoids the need for multiple digital receivers. Direct digital conversion of large bandwidths may simplify the RF and signal processing chain, leading to a reduction in receiver complexity, power consumption, volume and weight. System reliability may also be increased due to a significant reduction in component count. The wideband digitized architecture may enable the development of a compact and cost effective ES receiver, which could be used in all situations where a wideband receiver is required. It may offer a reduction in power, space and weight over current systems and may be beneficial in situations where portability and endurance are important.

Turning to further consideration of the configuration of thej system, the choice of the integer value L and, therefore, the degree of undersampling is governed by the required sensitivity of the system.

All sub-Nyquist sampling strategies lose sensitivity as a function of the degree of undersampling, due to noise folding (the noise appears not only in its original subband but also in all the aliased subbands). The ideal sensitivity loss is 3 dB per factor of 2 undersampling. For example, sampling a bandwidth of 16 GHz at 1 Gsps instead of the Nyquist rate of 32 Gsps would incur a sensitivity loss of approximately 15 dB. This will ultimately be a limiting factor to the degree of undersampling achievable.

The spurious free dynamic range of an ES system may be required to be more than 70 dB in order to provide high sensitivity and to improve the probability of detection for low probability of intercept (LPI) signals. In order to achieve a theoretical SFDR of more than 70 dB it is necessary to use an ADC with at least 12 bits resolution. The realizable SFDR depends on the physical hardware and the ability to compensate for gain, offset and timing mismatches.

An ADC device with a sub-Nyquist sample rate of 1 Gsps and from 12 to 16 bits in conjunction with a wideband track and hold circuit may digitize wide bandwidths at sub-Nyquist sample rates with high dynamic range and with high fidelity.

In the system of Figure 1 and process of Figure 2, all frequencies in the wide input band are monitored all of the time. In comparison with an RSSR receiver which must be rapidly swept across frequencies, the persistent wideband monitoring means that significantly longer analysis windows can be used providing associated processing gain. The system is compatible with a frequency selective filter front end that restricts the input to bands of interest. This can help improve the sensitivity of the signal detection.

In an alternative embodiment, the system of Figure 1 and Figure 2 adaptively trades off sensitivity for bandwidth by varying the frequency selective filter 16.

Through front end filtering it is possible for the receiver to adaptively trade off sensitivity for bandwidth. This may allow for both wideband low sensitivity surveillance and targeted narrowband surveillance. The frequency selective filter 16 may be used to restrict the bandwidth of the input signal x(t) to any given band, or even multiple bands of interest. As the sampling structure described above is capable of reconstructing the full band signal (for example, 2 to 18 GHz) it is equally able to reconstruct a signal with limited bandwidth.

A benefit of front-end filtering can come from the increased sensitivity since the receiver noise power is a function of total bandwidth. Therefore, the sampling and reconstruction system of Figure 1 and Figure 2 allows narrowband, even full Nyquist sampling at high sensitivity, and wideband sub-Nyquist sampling with reduced sensitivity (3dB per octave undersampling).

The mode used (Nyquist versus sub-Nyquist or alternative sampling rates) may be dependent on the type of signal expected, the number or density or type of the signals being detected, or other factors.

In one embodiment, when a particular individual signal component of interest is detected within the input signal when sampling at sub-Nyquist rate, the system may switch to Nyquist rate operation. For example, the system may reduce its input bandwidth through filtering to focus on the frequency band in which the individual signal component of interest was detected, which it will then sample at Nyquist rate.

As stated above, the analog multi-coset sampling component 4 has a similar form to existing interleaved designs and therefore can be implemented using existing technology. In common with the interleaved ADC architecture the performance of the multi-coset sampling scheme is sensitive to the gain and delay time mismatches between the channels. Any actual hardware will incur mismatches between the channels. Of particular importance may be the channel gain and offset, and errors in the relative delay between channels, the timing skew. If uncorrected, these could introduce significant distortion, increasing the noise in the system and therefore reducing the SFDR. However, with knowledge of the true values, the compressed sensing solution is robust to the actual channel parameters. Therefore, as long as the delays, gains and offsets can be determined though appropriate calibration, their effects can be compensated for digitally.

