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COMPRESSED SENSING FOR WIDEBAND COGNITIVE RADIOS

Zhi Tian

Dept. of Electrical & Computer Engineering

Michigan Technological UniversityHoughton, MI 49931 USA

Georgios B. Giannakis

Dept. of Electrical & Computer Engineering

University of MinnesotaMinneapolis, MN 55455 USA

ABSTRACT

In the emerging paradigm of open spectrum access, cog-

nitive radios dynamically sense the radio-spectrum environ-

ment and must rapidly tune their transmitter parameters to ef-

ciently utilize the available spectrum. The unprecedented

radio agility envisioned, calls for fast and accurate spectrum

sensing over a wide bandwidth, which challenges traditional

spectral estimation methods typically operating at or aboveNyquist rates. Capitalizing on the sparseness of the signal

spectrum in open-access networks, this paper develops com-

pressed sensing techniques tailored for the coarse sensing task

of spectrum hole identication. Sub-Nyquist rate samples are

utilized to detect and classify frequency bands via a wavelet-

based edge detector. Because spectrum location estimation

takes priority over ne-scale signal reconstruction, the pro-

posed novel sensing algorithms are robust to noise and can

afford reduced sampling rates.

Index Terms spectrum estimation, compressed sens-

ing, sub-Nyquist sampling, wavelet transform, cognitive radio

1. INTRODUCTION

The emerging paradigm of Dynamic Spectrum Access shows

promise to alleviate todays spectrumscarcity problemby ush-

ering in new forms of spectrum agile wireless networks [1].

Key to this new paradigm are cognitive radios (CRs) that are

aware of and can sense the environments, and perform func-

tions to best serve their users withoutcausing harmful interfer-

ence to other authorized users [2]. As such, therst cognitive

task preceding any form of dynamic spectrum management

is to develop wireless spectral detection and estimation tech-

niques for sensing and identication of available spectrum.

Spectrum sensing in the wideband regime faces consider-able technical challenges. The radio front-end can employ a

bank of tunable narrowband bandpasslters to search one nar-

row frequency band at a time. In each narrowband, existing

spectrum sensing techniques perform either energy detection

[3] or feature detection [2]. It requires an unfavorably large

number of RF components and the tuning range of each l-

ter is preset. Alternatively, a wideband circuit utilizes a sin-

gle RF chain followed by high-speed DSP to exibly search

over multiple frequency bands concurrently [4]. A major im-

plementation challenge lies in the very high sampling rates

required by conventional spectral estimation methods which

have to operate at or above the Nyquist rate. Meanwhile, due

to the timing requirements for rapid sensing, only a limited

number of measurements can be acquired from the received

signal, which may not provide sufcient statistic when tradi-

tional linear signal reconstruction methods are employed.

This paperaimsat fast spectrum sensing at affordablecom-

plexity. A couple of key premises are capitalized to alleviate

the stringent sampling requirements in the wideband regime.

First, we take a multi-resolution approach to decompose the

cognitive sensing task into two stages. The rst stage is coarse

sensing to detect non-overlappingspectrum bands and classify

them intoblack, gray or whitespaces, depending on whether

the power spectral density (PSD) levels are high, medium or

low [2]. Based on the spectrum sharing mechanism adopted

[1], the second stage ofne-scale spectral shape estimation

is performed only when needed, and mostly conned withinthe available (narrowband) white spaces to alleviate the sam-

pling requirements. Second, we recognize that the wireless

signals in open-spectrum networks are typicallysparsein the

frequency domain. This is due to the low percentage of spec-

trum occupancy by active radios a fact motivating dynamic

spectrum management. For sparse signals, recent advances in

compressed sensing have demonstrated the principle of sub-

Nyquist-rate sampling and reliable signal recovery via com-

putationally feasible algorithms [5, 6, 7, 8].

