High-Resolution Cyclic Spectrum Reconstruction

from sub-Nyquist Samples

Seyed Alireza Razavi∗, Mikko Valkama∗, Danijela Cabric†,∗Department of Electronics and Communications Engineering, Tampere University of Technology, Finland†Cognitive Reconfigurable Embedded Systems Lab, University of California, Los Angeles (UCLA), CA

Emails: alireza.razavi@tut.fi, mikko.e.valkama@tut.fi, danijela@ee.ucla.edu

Abstract—In this paper, the problem of reconstruction of SpectralCorrelation Function (SCF) from sub-Nyquist samples is studied.We will first propose a novel formulation for the problemand then employ two two-dimensional greedy like sparse signalrecovery algorithms, namely Compressive Sampling MatchingPursuit (CoSaMP) and Iterative Hard Thresholding (IHT), forthe recovery of the sparse SCF. The achievable resolution of theproposed methods is shown to be significantly higher than theexisting methods and therefore the methods can be applied to sig-nals with fine frequency components. Comprehensive simulationresults shows that the method can efficiently reconstruct the SCFof a signature-embedded OFDM signal, which has applicationsin cognitive radio systems.

I. INTRODUCTION

Cyclostationary signal analysis [1] is known as a powerfultool for analysis of signals exhibiting periodicity in mean,correlation, or spectral descriptors. The applicability of themethod is based on the observation that many man-madesignals contain hidden periodicities which can be exploited forsignal analysis. Among many other applications, the methodis recognized as a promising approach for modulation clas-sification in the presence of noise. Gardner et. al. in a two-part paper [2], [3] studied the cyclic features of analog anddigital modulated signals. They showed that the SCF of analogand digital modulations exhibits certain peaks which can beused for modulation classification. Since the rise of CognitiveRadio (CR) as a key research area in communications for highutilization of spectrum, modulation detection-based methods(also known as cyclic feature detection methods) have beenemployed as an important spectrum sensing (SS) technique,especially in presence of noise with unknown level; see, e.g.[4], [5]. It has also been exploited for coordination of CRnetworks based on cyclostationary signatures [6].

To avoid losing frequency details of a signal, Nyquist-Shannonsampling theorem asks to sample it in traditional Nyquistrate. However, in some applications (e.g. wideband spectrumsensing), due to the limitations of today’s analog-to-digitalconverter (ADC) front-ends which cannot support very highbandwidth and the need for excessive memory and prohibitiveenergy costs for implementing digital signal processing sys-tems [7], it is very costly and even impractical to sense thesignal based on Nyquist-rate samples. This has motivated theemployment of Compressive Sampling [8], [9] methods forreconstruction of sparse spectrum using sub-Nyquist samples.Compressive Sampling (CS) is a relatively new theory insignal processing which states that whenever a signal is sparsein some known basis, i.e. it can be represented as a linear

combination of only few basis functions, it is possible toreconstruct it from sub-Nyquist-rate samples.

Contributions and relation to previous works: In thispaper we introduce two new methods for reconstructing thespectral correlation of a signal from its sub-Nyquist samplesusing greedy compressive sensing algorithms. The proposedmethods support high resolution in SCF without assumingany knowledge about the location of sparse cyclic featuresin reconstruction of SCF. Such solutions are important e.g.in cognitive radio applications where various RF and otherimpairments can considerably complicate the feature extractiontask [10]. To the best of our knowledge, there are very fewworks devoted to the problem of reconstruction of SCF fromsub-Nyquist samples. As our literature review shows, the state-of-the-art method is the one proposed in [11] and later on in[12], but these methods cannot be employed for signals withsmall separation between frequency details in their SCF (likeOFDM signal) as they cannot support high resolution for SCFon reasonable computing platforms and therefore the frequencycomponents are not practically resolvable. This is because toobtain a resolution of order N ×N in cyclic spectrum plane,their methods need matrices of size proportional to N4 whichrequires a massive memory for big values of N .

In Section II we will provide some preliminaries and novelproblem formulation. In Section III we introduce two newmethods for reconstruction of sparse SCF and discuss theiradvantage over existing methods. A simulation example willbe studied in Section IV.

Notations: Throughout this paper matrices and vectors aredenoted by capital and small boldface letters, respectively. E isreserved for expected value and ∗ denotes complex conjugate.IP and 0P×Q represent, respectively, P × P identity matrixand P × Q full-zero matrix. diag(x) is the diagonal matrixwhose diagonal elements are the components of vector x, anddiag(X) is the diagonal matrix whose diagonal elements arethe diagonal elements of X.

