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.
Keywords:Spectrum efficiency; Nested sampling; Coprime sampling; Autocorrelation; Power spectral density
In recent years, spectrum efficiency has gained renewed interest in wireless communication system. From , we know that the performance of a particular communication system is often measured in terms of spectral efficiency (or bandwidth efficiency). Spectral efficiency describes the ability of a communication system to accommodate data within a limited bandwidth. It reflects how efficiently the allocated bandwidth is utilized and defined as the ratio of the throughput data rate per Hertz in a given bandwidth. Letting R to be the data rate in bits per second, and B the bandwidth occupied, the bandwidth efficiency η is expressed as
If we apply Shannon’s capacity to AWGN non-fading channel, i.e., , and with the knowledge that all communication rates are below channel capacity R ≤ C, we can get the fundamental upper bound  on achievable spectrum efficiency, for an arbitrarily small probability of error, where is the signal to noise ratio.
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)mT, 1 ≤ m ≤ N2 for one period, the autocorrelation Rc(τ) of the signal xc(t) could be estimated at all lags τ = kT, 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 Rc(kT) could be obtained from these two sets of samples of the signal xc(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.
In the simplest form, the nested array  has two levels of sampling density, with the level 1 samples at the N1 locations and the level 2 samples at the N2 locations.
Figure 1 shows an example of periodic sparse sampling using nested sampling structure with N1 = 3 and N2 = 5. The cross-differences are given by
Figure 1. Nested sampling with N1 = 3, N2 = 5.
The cross-differences  are in the following range with the maximum value (N1 + 1)N2 - 1, except the integers and the corresponding negated versions shown in (5).
For example, consider the example in Figure 1, where 1 ≤ m ≤ 5 and 1 ≤ l ≤ 3, the cross differences k = (N1 + 1)m - l will achieve these values
with 4,8,12,16 missing.
Besides these integers, the difference 0 is also missing, for the reason that m and l are nonzero. While, we notice that the self differences among the second array could cover all of the missing differences, as shown
The difference-co-array could be obtained from the cross-differences and the self-differences, which is a filled difference co-array as shown in (4). This means that using nested array structure, with sparse samples, we could obtain the degrees of freedom as
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.
Therefore, in (N1 + 1)N2T seconds, there are totally N1 + N2 samples. The average sampling rate is
Here, T = 1/fn, fn ≥ 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 fs 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.
Different with nested sampling, coprime sampling involves two sets of uniformly spaced samplers as shown in Figure 2.
Figure 2. Coprime sampling in the time domain.
The coprime sampling uniformly sample the wide-sense stationary (WSS) process xc(t) using two sub-Nyquist samplers, with sample spacing PT and QT, respectively, where P and Q are coprime integers with P < Q. 1/T Hz is the Nyquist rate for a bandlimited process, i.e., 1/T = 2fmax, fmax being the highest frequency.
Consider the product
where x(Pn1) and x(Qn2) comes from the first and the second sampler. Set the difference as
The authors in  have shown that k can achieve any integer value in the range 0 ≤ k ≤ PQ - 1, if n1 and n2 in the ranges 0 ≤ n1 ≤ 2Q - 1 and 0 ≤ n2 ≤ P - 1.
For coprime sampling, the two samplers collect P + Q samples in PQT seconds, the average sampling rate is
Same as in nested sampling, T = 1/fn, fn ≥ 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.
The autocorrelation  of a real and stationary signal xc(t) is defined by this averaging
Rc(τ) is always real-valued and an even function with a maximum value at τ = 0.
For sampled signal, define x(n)=xc(nT), for some fixed spacing T. For the autocorrelation samples, R(k) = Rc(kT), where Rc(·) as shown in (13). Therefore,
R(k) can be computed from samples of xc(t) taken at an arbitrarily lower rate using nested or coprime sparse sampling.
And here we only list some important autocorrelation properties which will be used in this article:
(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.,
(2) The autocorrelation function of a periodic signal is also periodic.
(3) The autocorrelation function of WSS process is a conjugate symmetric function of k:
The power spectral density (PSD) describes how the power of a signal or time series is distributed with frequency. The PSD is the Fourier transform of the autocorrelation function of the signal if the signal is treated as a wide-sense stationary random process . Therefore, the Fourier transform of Rc(τ) is the PSD S(f),
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.