Tailored to the above distinct nature of CR sensing, this

paper derives novel compressed sensing algorithms for the

coarse sensing task of spectrum band classication. Random

sub-Nyquist-rate samples are employed to formulate an op-timal signal reconstruction problem, which incorporates the

wavelet-based edge detector we recently developed in [10] to

recover the locations of frequency bands. Because spectrum

location estimation takes priority over ne-scale signal recon-

struction, the novel sensing algorithms are robust to noise and

can afford reduced sampling rates.

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2. SIGNAL MODEL AND PROBLEM STATEMENT

Suppose that a total ofBHz in the frequency range [f0, fN]isavailable for a wideband wireless network. A CR receives the

signalr(t)that occupiesNconsecutive spectrum bands, withtheir frequency boundaries located atf0 < f1 < fN. Thefrequency response ofr(t)is illustrated in Fig. 1. Dependingon whether the PSD level is high, medium or low, each fre-

quency segment can be categorized into black, gray or white

spectrum spaces [2]. White holes, and sometimes gray spaces,

can be picked by the CR for opportunistic transmission, while

the black holes are to be avoided for interference control.

PSD

wide band of interest

0 N1 2 nn-1

Bn

c,n

Fig. 1.Nfrequency bands with piecewise smooth PSD.

Suppose that the time window for sensing ist [0, M T0],whereT0 is the Nyquist sampling rate. Using Nyquist sam-pling theory,Msamples are needed to recover r(t) withoutaliasing. A digital receiver converts the continuous-domain

signalr(t)to a discrete sequence xt CK of lengthK. The

sampling process can be expressed in discrete-time domain in

the following general form:

xt= STrt (1)

where S is an M K projection matrix and rt representsthe M 1 vector with elements r t[n] = r(t)|t=nT0 , n =1, . . . , M . Columns {sk}

Kk=1 ofS can be viewed as a set

of basis signals or matched lters, while the measurements

{xt[k]}Kk=1 are in essence the projection ofr(t) onto the ba-

sis. The model in (1) subsumes all sampling schemes yield-

ing linear measurements. For example, S = IM representsNyquist-rate uniform sampling, where IMis the size-M iden-tity matrix; S= FM amounts to frequency-domain sampling,whereFMis theM-point unitary discrete Fourier transform(FT) matrix. When K < M, reduced-rate sampling arises.We focus on non-adaptive measurements where Sis preset.

The goal of CR sensing is to classify and estimate the

spectrum ofr(t) given the sample set xt, where K < Mis possible. Spectrum classication refers to identifying the

number of subbandsN and their locations {[fi, fi+1]}N1i=0 ,

and classifying them intoblack, gray or white spaces. Spec-

trum estimation, on the other hand, can have different objec-

tives: either to estimate the frequency response ofr(t)withinthe entire wideband, or conne the estimation to be within the

identied (narrowband) white spaces only. This paper primar-

ily concerns the coarse sensing task of spectrum classication.

3. MULTI-STEP COMPRESSED SENSING

Our rst approach to reduced-complexity spectrum sensing

takes the following four steps: i) compressed random sam-

pling to generate measurementsx t fromr(t); ii) reconstruc-tion of the frequency responser f = FMrt fromxt;iii) esti-mation of frequency band numberNand locations {fi}

N1i=1

based onrf; and,iv) estimation of the average amplitude ofrfwithin each identied band for spectrum classication. It

is worth emphasizing that Stepii) recovers the accurate ne-

resolution signal spectrum rf represented by M frequencysamples at the Nyquist rate, while the available measurement

setxtis of a reduced size ofK(< M)elements.

3.1. Sub-Nyquist-rate Sampling

Let Fdenote the non-zero frequency-domain support ofr(t)in the noise-free case. In open-spectrum networks, it gener-

ally holds that |F| B [4], indicating the sparseness natureofr(t). Equivalently speaking, theM 1frequency response

vector rf contains on average Kb := |F|M/B non-zeroelements when noise free, and Kb M. The key resultsin compressed sensing stated that the sparse vectorrfcan be

recovered asymptotically fromK ( M) samples ofr(t), aslong asK Kb. These samples xtcan be generated from (1)via universal non-uniform sampling [5] or random sampling

[6, 7], both of which can enable perfect recovery ofr f when

free of noise. To distinct, we denote a reduced-rate sampling

matrix asSc of dimensionM K, whereKb K M. Asimple example ofSc is a selection matrix that randomly re-

tainsKcolumns of the size-M identity matrix, which meansthatK M time instants on the sampling grid are skipped.