II. PROPOSED PROBLEM FORMULATION

A. Background in Cyclostationary

Signal x(t) is said to have second-order cyclostationary (in thewide sense) if its autocorrelation function (ACF) defined as

Rx(t, τ) , Ex(t+ τ/2)x∗(t− τ/2) (1)

is periodic in time t with some fundamental period T0. AsRx(t, τ) is periodic in t, it may be expressed by the Fourierseries expansion Rx(t, τ) =

∑

α Rαx (τ)e

j2παt, where the

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Fourier series coefficient Rαx is called Cyclic Autocorrelation

Function (CAF) which is computed as

Rαx (τ) =

1

T0

∫ T0/2

−T0/2

Rx(t, τ)e−j2παtdt. (2)

The frequencies α , nT0

n∈Z are called cyclic frequencies.Signal x is said to exhibit second-order cyclostationarity if atleast there is one nonzero cyclic frequency for which Rα

x (τ) 6=0. The Spectral Correlation Function (SCF), sometimes calledcyclic spectral density (CSD) of signal x is defined as theFourier transform of CAF with respect to the delay τ

Sαx (f) =

∫ ∞

−∞Rα

x (τ)e−j2πfτ , (3)

where f is called spectral (or angular) frequency.

B. Matrix-based formulation

In this paper, we are going to recover the SCF of signals whichare sparse in α−f plane from their sub-Nyquist samples usingcompressive sampling algorithms. In other words, we assumethat for a signal x(t), the SCF Sα

x (f) exhibits only few peaksin α − f plane. Furthermore we assume that due to practicalissues we sample the signal at sub-Nyquist rate. This can berepresented in matrix form as (see [11], [12] and referencestherein for more details about sub-Nyquist sampler)

z = Ax (4)

where M × 1 vector z consists sub-Nyquist sampled signal,N × 1 vector x denotes unavailable Nyquist sampled signal,and A is the M×N random measurement matrix with columnsnormalized to have norm one. M

N < 1 is called compressionratio. From (4) we have

Rz = ARxAH , (5)

where N × N matrix Rx , E(xxH ) = [rx(n, ν)]n,ν andsimilarly-defined M×M matrix Rz are data covariance matri-ces of x and z, respectively, and rx(n, ν) = E(x(n)x∗(n+ν)).The discrete SCF of finite-length signal x (and similarly forz) can be written as [11]

Sx(a, b) =1

N

N−1∑

ν=0

N−1−ν∑

n=0

rx(n, ν)e−j 2π

Na(n+ ν

2)

e−j 2π

Nbν ,

where a and b are discrete counterparts of α and f , respec-

tively. Remark that term e−j 2π

Na( ν

2) has been employed for

converting the Fourier transform of asymmetric covariancerx(n, ν) = E(x(n)x∗(n + ν)) to the Fourier transform ofsymmetric covariance E(x(n − ν/2)x∗(n + ν/2)) as definedin (1). We propose and introduce the following matrix-basedformulation for SCF (all matrices are of size N ×N ):

Sx = B(Rx)

,

N−1∑

ν=0

N−1−ν∑

n=0

GνDnRxJnDνF, (6)

where Gν = [ 1√Ne−j2πa(n+ν/2)/N ](a,n), F =

[ 1√Ne−j2πνb/N ](ν,b), Dν is a diagonal matrix with only

its (ν, ν) diagonal element being 1 and the rest being 0, and

Jn =

[0n×(N−n) 0n×n

IN−n 0(N−n)×n

]

.

As the inverse transform, Rx is calculated from Sx using thefollowing formula

Rx = B−1(Sx)

=N−1∑

ν=0

N−1−ν∑

n=0

(

DnGHν SxF

HDνJTn

+KnDνFSHx GνDn

)

, (7)

where

Kn =

[0(n+1)×1 0(n+1)×(N−n−1) 0(n+1)×n

0(N−n−1)×1 IN−n−1 0(N−n−1)×n

]

.

Remark that the role of the second term in the right side of (7)is copying the conjugate of elements above main diagonal tothe elements below it. This is required since Rx is Hermitian.

Inserting (7) in (5), the relationship between the two-dimensional (2-D) unknown sparse signal Sx and the 2-Dobservations Rz can be expressed as follows:

Rz = A(Sx)

,

N−1∑

ν=0

N−1−ν∑

n=0

(

ADnGHν SxF

HDνJTnA

H

+AKnDνFSHx GνDnA

H)

. (8)

Since (8) is an underdetermined system of equations, A doesnot have any inverse. Instead, its adjoint A⋆ can be expressedas

S⋆x = A⋆(Rz)

=

N−1∑

ν=0

N−1−ν∑

n=0

AHGνDnRzJnDνFA, (9)

where S⋆x = A⋆(A(Sx)) represents a proxy for Sx, in the sense

that if Sx is K-sparse and operator A has restricted isometryproperty (RIP) [8] with Kth restriced isometry constant smallenough, then the energy in each K entries of S⋆

x approximatesthe energy in corresponding entries in Sx [13].

