DETECTION AND ESTIMATION OF MULTIPLE WEAK SIGNALS IN

Fluctuation and Noise Letters Vol. 0, No. 0 (2001) 000–000 c World Scientific Publishing Company 

DETECTION AND ESTIMATION OF MULTIPLE WEAK SIGNALS IN NON-GAUSSIAN NOISE

D.J. Nelson U.S. Department of Defense Fort Meade, Maryland Received (14 August, 2007) Revised (17 September, 2007) Accepted (accepted date) We address the problem of efficient detection and estimation of multiple weak signals in severe noise. To address this problem, we propose a concentrated peak representation (CPR) in which the spectral energy is concentrated in spectral peaks, and only the magnitudes and locations of the peaks are retained. We base our process on the cross spectral representation we have previously applied to other problems. The advantage of this representation is that it preserves the energy in signal components while significantly reducing the data volume. We demonstrate the method on a composite signal consisting of one billion samples, reducing the data volume by a factor of 1000 or more, and we demonstrate the detection of weak signals in the reduced representation. Keywords : sentations

1.

signal detection, signal estimation, transient signals, time-frequency repre-

Introduction

A common signal processing problem is the detection, isolation and estimation of weak signals. This problem arises both in communications and in search. In RF communications, even though the receiver may know the parameters of the transmitter, the receiver must detect that the transmitted signal is present and then detect and estimate the baud rate, carrier and other information needed to demodulate the signal. In search, the parameters of the signal may or may not be known. Typical search situations are the detection of rescue beacons or the detection and estimation of signals in the environment for purposes of threat assessment or geolocation. In many of these situations, the signal may be weak and may fade in and out due to multipath and other effects. In addition, the environmental noise and interference may not be assumed stationary or white. We address the problem of detection and estimation of multiple weak signals severe noise. For a signal model, we assume a digitally recorded broadband composite signal consisting of the superposition of weak, unknown, narrowband signals

Detection and Estimation of Multiple Weak Signals in Non-Gaussian Noise

in noise. We assume that a data segment consists of a billion or more samples. The challenge in this problem is to preserve the energies of the narrowband signals while reducing the data to a volume that can be displayed and efficiently searched. We propose a concentrated peak representation (CPR) in which the spectral energy is concentrated in spectral peaks, and only the magnitudes and locations of the peaks are retained. We base our process on the cross spectral representation we have previously applied to other problems [1]-[10]. In selecting this method, we have considered other representations and estimation methods such as the Wigner distribution [11, 12] and Welch’s method [13]. We compare our method to these methods. The spectral estimation method we propose is a variation of Welch’s method and the cross-power spectral (CPS) estimator that was first applied to signal estimation and detection in the mid 1980’s [1]. The CPS algorithm and the method presented here are based on phase differentiation of the short time Fourier transform. The process is similar to the modified moving window method (MMWM)first proposed by Kodera et al. [14, 15]. 2.

Basic Assumptions and Issues

We assume a signal environment consisting of the superposition of narrowband and broadband signals with additive noise which need not be white or Gaussian  s(t) = sk (t) + η(t) (1) k

sk (t)

=

ak eiφk (t) ,

(2)

where η(t) is the noise function. In this environment, we assume that there are several weak unknown signals we would like to detect and parameterize. For simplicity, we assume that the desired signals are narrowband in the sense that their instantaneous frequencies, ωk (t) = φk (t), are slowly changing with time and their instantaneous bandwidths are small compared to the bandwidth that must be searched. This model equally well describes the underwater acoustic problem of detecting weak machine noise or the search problem in which a broad RF band must be searched for unknown signals. The constraints on the solution to this problem are that it must be a high gain process to detect multiple signals at very low SNR/SIR. Since we do not assume that the noise is white or stationary, the process must somehow adapt to time and frequency variations in the noise. Finally the process must provide a low density, high resolution representation from a potentially massive amount of data. The problem is somewhat simplified by not requiring detection of extremely short transients. 2.1.