3.2. Spectrum Reconstruction

With theK measurementsxt = STcrt, we now estimate the

frequency response ofr(t) in the form ofrf = FMr. Fora given linear sampler Sc : C

M CK, we seek a nonlinearreconstruction functionR() : CK CM that offers an ap-proximate reconstruction ofrf C

M from xt CK based on

the linear transformation equality x t = (STcF

1

M)rf; c.f. (1).This is a linear inverse problem with sparseness constraint,

which is NP-hard. A conceptually intuitive approach to signal

reconstruction is the Basis Pursuit (BP) technique [9], which

transforms the sparseness constraint on rfinto a convex opti-

mization problem solvable by linear programming:

rf =arg minrf

||rf||1, s.t. (STcF

1

M)rf =xt. (2)

Besides BP, a number of efcient reconstruction methods ex-

ist, including orthogonal matching pursuit (OMP) algorithm

and tree-based OMP (TOMP) algorithm [8]. Since the mea-

surements can be complex-valued, wend it convenientto use

TOMP in our simulations, but for illustration, formulate our

signal reconstruction problem based on BP, as in (2).

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3.3. Band Location Estimation

Having estimated rf, we turn to the wavelet-based edge de-

tector in [10] for detecting the number and frequency loca-

tions of spectrum spaces. The basic idea is to view the entire

wideband under scrutiny as a train of consecutive frequency

subbands, where the PSD is smooth within each subband, but

exhibits a discontinuous change between adjacent subbands.These irregularities are in fact edges in PSD, which carry key

information on the locations and intensities of spectrum holes.

To further simplify and expedite the coarse sensing stage, we

approximately treat the spectral amplitudes within each sub-

band to be almost at, at an unknown level n over then-thband. These modeling approximations are invoked to reduce

the overall wideband sensing complexity. If needed, the sens-

ing quality can be rened after spectrum holes are identied.

Based on these modeling assumptions and with reference

to Fig. 1, wideband sensing can be viewed as an edge detec-

tion problem in an image depicted by rf in frequency. Edgesin this image correspond to the locations of frequency discon-

tinuities {fi}N1i=1 , which are to be identied. The waveletapproach is well motivated for edge detection [11]. Given rf,we re-cast the edge detector in [10] in discrete form.

Let(f)be a wavelet smoothing function with a compactsupport. The dilation of(f)by a scale factorsis given by

s(f) =1

s

f

s

. (3)

For dyadic scales,s takes values from powers of 2, i.e.,s =2j, j = 1, 2, . . . , J . Let s() := F

1{s(f)} = (s)represent the inverse FT of the wavelet function. The contin-

uous wavelet transform ofR(f)( rf)is given by [10]

WsR(f) = F {Wsr()}= F{r() (s)}. (4)

Mapping WsR(f),r()and(s)to their length-Mdiscretecounterparts ys, rt and s respectively, (4) is equivalent to

ys = FMdiag{s}rt. (5)

Replacingrt in (5) by its estimate rt =F1Mrf, we reach

the estimated wavelet transform

ys = FMdiag{s}F1

Mrf. (6)

The derivative wavelet ofrfat scalesis given by zs withelements {zs[n]}

Mn=1in the form

sd

df (WsR(f)) zs : zs[n] = ys[n] ys[n 1]. (7)

The boundaries {fn}N1n=1 can thus be acquired by picking the

local maxima of the wavelet moduluszs, while the band num-

berN is determined by the number of local peaks [10, 11].

