- Open Access
Spectrum efficiency of nested sparse sampling and coprime sampling
© Chen et al.; licensee Springer. 2013
Received: 27 January 2013
Accepted: 5 February 2013
Published: 22 February 2013
This article addresses the spectrum efficiency study of nested sparse sampling and coprime sampling in the estimation of power spectral density for QPSK signal. The authors proposed nested sampling and coprime sampling only showed that these new sub-Nyquist sampling algorithm could achieve enhanced degrees of freedom, but did not consider its spectrum efficiency performance. Spectral efficiency describes the ability of a communication system to accommodate data within a limited bandwidth. In this article, we give the procedures of using nested and coprime sampling structure to estimate the QPSK signal’s autocorrelation and power spectral density (PSD) using a set of sparse samples. We also provide detailed theoretical analysis of the PSD of these two sampling algorithms with the increase of sampling intervals. Our results prove that the mainlobe of PSD becomes narrower as the sampling intervals increase for both nested and coprime sampling. Our simulation results also show that by making the sampling intervals, i.e., N1 and N2 for nested sampling, and P and Q for coprime sampling, large enough, the main lobe of PSD obtained from these two sub-Nyquist samplings are much narrower than the original QPSK signal. That is, the bandwidth B occupancy of the sampled signal is smaller, which improves the spectrum efficiency. Besides the smaller average rate, the enhanced spectrum efficiency is a new advantage of both nested sparse sampling and coprime sampling.
From (1), if we hope to improve the spectrum efficiency, we should either increase the data rate R or efficiently use the bandwidth B. Lots of efforts have been made to increase the spectrum efficiency. For example, power and spectral efficient family of modulations for wireless communication systems were introduced in . The author in  proposed a high spectrum efficient multiple access code. Cognitive radios have been proposed as a method to efficiently reuse the licensed limited spectrum. And in general, the spectral efficiency can be improved  by frequency re-use, spatial multiplexing, OFDMA, or some radio resource management techniques such as efficient fixed or dynamic channel allocation, power control, link adaptation etc. As stated in , time is also a factor in determining overall spectrum efficiency, because most applications do not use spectrum on a continuous basis and users typically share resources on a time basis.
A new approach to super resolution spectral estimation using nested sparse sampling is provided by [7, 8]. In , a two user case of sparse sampling, coprime sampling is also introduced. The authors has already proved that these two new sub-Nyquist sampling algorithms could achieve enhanced degrees of freedom. While in this article, we will show that both nested sparse sampling and coprime sampling are much more spectrum efficiency, i.e., they occupy much narrower bandwidth than the original non-sampled signal.
Traditional sampling methods are based on Nyquist rate sampling, which will have poor efficiency in terms of both sampling rate and computational complexity. Nowadays, more and more techniques are proposed to overcome the Nyquist sampling. Compressive sensing  provides us a new point of view, which could only use much less samples to perfectly recover the original signal at a high compression ratio. The authors give a new idea of coprime sampling in , which uses two uniform sampling to estimate the autocorrelation for all lags.
Differently, nested sparse sampling is an non-uniform sampling, using two different samplers in each period. Although the signal is sampled sparsely and nonuniformly at 1 ≤ l ≤ N1T and (N1 + 1)m T, 1 ≤ m ≤ N2 for one period, the autocorrelation R c (τ) of the signal x c (t) could be estimated at all lags τ = k T, k, l, and m are all integers. Hence, nested sparse sampling can be used to estimate power spectrum even though the samples in the time domain can be arbitrarily sparse . While coprime sampling uses two uniform samplers, with sample spacings PT and QT, respectively, where P and Q are coprime integers. Similar as nested sparse sampling case, the authors in  proved that the estimates of all lags of autocorrelation R c (k T) could be obtained from these two sets of samples of the signal x c (t), both of the samples are taken at much smaller rates than Nyquist sampling rate, which results in a much less time consumption.
In this article, we give the principle of nested sparse sampling and coprime sampling first and provide the procedures of using these two sparse sampling structures to estimate the QPSK signal’s autocorrelation and power spectral density (PSD). We give the theoretical analysis of how these two sparse sampling methods effect the power spectral density as well. Our simulation results also show that with if we choose the sampling spacings larger, the main lobe of PSD obtained from these two sampling will be much narrower than the original QPSK signal. That is, besides the much less time consumption, the occupied bandwidth B in expression (1) is smaller, which makes the spectrum efficiency higher. Besides the smaller average rate, the increased spectrum efficiency is a new advantage of these two sparse sampling algorithms, which is studied for the first time.
The rest of this article is organized as follows. In Section 2, we give a brief overview of nested sparse sampling. An introduction of coprime sampling is in Section 3. Spectrum estimation based on the difference sets obtained from both nested sampling and coprime sampling structures is detailed in Section 4. In Section 5, we give the theoretical analysis of these two sparse sampling and how they will effect the power spectral density. In Section 6, we provide the numerical results of the power spectral density estimation. Conclusions are presented in Section 7.
Nested sparse sampling
The nested array was introduced in  as an effective approach to array processing with enhanced degrees of freedom . The time domain autocorrelation could also be obtained from sparse sampling with nested sampling structure . And the samples of the autocorrelation can be computed at any specified rate, although the samples from this nested sparse sampling are sparsely and nonuniformly located.
with 4,8,12,16 missing.
Using the above principle, we could get a sparse sampling using nested sampling structure as shown in Figure 1. We have two levels of nesting, with N1 level-1 samples and N2 level-2 samples in each period, with period (N1 + 1)N2. This shows that nested sampling is non-uniform and the samples obtained are very sparse.
