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# Sparse correlation matching-based spectrum sensing for open spectrum communications

- Eva Lagunas
^{1}Email author and - Montse Nájar
^{1, 2}

**2012**:31

https://doi.org/10.1186/1687-6180-2012-31

© Lagunas and Nájar; licensee Springer. 2012

**Received: **15 September 2011

**Accepted: **15 February 2012

**Published: **15 February 2012

## Abstract

To deal with the current spectrum scarcity problem and exploiting the fact that exclusive access through tightly regulated licensing leads to idle spectrum, cognitive radio has been proposed as a way to reuse this underutilized spectrum in an opportunistic manner, i.e., allowing the use of temporarily unused licensed spectrum to secondary users who have no spectrum licenses. To protect the licensed users from the cognitive users' interference, the opportunistic user requires knowledge of the original license holder activity. In this article, a feature-based approach for spectrum sensing based on periodic non-uniform sampling is addressed. In particular, we face the compressed-sampling version of detecting predetermined spectral shapes in sparse wideband regimes by means of a correlation-matching procedure.

## Keywords

## 1 Introduction

Current spectrum division among users in wireless communication systems is assigned by regulatory and licensing bodies like the Federal Communication Commission^{a} (FCC) in the US or the European Telecommunications Standard Institute^{b} (ETSI) in Europe. In the usual spectrum management approach, the radio spectrum is divided into fixed and non-overlapping blocks, which are assigned to different services and wireless technologies. The recent proliferation of wireless communications services together with the inflexible spectrum regulations have resulted in a crowded radio frequency (RF) spectrum. This spectrum congestion becomes a bottleneck for the increasing demand of new transmission bands, which can rarely be satisfied using permanent allocation.

The scarcity of electromagnetic spectrum is obvious, but the real problem is not a dearth of radio spectrum; it's the way that spectrum is used. The radio spectrum is actually poorly utilized in many bands in the sense that large portion of the assigned bands are not used most of the time [1]. A solution to this inefficiency is to allow opportunistic unlicensed access to the poorly utilized frequency bands that have been already allocated. This more flexible allocation approach is known as cognitive radio (CR) [2].

In CR, radios opportunistically look for holes (non-used spectrum gaps) in the licensed spectrum, which can subsequently be exploited for setting up a communication link. However, the approach described previously requires knowledge of the primary (licensed) user spectrum activity in order to avoid causing interference. Protecting the non-cognitive users is mandatory, since they have the priority of service. The task of accurately detecting the presence of licensed user is encompassed in spectrum sensing. The signal-processing fundamentals specific to spectrum sensing implementation have been investigated in [3]. Among the implementation challenges mentioned in [3], the most critical design problem is the need to process very wide bandwidth (regardless of operating frequency range) and reliably detect presence of primary users. Identifying unoccupied frequencies is a complicated problem which involves sampling many points on the radio spectrum. Moreover, with the current analog-to-digital converters (ADC) technology, wideband RF signal digitising is a quite demanding task. Consequently, each CR node can only sense a relatively narrow band.

Sampling at the Nyquist rate is shown to be inefficient when the signals of interest contain only a small number of significant frequencies relative to the bandlimit [4]. To alleviate the sampling bottleneck, a promising alternative for this type of sparse signals is the use of sub-Nyquist sampling techniques.

### 1.1 Background and prominent related work

This article blends two topics: Spectrum sensing and sub-Nyquist sampling. The goal of this section is to present a review of the most prominent published works related to spectrum sensing and sub-Nyquist sampling techniques.

#### 1.1.1 Beyond Nyquist sampling rate

Signal acquisition is a main topic in signal processing. Sampling theorems provide the bridge between the continuous and the discrete-time worlds. The most famous theorem is often attributed to Shannon [5, 6] (but usually called Nyquist rate) and says that the sampling rate must be twice the maximum frequency present in the signal in order to perfectly recover the signal. However, sampling at twice as high as the upper frequency of the signal spectra might be problematic when the band limit of the signal is large. The usual method of sampling at equally spaced instants of time permits unambiguous reconstruction of the original signal if and only if the spectra of the signal is known in advance to lie in the Nyquist band.

Shapiro and Silverman [7] were the first who noticed these problems back in 1960. In order to avoid aliasing they proposed unequally spaced instants of sampling time. In fact, they showed that random sampling schemes succeed in eliminating aliasing, while others do not. One example proposed in [7] was to take the sampling time the occurrence times of the events of some Poisson process.

