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Frequency Estimation from Compressed Measurements of a Sinusoid in MA Colored Noise

Frequency estimation of a single sinusoid in colored noise has received a considerable amount of attention in the research community. Taking into account the recent emergence and advances in compressive covariance sensing (CCS), the aim of this work is to combine the two disciplines by studying the effects of compressed measurements of a single sinusoid in moving-average (MA) colored noise on its frequency estimation accuracy. CCS techniques can recover the second-order statistics of the original uncompressed signal from the compressed measurements, thereby enabling correlation-based frequency estimation of single tones in colored noise using higher-order lags. Acceptable accuracy is achieved for moderate compression ratios and for a sufficiently large number of available compressed signal samples. It is expected that the proposed method would be advantageous in applications involving resource-limited systems such as wireless sensor networks.


Introduction
Frequency estimation of a sinusoid in noise is one of the major problems in signal processing and communications with various applications in radar, sonar, carrier synchronization and signal detection, among many others. Many frequency estimation methods have been devised for the white noise case such as the maximum likelihood (ML) estimation method which involves locating the peak of the periodogram [1]. This efficient method attains the Cramer-Rao lower bound (CRLB) asymptotically but is computationally complex even when the fast Fourier transform (FFT) is used. In addition, a number of fast and accurate frequency estimators of a sinusoid in white noise have been developed as linear prediction estimators, also achieving the CRLB on variance asymptotically [2]. A more computationally efficient method than the ML method is the correlation method [3,4] in which an estimate of the frequency is obtained from knowledge of one or more values of the sinusoid-plus-noise autocorrelation sequence.
Improved-performance frequency estimators have been obtained using multiple autocorrelation lags as in the P-estimator [5] and the modified Pisarenko harmonic decomposer (MPHD) [6]. These, however, involve phase unwrapping to overcome the resulting frequency ambiguities, thereby increasing complexity. Digital phase locked loops (DPLL) offer low-complexity sample-by-sample frequency estimation for real-time applications in contrast to the above-mentioned batch estimators [7][8][9]. Disadvantages of DPLL estimators are overshoot and long settling times [10]. In [11], unbiased estimation of a sinusoid in colored noise is achieved via adapted notch filters. Colored noise is practically more relevant and appears in many applications [12].
The purpose of the present work is to perform single-tone frequency estimation in colored noise using the correlation-based methods as they allow frequency estimation from compressed measurements using compressive covariance sensing (CCS). CCS is akin to compressive sensing (CS) [13]. As asserted in [14], CS has great potential for energy-efficient data processing and communication in wireless sensor networks (WSN) that have become commonplace due to their major role is various applications such as industrial monitoring, health care monitoring, and environmental sensing, etc. CS compresses data by sampling randomly at a sub-Nyquist rate to gain numerous compression advantages such as limitation of sensing, storage and communication costs, thereby making CS ideal for resource-limited systems such as WSNs. The original uncompressed signal can be recovered from its compressed version on the condition that the signal is sparse in a certain transform domain. In contrast, CCS techniques [15] can recover the second-order statistics of an original signal that has been compressed rather than the signal itself without the exertion of the sparsity condition, and at the same time retaining all the above-mentioned advantages of compression. CCS relies on certain structure forms of the covariance matrix to be captured during compression enabling the recovery of covariance (or correlation) information of wide sense stationary (WSS) signals, and even, more recently, nonstationary signals via online CCS [16]. One of the major applications of CCS is correlation-based frequency estimation from compressed measurements.
Other applications of CCS include power spectrum estimation [17,18] and direction-of-arrival estimation in array signal processing [19].
Frequency estimation of a sinusoid from compressed measurements given the additive white Gaussian noise (AWGN) case has been addressed in the literature [20][21][22][23]. In [20], ML estimation by grid search optimization is proposed such that the frequency estimate is that which maximizes a cost function. In [21], a Newton-like algorithm is used to further refine the estimate after grid search, whereas the ML estimator in [22] requires a course-fine grid search. In [23], frequency estimation from compressed measurements is cast as a linear least squares problem.
Correlation-based frequency estimation of a real-valued single tone in colored noise has been dealt with in [24] using higher-order lags, thereby restricting the treatment to moving-average (MA) noise and achieving low complexity. We focus on extending the work of [24] to achieve frequency estimation from compressed sinusoids in colored noise using CCS. This approach in the compressed measurements context has the advantage of reduced computational complexity in comparison with the ML approaches of [20][21][22][23] and even with the correlation-based approaches of [5,6] since no phase unwrapping is involved, as well as the applicability to the colored noise case.
The rest of the paper is organized as follows. Section 2 summarizes the frequency estimation method of [24]. CCS is explained in Section 3 relating to the present frequency estimation problem. Simulation results are given in Section 4, and finally, Section 5 concludes the paper.

