# Spectrum efficiency of nested sparse sampling and coprime sampling

- Junjie Chen
^{1}Email author, - Qilian Liang
^{1}, - Baoju Zhang
^{2}and - Xiaorong Wu
^{2}

**2013**:47

https://doi.org/10.1186/1687-1499-2013-47

© Chen et al.; licensee Springer. 2013

**Received: **27 January 2013

**Accepted: **5 February 2013

**Published: **22 February 2013

## Abstract

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., *N*_{1} and *N*_{2} 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

## Introduction

*R*to be the data rate in bits per second, and

*B*the bandwidth occupied, the bandwidth efficiency

*η*is expressed as

*R*≤

*C*[2], we can get the fundamental upper bound [1] on achievable spectrum efficiency, for an arbitrarily small probability of error, where $\frac{S}{N}$ 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 [3]. The author in [4] 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 [5] 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 [6], 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 [8], 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 [9] 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 [8], 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* ≤ *N*_{1}*T* and (*N*_{1} + 1)*m* *T*, 1 ≤ *m* ≤ *N*_{2} 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 [8]. 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 [8] 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 [7] as an effective approach to array processing with enhanced degrees of freedom [10]. The time domain autocorrelation could also be obtained from sparse sampling with nested sampling structure [11]. 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.

*N*

_{1}locations and the level 2 samples at the

*N*

_{2}locations.

*N*

_{1}= 3 and

*N*

_{2}= 5. The cross-differences are given by

*N*

_{1}+ 1)

*N*

_{2}- 1, except the integers and the corresponding negated versions shown in (5).

*m*≤ 5 and 1 ≤

*l*≤ 3, the cross differences

*k*= (

*N*

_{1}+ 1)

*m*-

*l*will achieve these values

with 4,8,12,16 missing.

*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

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 *N*_{1} level-1 samples and *N*_{2} level-2 samples in each period, with period (*N*_{1} + 1)*N*_{2}. This shows that nested sampling is non-uniform and the samples obtained are very sparse.

*N*

_{1}+ 1)

*N*

_{2}

*T*seconds, there are totally

*N*

_{1}+

*N*

_{2}samples. The average sampling rate is

Here, *T* = 1/*f*_{
n
}, *f*_{
n
} ≥ 2*f*_{max} 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 *N*_{1} and *N*_{2} larger, the average sampling rate *f*_{
s
} would be arbitrarily smaller. In the theoretical and numerical results sections, we will show that with *N*_{1} and *N*_{2} becoming larger, the bandwidth of the power spectrum density goes narrower, i.e., the spectrum gets more efficiently used.

## Coprime sampling

*x*

_{ c }(

*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*= 2

*f*

_{max},

*f*

_{max}being the highest frequency.

*x*(

*P*

*n*

_{1}) and

*x*(

*Q*

*n*

_{2}) comes from the first and the second sampler. Set the difference as

The authors in [11] have shown that *k* can achieve any integer value in the range 0 ≤ *k* ≤ *P* *Q* - 1, if *n*_{1} and *n*_{2} in the ranges 0 ≤ *n*_{1} ≤ 2*Q* - 1 and 0 ≤ *n*_{2} ≤ *P* - 1.

*P*+

*Q*samples in

*PQT*seconds, the average sampling rate is

Same as in nested sampling, *T* = 1/*f*_{
n
}, *f*_{
n
} ≥ 2*f*_{max} 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.

*x*

_{ c }(

*t*) is defined by this averaging

*R*_{
c
}(*τ*) is always real-valued and an even function with a maximum value at *τ* = 0.

*x*(

*n*)=

*x*

_{ c }(

*n*

*T*), for some fixed spacing

*T*. For the autocorrelation samples,

*R*(

*k*) =

*R*

_{ c }(

*k*

*T*), where

*R*

_{ c }(·) as shown in (13). Therefore,

*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.,$R\left(0\right)\ge \mid R\left(k\right)\mid ,k\ne 0$(15) - (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*:$R\left(k\right)={R}^{\ast}(-k)$(16)

*R*

_{ c }(

*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*.

Next, we will separately describe how nested sampling and coprime sampling estimate the autocorrelation function.

### For nested sampling

*x*(

*n*

_{1})

*x*

^{∗}(

*n*

_{2}), with

*n*

_{1}and

*n*

_{2}belong to the first period in Figure 1. We will get the samples at the following locations

The set of differences *n*_{1} - *n*_{2} are exactly the difference-co-array described in (4), that is, *n*_{1} - *n*_{2} will achieve all integer values in (4).

We can see that although the signal is sampled sparsely and nonuniformly at 1 ≤ *l* ≤ *N*_{1} and (*N*_{1} + 1)*m*, 1 ≤ *m* ≤ *N*_{2} for one period, the autocorrelation *R*_{
c
}(*τ*) of the signal *x*_{
c
}(*t*) could be estimated at all lags *τ* = *k*.

*k*could be obtained [11] by averaging the products

*x*(

*n*

_{1})

*x*

^{∗}(

*n*

_{2}) over

*L*periods,

### For coprime sampling

*P*and

*Q*are coprime, there exist integers 0 ≤

*n*

_{1}≤ 2

*Q*- 1 and 0 ≤

*n*

_{2}≤

*P*- 1, such that the difference in Equation (11) can achieve any integer value

*k*=

*P*

*n*

_{1}-

*Q*

*n*

_{2}in the range of 0 ≤

*k*≤

*P*

*Q*- 1. Since

*k*=

*P*(

*n*

_{1}+

*Q*

*l*) -

*Q*(

*n*

_{2}+

*P*

*l*) for any

*l*, we can average

*l*to obtain an estimate of the autocorrelation

*R*(

*k*), that is,

## Theoretical analysis

As nested sampling and coprime sampling are similar, in this part, I will use nested sparse sampling to state the theoretical analysis.