Later in the 1970s, Beutler [8] generalized the formulation of the alias-free sampling problem and studied special cases depending on the spectral distribution of the signals. In this context, Masry [9] studied the random sampling in a more general framework. In the 1990s, Bilinskis [10] presented his breakthrough study in digital alias-free signal processing (DASP) which was summarized in 2005 in a book with the same name [11]. As the term suggests, DASP is focussed on the problem of aliasing prevention, as well as all the previous mentioned methods.

All these researchers realized that the restrictions defined by Shannon-Nyquist do not have to be always satisfied. Of course, the obvious way (and the simplest way) to avoid aliasing when there is no extra information available is to require two times the maximum frequency present in the signal. This approach is very conservative but ensures perfect recovery of the signal. The strong requirements of the ADCs can be reduced by exploiting prior knowledge on the signal model. Due to the low occupancy of many communication systems, whose frequency support is much smaller than the band limit, the spectrum can be considered sparse and the uniform sampling becomes very redundant.

Following this vision, a clever way of sampling the signal is the periodic non-uniform sampling. This method, called multi-coset sampler and originally proposed by Feng and Bresler [12], shares many aspects with the recent compressed sensing (CS) theory. CS [13, 14] provides a robust framework for reducing the number of measurements required to summarize sparse signals allowing to compress the data while is sampled. Although multi-coset sampling can be casted into a CS framework, its implementation becomes simpler: while usually CS considers an analog to Information converter (AIC), in the multi-coset approach only a limited number of parallel ADCs operating at low sampling rate are needed. In this context, Mishali and Eldar [15] proposed a sub-Nyquist analog-to-digital converter of wideband inputs, the first reported wideband hardware for sub-Nyquist conversion based on the multi-coset technique (as the authors claim).

#### 1.1.2 Spectrum sensing

A CR monitors the available spectrum bands, captures their information, and then detects the spectrum holes where is possible to transmit in an opportunistic manner in order to avoid possible interferences with the primary or licensed users.

The identification procedure of available spectrum is quite a difficult task due to the strict requirements imposed to guarantee no harmful interference to the licensed users. In general, the minimum signal-to-noise ratio (SNR) at which the primary signal may still be accurately detected required by the sensing procedure is very low. Thus, low SNR levels must be sensed which translates into a high detection sensitivity.

A second constraint is the required detection time [16]. The longer the time that we sense, the better the signal processing gain. However, the spectrum behaves dynamically, changing all the time, and cognitive users need to be aware of these fast changes. Another desirable feature is that the primary user detector has to provide an accurate power level for the primary user. The estimated power level can be used to obtain information about the distance at which the primary user is located providing the level of interference that unlicensed users represent.

A number of different methods are proposed for primary user detection. According to the a priori information required to detect the primary user and the resulting complexity and accuracy, general spectrum sensing techniques can be categorized in the following types: blind sensing and feature-based sensing techniques. One of the most popular blind detection strategy is energy detector (ED) [17]. However, ED is unable to discriminate between the sources of received energy. On the other hand, the most famous feature-based method is the matched filter. If the full structure of the primary signal is known (together with time and carrier synchronization), the optimal detector is the matched filter detector. Unfortunately, the complete knowledge of the primary signal is not usually available. If only some features of the primary signal are known, feature-based detectors such cyclostationary detector [18] are more suitable. In feature-based approaches, the secondary users are considered as interference. A survey of the most common spectrum sensing techniques, both non-feature and feature-based detectors, has been published in [19].

As it was mentioned before, the design of the analog front-end is critical in the case of CR. The worst problem is the high sampling rate required to process very wide bandwidth. The present literature for sparse spectrum sensing is still in its early stages of development. The traditional way for detecting holes in a wide-band spectrum is channel-by-channel scanning. In order to implement this, an RF front-end with a bank of tunable and narrow bandpass filters is needed. Some alternative methods have been proposed in the literature to facilitate the wide-band sensing process [16, 20, 21]. In [16], a compressive sensing approach is used to reconstruct the spectrum of a wide-band signal using time samples, which studies for special signals whose Fourier transform is real. In [20], the received analog signal is sampled at the information rate of the signal using an AIC. An estimate of the original signal spectrum is then made based on CS reconstruction using a wavelet edge detector. Wang et al. [21] proposed a two-step compressed spectrum sensing method which first quickly estimates the actual sparsity order of the wide spectrum of interest, and adjusts the total number of samples collected according to the estimated signal sparsity order.

