Twenty Fourth National Conference on Communications (NCC) Co-Prime Sampling and Estimation Usham V. Dias and Seshan Srirangarajan Department of Electrical Engineering Bharti School of Telecommunication Technology and Management Indian Institute of Technology Delhi, New Delhi, India Abstract Research in the field of co-prime arrays and samplers has been mainly focused on reconstructing the autocorrelation and the spectral content of a signal at the rate from sub- data. This has found applications in power spectrum estimation, beamforming, direction-of-arrival estimation, and system identification. However, the use of coprime samplers for cross-correlation estimation has not received much attention. We describe cross-correlation estimation using co-prime samplers and consider two scenarios. In the first, both signals are acquired using co-prime samplers, while in the second scenario we assume one of the signals to be a known signal and thus available at the rate, and the second signal is acquired using a co-prime sampler. We determine the number of contributors available for cross-correlation estimation at each difference value as this is a key parameter in determining the estimation accuracy. The work presented in this paper will have applications in time-delay, range, velocity, acceleration, and crossspectrum estimation, which require cross-correlation estimation. I. INTRODUCTION The co-prime array and its variants were introduced as efficient structures to estimate the second order statistics at the rate from sub- data [] [3]. Most of the work in this field has focused on autocorrelation estimation with applications to power spectrum estimation [], [5], system identification [], beamforming [7], and direction-of-arrival estimation []. However, application of these structures for cross-correlation and cross-spectral estimation has not received much attention. Cross-correlation estimation is of importance in many applications including radars for time-delay, velocity and acceleration estimation, and underwater monitoring. Underwater acoustic sensor networks (UASN) play an important role in communication and data collection in applications such as tsunami warning, environmental pollution and oil spill monitoring, as well as underwater surveillance and assisted navigation. In these applications it is vital to estimate the number of active nodes required to guarantee proper network operation, also known as cardinality estimation. In the underwater environment the nodes tend to drift from their equally spaced geometry and hence the work in [9] analyzes the effect of unequal sensor separation on cardinality estimation using a cross-correlation based method for sensorin-line (SL) geometry. Chirp signals are a popular class of signals that have been used in automotive radar ranging [], multifunctional communication and radar systems [], measurement of sea ice thickness [], and multiple-input multiple-output (MIMO) radars [3]. The work in [] describes a practical approach for Usham Dias is supported through the Visvesvaraya PhD Scheme fellowship from the Ministry of Electronics and Inf. Tech. (MEITY), Govt. of India. implementing a receiver that can handle high linear chirp rates and a time-of-arrival (ToA) estimator that can detect and measure stationary radio frequency pulses and linear chirp rates of up to. GHz in ns. [5] describes a cross-spectral singular value decomposition (SVD) method for time-delay estimation of a chirp signal and compares it with the correlation method. Time-delay estimation of wideband signals using the cross-correlation method is analyzed and a generalized maximum-likelihood estimate based on the main peak and spill over cells of the cross correlation is presented in []. It relates the estimation accuracy to the sampling period, signal-to-noise ratio, and the number of samples in each observation interval. Motion estimation of targets using a stepped frequency chirp signal and an extended cross-correlation method to estimate the radial velocity and acceleration by using echoes of the sub-pulses within a burst is presented in [7]. The difference set for co-prime arrays and samplers was studied in []. The co-prime array-based acquisition models can also be employed for applications that rely on the cross-correlation between signals. We consider this for two scenarios. The first is referred to as fully sub- with both signals being acquired using co-prime samplers. The second scenario is referred to as partially sub- with one co-prime sampled signal and the other being a rate signal. The partial sub- scenario is encountered in applications where a pilot or known template signal is employed. Consider for example the estimation of range and velocity by transmitting predefined pulses and listening to the echoes. Since a known signal is transmitted we can generate this rate signal at the receiver. The remainder of this paper is organized as follows. In Section