Taking discrete Fourier transform (DFT) of these lags of autocorrelation values, we could obtain the power spectral density as
Next, we will separately describe how nested sampling and coprime sampling estimate the autocorrelation function.
For nested sampling
For the samples obtained from nested sparse sampling, consider the product x(n1)x∗(n2), with n1 and n2 belong to the first period in Figure 1. We will get the samples at the following locations
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 Rc(τ) of the signal xc(t) could be estimated at all lags τ = k.
An estimate of the autocorrelation samples for all k could be obtained  by averaging the products x(n1)x∗(n2) over L periods,
For coprime sampling
As P and Q are coprime, there exist integers 0 ≤ n1 ≤ 2Q - 1 and 0 ≤ n2 ≤ P - 1, such that the difference in Equation (11) can achieve any integer value k = Pn1 - Qn2 in the range of 0 ≤ k ≤ PQ - 1. Since k = P(n1 + Ql) - Q(n2 + Pl) for any l, we can average l to obtain an estimate of the autocorrelation R(k), that is,
As nested sampling and coprime sampling are similar, in this part, I will use nested sparse sampling to state the theoretical analysis.
From the property of DFT, we know that, if are real, then S(N - n) and S(n) are related by
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.
In our simulation, we always consider the amplitude of the PSD, which indicates that
This gives the reason of why the PSD figure is always symmetric.
From the simulation, we observe the absolute values of the autocorrelation are the same for the QPSK signal, which obtain positive or negative of a fixed value as shown in Figure 3 for different N1 and N2 of nested sampling, and Figure 4 for different P and Q of coprime sampling. This could make the calculation of the PSD easier. In our analysis, for simplicity, we assume all the have the same absolute value R, i.e., . Therefore, we set . The estimated autocorrelation satisfies those properties we stated before, i.e., , and as the QPSK signal we used is periodic, the estimated autocorrelation function is also periodic.
Figure 3. Nested sampling estimated autocorrelation.
Figure 4. Coprime sampling estimated autocorrelation.
Let’s see how the PSD changes with the increase of N1 and N2. As stated in the principle of nested sparse sampling, k falls in the range of (4). Here we only use those positive, i.e., k = 0, 1, …, (N1 + 1)N2 - 1, that is, N = (N1 + 1)N2. We could get the PSD by taking the Fourier transform of the estimated autocorrelation,
As stated in the principle of nested sparse sampling, k falls in the range of (4), N = (N1 + 1)N2. For coprime sampling, we show that k = P(n1 + Ql) - Q(n2 + Pl) can achieve any integer value in the range of 0 ≤ k ≤ PQ - 1, i.e., in this case N = PQ. N could either be even or odd. For example, for nested sampling, if N1 = 2 and N2 = 5, then N = (N1 + 1)N2 = 15 is odd, while if N1 = 3 and N2 = 5, then N = (N1 + 1)N2 = 20 is even. For coprime sampling, if P = 2, Q = 5, then N = PQ = 10 is even, while if P = 3, Q = 5, then N = PQ = 13 is odd. First, we assume N = (N1 + 1)N2 is odd,
And if N = (N1 + 1)N2 is even,
The corresponding amplitude of the PSD is
We could draw these two expressions in (27) as shown in Figures 5, 6, 7, and 8. It’s obvious that no matter N is odd or even, with the increase of N, the mainlobe becomes narrower and the number of sidelobes increases. In next paragraph, we will prove the central of the mainlobe represents the central frequency.
In the simulation, if we take N-point fast Fourier transform (FFT), we will get N PSD values. Let fn represents the Nyquist sampling frequency, fn = 2fc (fc is the carrier frequency), using f = fn · (0 : N - 1)/N, we could map these PSD values to the frequency. It is obvious that when n=N/2, the PSD gets its central value of at . This could be represent as
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 = PQ for coprime sampling, which will result in a better spectrum efficiency for nested sampling.
This section presents some numerical results for the autocorrelation and power spectrum density estimation using nested sampling structure. We use QPSK modulated signal with carrier frequency fc = 400 Hz, which could be expressed as 
where Ts is the symbol duration. In our simulation, we set Es = 1 and Ts = 1/50.