Calibration schemes used in interleaved ADCs are equally applicable to the multi- coset sampling system. These calibration solutions can be analog (K Dyer, D Fu, S Lewis and P Hurst, 'An analog background calibration technique for time-interleaved analog-to-digital converters', IEEE J. Solid-State Circuits, vol. 33, no 12, p1912-1919, 1998), digital (D. Fu, K Dyer, S Lewis and P Hurst, Ά digital background calibration technique for time-interleaved analog-to-digital converters', IEEE J. Solid-State Circuits, vol. 22, no 12, pp 1904-1911 , 1998) or even mixed signal and can run online, in the background, without interrupting the receiver.

In one embodiment, a pilot signal is injected into the receiver input 4 within a known subband. The frequency, timing and amplitude characteristic of the pilot signal are known. The pilot signal is detected and transformed on each channel and the resulting signals are compared. This will indicate whether the analog time delay and the gain of each channel are as expected. Even if a mismatch is found which relates to an analog component, it is not necessary to alter the analog component itself. Instead the observed difference may be compensated in the digital section 8.

The calibration process may be programmed into the system to occur automatically. Alternatively calibration may be performed by an operator. The results of calibration may be stored in the apparatus or may be transmitted to a user. Although in this embodiment, calibration is performed while the system is operational, in further embodiments it may be performed while the system is not operating.

The mathematical grounding for the process of Figure 2 as described above is now outlined. The process of Figure 2 may be described as a Low Complexity Multi-Coset sampling technique (LoCoMC). Consider an input signal, x(t) . Following Feng and Bresler as cited above, examination of the z^'th channel shows that the input signal x(t) , when sampled on the z^'th channel at a rate of l/LT , can be written as y_i[n] = x((nL + c_i)T) , where L , c, and Tare as described above.

For simplicity of notation it is assumed that the support of the Fourier transform of x(t) , Χ{ω) , is essentially bandlimited to [0,2π /Τ] . Generalizing to other frequency supports is straightforward.

The Discrete Time Fourier Transform (DTFT) of >>,[«] (the sampled signal on the z^'th channel) can therefore be written as

-jc_tT(co-2xll LT) (Equation 1)

^L1 z=o where T is the Nyquist interval, c, is the time delay applied to the z^'th channel as described above, LT is the sampling rate on each channel z , and X(co) is the continuous Fourier transform of input signal x{t) (and therefore Χ(ω - 2πΙ I LT) is a frequency-shifted copy, or alias, of the Fourier transform of input signal x(t) ).

Applying a reversed digital fractional delay of -c_tT seconds to y^n] yields z_f [n] .

This compensates for the analog time delay at stage 34 and brings the channels into the same time frame.

The DTFT of

is therefore

Z e^jaL' ) (Equation 2)

This shows the presence of aliases in the DTFT of the sampled signal. The frequency support is divided into L aliased subbands, each of size 2πΙ LT . The output z_t\ri is the superposition of L terms: each component in x(t) results in L aliased components in the DTFT of z n . Each term comprises the subband components of x(t) multiplied by a phase shift that depends on the aliased band number / and the channel delay c_i

It is of interest to obtain a time-frequency representation of x(t) . In this exposition a general Gabor transform is considered. However, other time-frequency transforms may be used. Initially, the continuous time time-frequency transform is considered, followed by the discrete time transform that is used on the sampled input signal.

Input signals of interest are in some sense sparse in a Gabor-like time-frequency (TF) representation.

Specifically, consider a TF atom of the form g_{m k}(f) = g(t - πιτ₀)β^Μα'

Here, g{t) defines the window function (see, for example, S Mallet, Ά Wavelet Tour of Signal Processing', Academic Press, 1999). The window function g(t) is assumed to be normalised, ||g|| = 1 , essentially bandlimited to ω e [Q,2 I LT) and having a temporal support in the interval 0≤t < LNT (where N is an integer value that defines the length of the window).