### 1.2 Outline and contributions

Many research studies such as Viberg [22] or Lexa [23] use the sub-Nyquist methods to obtain information of the unknown power spectrum from the compressed samples avoiding the signal reconstruction. In particular, in [23], the estimator does not require signal reconstruction and can be directly obtained from a straightforward application of nonnegative least squares. In [22], the estimation of the signal spectrum is skipped, and the occupied channels are directly detected from the sampled data in the time domain. Others such as Giannakis [16] or Leus [20] look for an estimate of the original signal spectrum based on CS reconstruction using a wavelet edge detector. Here, a more particular problem is studied. In this article, the problem of detecting predetermined spectral shapes present in the spectrum of the wide-band signal received at the CR detector is addressed. The final goal of this proposal is to determine the spectrum occupancy of the licensed system. Taking advantage of the sparsity of the signals sent out over the spectrum, a sub-Nyquist periodic non-uniform sampling is used to reduce the amount of data needed to find the white space and still maintain a high degree of accuracy.

The procedure is developed following a correlation matching framework, changing the traditional single frequency scan to a spectral scan with a particular shape. The spectrum sensing scheme considered here was first presented in [24] without solving the sampling bottleneck. In [24], the data autocorrelation matrix was estimated from the Nyquist samples of the analog received signal due to the traditional assumption that the sampling state needs to acquire the data at the Nyquist rate, corresponding to twice the signal bandwidth. There are two drawbacks in [24]: (1) due to the timing requirements for rapid sensing, only a limited number of measurements can be acquired from the received signal; and (2) the implementation quickly becomes untenable for wideband spectrum sensing. Here, we take advantage of the sparsity of the spectrum to alleviate the sampling burden. Sensing and compressing in a single stage allows fast spectrum sensing while simplifying the implementation. In this article, the estimate of the data autocorrelation matrix is directly obtained from the compressed samples. Three procedures are derived depending on the criteria used to compare the estimated matrix with the predetermined one. We evaluate the resulting detector with particular examples, we derive simulated ROCs and the performance is evaluated with the RMSE and compared with classical filter-bank approaches as well as with the non-compressed version of the procedure.

This article is organized as follows. The following section states the signal model and problem formulation introducing the periodic sub-Nyquist sampling notation. Then, the following section introduces the spectrum sensing method paying special attention to the data autocorrelation matrix estimation. Finally, the last section shows the simulation results and the performance evaluation. The concluding remarks of this article are given in the very last section.

## 2 Signal model, definitions and problem statement

We consider a wideband signal *x*(*t*) which may represent the superposition of different primary services in a CR network. This signal is assumed to be multi-band signal, i.e, a bandlimited, continuous-time, squared integrable signal that has all of its energy concentrated in one or more disjoint frequency bands.

*x*(

*t*) as

*X*(

*f*), the spectral support

*F*⊂ [0,

*f*

_{max}] of the multiband signal

*x*(

*t*) is the union of the frequency intervals that contain the signal's energy:

*F*) is the Lebesgue measure of the frequency set

*F*which, in this particular case, is equal to ${\sum}_{i=1}^{N}\left({b}_{i}-{a}_{i}\right)$. For the set of sparse multiband signals Ω ranges from 0 to 0.5 (see Figure 1). In the spectrum sensing framework, the spectral support

*F*is unknown but the total bandwidth under study is assumed to be sparse.

## 3 Sparse-based sample acquisition

*T*. The inverse of this period (1

*/T*) will determine the base frequency of the system, being 1

*/T*at least equal to the Nyquist rate so that sampling at 1

*/T*ensures no aliasing. Given the received multiband signal

*y*(

*t*), the periodic nonuniform samples are obtained at the time instants,

where *L* > 0 is a suitable integer, *i =* 1, 2,... *,p* and *n* ∈ ℤ. The set {*c*_{
i
}} contains *p* distinct integers chosen from {0,1,..., *L* - 1}. The reader can notice that the multi-coset sampling process can be viewed as a classical Nyquist sampling followed by a block that discards all but *p* samples in every block of *L* samples periodically. The samples which are not thrown away are specified by the set {*c*_{
i
}}.

*c*

_{ i }. The period of each one of these sequences is equal to

*LT*. Therefore, one possible implementation consists of

*p*parallel ADCs, each working uniformly with period

*LT*. Another widely-used notation for the multi-coset sampling is to express each

*i*th sampling scheme as follows,

*y*(

*t*) denotes the received signal, which contains the multiband signal

*x*(

*t*), plus an interference

*i*(

*t*), plus a double-side complex zero-mean AWGN

*w*(

*t*) with spectral density

*N*

_{0}/2,

The interference is assumed independent of the noise and desired signal, and its spectral shape is different from that of the desired.