In the next section we will show how this proposed formulationwill be employed for SCF reconstruction from sub-Nyquistsamples.

III. SUB-NYQUIST RECONSTRUCTION OF SPECTRAL

CORRELATION FUNCTION

A. SCF Reconstruction Using CoSaMP

In this part we will show how CoSaMP [13] can be formulatedto solve the 2-D underdetermined regression problem of (8)for reconstructing a K-sparse 2-D signal. Remark that hereK is the number of unique nonzero features in cyclic powerspectrum. The iterative algorithm starts with the followinginitialization:

Initialization: Set initial estimate for SCF as S0x = 0N×N ,

residual matrix Υ = Rz , support set E0 = , and iterationi = 0.

Then in each iteration i, put i ← i + 1, and then do thefollowing steps untill some stopping criterion is satisfied:

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Step 1 - Identification: Use (9) to form a proxy for theresiduals as A⋆(Υ) and identify the support of its 2Kbiggest (in magnitude) entries and put it in set E i =(r1, c1), (r2, c2), . . . , (r2K , c2K), where (rk, ck) denotes therow and column indices of k-th biggest element.

Step 2 - Support merger: Merge the support identified in

Step 2 with the support of Si−1x as E i = E i∪E i−1. We denote

the cardinality of set E i by K . Remark that 2K ≤ K ≤ 3K .

Step 3 - Least-square estimation: The goal of this step is tofind the sparse matrix with support set E i whose transform tomeasurements covariance domain computed by (8) is closest toRz in the least-squares (LS) sense. As was mentioned earlier,the second term in the right hand side of (8) is for copyingthe elements above the main diagonal to the elements belowit. This suggests that the elements below the main diagonal

are redundant and the M ×M matrix Rz has onlyM(M+1)

2unique terms. Keeping only the elements on and above maindiagonal, and therefore only the first term in the rights handside of (8), we will have

Rz =

N−1∑

ν=0

N−1−ν∑

n=0

Tn,νΩTTn,ν , (10)

where Tn,ν = ADnGHν Er and Tn,ν = (ET

c FHDνJ

TnA

H)T

are both M × K matrices, Rz is Rz with the elements belowthe main diagonal set to zero, Ω is a K ×K diagonal matrixwhose unknown diagonal elements ωkKk=1 are what we aretrying to estimate in this step, and

Er = [er1 , er2 , . . . , erK ], Ec = [ec1 , ec2 , . . . , ecK ],

where ek is the k-th column of identity matrix IN . From theproperties of Kronecker product and vec operator [14, Chapter11], we will have then

rz = Tω (11)

where rz = vec(Rz), T =∑N−1

ν=0

∑N−1−νn=0 Tn,ν , and

Tn,ν =[

t1 ⊗ t1, . . . , tK ⊗ tK

]

, (12)

with tk and tk denoting the k-th columns of Tn,ν and Tn,ν ,respectively. Denoting the set of nonzero components of rz byN , (11) can be rewritten now as

rz,N = TNω, (13)

where rz,N and TN are rz and T, respectively, with the rowscorresponding to N kept and the rest discarded, and ω =[ω1, . . . , ωK ]. One-dimensional regression problem (13) hasthen the following LS estimate for unknown vector ω

ω = T†N rz,N . (14)

Finally, denoting Ω = diag(ω), the K-sparse LS estimate ofSx at iteration i is obtained as

Six,K

= ErΩETc . (15)

Step 4 - Support pruning: We prune the support by keeping

only K largest (in magnitude) entries of Six,K

and zeroing the

other K−K . The resulting K-sparse matrix is denoted by Six,

and its support set is denoted by E i.

Step 5 - Update residue: Finally, we update the residual as

Υ = Rz −A(Six). (16)

B. SCF Reconstruction Using IHT

The second greedy method which is employed in this paperfor SCF reconstruction is IHT [15]. For a 1-D K-sparse signals measured through y = Φs, the IHT algorithm tries tominimize the objective function ‖y − Φs‖2 for finding thesolution s. The minimization is accomplished using gradientdescent algorithm. The update equation for IHT algorithm thentakes the form

si+1 = HK(si+1 + µiΦT (y −Φsi)), (17)

where µi is the step size at iteration i and HK is thenonlinear operator which sets all but the largest K elements(in magnitude) of its argument to zero.