II we describe the fully sub- cross-correlation estimation. Section III analyzes the partial sub- scenario and a comparison of the two schemes is presented in Section IV. Co-prime based cross power spectral density is discussed in Section V while applications of cross-correlation estimation to time-delay, range, velocity, and acceleration estimation, are considered in Section VI. Section VII concludes the paper. II. FULLY SUB-NYQUIST CO-PRIME CROSS-CORRELATION In this section we analyze the cross-correlation estimation process for signals that are independently acquired using sub- co-prime samplers operating at rates M and N times the rate. Let and represent the two signals whose crosscorrelation needs to be estimated. In the first case, we assume and are of duration or length equal to one co- 97--53--//$3. IEEE
Twenty Fourth National Conference on Communications (NCC)............ MN (a) Case MN (b) Case 3 3MN (c) Case 3 Fig.. Signal for fully sub- and partial sub- scenarios..... MN.... (a) Case and Case MN (b) Case 3 Fig.. Signal for the fully sub- scenario. prime period, i.e. MNT s where T s is the sampling period. In the second case, is assumed to have a duration equal to one co-prime period, while is a multiple of the co-prime period. Finally, we describe the case where both signals have lengths that are an integer multiple (greater than one) of the co-prime period. We refer to these as Cases,, and 3, respectively, and (M, N) = (, 3) for all the examples and illustrations in this paper. A. Case Consider the signals shown in Fig. (a) and Fig. (a), where each signal has a duration equal to the co-prime period and is sampled using co-prime samplers with (M, N) = (, 3). Let i denote the sample index which lies in the range [, MN ]. When two signals of the same length are sampled at the rate, the cross-correlation length will be twice the signal length minus one and the number of samples that contribute to the estimate at each difference value (popularly known as lag) has a maximum value equal to the signal length at difference value zero and decreases linearly for both positive and negative difference values. The co-prime sampled signals do not exhibit such uniform behavior. Cross-correlation operation for the co-prime sampled signals is described in Fig. 3 where represents the number of samples that contribute to the estimate at difference value l. is also known as the weight function and is shown in Fig. (a) for the co-prime and traditional sampling schemes. A larger value of can improve the estimation accuracy. = for l {,, 7, 7,, } for the co-prime scheme and thus the cross-correlation cannot be estimated for these difference values. The weights for the sub- cross-correlation estimation equal the weights for the autocorrelation estimation, presented as Proposition-III in [], which also contains missing values. For real signals, the autocorrelation estimate at difference value l is equal to the estimate at difference value l. However, this does not hold for the cross-correlation estimation since the cross-correlation function is not symmetric. In Fig. (a), the normalized cross-correlation estimate for Case is compared with the scheme estimate. The root mean square error (RMSE) between the two crosscorrelation estimates without normalization is.33. It may also be noted that the co-prime based cross-correlation has a maximum at l =, while the framework estimates it at l =, i.e. an erroneous delay of two samples. B. Case In practice two signals to be correlated may not have the same length. For example, an incoming radar echo signal is typically a long signal in which a short length transmitted pulse is to be detected. Consider the signals shown in Fig. (b) and Fig. (a) which are sampled using co-prime samplers. Signal is of length MN and has length equal to the co-prime period MN. The number of contributors for the cross-correlation estimation at each difference value is shown in Fig. (b) along with the weight function. A comparison between the and sub- normalized cross-correlation estimates is also shown in Fig. (b) and both schemes result in maximum correlation at difference value l =. In this case, the sub- technique accurately estimates the time delay and this is mainly due to the fact that at the true cross-correlation peak (l = ) the co-prime scheme has a large number of contributors. Thus the weight function is an important parameter in the cross-correlation estimation. The overall RMSE compared to the scheme is 3.33. C. Case 3 Now consider the case where both signals have lengths that are greater than the co-prime period. The co-prime sampled signals and are shown in Fig. (c) and Fig. (b), where is of length 3MN while is of length M N. The weight function in this case for cross-correlation estimation is shown in Fig. (c) for the sub- and schemes. The RMSE for cross-correlation estimation is 5.37 and the location of maximum correlation is l = for both schemes.