*c*

_{ i }} is referred to as an (

*L,p*) sampling pattern and the integer

*L*as the period of the pattern. Figure 3 shows a scheme of how the

*p*cosets are obtained.

*M*blocks of

*p*nonuniform samples notated as

**y**

_{ m }. Thus, the notation can be compacted in

**Y**as follows,

The sub-Nyquist data matrix **Y** has dimension *p* × *M.*

### 3.1 Relation between multi-coset sampling and CS theory

**z**

_{ m }as the

*m*th block of

*L*uniform Nyquist samples of

*y*(

*t*),

The problem in CS consist of designing a convenient measurement matrix Φ_{
m
}such that salient information in any compressible signal is not damaged by the dimensionality reduction. A low value of coherence between Φ_{
m
}and the basis where the signal becomes sparse (Fourier in our case) is desirable in order to ensure mutually independent matrices and therefore better compressive sampling. The incoherence is defined as the maximum value amongst inner product of the orthonormal basis where the signal becomes sparse, and the orthonormal measurement matrix Φ_{
m
}. In our example, the maximal incoherence associated to the Fourier basis is given by the canonical or spike basis *φ*_{
k
}(*t*) *= δ*(*t -- k*). Thus, Φ_{
m
}must be a matrix that randomly selects *p* samples of **z**_{
m
}, where *p* < *L*. This matrix Φ_{m} is given by randomly selecting *p* rows of the identity matrix **I**_{
L
}.

_{ m }is obtained from the identity matrix and it remains the same whatever the

*m*th block is considered, then the signal

**y**

_{ m }is the same than the one obtained using the multi-coset sampling. As the matrix notation is much more clear, we will proceed following this notation,

where ${\widehat{\mathbf{R}}}_{x}\in {\u2102}^{L\times L}$ indicates the estimated autocorrelation matrix of the primary user that we want to detect, and ${\stackrel{\u0303}{\mathbf{R}}}_{i}\in {\u2102}^{p\times p}$ and ${\stackrel{\u0303}{\mathbf{R}}}_{w}\in {\u2102}^{p\times p}$ denotes the sub-Nyquist interference and the sub-Nyquist noise estimated autocorrelation matrix, respectively. As Φ comes from the identity matrix, ${\stackrel{\u0303}{\mathbf{R}}}_{w}$ is expected to be *σ*^{2} **I**_{
p
}. This notation simplifies the notation presented in [22], where the coset samples are fractional shifted and used to compute the correlation matrix of the signal.

## 4 Sparse correlation matching-based spectrum sensing

The proposed procedure consists of detecting the presence of a licensed user whose power spectral shape (called candidate spectral shape henceforth) is the only prior knowledge we have. Based on a feature-based detector perspective, a correlation matching approach is used with the candidate spectral shape as a reference. The baseband candidate autocorrelation matrix **R**_{
b
}, which depends only on the basic pulse used by the modulation transport, can be easily obtained from the candidate spectrum shape.

**R**

_{ b }is modulated by a rank-one matrix formed by the steering frequency vector at the sensed frequency

*w*as follows,

where ⊙ denotes the elementwise product of two matrices, $\mathbf{s}={\left[1\phantom{\rule{2.77695pt}{0ex}}{e}^{jw}\dots {e}^{j\left(L-1\right)w}\right]}^{T}$. Note that in (11) the dependency on *w* has been removed to clarify notation.

where **R**_{
n
}is the randomly sampled AWGN plus interference autocorrelation matrix and *γ*(*w*_{
s
}) is the power level at frequency *w*_{
s
}, which denotes the tentative frequency of the active primary user.

Summarizing, the problem to solve consists in finding the frequency that the compressed candidate correlation has to be modulated to best fit the data autocorrelation matrix ${\widehat{\mathbf{R}}}_{y}$ and to find the contribution of this modulated candidate autocorrelation contained in ${\widehat{\mathbf{R}}}_{y}$. Thus, the procedure not only provides the frequency location of the desired user but also an estimation of its transmitted power.

*γ*can be formulated as,

where Ψ(⋅, ⋅) is an error function between the two matrices. Note that the solution to (13) will be clearly a function of the steering frequency.