The values r₀ and ξ₀ define the discrete TF lattice of the TF representation,

(τ,ξ)€ {(mr₀,£ ) \ (m,k) e Z² }.

For convenience, attention is restricted to TF lattices such that r₀ = MLT for some integer M and ξ₀ = 1/ KLT for some integer K .

The frame coefficients of x(t) , s_{m k} , can be each calculated as the inner product of the input signal x(t)with the TF atom g_{m k}(t) . = jxWgit-mr^e-^'^dt (Equation 3)

=— f Χ(ω + 2τάΙ ΚΙΤ)0*(ω)ε- ^ΜΤάω

2π Ja>e[-2nk/KLT,2 Q-k/KL)T where the last line follows from Parseval's Theorem. k - 0,...,LK-1 and g_mk(t) is the time-frequency atom at frequency ω- Ink I KLT at time m . This frequency discretization divides the full Nyquist band into LK frequency bins, or, equivalently, divides each of L aliased subbands into K frequency bins.

Using the essentially bandlimited assumption on g(t), such that G(ai) is essentially bandlimited to οε [0,2π / LT) , s_mk can be accurately approximated by ^sm,k *— \^χ(^ω + ^{27jk 1} KLT)G * (a⁾)e-^JmnMLT dco (Equation 4)

In practice, the signal being transformed into a TF representation is not the continuous-time input signal x(t) but its sub-Nyquist sampled version (with the appropriate reversed fractional time delay). Therefore it is necessary to consider the discrete TF representation for a single sub-Nyquist coset channel signal,

. Since g(t) is essentially bandlimited, one can use the discrete time sampled atoms, _{gm k} [n] = g[n - πιΜγ^{2 ιΙΚ} = g(m(n - mM)LT)e^j2nin/K .

The discrete time TF coefficients r^_k are given by the inner product of the sampled input signal and the discrete time sampled atom:

N+mM-l

= ∑z ] *[«-w ]<r^y2¾te* (Equation^

n=mM

LT

2π Ιω ,ϊπΙΙΤ) ' ' where G_d(e^J<aLT) is the discrete time Fourier transform of g[n\ and satisfies:

G_d {e^j0>LT ) =—∑ G(o) - 2τά I LT) *— G(t») , 0 < ω < 2π I LT

LT _k=→13 LT

where the approximation follows the essentially bandlimited assumption.

From Equation 2, it is possible to shift the frequency window over which Z_t(e^J ) is evaluated and therefore write:

+ 2dlLT) (Equation 6)

gives: )G * (G))e-^JcmMLTda>

(Equation 7) where the final approximation assumes that out-of-band aliasing effects are negligible. The transform atoms are assumed to be essentially bandlimited to within the subbands. That is to say that only a small amount of their energy falls outside the defined subband. The part of the signal that falls outside the subband will be aliased. However, as the amount of energy is assumed small, the alias effects are assumed to be negligible.

The discrete TF coefficients r^_k are therefore well approximated as the sum of full band TF coefficients s_{m k} , weighted by a phase term that is dependent on the coset channel .

The summation may extend over L + l subbands. This is because the TF representation under consideration exceeds the essential bandwidth of the input signal. Here there is a choice available. If the aim is to reconstruct the analog TF representation associated with the atoms g_{m k}(t) then L + l subbands must be considered. If however it is only necessary to reconstruct the full band discrete time representation associated with atoms g_{m k} (nT) , n = 0,..., LN - I , then it is only necessary to consider L subbands. This is because the first (/ = -1) and last (/ = L - 1) subbands are equivalent in the full band discrete representation as the representation has been automatically constrained to be bandlimited to ω e [0,2π /Τ) by the sampling process. In the following discussion it will be assumed that the discrete time representation is being considered, and therefore only L different subbands need to be considered.