Interference and nonlinear representations

In [16], the problem of detection of transients was addressed. In that paper, the problem of interference was not an issue since the transient signals were sparse and of such short duration that there was no significant interference. As a result, a detection method based on the Wigner distribution was proposed. One may ask

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whether the Wigner distribution or a related bilinear distribution is a reasonable representation for the problem we consider. If we only consider the volume of data, the answer is no. For a signal consisting of a million (106 ) samples,the Wigner distribution consists of 1012 elements, which is clearly an intractable computational problem. One may, however compute a short time Wigner distribution with fewer coordinates. The question then still remains whether the Wigner or any related bilinear distribution is a good representation for the task. If we do not assume short duration signals and allow multiple signals to be active at any given time, the situation changes dramatically. The Wigner distribution [11]  ∞ s(t + τ /2)s∗ (t − τ /2)e−iωτ dτ (3) W (ω, T ) = −∞

and many other representations are based on a quadratic (bilinear) process, resulting in cross-terms in the distribution. These cross-terms result from the nonlinear interaction of signal components and contribute “energy” to the TF distribution not present in any of the components of the original signal. Like noise, these cross-term components obscure the signals we are trying to find, but worse these components are not random. In Fig. 1, we present a simple demonstration that the Wigner distribution does not perform well for multi-component signals in noise. Fig. 1 represents a comparison of the Wigner distribution and the concentrated CIF surface from which we will compute the CPR. In this experiment, we compute each distribution of the sum of two sine waves in noise and increase the noise level until one of the methods fails to detect the signals. At a critical SNR of -5 dB in the test represented by Fig. 1, the signals could not be recovered from the Wigner distribution. The critical SNR for the concentrated CIF representation was -12 dB. The comparatively poor poor performance of the Wigner distribution in this test was attributed to the cross-terms. the tests were consistent for a variety of multi component signals, with some variation in the critical SNR. In processing multiple signals in deep noise, linearity of the representation is important. In nonlinear representations, the SNR/SIR is degraded by nonlinear interaction of the signals in the construction of the representation. It may be argued that the spectrogram is a nonlinear representation. However, the spectrogram is computed as the squared magnitude of the short time Fourier transform (STFT) [17], which is linear. Nonlinear mixing of signal components in the spectrogram is local in the sense that mixing occurs at a particular time-frequency coordinate, (T0 , ω0 ) only if more than one signal component has significant energy at that particular coordinate. If signal components are separated by the STFT, they do not significantly interfere with each other in the spectrogram. 2.2.

Welch’s Method and the CPS

In the context of TF processing, Welch’s method may be described as a convolution of the spectrogram in time by a smoothing window [13]. The STFT is first computed [17]  ∞

Sh (ω, T ) =

−∞

s(t + T )h(−t)e−iωt dt,

(4)

Detection and Estimation of Multiple Weak Signals in Non-Gaussian Noise

Fig 1. The performance of the short time Wigner distribution and the STFT for two tones at -5 dB SNR. Both representations are based on a 512 point zero filled transform computed with a 257 long window. The signals are completely lost in the STWD but have significant strength in the STFT. a: superimposed STWD spectra; b: log magnitude of the STWD; c: superimposed STFT magnitude spectra; d: squared magnitude of the STFT (spectrogram) dB.

where the windowing function, h(−t), is of relatively short length. Again, in a TF context, Welch’s representation may be expressed as Ws,h,w (ω, T ) = |Sh (ω, T )|2 ∗T w(T ),

(5)

where ∗T represents convolution with respect to the variable, T , and w(T ) is a smoothing window. The advantage of Welch’s method is that the noise variance is reduced, resulting in improved detection of stationary signals. The disadvantage is that the method provides poor frequency resolution and poor frequency accuracy due to the short window used in the computation. An improvement of Welch’s method is the cross-power spectral estimator (CPS) [1]. Again in the TF context, this method reduces to a time smoothing of a crossspectrum computed from the STFT and time delayed STFT Cs,h, (ω, T )

=

CPSs,h,,w (ω, T )

=

Sh (ω, T + /2)S∗h (ω, T − /2) Csh (ω, T ) ∗T w(T ),

(6) (7)

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for a small delay, , typically equal to one sample for discrete signals. The CPS has been demonstrated to perform better than Welch’s method in detecting weak stationary signals, but its main advantage is that the frequency of a signal detected at (ω0 , T0 ) may be accurately estimated as arg{CPSs,h,,w (ω0 , T0 )}/. 3.