Here, T = 1/f n , f n ≥ 2fmax is the Nyquist sampling frequency, which is greater than twice of the maximum frequency. As the Nyquist sampling rate is 1/T, the average sampling rate of nested sampling is smaller than the conventional Nyquist sampling rate.
If we set N1 and N2 larger, the average sampling rate f s would be arbitrarily smaller. In the theoretical and numerical results sections, we will show that with N1 and N2 becoming larger, the bandwidth of the power spectrum density goes narrower, i.e., the spectrum gets more efficiently used.
The authors in  have shown that k can achieve any integer value in the range 0 ≤ k ≤ P Q - 1, if n1 and n2 in the ranges 0 ≤ n1 ≤ 2Q - 1 and 0 ≤ n2 ≤ P - 1.
Same as in nested sampling, T = 1/f n , f n ≥ 2fmax is the Nyquist sampling frequency. We could notice the average sampling rate of coprime sampling is much smaller than the conventional Nyquist sampling rate of 1/T.
Similar as stated in nested sampling, if we set P and Q larger, the average sampling rate would be arbitrarily smaller. We will show that with P and Q becoming larger, the bandwidth of the power spectrum density goes narrower.
Power spectral density estimation based on nested & coprime sampling
In this part, we will detail the estimation of PSD using nested and coprime sampling structure. In signal and systems analysis, the autocorrelation plays a very important role. The autocorrelation function of a random signal describes the general dependence of the values of the samples at one time on the values of the samples at another time.
R c (τ) is always real-valued and an even function with a maximum value at τ = 0.
R(k) can be computed from samples of x c (t) taken at an arbitrarily lower rate using nested or coprime sparse sampling.
- (1)Maximum value: The magnitude of the autocorrelation function of a wide sense stationary random process at lag m is not greater than its value at lag m = 0, i.e.,(15)
The autocorrelation function of a periodic signal is also periodic.
- (3)The autocorrelation function of WSS process is a conjugate symmetric function of k:(16)
S(f) is a real-valued, nonnegative function. Definition (17) shows that S( - f) = S(f), i.e., the PSD is an even function of frequency f.
Next, we will separately describe how nested sampling and coprime sampling estimate the autocorrelation function.
For nested sampling
The set of differences n1 - n2 are exactly the difference-co-array described in (4), that is, n1 - n2 will achieve all integer values in (4).
We can see that although the signal is sampled sparsely and nonuniformly at 1 ≤ l ≤ N1 and (N1 + 1)m, 1 ≤ m ≤ N2 for one period, the autocorrelation R c (τ) of the signal x c (t) could be estimated at all lags τ = k.
For coprime sampling
As nested sampling and coprime sampling are similar, in this part, I will use nested sparse sampling to state the theoretical analysis.
for n=0,1,…,N-1, where denotes the complex conjugate. This also means that the component S(0) is always real for real data.
This gives the reason of why the PSD figure is always symmetric.
From this derivation, we also notice that with the increase of N, besides the mainlobe becomes narrower, the central value of the PSD gets higher, which results in a higher spectrum efficiency.
In the numerical results part, we will show that with the same sampling spacings chosen for both nested and coprime sampling, i.e., N1 = P, N2 = Q, we could achieve N = (N1 + 1)N2 for nested sampling will be larger than that of N = P Q for coprime sampling, which will result in a better spectrum efficiency for nested sampling.
where T s is the symbol duration. In our simulation, we set E s = 1 and T s = 1/50.
Another interesting observation is that the bandwidth of the PSD estimated using coprime sampling is a little larger than that estimated by nested sampling, as shown in our example, the bandwidth for coprime estimated PSD is 411-389≈22 Hz, while it is 409-391≈18 Hz for nested sampling. This is because for the same number of P and Q (or N1 and N2), the nested sampling could achieve N = (N1 + 1)N2, while coprime sampling could only get N = P Q. If N1 = P and N2 = Q, it is obvious that the nested sampling estimate a larger number of N than coprime sampling. Refer to the theoretical analysis, we could conclude that larger N results in narrower bandwidth, which indicates that if N1 = P and N2 = Q for nested and coprime sampling, nested sampling would have a more efficient spectrum performance.
From the results got from Figures 17 and 18, we conclude that in the nested sparse sampling process, besides its advantage of less samplers, with N1 and N2 chosen larger, the bandwidth of the PSD occupied will becomes narrower, which increases the spectrum efficiency.
From Figures 17, 18, 19, 20, we could observe nested sparse sampling and coprime sampling could obtain similar estimated PSD and both are spectrum efficient as the sampling intervals increase, i.e., N1 and N2 for nested sampling, and P and Q for coprime sampling.
In this article, the estimated power spectrum density is analyzed and simulated using both nested sampling and coprime sampling structures, which provide us a new way to efficiently use the spectrum.
We give the principle of nested arrays and coprime samplers, and the procedure of how to estimate the autocorrelation and PSD with the sparse samples using nested and coprime sampling for QPSK signal. We give detailed theoretical analysis of how these two sparse sampling effects the PSD and why it is spectrum efficiency with the increase of sampling intervals, i.e., N1 and N2 for nested sampling, P and Q for coprime sampling.
Our simulation results show that with the proper choice of sampling intervals, i.e., making them large enough, the main lobe of PSD obtained from both nested sampling and coprime sampling is much narrower than the original QPSK signal. If we choose the sampling intervals larger, the bandwidth occupied will be narrower, which improves the spectrum efficiency. Besides the smaller average rate, the increased spectrum efficiency is a new advantage of both nested sparse sampling and coprime sampling.
This study was supported in part by U.S. Office of Naval Research under Grants N00014-13-1-0043, N00014-11-1-0071, N00014-11-1-0865, and U.S. National Science Foundation under Grants CNS-1247848, CNS-1116749, and CNS-0964713.
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