In order to proceed the simplifying assumption of Approximate Disjoint Aliased Support (ADAS) is made. This concept is similar to the concept of approximate disjoint orthogonality used in blind source separation (see, for example, A Jourjine, S Rickard and O Yilmaz, 'Blind separation of disjoint orthogonal signals: Demixing n sources from 2 mixtures', Proc IEEE Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP), vol. V, 2000, pp 2985-2988) although unlike the DUET algorithm here the phase shift is constant across the aliased band and therefore is applicable to a much broader class of time-frequency representations.

In mathematical terms, x(t) has approximate disjoint aliased support in a given time-frequency representation if each sub-Nyquist discrete time-frequency coefficient is dominated by a single full-band time-frequency coefficient such that r^ * e^J2M^^{> IL}s_{k+lm iK} (Equation 9) where / = l_{m k} is a function of the sub-Nyquist time-frequency position (m,k ), where m = 0,1,... and k = 0,..., K - \ .

A priori, the subband to which r^_k should be assigned is not known. The problem of assigning jj'j. to a subband can be solved by comparing the discrete time-frequency representations for each coset channel .

In order to analyze or reconstruct the input signal it is necessary to detect the significant coefficients containing signal components and then to determine the correct subbands I _k with which to associate them. Both tasks can be accomplished considering the cross channel vector of time-frequency coefficients,

₌ r_r0⁾ ι γ

Consider first the question of joint detection. It is necessary to identify which time- frequency components are significant. Under the ADAS assumption, it may be expected that the time-frequency coefficients for each channel would have a similar magnitude, « |r ¾ \ ... « |rj¾ | .

It is reasonable to assume that any noise on different channels is independent since the samples are temporally distinct. Therefore one possible detection strategy is to define the coefficient as significant as long as the magnitude ||r_{m A} ||₂ > τ , where the threshold value τ can be chosen to achieve a constant false alarm rate.

Once a coefficient vector has been detected as significant it then needs to be assigned to a subband. It is necessary to know which out of the possible aliased subbands is the correct one.

This can be considered as a classification or decoding task. Under a Gaussian noise assumption, the optimal classifier is achieved by maximizing the inner product between r_{m k} and the phase vector 0(/) for the / th subband

(Equation 9)

This classification is uniquely defined as long as the L phase vectors θ(1) , l = 0,..., L - \ are all distinct. This can be achieved with as few as q = 2 coset channels.

Combining the detection and subband classification, the full band time-frequency representation s _k can finally be estimated as follows. tion 10)

^Sm,k+pK ^{if p} = L, HK > r (Equa

otherwise for /M = 0,1,... , k = 0,...,K - l and p = 0,...,Z - 1 . Here, averaging the different channel outputs reduces the noise and therefore increases the overall Signal-to- Noise ratio (SNR) of the system.

Full band time reconstruction, if desired, can then be achieved by applying an inverse frame operator to the coefficients s_{m k} .

Although the discussion above has been focused on a STFT representation, the technique is applicable to a wide variety of time-frequency representations. The main requirements are that the time-frequency atomic structure tiles across the aliased subbands and that the individual atoms g_{m k} (t) have an essential bandwidth less than the subband bandwidth InILT . For example, by an appropriate choice of function g(t) , the above time-frequency representation can be a fractional Fourier transform. Similarly, it is possible to use the above approach with redundant transforms such as the chirplet transform (F Millioz and M Davies, 'Sparse detection in the chirplet transform: application to FMCW radar signals', Signal Processing, IEEE Transactions on, vol. 6. pp 2800-2813, 2012), so long as the chirplets obey the essential bandlimited property and the local time frequency components of the aliased signal are dominated by the contribution from a single subband. This may make the proposed solution simpler to use with redundant and more exotic representations than traditional compressed sensing strategies.