The Basis of the Method

As in the CPS algorithm, we base our method on the cross-spectrum, Cs,,h (ω, T ), computed from the STFT. The instantaneous frequency surfaces may be estimated as Ωs,h (ω, T )

≡ ≈

˜ s,h,w (ω, T ) Ω

≡ ≈

∂ arg{Sh (ω, T )} ∂T 1 arg{Cs,h, (ω, T )}  ∂ arg{Sh (ω, T )} ∂T 1 arg{CPSs,h,,w (ω, T )}, 

(8) (9) (10) (11)

for small values of . For small values of , the magnitude of Ωs,h (ω, T ) is essentially the spectrogram, |Sh (ω, T )|2 . The representation, Eq. (7), conveniently encodes both the energy at each time and frequency and the estimated instantaneous frequency of the dominant signal component at each time and frequency [1]-[10]. We will use these two properties to accurately estimate the frequencies of individual signal components and to estimate and encode signal components. For digital implementations, Cs,h, (ω, T ) can be easily computed. We first compute the STFT of the signal (Eq. (4)). We then delay the signal by one sample and compute the STFT of the delayed signal. The surface Cs,h, (ω, T ) is computed as the product of the STFT of the un-delayed signal and the complex conjugate of the STFT of the delayed signal. Calculated in this manner and setting  = 1, the IF, Ωs,h (ω, T ), is calibrated in units of angular frequency divided by the sample rate of the signal. An important feature of this process is that, even though the signal and STFT surfaces are sampled in time (and frequency), the IF, Ωs,h (ω, T ), estimated in this manner is not quantized and represents an accurate estimate of the instantaneous frequency of the dominant signal component (if any) at the time and frequency at which the estimate is computed (c.f. [2, 10]). Equally important is that the representation encodes both energy and instantaneous frequency in a single complex representation. 4.

Concentration of Cs

As they are normally implemented, Welch’s method and the CPS algorithm, described above, only work for stationary signals. This is because the windowing function, w(T ), used in these representations is long. The concentration process we now describe may be applied to the surfaces, Cs,h, (ω, T ) or CPSs,h,,w (ω, T ). We will describe the process by applying it to the original surface, C,h,s (ω, T ), since application to CPSs,h,,w (ω, T ) is essentially identical. This concentration process preserves angular instantaneous frequency and results in spectral “energy” locally

Detection and Estimation of Multiple Weak Signals in Non-Gaussian Noise

concentrated near the IF of individual signal components. While we apply only the instantaneous frequency in this process, we could, in addition, apply the local group delay as first suggested by Kodera and possibly realize some additional gain [14]. We select a frequency quantization value, ΔΩ, and quantize the angular frequency into frequency intervals In = [(n − 1/2)ΔΩ, (n + 1/2)ΔΩ).

(12)

We then redistribute the values of the surface, Cs,h, (ω, T )  Cs,h, (ω, T ) Q(n, T ) =

(13)

Ωs,h (ω,T )∈In

While the representation, Eq. (13), is sufficient for our needs, there are at least two modifications which may improve the quantization process. We may interpolate to distribute the contribution of each element Cs,h, (ω0 , T0 ) over the two closest intervals, and we may weight the contribution of each element Cs,h, (ω0 , T0 ) to reflect the consistency of Ωs,h near (ω0 , T0 ). These modifications are discussed in [10], where the correct weighting function is derived. In any event, the surface Q(n, T ) encodes energy and instantaneous frequency as magnitude and phase of each surface component. The process represented by Eq. (13) results in a surface on which the “energy” of each narrowband signal component is concentrated near a curve representing the IF of that component (c.f. [4, 10]). The performance and accuracy of this process are discussed in [10]. 5.