In 2-D case, this minimization is written as

minSx

‖Rz −A(Sx)‖2F, s.t. ‖Sx‖0 ≤ K, (18)

where ‖Sx‖0 denotes number of nonzero elements of Sx. Solv-ing (18) is equivalent with solving the following optimizationproblem for Rx, and then inserting it in (6) to find Sx:

minRx

‖Rz −ARxAH‖2F

︸ ︷︷ ︸

J(Rx)

, s.t. ‖B(Rx)‖0 ≤ K (19)

From [14, Theorem 4.80], we notice that the cost function in(19) can be rewritten as

J(Rx) = trace(

(Rz −ARxAH)H(Rz −ARxA

H))

= trace(RHz Rz)− 2trace(RH

z ARxAH)

+trace(ARxAHARxA

H). (20)

Since Rx is Hermitian, from [14, Chapter 17] we have

∂trace(RHz ARxA

H)

∂Rx= 2AHRzA−diag(AHRzA), (21)

and

∂trace(ARxAHARxA

H)

∂Rx= 4AHARxA

HA

−2diag(AHARxAHA). (22)

Inserting (21) and (22) in (20), and since∂trace(RH

zRz)

∂Rx= 0,

we will have

Q , −∇J(Rx) = 4AH(Rz −ARxAH)A

−2diag(

AH(Rz −ARxAH)A

)

(23)

which gives the following iterative formula for updating theestimate

Ri+1x = B−1

(

HK

(

B(Rix + µiQ)

))

. (24)

Similar to the ordinary IHT algorithm [16], the optimum stepsize here is the one which maximally reduces the error in eachiteration, that is

µi =‖Q‖2F

‖AQAH‖2F(25)

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(a) The SCF of noisy OFDM signal x in α− f plane

with signature peaks around ( αFs

, fFs

) = (0.25, 0.25)

and its image around (0.75, 0.75).

(b) The SCF of sub-Nyquist sampled signal z. (c) Reconstructed SCF of x from sub-Nyquist samplesby using 2-D IHT algorithm. The signature has beenidentified correctly.

Fig. 1. Reconstruction of a signature-embedded OFDM signal using IHT algorithm. The IHT method has successfully detected the signature locations.

C. Comparison to [11] and [12]

Both methods introduced in this section can obtain a signif-icantly higher resolution comparing to the method proposedin [11] and [12]. This is because in [11] and [12] the 2-Dsparse reconstruction method has been changed to a 1-D oneby vectorizing the measurements as well as the desired sparsesignal. This vectorization requires matrices of size proportionalto N2 × N2 (see, e.g., [11, Equation (11)]) which needsmassive memory for large values of N . On ordinary computingplatforms their method is only practical for N in the range offew tens (for generating simulation results, N was set to 32in [11] and 36 in [12]).

The CoSaMP-based approach does not have this limitationsince the major part of operations have been done in 2-D space.The only place that we have used the vectorization is in (11),but the matrix in (12) is of size M2×K2, which does not needa huge memory since K ≪ N . The other matrices are at mostof size N×N . The method can therefore support the range ofup to one thousand for N on ordinary computing platforms.

On the other hand, IHT involves only matrices of size up toN ×N which do not pose any trouble in this regard.

IV. SIMULATION RESULTS

In this section we study the ability of the proposed methodsfor OFDM signature identification from sub-Nyquist samples.The signature embedded in the OFDM signal can be used,for example, for network coordination [6] or spectrum sens-ing [10]. For the first experiment, we consider an OFDMsignal with NFFT = 256 subcarriers and cyclic prefix lengthof NCP = NFFT/4. A signature [6] is embedded in OFDMsignal by copying the symbol carried by subcarrier numberk1 = 32 to the subcarrier separated by p = 64 subcarriers,i.e. subcarrier number k2 = 32 + 64 = 96. Denoting thesampling frequency by Fs, this results in peaks around the

point ( αFs

, fFs

) = ( pNFFT

, k1+p/2NFFT

) = (0.25, 0.25) and its image

around (1− pNFFT

, k1+p/2NFFT

+ 0.5) = (0.75, 0.75); see [17], [6],[18]. The signal-to-noise ratio (SNR) is 0 dB. Remark that inthe ideal case when the noise is purely white and we haveused infinite number of blocks for computing the SCF, noisewill not affect the SCF unless at α = 0. The SCF of thissignature-embedded OFDM signal is shown in Figure 1(a). Tocompute the SCF we use L = 1000 blocks of data each of

length N = 256, compute the SCF for each block from (6),and average over all blocks to obtain Sx. The correspondingcovariance matrix is then computed as Rx = B−1(Sx).Remark that this is the minimum value of N , since we needto have N ≥ NFFT to be able to resolve NFFT subcarriers. Ofcourse the bigger is N , the higher is the resolution in (α, f)plane. As it can be observed from Figure 1(a), the peakscorresponding to the first case of [6, equation (10)] (or [17,equation (18)]) cannot be seen in this picture unless for α = 0.This is because here the separation between two consequtivepoints along α axis is 1