The multi-coset delays can be chosen to minimize the probability of incorrect subband classification. Under the ADAS assumption, optimal subband classification is achieved by sampling delays that are associated with harmonic frames with minimal coherence, , defined as:

(Equation 11)

For small numbers of channels the delay sequence can be determined through an exhaustive search. A benefit of using more multi-coset channels is that the coherence, μ , can be reduced towards the optimal Welch bound (L Welch, 'Lower bounds on the maximum cross-correlation of signals', Information Theory, IEEE Transactions on, vol. 20, no. 3, pp 397-399, 1974) associated with equiangular tight frames. For the simplest case of q - 2 multi-coset channels, any integer delay for c₂ (fixing C_j = 0 ) achieves minimal coherence as long as the greatest common divider of c₂ and L is c₂L . For example this is trivially met by choosing c₂ = 1 .

There is a further trade-off, however, as larger delays will be more sensitive to sampling jitter. Therefore the best delay set, {c, }, will generally be a compromise between maximizing coherence (or minimizing incoherence) while controlling classification errors due to delay jitter.

In this way the multi-coset sampling sequence can be explicitly optimized to maximize the probability of success.

It has been found that, for a modest range of downsampling (for example, downsampling by a factor in the range of 2 to 32), the output Signal-to-Noise ratio (SNR) may be related to the implementation of the digital fractional delay (DFD) filter 18 at stage 40. The ideal DFD filter would be a shifted sine function, which is non- causal and infinitely long. Any truncation of filter coefficients for practical purposes may generate artefacts in the output.

It is possible to implement the digital fractional delay shift using a frequency domain implementation (that is, a linear phase shift). The frequency domain implementation of the digital fractional delay shift may be referred to as a phase shifting technique or a Fourier based technique.

While a frequency domain implementation may seem to be a good solution, a frequency domain implementation may introduce delay distortion for large fractional delays. The reason is that such a fractional delay has to be implemented using a discrete Fourier transform of a finite length. A frequency domain implementation may use a circularly periodic sine function, which may introduce time delay distortion.

An implementation of a fractional delay is described below in which the DFD filter and TF filter are combined. The implementation described below may in some cases result in less output distortion and be computationally more efficient when using a class of TF transforms for which the continuous kernel is known. The digital part of the receiver apparatus 2 has two linear operators in each channel, before the subband classifier 22. The two linear operators are a digital fractional delay and a TF transform. In the embodiment of Figure 1 , the linear operators are implemented as digital fractional delay filter 18 and time-frequency transform component 20.

The linear operators (digital fractional delay and TF transform) may be implemented using linear filtering. The implementation of channel i can be formulated as follows: Zi[n] = h_c. [n] * y_l[n] (Equation 12) and

r_T ⁽ = [a_k,MM_k(g[n] * Zi )]_1≤fc≤/f

= * ^zi[ⁿ] ]_1≤fc≤K (Equation 3)

where τ := mM, a_k>x ·■= e^2nikT/K and

g_k[n] := M_k(g[n])

= g(nLT)e^2niknl^K .

By substituting Z_jfn] from Equation 12 into Equation 13, the following equation may be derived:

rm,k = ^α τ^Ότ (¾W * (¾_C| [n] * yj[n]))

= ((_¾[n] * _c.[n]) * y_t[n])

where the second and third equations are respectively derived using the associativity and the commutativity of the convolution operator.

The operation h_c. [n] * g_k[n] is fractionally delaying

The implementation of the operation h_c.[n] * g_k[n] may be expected to have a similar difficulty to before in circumstances for which g_k t) is only known at discrete values (a similar difficulty to the difficulty that may occur when using separate operations for fractional delay and TF transform, in which the fractional delay filter may not be uniformly good across all frequencies). However, for the class of windows used in this embodiment (which may be any TF transform with a bandlimited kernel), g_k(t) is known over the period of [0, N]. To apply the fractional delay in this case, it is only necessary to sample g_k(t) at the delayed locations of interest. The sampling process can be formulated as follows:

* g_k[n] = h_c.[n] * g{nLT)e ^niknl^K

(Equation 14)

The relation between the formulation of Equation 14 and the phase shifting technique may be noted. In the phase shifting technique, only the right hand term, e^2nikCi/NK, is used. In the phase shifting technique, the window shift is ignored.