The Concentrated Peak Representation (CPR)

To reduce the data volume, we now reduce the surface Q(n, T ) to a peak representation (c.f. [8, 10]). In this process, each surface component at time, T0 , is discarded if its magnitude is not greater than the adjacent components PEAKS(T ) = {Q(n, T ) : (|Q(n, T )| > |Q(n − 1, T )|) ∩ (|Q(n, T )| > |Q(n + 1, T )|)} (14) For each value T0 , PEAKS(T0 ) is a vector, but not all such vectors are of the same length. We enforce a uniform vector length, N , by sorting, for each time, T0 , the peaks in PEAKS(T0 ) by descending magnitude and discarding peaks or zero filling the resulting peak vectors at each time so that each vector has length N , resulting in the peak representation P(n, T ) , n = 1 . . . N

(15)

The representation P(n, T ) essentially preserves the time, energy and IF information of the strongest components of the original STFT surface. The assumption implicit in this representation is that each narrowband component, sk , may be represented by a time-parameterized ridge, P(nk (T ), T ), on the concentrated and frequency quantized surface, Q(n, T ), where the the ridge index, nk (T ), of a particular ridge can not be assumed to be time independent. For stronger components, one would expect this to be true. For component signals at very low negative SNR, the expectation is that the concentrated signal will add enough energy to the spectrum

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Fig 2. Peak representation of one billion data samples. The TF representation was computed with a 131072 point FFT using a 65537 long Hanning window and a 50% overlap. The size of the original image was 64k x 16k. In this rendering, the 100 strongest peaks at each time are displayed

Fig 3. A sub-image of the peak representation of the previous figure representing 50 million data samples or approximately 5% of the previous figure. This image has 7 signals indicated by the arrows along the left margin.

Detection and Estimation of Multiple Weak Signals in Non-Gaussian Noise

that a sufficient number of peaks will be observed near the IF of the signal that the signal can be detected and estimated. The success of this process is dependent on SNR. However, it has the advantage that it is insensitive to time and frequency variations of non-uniformly distributed noise. To further improve the representation and reduce the number of noise related peaks, a peak tracking algorithm was implemented. In this process, peaks are removed from the peak vectors if there are not a sufficient number of peaks at a similar frequency observed within a prescribed time interval. The peak removal process is implemented by setting the amplitudes of the unwanted peaks to zero and resorting the peak vectors. This process greatly reduces the amount of noise speckle, making identification of causal components easier. The peak representation can greatly reduce the data volume, preserving critical signal information. The process has been applied to a variety of signals, including a broadband environment simulating multiple very weak tones in very heavy noise. In blind testing, an STFT was computed from a billion (109 ) data samples using a Hanning window of length 65537 and a transform length of 131072 and an overlap of 32768 samples. The proposed method was applied to the surface, CPSs,h,,w (ω, T ), by convolving the cross-spectral surface in time with a Hanning window of length 17 and decimating the resulting surface in time by a factor of 8. The peak representation, P(n, T ), was then computed, retaining a maximum of 200 peaks at each time and resulting in a reduction of the original data by a factor of nearly 800. By peak tracking, the number of peaks could be reduced to approximately 50 peaks at each time, further reducing the data volume to a combined data reduction factor of 3200. The 50-peak representation of the entire surface is represented by Fig. 2, and a portion of the surface representing approximately 50 million original data samples or 5% of the data is represented by Fig. 3. In Fig. 2, one weak signal is easily observed. In the expansion, Fig. 3, 5 very weak signals are apparent. 6.