The power spectrum density  of a QPSK signal using rectangular pulses can be expressed as
Figure 9 shows the PSD of a QPSK signal for rectangular and raised cosine filtered pulses. The x-axis refers to the frequency in Hz, and the y-axis are the normalized power spectral density in dB. It can be observed the PSD centers at fc=400 Hz with symmetric sidelobes on both sides.
Figure 9. PSD of the QPSK signal.
Figure 10. Zoom in the main lobe of PSD for QPSK signal.
The estimated autocorrelation using nested sampling and coprime sampling structures are plotted in Figures 11 and 12. In the simulation, for nested sampling, we use N1 = 7, N2 =11 , and L = 10. Therefore, can be estimated for ∣ k ∣≤ (N1 + 1)N2 - 1. For each period, we totally get (N1 + 1)N2 = (7 + 1) × 11 = 88 samples. While for coprime sampling, we set P = 7 and Q = 11, for each period, we get PQ = 7 × 11 = 77 lags of .
Figure 11. Nested sampling estimated autocorrelation of the QPSK signal, when N1 = 7, N2 = 11.
Figure 12. Coprime sampling estimated autocorrelation of the QPSK signal, when P = 7, Q = 11 .
Using the relationship of autocorrelation and the PSD described in Section 3, we could obtain the estimated PSD using nested sampling structure for this QPSK signal as shown in Figure 13. In the simulation, we use 1024 point fast Fourier transform and normalize the PSD. We can see that the estimated PSD is also centered at fc=400 Hz with symmetric sidelobes on both sides. As stated in Section 3, we can see the PSD is an even function.
Figure 13. PSD of nested sampling QPSK signal (N1 = 7, N2 = 11).
Similarly, if we zoom in this PSD around the central frequency fc, in Figure 14, we could find the main lobe, i.e., the bandwidth occupied is approximately 409-391≈18 Hz, which is much narrower than that 32 Hz of the PSD of the original QPSK signal. Hence, the spectrum efficiency is improved in the estimation using nested sampling structure.
Figure 14. Zoom in the main lobe of PSD for nested sampling QPSK signal (N1 = 7, N2 = 11).
We could also get the estimated PSD using coprime sampling structure for this QPSK signal as shown in Figure 15. In the simulation, we use 1024 point fast Fourier transform and normalize the PSD. We can see that the estimated PSD is also an even function centered at fc = 400 Hz with symmetric sidelobes on both sides.
Figure 15. PSD of coprime sampling QPSK signal (P = 7, Q = 11) .
If we zoom in this PSD around the central frequency fc, in Figure 16, we could find the main lobe, i.e., the bandwidth occupied is approximately 411-389≈22 Hz, which is near to that estimated using nested sampling and is much narrower than that 32 Hz of the PSD of the original QPSK signal. Hence, the spectrum efficiency is improved in the estimation using coprime sampling structure as well.
Figure 16. Zoom in the main lobe of PSD for coprime sampling QPSK signal (P = 7, Q = 11) .
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 = PQ. 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.
By changing different N1 and N2 pairs, as shown in Figure 17, it is obvious that for N1 fixed to N1 = 3, with the increase of the value of N2 from 5,7 to 13, the main lobe of the estimated PSD using nested sampling structure becomes narrower significantly, i.e., the bandwidth occupied gets smaller. Here, in the simulation, we normalize the PSD values.
Figure 17. PSD of nested sampling QPSK signal with different N2.
Similarly, Figure 18 shows that with the increase of N1 from N1 = 3, 5 to N1 = 11, while N2 fixed to N2 = 13, the main lobe also gets narrower, which also results in the increase of spectrum efficiency.
Figure 18. PSD of nested sampling QPSK signal with different N1.
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.
Similar as nested sampling, as P and Q increase for coprime sampling, the mainlobe of the estimated PSD narrows down as well, which also indicates smaller bandwidth and higher spectrum efficiency as shown in Figure 19, where we increase the second sampler’s sampling interval of Q from 5, 7, to 13, and in Figure 20, where we increase the first sampler’s sampling interval of P from 3, 5, to 11.
Figure 19. PSD of coprime sampling QPSK signal with different Q .
Figure 20. PSD of coprime sampling QPSK signal with different P .
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.
The authors declare that they have no competing interests.
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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