*S*(

*N*-

*n*) and

*S*(

*n*) are related by

for *n*=0,1,…,*N*-1, where $\stackrel{\u0304}{S}\left(n\right)$ 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.

*N*

_{1}and

*N*

_{2}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 $\widehat{R}\left(k\right)$ have the same absolute value

*R*, i.e., $R=\left|\widehat{R}\right(k\left)\right|=\widehat{R}\left(0\right)=-\widehat{R}\left(1\right)=\widehat{R}\left(2\right)=\cdots \phantom{\rule{0.3em}{0ex}}$. Therefore, we set $\widehat{R}\left(k\right)={(-1)}^{k}R$. The estimated autocorrelation satisfies those properties we stated before, i.e., $\widehat{R}\left(0\right)\ge \left|\widehat{R}\right(k\left)\right|,k\ne 0$, and as the QPSK signal we used is periodic, the estimated autocorrelation function $\widehat{R}\left(k\right)$ is also periodic.

*N*

_{1}and

*N*

_{2}. 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, …, (

*N*

_{1}+ 1)

*N*

_{2}- 1, that is,

*N*= (

*N*

_{1}+ 1)

*N*

_{2}. We could get the PSD by taking the Fourier transform of the estimated autocorrelation,

*k*falls in the range of (4),

*N*= (

*N*

_{1}+ 1)

*N*

_{2}. For coprime sampling, we show that

*k*=

*P*(

*n*

_{1}+

*Q*

*l*) -

*Q*(

*n*

_{2}+

*P*

*l*) can achieve any integer value in the range of 0 ≤

*k*≤

*P*

*Q*- 1, i.e., in this case

*N*=

*P*

*Q*.

*N*could either be even or odd. For example, for nested sampling, if

*N*

_{1}= 2 and

*N*

_{2}= 5, then

*N*= (

*N*

_{1}+ 1)

*N*

_{2}= 15 is odd, while if

*N*

_{1}= 3 and

*N*

_{2}= 5, then

*N*= (

*N*

_{1}+ 1)

*N*

_{2}= 20 is even. For coprime sampling, if

*P*= 2,

*Q*= 5, then

*N*=

*P*

*Q*= 10 is even, while if

*P*= 3,

*Q*= 5, then

*N*=

*P*

*Q*= 13 is odd. First, we assume

*N*= (

*N*

_{1}+ 1)

*N*

_{2}is odd,

*N*= (

*N*

_{1}+ 1)

*N*

_{2}is even,

*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.

*N*-point fast Fourier transform (FFT), we will get

*N*PSD values. Let

*f*

_{ n }represents the Nyquist sampling frequency,

*f*

_{ n }= 2

*f*

_{ c }(

*f*

_{ c }is the carrier frequency), using

*f*=

*f*

_{ n }· (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 $S\left(\frac{N}{2}\right)$ at $f=\frac{1}{2}{f}_{n}={f}_{c}$. 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., *N*_{1} = *P*, *N*_{2} = *Q*, we could achieve *N* = (*N*_{1} + 1)*N*_{2} 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.

## Numerical results

*f*

_{ c }= 400 Hz, which could be expressed as [1]

where *T*_{
s
} is the symbol duration. In our simulation, we set *E*_{
s
} = 1 and *T*_{
s
} = 1/50.

*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

*f*

_{ c }=400 Hz with symmetric sidelobes on both sides.

*N*

_{1}= 7,

*N*

_{2}=11 , and

*L*= 10. Therefore, $\widehat{R}\left(k\right)$ can be estimated for ∣

*k*∣≤ (

*N*

_{1}+ 1)

*N*

_{2}- 1. For each period, we totally get (

*N*

_{1}+ 1)

*N*

_{2}= (7 + 1) × 11 = 88 samples. While for coprime sampling, we set

*P*= 7 and

*Q*= 11, for each period, we get

*P*

*Q*= 7 × 11 = 77 lags of $\widehat{R}\left(k\right)$.

*f*

_{ c }=400 Hz with symmetric sidelobes on both sides. As stated in Section 3, we can see the PSD is an even function.

*f*

_{ c }, 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.

*f*

_{ c }= 400 Hz with symmetric sidelobes on both sides.

*f*

_{ c }, 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.

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 *N*_{1} and *N*_{2}), the nested sampling could achieve *N* = (*N*_{1} + 1)*N*_{2}, while coprime sampling could only get *N* = *P* *Q*. If *N*_{1} = *P* and *N*_{2} = *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 *N*_{1} = *P* and *N*_{2} = *Q* for nested and coprime sampling, nested sampling would have a more efficient spectrum performance.

*N*

_{1}and

*N*

_{2}pairs, as shown in Figure 17, it is obvious that for

*N*

_{1}fixed to

*N*

_{1}= 3, with the increase of the value of

*N*

_{2}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.

*N*

_{1}from

*N*

_{1}= 3, 5 to

*N*

_{1}= 11, while

*N*

_{2}fixed to

*N*

_{2}= 13, the main lobe also gets narrower, which also results in the increase of spectrum efficiency.

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 *N*_{1} and *N*_{2} chosen larger, the bandwidth of the PSD occupied will becomes narrower, which increases the spectrum efficiency.

*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.

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., *N*_{1} and *N*_{2} for nested sampling, and *P* and *Q* for coprime sampling.

## Conclusions

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., *N*_{1} and *N*_{2} 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.

## Declarations

### Acknowledgements

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.

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.