CONCLUSIONS

We have addressed the problem of detection and estimation of multiple narrowband signals deep in non-stationary non-Gaussian noise. The proposed method is based on the cross-power spectrum variation of Welch’s and has the advantage that it provides a sparse representation in which most of the effects of noise are removed, preserving most of the signal. The process may be applied to massive amounts of data to detect and display very weak signals. References [1] D. Nelson, “Special Purpose Correlation Functions, internal report, 1989, (republished in part in: “Special Purpose Correlation Functions for Improved Parameter estimation and Signal Detection, Proceedings of the IEEE Conference on Acoustics, Speech and Signal Processing, Minneapolis, pp. 73-76, April,1993). [2] D.Nelson, “Cross-spectral methods for processing speech, in J. Acoust. Soc. Am., vol.110, num. 5, pt. 1, pp. 2575-2592, Nov., 2001. [3] D.J. Nelson and D.C. Smith, “A New Linear Time-Frequency Paradigm, in Proc. 6th Int. Conf. Sig. and Im. Proc., pp. 451-456, Honolulu, Aug.23-25, 2004. [4] D.J. Nelson, D.C. Smith and R.C. Masenten, “Linear Distribution of Signals, in Proc. SPIE Adv. Sig. Proc. Algor., Arch. and Imp. XIV, pp. 245-257, Denver, Aug.4-6, 2004.

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[5] D.J. Nelson, “CROSS-SPECTRAL BASED FORMANT ESTIMATION AND ALIGNMENT ,in IEEE Proc. ICASSP, Montreal, April, 2004. [6] Nelson, D.J. and Smith D.C., "Concentrating the Short-time Fourier Transform Using Its Higher Order Derivatives", Proc. 7th IASTED Int. Conf. Sig. and Im. Proc., pp. 274-279, Honolulu, 2005 [7] Nelson D.J. and Smith D.C., “A New Linear Time-Frequency Paradigm, [8] D.J. Nelson, “Signal Reconstruction from Concentrated STFT Peaks, in IEEE Conf. on Acoust. Speech and Signal Proc., Philadelphia, 2005. [9] Nelson D.J. and Smith D.C., "A higher order method for concentrating the STFT", Proc. SPIE Adv. Sig. Proc. Algor., Arch. and Imp. XV, pp. OG1-OG8, San Diego, Aug. 2-4, 2005 [10] Nelson, D.J., and Smith D.C., "A Linear Model for TF Distribution of Signals", IEEE Transactions on Signal Processing, Sept, 2006. [11] E. Wigner, “On the Quantum Correction For Thermodynamic Equilibrium, in Physical Review, volume 40, pp. 749-759, June, 1932. [12] J.Ville, “Theorie et applications de la notion de signal analytique, in Cables et Transmissions, vol. 2A, num.1, pp.61-74, 1948. Translated from French by I. Selin, “Theory and applications of the notion of complex signal, RAND Corporation Technical Report T-92, Santa Monica, CA, 1958. [13] Welch, P.D., "The use of the fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms," IEEE Trans. Audio Electroacoustics, AE-15:226-232, June 1967. [14] K. Kodera, R. Gendrin and C. de Villedary, “Analysis of Time-Varying Signals with Small BT Values, in IEEE Trans. Acoust., Speech and Sig. Proc., vol.ASSP-26, no. 1, Feb. 1978. [15] F. Auger and P. Flandrin, “Improving the Readability of Time-Frequency and TimeScale Representations by the Reassignment Method, in IEEE Trans. Sig. Proc., vol. 43, no. 5, May 1995. [16] L. Galleani, Leon Cohen and D.J. Nelson, “Detection of False Transients, Proc. SPIE Adv. Sig. Proc. Algor., Arch. and Imp. XV, pp. OC1-OC11, San Diego, Aug. 2-4, 2005 [17] D. Gabor, “Theory of communication,, in J. IEE, vol. 93, 429-457, 1946. [18] D.C. Smith and D.J. Nelson, “Detection and Resolution of Narrow Band Signal Components by Concentrating the DFT, in Proc. of the 6th int. Workshop on International Optics, Toledo Spain, June 2006.