NFFTand therefore multiple integers

of 1NFFT+NCP

, which are the locations of those peaks alongα axis, are not distinguishable. This may look problematicat the first glance, but in fact it is an advantageous pointfor the purpose of this paper since in signature-based OFDMidentification these peaks make the signal less sparse (denser)and therefore deteriorates the performance of CS algorithmsfor reconstruction of SCF. Then we set the compression ratioto M

N = 0.75 by choosing appropriate dimension for A in (4).The iid entries of A are taken from Gaussian distribution andits columns are normalized to have norm one. The SCF ofthis sub-Nyquist sampled OFDM signal can be seen in Figure1(b). As it is observed, the SCF does not exhibit the peakscorresponding to embedded signature.

Figure 1(c) demonstrates the reconstructed SCF using 2-D IHTalgorithm. As it can be seen the algorithm has successfullyrecovered the signatures. As a preliminary step before applyingIHT, in order to make the signal sparse, we first estimatedS0x(f) by adopting two assumptions: (i) S0

x(f) = C, ∀fand (ii) Sα

x (f) = 0, ∀α 6= 0. The value of C can be

estimated as C = ‖Rz‖F

‖A(Sx)‖F

, where Sx is an N × N matrix

with the elements in the first row (corresponding to α = 0)equal to one and the rest zero. After estimating S0

x(f), wesubtracted its effect from the 2-D observation Rz and applied

the IHT algorithm to the residual observation Rz −A(CSx).Remark that assumption (ii) implies that the signature peakshave been considered as noise in estimating C. This does notaffect the estimate heavily as the number of peaks is muchsmaller than the whole number of subcarriers. Assumption (i)is also reasonable with the settings of our simulation. However,in general S0

x(f) is the spectrum of noisy OFDM signal. Sincethe focus of this paper is on reconstruction of sparse SCF ratherthan dealing with non-sparse parts, we do not elaborate more

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0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Compression Ratio

Pro

babi

lity

of c

orre

ct s

igna

ture

det

ectio

n

2−d IHT

2−d CoSaMP

Fig. 2. Probability of signature detection versus compression ratio MN

.Number of blocks is L = 1000 and signal is noise-free.

200 300 400 500 600 700 800 900 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of blocks L

Pro

babi

lity

of c

orre

ct s

igna

ture

det

ectio

n

2−d IHT

2−d CoSaMP

Fig. 3. Probability of signature detection versus number of blocks L.Compression ratio is M

N= 0.75 and signal is noise-free.

on this issue and leave it for an extended version of this paper.

We re-emphasize here that while the methods proposed in thispaper are indeed capable of reconstructing a signal with suchfrequency details using reasonable computational resources,the methods proposed in [11], [12] need matrices of size 2564

which cannot be supported by ordinary computing platforms.

In the second experiment, the effect of compression ratioMN on signature detection of a noise-free OFDM signal isinvestigated. Remark that although the OFDM signal is noise-free, however, because of the finite-sample effect, (8) becomesa noisy CS problem. For this experiment we take N = NFFT =128 and the place of signature is determined randomly ineach experiment. The number of Monte Carlo runs is 500and the number of blocks for estimating the SCF from sub-Nyquist data is L = 1000. Figure 2 depicts the probability ofsignature detection (support detection of sparse SCF) versuscompression ratio. It can be seen that the IHT algorithmoutperforms the CoSaMP algorithm.

The third experiment is devoted to the investigation of theeffect of number of blocks L. For this simulation the com-pression ratio is fixed and equal to M

N = 0.75. The values ofN , NFFT, and number of random runs is the same as in secondexperiment. Again it can be seen from Figure 3 that the IHTalgorithm is superior to the CoSaMP algorithm.

V. CONCLUSION

The problem of reconstructing SCF from sub-Nyquist samplesusing 2-D IHT and CoSaMP was studied. The proposed

methods stemming from novel problem formulation, supportmuch higher resolution in SCF than the existing methods, andtherefore can be applied for sensing signals with fine frequencydetails like OFDM signals. We showed how the methods canbe used efficiently for detection of embedded signature in anOFDM signal, with direct applications in emerging cognitiveradio systems and networks.

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