Frequency Estimator of Real-Valued Single Tone in Colored Noise using Multiple Autocorrelation Lags
The estimator proposed in [24] can be summarized as follows. A real-valued sine wave ) (n (1) The colored noise ) (n u is zero-mean, WSS, Gaussian and independent of ) (n s . It is assumed to be generated by passing zero-mean white Gaussian noise of variance 2  through an order-Q FIR filter. The normalized noise autocorrelation The autocorrelation function of ) (n v is denoted by ) (k r and is given by: Using trigonometric relations, it is straightforward to show that an estimate of the frequency o  , denoted by o  , can be expressed as: for any integers is the unbiased estimation of autocorrelation given by: This method of single-tone frequency estimation has been shown in [24] to outperform the P-estimator and the MPHD methods in terms of computational load and accuracy and for the colored noise case. We next discuss the application of the CCS method to the frequency estimator discussed above while considering compressed measurements of the noisy sinusoid.

The Compressive Covariance Sensing Method and its Application to Frequency Estimation
Autocovariance of a signal is equivalent to its autocorrelation when the signal mean is zero [27], and it will be referred to throughout the rest of this paper as covariance for simplicity. CCS recovers the covariance of a signal form a compressed version of it when compression is carried out by further sampling the signal using a linear sparse ruler (LSR) [15]. A LSR of length N-1, N=11, is shown in Figure 1.

R y
is: is a sparse sampling matrix that performs the function of the LSR.
Each of the M rows of  has at most one non-zero value.

Simulation Results
Simulations are carried out in MATLAB. The frequency estimator of [24] is first simulated. The length L of the real-valued noisy sinusoid is first chosen to be L=110   = dB SNR (12) Zero-mean white Gaussian noise (WGN) of variance 2  is passed through a MA filter of order Q=2, and having a system transfer function To make a meaningful plot of estimation mean square error (MSE) in dB vs. SNR (dB), we need to plot the CRLB as well. The CRLB provides the minimum variance of the unbiased estimator as SNR increases. For the problem of frequency estimation of a single sine in white noise, the CRLB was found in [31] to be: where L is the number of signal samples. To take into account MA colored noise [32], we take into account the noise spectral density at the sinusoidal position The above holds provided that the noise spectrum is relatively smooth over frequency intervals corresponding to  2 times the reciprocal of the number of signal samples [32], which is the case in the present settings. The larger the number of samples, the better the approximation is.
Next, a compressed version of the length-110 noisy sinusoid is considered by first dividing the signal vector into 10 blocks (B=10), each of length N=11. Each block is compressed using an LSR with a compression ratio M/N=6/11 as in Figure 1. CCS is then applied as explained in Algorithm 1 of Section 3 to the compressed vector to recover the correlation values to be substituted in Equation 3. The process is repeated using M/N=9/11 for which the LSR is {0, 1, 3, 4, 6, 7, 8, 9, 10}.
All simulations are averaged over 500 independent runs. The results of the above procedure are demonstrated in Figure 2. It is a plot of the MSE vs. SNR for the estimator with no compression corresponding to [24], the estimator of compressed noisy sinusoid using CCS with compression ratio 9/11, and finally its 6/11 counterpart. The CRLB is also shown. Clearly, as compression increases, performance is degraded. Figure 3 demonstrates the MSE vs. SNR for only one compression ratio (9/11) but for different values of the number of blocks B. The improvement is clear as B increases. Increasing B may appear to be contradictory to the concept of compression, but in fact, it is not since the average sampling rate is reduced by compression regardless of the length L (=B.N) of the signal, and it is the sampling rate that determines hardware complexity and consequently cost-effectiveness [15]. For the case of compression ratios equal to 8/11 and 9/11, the estimation is acceptable as may be discerned from Figure

Conclusion
Compressed measurements of a single sinusoid in MA colored noise are used to reconstruct the second-order statistics for the purpose of frequency estimation of the original uncompressed noisy sinusoid. Estimation accuracy is acceptable for moderate compression ratios, and was found to degrade for lower (better) compression ratios generally resulting in biased estimates. For a fixed compression ratio, the mean square estimation error was found to improve (decrease) as the number of available compressed signal samples increases. The advantages gained by compression may outweigh the disadvantage of decreased estimation accuracy depending on the application.