Twenty Fourth National Conference on Communications (NCC).................... - -5 5 5 = l= - - -5 5 5 = l= - - -5 5 5 = l= -9 - -5 5 5 = l= - - -5 5 5 = l= -7.................... - -5 5 5 = l= - - -5 5 5 = l= -5 - -5 5 5 = l= - - -5 5 5 =3 l= -3 - -5 5 5 = l= -.................... - -5 5 5 = l= - - -5 5 5 = l= - -5 5 5 = l= - -5 5 5 = l= - -5 5 5 =3 l= 3.................... - -5 5 5 = l= - -5 5 5 = l= 5 - -5 5 5 = l= - -5 5 5 = l= 7 - -5 5 5 = l=............ - -5 5 5 = l= 9 - -5 5 5 = l= - -5 5 5 = l= Fig. 3. Cross-correlation process for the fully sub- scenario with and of length equal to one co-prime period. 5 5 5 - -5 5 - -5 5 5.... (a) Case - - 3.... (b) Case.... (c) Case 3 RMSE=.33 - -5 5 RMSE=3.33 - -5 5 5 RMSE=5.37 - - 3 Fig.. Contributors per difference value or weight function (left) and crosscorrelation estimate (right) for the fully sub- scenario. III. PARTIAL SUB-NYQUIST CROSS-CORRELATION In many applications where cross-correlation estimation is required, one of the signals is predefined and the system does not have to acquire it via sampling. Therefore, only the incoming signal is sampled using the co-prime scheme, denoted by (Fig. ), while the rate signal (or pattern) is known (Fig. 7). A classic example of this is a radar system that tracks a target by estimating the timedelay profile. The transmitted pulse could be a sinusoid or a mixture of frequencies similar to a chirp signal, which can be synthetically generated at the rate. A. Case The co-prime sampled signal is shown in Fig. (a), while the known rate signal is shown in Fig. 7(a). In this case we consider signals having the same length equal to the co-prime period. The cross-correlation process between these two signals is described in Fig. 5. The weight function and cross-correlation estimates are shown in Fig. (a). Here the number of contributors at each difference value is higher compared to the fully sub- scenario resulting in cross-correlation estimation with higher accuracy (RMSE of.3 as compared to.33 for the fully sub- Case ). In addition, the maximum correlation is correctly detected at l =, while this detection was erroneous in the fully sub- scenario. B. Case Now consider signal of length MN as shown in Fig. 7(a), while is acquired over two co-prime periods MN and is shown in Fig. (b). The weight function and the cross-correlation estimates are shown in Fig. (b). The crosscorrelation peak occurs at l = which matches the result in the case, and the RMSE is.7 as compared to 3.33 for the fully sub- scenario. C. Case 3 Next we consider the case where both the acquired signal and the synthetically generated rate signal
Twenty Fourth National Conference on Communications (NCC).................... - -5 5 5 = l= - - -5 5 5 = l= - - -5 5 5 = l= -9 - -5 5 5 = l= - - -5 5 5 =3 l= -7.................... - -5 5 5 =3 l= - - -5 5 5 = l= -5 - -5 5 5 = l= - - -5 5 5 =5 l= -3 - -5 5 5 = l= -.................... - -5 5 5 = l= - - -5 5 5 = l= - -5 5 5 =5 l= - -5 5 5 =5 l= - -5 5 5 =5 l= 3.................... - -5 5 5 = l= - -5 5 5 =3 l= 5 - -5 5 5 =3 l= - -5 5 5 = l= 7 - -5 5 5 = l=............ - -5 5 5 = l= 9 - -5 5 5 = l= - -5 5 5 = l= Fig. 5. Cross-correlation process for the partial sub- scenario with and of length equal to one co-prime period. 5 5 5 - -5 5 - -5 5 5.... (a) Case - - 3.... (b) Case.... (c) Case 3 RMSE=.3 - -5 5 RMSE=.7 - -5 5 5 RMSE=3.5 - - 3 Fig.. Contributors per difference value or weight function (left) and crosscorrelation estimate (right) for the partial sub- scenario. have lengths that are an integer multiple of the coprime period MN. is shown in Fig. (c) with length 3MN and with length MN is shown in Fig. 7(b). The weight function and the cross-correlation estimates are shown in Fig. (c). The RMSE is 3.5 which is significantly lower.... MN.... (a) Case and Case MN (b) Case 3 Fig. 7. Signal for the partial sub- scenario. than the RMSE of 5.37 obtained when both the signals were acquired using co-prime samplers. The cross-correlation peak occurs at l = matching the result from the case. IV. COMPARISON BETWEEN SUB-NYQUIST AND NYQUIST SCENARIOS In this section we compare the proposed sub- schemes for cross-correlation estimation with the traditional scheme. We begin with a set of definitions to aid our discussion. The cross-correlation of signals and