The difference between the formulation of Equation 14 and the phase shifting technique may be more important when the window has a wide bandwidth, for example in embodiments for which a chirplet window is used.

A diagram which demonstrates the process of combining a DFD filter with a TF filter is shown in Figure 3. The first line of Figure 3 shows a separate digital fractional delay filter 18 and TF transform component 20. The second line shows the combination of the digital fractional delay filter 18 and TF transform component 20 into a single unit. The third line shows a combined filter 50 which combines the two linear operators.

As an example of combining a DFD filter with a TF filter, a chirped Hanning window and its discrete versions with and without delay are shown in Figures 4a, 4b and 4c. Figure 4a shows a chirped Hanning window, its discrete window without delay (dots) and its discrete window with a delay of 4/13 (plus signs). Figure 4b shows a discrete chirped Hanning window without delay (the line in Figure 4b is constructed from the dots of Figure 4a). Figure 4c shows a discrete chirped Hanning window with a delay of 4/13 (the line in Figure 4c is constructed from the plus signs of Figure 4a).

It may be seen from Figures 4a, 4b and 4c that the delayed window is not only shifted with the delay factor but also has some deformations.

Another potential application of the implementation of Equation 14 is that the fractional delay may be done precisely. Precision may be necessary for the digital calibration of the analog delays. Digital calibration of the analog delays may be required as it is normal to have slight clock synchronisation errors in the fabrication process, caused by the fabrication tolerance.

Simulations of an embodiment were performed using simulated radar ES signals with an active band between 10 and 11.2 GHz. The signal was preprocessed by downconverting to baseband. A 4-channel multicoset sampling structure was used, with 13 times undersampling in each track and hold, 1200/13 « 92 MHz.

The delay factors were c = [6,7,10,12] and the STFT was used as the TF transform.

The signal was reconstructed using LoCoMC with the Fourier based technique. The signal was also reconstructed using LoCoMC with the DFD technique described above with reference to Figure 3, in which the DFD filter and TF filter are combined.

The simulation results for an approximately 30 dB SNR noisy input signal is presented in Figures 5a, 5b and 5c. Figure 5a shows a spectrogram of the noisy input signal. Figure 5b shows a reconstruction using the method of Figure 2 with a Fourier domain phase shift. Figure 5c shows a reconstruction using the modified TF transform in which DFD and TF transform are combined.

Although both techniques were found to be successful in the recovery of the input pulses and chirps, the SNR of the output using combined DFD and TF transform was found to be more than 1.5 dB higher than the Fourier based method.

The simulation was repeated 100 times for an average case analysis. Different additive noise was used for each repetition, with the noisy signals having roughly the same SNR level. The average output SNR was 34.07 dB for the technique with combined DFD and TF transform, while it was 32.44 dB for the Fourier based DFD.

In a further experiment, the LoCoMC technique using combined DFD and TF transform was compared with a RSSR technique. The RSSR technique for channelised receivers is a popular technique for sub-Nyquist radar ES. The RSSR technique is based on time-sharing and resource allocation principles. There is a bank of band pass filters (BPF) at the front-end of the receiver, such that the output of each filter falls in the frequency bandwidth which is able to be sampled at the Nyquist rate. The signals are then down-converted to the baseband or IF (intermediate frequency) band using LOs (local oscillators). Whether the conversion is to the baseband or to the IF band depends on the structure of the digitiser.

The channelised receiver can be time-division multiplexed and digitised with a single ADC. There are some dual channel ADCs which sample the signal and its 90° phase shifted version, to double the instantaneous bandwidth.

To minimise the switch-over period artefact, a second ADC may also be used with an analysis window that overlaps the analysis window of the first ADC by 50% in time. With this implementation, at any time, one ADC is always monitoring some part of the spectrum. The period of monitoring each frequency band is related to the shortest pulse width. For the simulations described below, the RSSR technique with two ADCs was used for comparison, as the comparison in SNR with the technique of Figure 2 is fairer. No windowing edge artefact is caused by band switch-over in the RSSR in the technique with two ADCs.