The different estimates result from the proper choice of the aforementioned error function can be divided in two groups: (1) error functions based on the distance between the two matrices and (2) error functions based on the positive definite character of the difference $\left({\widehat{\mathbf{R}}}_{y}-\gamma \Phi {\mathbf{R}}_{cm}{\Phi}^{H}\right)$.

### 4.1 Derivation of different methods depending on the choice of Ψ (•, •)

Three different candidate methods were defined in [24] for the non-compressed case. Here comes a brief review of the three procedures adapted to the sparse signal acquisition case.

However, this estimate does not preserve the positive definite property of the difference.

**R**

_{1}and

**R**

_{2}is given by,

**R**

_{1}=

*γ*Φ

**R**

_{ cm }Φ

^{ H }and ${\mathbf{R}}_{2}={\widehat{\mathbf{R}}}_{y}$ and minimizing (16), the power level estimate and the resulting minimum geodesic distance can be derived (18).

_{ q }(

*q =*1,...,

*Q*) denotes the

*Q*generalized eigenvalues of the pair $\left({\widehat{\mathbf{R}}}_{y},\Phi {\mathbf{R}}_{cm}{\Phi}^{H}\right)$. That is,

In interesting, the power level estimate *γ*_{
G
}does not depend on the frequency *w* of the candidate. Thus, the power level estimate *γ*_{
G
}does not require frequency scanning. The frequency location is obtained detecting the maximum of the inverse of the minimum geodesic distance (18b) versus frequency.

Finally, a third power level estimate can be derived by forcing a positive definite difference between the data autocorrelation matrix and the candidate matrix, as it was done in [24]. Here, we propose a different way to get to the same result following a minimum mean square error between the received signal and the candidate signal.

where **x** denotes here the candidate signal, $\sqrt{\gamma}$ its amplitude and **n** the noise plus interference.

**A**, which is applied to the received signal

**y**in order to obtain an approximation of the desired signal

**x**. The resulting error

**e**is defined as,

Matrix *ξ* is positive semi-definite by definition. If so, $\mathbf{I}-\gamma {\mathbf{R}}_{yy}^{-1}{\mathbf{R}}_{xx}$ must be too.

**U**Λ

**U**

^{ H },

^{-1}is a diagonal matrix whose diagonal elements are the corresponding eigenvalues of the matrix $\left({\mathbf{R}}_{xx}^{-1}{\mathbf{R}}_{yy}\right)$. In the worst case we assume that the minimum eigenvalue of (Λ

^{-1}-

*γ*

**I**) is equal to zero,

*γ*can be obtained as,

The last procedure is denoted as CANDIDATE-M because it looks for the minimum eigenvalue of ${(\Phi {R}_{cm}{\Phi}^{H})}^{-1}{\widehat{R}}_{y})$.

## 5 Numerical results

This section is divided in two parts. The first part concentrates on the general performance of the candidate spectrum sensing method proposed in the previous section. In the first section, scenarios with high SNR are used for the sake of figure clarity. The second part gives the ROC results for low SNR scenarios.

### 5.1 High SNR scenario

*w*

_{0}= 0.2. The size of the observation

**x**

_{ m }in

*L =*33 samples. The sampling rates of

**y**

_{ m }and

**x**

_{ m }are related through the compression rate $\rho =\frac{p}{L}$. To strictly focus on the performance behavior due to compression and remove the effect of insufficient data records, the size of the compressed observations is forced to be the same for any compression rate. Therefore, we set

*M = 2Lϵρ*

^{-1}where

*ϵ*is a constant (in the following results

*ϵ*= 10). Thus, for a high compression rate, the estimator takes samples for a larger period of time. The spectral occupancy Ω for this particular example is 0.25. The simulation parameters are summarized in Table 1.

Simulation parameters

$\rho =\frac{p}{L}$ | 1 | 0.76 | 0.52 | 0.24 |
---|---|---|---|---|

| 33 | 25 | 17 | 8 |

| 660 | 871 | 1281 | 2723 |

Acquisition time (ms) | 2.2 | 2.9 | 4.2 | 9.0 |

*γ*

_{ F }estimate, which is shown in Figure 4a, presents lower resolution and higher leakage compared with

*γ*

_{ M }and ${d}_{geo,\phantom{\rule{2.77695pt}{0ex}}\text{min}}^{-1}$, which are plotted in Figures 4b and 4c1, respectively. From Figure 4 it can be concluded that the best power estimate in terms of resolution is given by

*γ*

_{ M }. Moreover, the range of ${d}_{geo,\phantom{\rule{2.77695pt}{0ex}}\text{min}}^{-1}$ is smaller than the range of

*γ*

_{ M }, which is longer than 15 dB when there is no compression and decreases when the compression rate increases. This robustness makes us think that CANDIDATE-M may still work in scenarios with low SNR, where CANDIDATE-G probably fails (it is confirmed in the following section). On the other hand, the independence of

*γ*

_{ G }with respect to the carrier frequency may be observed in Figure 4c2.