Twenty Fourth National Conference on Communications (NCC) is defined as: y cc (l) = ( i) = i= (i + l) () where represents the convolution operation. Set containing the possible difference values for cross-correlation estimation when and have lengths equal to r A MN and r B MN, respectively, can be defined as: R l = {l l [ r B MN +, r A MN ]} () The maximum cross-correlation value (m cc ) and the corresponding difference value (l cc ) are defined as: m cc = max l R l y cc (l) and l cc = {l y cc (l) = m cc } (3) The mean square error of the cross-correlation estimation using the sub- and schemes is computed as: RMSE = (y cc (s) (l) y cc (n) (l)) R l () l R l where y (n) cc (l) and y (s) cc (l) represent the cross-correlation estimate for the and sub- schemes respectively, and R l represents the cardinality of the set R l. Table I compares the and the proposed sub- schemes. The fully sub- scenario results in a relatively higher RMSE but is able to estimate the cross-correlation peak location correctly except in Case. RMSE increases from Case through Case 3 since the number of contributors for estimation do not increase proportionately to the increase in the number of difference values in R l. For the partial sub- scenario, a lower RMSE is observed and the crosscorrelation peak location l cc is estimated correctly in all the cases. The peak value of the cross-correlation estimate m cc is generally not of significance in applications that are trying to estimate the time-delay. However, it could be of significance in applications where different signals or patterns have to be classified. V. CROSS POWER SPECTRAL DENSITY ESTIMATION Cross power spectral density is the Fourier transform of the cross-correlation function and is expected to have spectral peaks at frequencies present in the cross-correlation signal. A relatively large peak will occur at frequencies that are common to both signals. Consider signals (t) and (t), where (t) has frequencies of 5 Hz and Hz, and (t) contains Hz and 35 Hz. Both signals are corrupted by additive white Gaussian noise with signal-to-noise ratio (SNR) of db. Consider these signals of s duration sampled at a rate of f s = Hz. The cross power spectral density is estimated using the cpsd function in Matlab for the scheme. A Bartlett window of size 5 with an overlap of samples is used for estimation with -point FFT. We employ a pair of co-prime samplers with M = and N = 3 to sample both (t) and (t), i.e. the fully sub- scenario. Next, we consider the partial sub- scenario where (t) is sampled using the co-prime sampler while (t) is sampled at the rate. The simulation results are shown in Fig.. The peak at Hz shows the ability of the proposed scheme to detect the frequencies common to both the signals. It may be noted that the results are based on a single realization of the signals (t) and (t), and the Power (db) Power (db) - - -3 - -5 3 5 Frequency (Hz) - - -3 - (a) Fully sub- scenario. -5 3 5 Frequency (Hz) (b) Partial sub- scenario. Fig.. Cross power spectral density estimation. fully sub- scenario does not result in a strong peak at Hz which needs further investigation. VI. TIME-DELAY, RANGE, VELOCITY, AND ACCELERATION ESTIMATION Consider an underwater sonar system that is used for tracking targets by transmitting chirp signal pulses with frequencies in the range [.5MHz,.5MHz] and a pulse width of µs. The echoes received are sampled at a rate f s = MHz, which is then compared with the template transmit pulse. For the sub- co-prime sampling scheme, we use (M, N) = (, 3). The difference value or lag at which the cross-correlation peak occurs gives the time-delay estimate. This delay depends on the distance or the range of the target or object. We consider an object moving away from the transmitter at a constant velocity along the axis of the transmitted signal. The received echo is assumed to be corrupted by additive white Gaussian noise with an SNR of db. The velocity and acceleration estimate is obtained by computing the change in the range and velocity estimates, respectively. Fig. 9 compares the fully sub- and partial sub- schemes with the traditional framework. Since the object is assumed to be moving away from the transmitter at a constant velocity, the time-delay and range profiles are linear. The velocity is constant and the acceleration is zero. Both the fully sub- and partial sub- schemes are found to work well. The partial sub- estimates the acceleration with a higher accuracy. In these simulations chirp pulses were transmitted and their received echo indices are denoted along the x-axis. VII. CONCLUSION The cross-correlation estimation procedure using sub- co-prime samplers is presented under two scenarios. In the first, both signals are sampled using co-prime samplers which is referred to as the fully sub- scenario. This relates to situations where both signals have to be physically acquired. The second scenario is referred to as the partial sub- scenario in which one signal is available at the rate while the other is acquired using a co-prime sampler. This