In the simulation, two ADCs were used which were operating with a 1200/6 = 200 MHz sampling rate. Such a setting was chosen to have roughly the same average sampling rate (undersampling factor) over the number of ADCs as in the multi-coset scheme, 13/4 « 6/2.

Simulation results are presented in Figures 6a to 6f. Figure 6a shows a clean version of the input signal used. Figure 6b shows the noisy input signal. Figure 6c shows a downsampled signal with a downsampling factor of 13.

The output of the LoCoMC technique using the combined DFD and TF transform is shown in Figure 6d. Figure 6e shows the output of a multi-coset technique in which reconstruction is performed using a windowed MUSIC algorithm. Figure 6f shows RSSR reconstructed signals.

It may be observed that the SNR of the RSSR is very low in comparison with the results of the multi-coset methods.

Figures 7a to 7f show zoomed plots of the results for LoCoMC with combined DFD and TF transform; multi-coset technique with windowed MUSIC algorithm; and RSSR reconstruction. Figure 7a, 7b and 7c show zoomed versions of the stream of pulses from Figures 6d, 6e and 6f respectively. Figures 7d, 7e and 7f show zoomed versions of the first chirp of each of Figures 6d, 6e and 6f respectively.

Looking at the zoomed plots of Figures 7a to 7f, some of the pulses are seen to be missing in the RSSR (Figure 7c) because of the multiplexing. There is also some processing gain loss in the chirps (Figure 7f) because some parts of each chirp are missing for the same region. The multi-coset methods appear to behave much the same, but the multi-coset technique with combined DFD and TF transform is computationally much simpler than a subspace method like MUSIC.

The above embodiments relate to Electronic Surveillance and the sensing of radio- frequency signals. However, the method described above may be used in any application for which the assumption may be made that an input signal has an approximate disjoint aliased TF support.

It may be understood that the present invention has been described above purely by way of example, and that modifications of detail can be made within the scope of the invention.

Each feature disclosed in the description and (where appropriate) the claims and drawings may be provided independently or in any appropriate combination.

Claims

1. A method of analyzing a radio-frequency input signal using sub-Nyquist sampling, comprising

receiving the input signal;

applying a relative time delay between a first channel and a second channel; sampling at a respective sampling rate each of the first channel and the second channel, wherein for each channel the sampling rate is less than the Nyquist rate;

applying a respective time-frequency transform to each of the first channel and the second channel;

identifying at least one time-frequency component in the first channel and the second channel; and

for the or each time-frequency component, obtaining a comparison of the time-frequency component from the first channel and the time-frequency component from the second channel and

distinguishing the time-frequency component from a plurality of aliases of the time-frequency component in dependence on the comparison.

2. A method according to Claim 1 , wherein the comparison comprises a phase comparison of the first channel and the second channel.

3. A method according to Claim 1 or 2, wherein obtaining the comparison comprises maximising an inner product between a phase vector and a cross-channel vector of time-frequency coefficients.

4. A method according to any preceding claim, wherein distinguishing the or each time-frequency component comprises allocating the or each time-frequency component to a subband.

5. A method according to any preceding claim, further comprising calculating a cross-channel vector of time-frequency coefficients.

6. A method according to any preceding claim, further comprising calculating a magnitude of time-frequency coefficients on each channel.

7. A method according to any preceding claim, further comprising comparing time-frequency coefficients obtained from the time-frequency transforms to a threshold value.

8. A method according to Claim 7, wherein the threshold value is representative of a threshold noise level.

9. A method according to any preceding claim, further comprising processing the input signal on at least one further channel, wherein processing the input signal on the further channel comprises:

applying a relative time delay to the further channel with respect to the first channel and to the second channel

sampling the further channel at the sampling rate;

applying the time-frequency transform to the further channel;

and identifying the or each time-frequency component in the or each further channel; wherein distinguishing the or each time-frequency component from a plurality of aliased time-frequency components is based on a comparison of the or each time- frequency component from the first channel, the or each time-frequency component from the second channel and the or each time-frequency component from the or each further channel.