*γ*

_{ G }is not able to provide two power level estimates because of the non-dependency on the frequency of the parameter

*γ*

_{ G }.

*γ*

_{ M }remains practically unfazed, CANDIDATE-G's performance has suffered a slight degradation. CANDIDATE-F clearly works as a ED. This sensitivity to interference suggests to discard CANDIDATE-F in favor of the two other candidate methods. Although both figures (Figures 4 and 6) make evident the degradation of the correlation-matching based spectrum sensing techniques in terms of detection capability due to the effect of the compression, it is interesting to note that the frequency and power level estimation do not suffer from the compression.

*w*

_{0}

*=*0.2 and the size of the observation

**x**

_{ m }in

*L =*33 samples. Figures 7 and 8 show the normalized root mean squared error (RMSE) of the estimated power level (this is the RMSE divided by the SNR) and the normalized RMSE of the estimated frequency location (this is the RMSE divided by

*w*

_{0}) of the desired user, respectively, for different compression rates. From Figures 7 and 8 it can be conclude that both the power level estimation accuracy and the frequency estimation accuracy remain almost constant whatever the compression rate we consider.

*γ*

_{ G }does not depend on the frequency and therefore it only works properly when only one desired user is present. Moreover, Figure 9b makes evident that CANDIDATE-G provides better results for the frequency location estimation. In conclusion, CANDIDATE-M seems to be the most complete technique of the three proposed methods because it provides good frequency and power estimations and it works in scenarios where more than one desired user is present.

*ρ*= 0.76 and in Figure 12 for

*ρ*= 0.52. It can be observed that the multipath causes significant losses regarding detection capabilities and also a deterioration in the frequency estimation due to the appearance of a bias into the frequency location. In any case, if these losses imply serious problem, it is reasonable to assume channel equalization at the sensing station.

### 5.2 Low SNR scenario-ROC curves

This section evaluates the performance in low SNR scenarios by means of the ROC curves (receiver operating characteristic) in order to illustrate the proposed candidate spectrum sensing method robustness against noise.

The primary user is located at normalized frequency 0.2. The primary user is a BPSK and its SNR is indicated in each plot. The detection methods under study will be CANDIDATE-M and CANDIDATE-G.

Both Figures 13 and 14 make evident that the CANDIDATE-M performance is much better than that of CANDIDATE-G.

## 6 Summary and conclusions

A feature-based approach for spectrum sensing based on periodic non-uniform sampling is addressed. In particular, the compressed-sampling version of detecting predetermined spectral shapes in sparse wideband regimes is faced by means of a correlation-matching procedure. The main contribution of the new sub-Nyquist sampling approach is that it allows to alleviate the amount of data needed in the spectrum sensing process. Once the sampling bottleneck is solved, the data autocorrelation matrix is obtained from sub-Nyquist samples. Following the correlation matching concept, the method is able to provide an estimate of the frequency location and a power level estimation of the desired user. Three different methods are proposed: The first one, which is based on the Euclidean distance, is discarded because of its low rejection to interference. The second one, which is based on the geodesic distance, works well in terms of interference rejection but the power level estimate that this method provides does not depend on the frequency parameter and therefore, it is not indicated when detecting more than one desired user. The third method, which is based on the positive semidefinite difference between matrices, is the one which works the better, both in terms of accuracy of the estimated parameters and in terms of robustness against noise. As it was expected, simulation results have shown that the compression affects the detection capabilities of all the correlation-matching methods. However, we have also shown that the accuracy of the frequency estimation and the accuracy of the power level estimation is not affected by the undersampling technique.

## Acknowledgements

This study was partially supported by the Catalan Government under grant 2009 SGR 891, by the Spanish Government under project TEC2008-06327-C03 (MULTI-ADAPTIVE), by the European Cooperation in Science and Technology under project COST Action IC0902. Eva Lagunas is supported by the Catalan Government under grant FI-DGR 2011.

## Endnotes

^{a}http://www.fcc.gov/.

^{b}http://www.etsi.org/.

## Declarations

## Authors’ Affiliations

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