Twenty Fourth National Conference on Communications (NCC) TABLE I COMPARISON OF THE PROPOSED SUB-NYQUIST SCHEMES WITH THE TRADITIONAL NYQUIST SCHEME Fully sub- Partial sub- m cc l cc RMSE m cc l cc RMSE m cc l cc Case.5.33. -.3 5.3 - Case 3.7 3.33 3.7.7 7. Case 3. 5.37. 3.5.5 Time-Delay (sec)..5..5 Range (m) 57 5.5 5 Velocity (m/sec)57.5 (a) Fully sub- scenario. Acceleration (m/sec ).5 -.5 - Time-Delay (sec)..5..5 Range (m) 57 5.5 5 Velocity (m/sec) 57.5 (b) Partial sub- scenario. Fig. 9. Application of cross-correlation for time-delay, range, velocity, and acceleration estimation. Acceleration (m/sec ).5 -.5 - occurs in applications where a predefined signal is transmitted and thus can be generated at the rate at the receiver for cross-correlation estimation. We also describe how the signal can be acquired over multiple co-prime periods, and its effect on the number of contributors for estimation and hence the estimation accuracy. We present some results for cross power spectral estimation to detect common frequencies between signals. Finally, we demonstrate the application of co-prime samplers for time-delay, range, velocity, and acceleration estimation using the fully sub- and partial sub- schemes. REFERENCES [] P. P. Vaidyanathan and P. Pal, Sparse sensing with co-prime samplers and arrays, IEEE Trans. Signal Process., vol. 59, no., pp. 573 5, Feb.. [] Q. Si, Y. D. Zhang, and M. G. Amin, Generalized coprime array configurations, in IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM),, pp. 59 53. [3], Generalized coprime array configurations for direction-of-arrival estimation, IEEE Trans. Signal Process., vol. 3, no., pp. 377 39, Mar. 5. [] S. Ren, Z. Zeng, C. Guo, and X. Sun, Wideband spectrum sensing based on coprime sampling, in nd Int. Conf. Telecommunications (ICT), 5, pp. 3 35. [5] P. Pal and P. P. Vaidyanathan, Soft-thresholding for spectrum sensing with coprime samplers, in IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM),, pp. 57 5. [] P. P. Vaidyanathan and P. Pal, System identification with sparse coprime sensing, IEEE Signal Processing Letters, vol. 7, no., pp. 3, Oct.. [7] Y. Gu, C. Zhou, N. A. Goodman, W. Z. Song, and Z. Shi, Coprime array adaptive beamforming based on compressive sensing virtual array signal, in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Mar., pp. 9 95. [] Y. D. Zhang, M. G. Amin, and B. Himed, Sparsity-based doa estimation using co-prime arrays, in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), May 3, pp. 397 397. [9] B. K. Dash, S. A. H. Chowdhury, A. H. M. M. Kamal, M. S. Anower, and A. Halder, Underwater network cardinality estimation using crosscorrelation: Effect of unequal sensor spacing, in International Workshop on Computational Intelligence (IWCI), Dec., pp.. [] M. Kronauge and H. Rohling, New chirp sequence radar waveform, IEEE Transactions on Aerospace and Electronic Systems, vol. 5, no., pp. 7 77, Oct.. [] G. N. Saddik, R. S. Singh, and E. R. Brown, Ultra-wideband multifunctional communications/radar system, IEEE Transactions on Microwave Theory and Techniques, vol. 55, no. 7, pp. 3 37, Jul. 7. [] P. Kanagaratnam, T. Markus, V. Lytle, B. Heavey, P. Jansen, G. Prescott, and S. P. Gogineni, Ultrawideband radar measurements of thickness of snow over sea ice, IEEE Transactions on Geoscience and Remote Sensing, vol. 5, no. 9, pp. 75 7, Sep. 7. [3] W. Q. Wang, Large time-bandwidth product mimo radar waveform design based on chirp rate diversity, IEEE Sensors Journal, vol. 5, no., pp. 7 3, Feb. 5. [] S. Benson, C. i. H. Chen, D. M. Lin, and L. L. Liou, Digital linear chirp receiver for high chirp rates with high resolution time-of-arrival and time-of-departure estimation, IEEE Transactions on Aerospace and Electronic Systems, vol. 5, no. 3, pp. 5, Jun.. [5] X. Yu, Y. Shi, and Y. Zhang, A cross-spectral svd method for chirp time delay estimation, in IEEE International Symposium on Communications and Information Technology (ISCIT), vol., Oct. 5, pp. 35 3. [] Y. Bar-Shalom, F. Palimieri, A. Kumar, and H. M. Shertukde, Analysis of wide-band cross correlation for time-delay estimation, IEEE Transactions on Signal Processing, vol., no., pp. 35, Jan. 993. [7] W. Zhai and Y. Zhang, Motion parameters estimation of high-speed moving target for stepped frequency chirp signal, in International Radar Conference, Oct., pp. 5. [] U. V. Dias and S. Srirangarajan, Co-prime arrays and difference set analysis, in 5th European Signal Processing Conference (EUSIPCO), 7, pp. 9 95.