10. A method according to any preceding claim, wherein distinguishing the or each time-frequency component from the or each plurality of aliased time-frequency components is based only on a comparison of a time-frequency component from the first channel and a time-frequency component from the second channel, and wherein no time-frequency component from any further channel is included in the

comparison.

11. A method according to any preceding claim, wherein the input signal comprises at least M active subbands, and wherein distinguishing the or each time- frequency component is based only on a comparison of time-frequency components on N channels, wherein M is greater than N.

12. A method according to any preceding claim, further comprising reconstructing at least part of the input signal.

13. A method according to any preceding claim, further comprising continuously monitoring a frequency band, wherein the received input signal is received within the frequency band.

14. A method according to Claim 13, wherein the Nyquist rate comprises at least one of the Nyquist rate for the maximum frequency of the frequency band, the Nyquist rate for the bandwidth of the frequency band.

15. A method according to Claim 14, wherein the monitored frequency band is between 100MHz and 20 GHz, optionally between 2 GHz and 18 GHz.

16. A method according to any preceding claim, where the relative time delay is selected for reduction of sampling jitter.

17. A method according to any preceding claim, wherein a sampling interval of the sampling comprises an integer multiple of the Nyquist interval.

18. A method according to any preceding claim, wherein at least one of:- a sampling interval of the sampling is a periodic sampling interval;

the sampling has a sampling pattern that is neither random nor

pseudorandom;

the sampling has a sampling pattern that comprises a periodic non-uniform sampling pattern.

19. A method according to any preceding claim, wherein the sampling rate for the first channel is the same as the sampling rate for the second channel.

20. A method according to any preceding claim, further comprising applying a filter to the input signal, optionally before applying the relative time delay, performing the sampling, or applying the time-frequency transform.

21. A method according to Claim 20, wherein the filter comprises at least one of: a frequency selective filter, a lowpass filter, a highpass filter, a bandpass filter.

22. A method according to any preceding claim, further comprising performing a relative calibration of the first channel and the second channel.

23. A method according to Claim 22, wherein performing the relative calibration comprises injecting a pilot signal within a known subband.

24. A method according to Claim 23, further comprising

receiving the pilot signal on each of the first channel and the second channel; and

measuring at least one of the amplitude, frequency and phase of the injected signal on each of the first channel and the second channel.

25. A method according to any preceding claim, wherein the input signal comprises an analog signal.

26. A method according to any preceding claim, wherein the input signal is approximately sparse in at least one time-frequency representation.

27. A method according to any preceding claim wherein the input signal comprises at least one of: a radar signal and a communication signal.

28. A method according to any preceding claim, wherein the time-frequency transform comprises at least one of: a Gabor transform, a Gabor-type transform, a short-time Fourier transform, a short-time fractional Fourier transform, a wavelet transform, a chirplet transform, a polyphase filterbank.

29. A method according to any preceding claim, wherein applying a respective time-frequency transform to each of the first channel and the second channel comprises modifying each time-frequency transform to implement a digital fractional delay, the digital fractional delay corresponding to the relative time delay between the first channel and the second channel.

30. A radio-frequency receiver apparatus comprising:

receiving means for receiving an input signal;

time delay means for applying a relative time delay between a first channel and a second channel;

sampling means for sampling at a respective sampling rate each of the first channel and the second channel, wherein for each channel the sampling rate is less than the Nyquist rate; transform means for applying a respective time-frequency transform to each of the first channel and the second channel; and

processing means for identifying at least one time-frequency components in the first channel and the second channel, and, for the or each time-frequency component, obtaining a comparison of the time-frequency component from the first channel and the time-frequency component from the second channel and distinguishing the time-frequency component from a plurality of aliases of the time- frequency component in dependence on the comparison.

31. A method of analyzing a radio-frequency input signal substantially as described herein with reference to the accompanying drawings.

32. A radio-frequency receiver apparatus substantially as described herein with reference to the accompanying drawings.