3.4. Frequency Response Amplitude Estimation

The estimated boundaries {fn}N1n=1 correspond toN 1se-

lected indices {In : fn = f0 + In, = B/M} in thefrequency response vectorrf. Elements ofrfbetween a pair

of adjacent indices belong to the same frequency band. The

average frequency response amplitude n of the n-th band,

can thus be computed as

n 1

In In1+ 1

Ini=In1

|rf[i]| , n= 1, . . . , N . (8)

This simple and coarse estimator in (8) allows us to categorize

the detected frequency bands intoblack, gray,orwhitespaces

[2], depending on whether {n} are high, medium or low.

4. ONE-STEP COMPRESSED SENSING

To further reduce the implementation complexity of coarse

spectrum sensing, we now ask: can we directly detect and

estimate the frequency band locations from the compressedmeasurementsxt in (1), without having to recover the detailed

frequency response rf? We address this question by deriving

signal recovery formulation for wavelet-based edge detection.

Recall from Section 3.3 that the band locations can be re-

covered from the (N1)peaks of the derivative wavelet mod-uluszs C

M. WhenN M (which is generally the case),zs can be treated as a sparse vector, with only a few non-

trivial elements located at frequency band boundaries; c.f., see

Fig. 3 for graphical validation. As such, z s can be recovered

under the sparseness constraint, provided that we can nd a

linear transformation equality linking zs to the compressed

measurement vector xt.

To this end, we rewrite (7) in matrix-vector form as z s =ys, whereis the differentiation matrix given by

=

1 0 0

1 1 . . . 0

0 . . .

. . .

0 1 1

MM

. (9)

Putting (9) and (5) together, we obtain:

rt=F1Mdiag{s}

1ys =

F1Mdiag{s}

1 1

:=G

zs.

(10)

Noting that xt= STcrt, and that zsis sparse, we reach thefollowing BP-based optimization formulation:

zs = arg minzs

||zs||1, s.t. xt=STc G

zs. (11)

Subsequently, the band boundaries {fn}can be acquiredfrom the locations of those non-zero elements in zs, obviating

the involved step of frequency response estimation on r f.

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5. SIMULATIONS

We consider a wide band of interest in the range off0 +[50, 150]Hz, where is the frequency resolution. Fig. 2illustrates the spectral amplitude |R(f)| observed by a CR.During the observed burst of transmissions in the network,

there are a total ofN = 6bands, with frequency boundariesat{fn}6n=1 = f0+ [60, 68, 83, 119, 123, 150]Hz. Amongthese bands (marked in Fig. 2),B1, B3 and B5 have rela-tively high signal amplitude at levels 16, 20, and 24, respec-

tively, while B2has low amplitude at a level of 2. The rest twobands,B4 andB6 are not occupied and are thus white spec-trum holes. The sampling lower bound is thusKb/M 40%.

For compressed sensing, the compression ratio K/M isset to vary from 50% to 100% with reference to the Nyquistrate. The noise level isn2w= 8dB. The samplerS= Scusedin (1) is uniformly random. Fig. 2 indicates that the signal

recovery quality (via TOMP) improves as K/M increases.In the wavelet-based edge detector, Gaussian wavelets are

used at four dyadic scaless = 2

j ,j = 1, 2, 3, 4

. Fig. 3 de-

picts the multiscale wavelet products computed from (7) [10].

Edges in theR(f) are clearly captured by the wavelet trans-form in all curves. As the scale factor sj increases, the wavelettransform becomes smoother within each frequency band, re-

taining the lower-variation contour of the noisy PSD.

For frequency band location estimation, Fig. 4 depicts

the normalized root mean-square estimation errors (RMSE)

B1N1

n=1 |fn fn|2 with respect to both the compres-

sion ratioK/Mand the inverse noise level n2w . When eitherthe numberof samples is very small or thenoise is very strong,

there exhibits an estimation error oor. Nevertheless, the at-

tained degree of estimation accuracy is benecial to effecting

CR agility at affordable sampling cost. Robustness to samplequantization errors is also illustrated.

6. REFERENCES

[1] Facilitating Opportunities for Flexible, Efcient, and Reli-

able Spectrum Use Employing Cognitive Radio Technologies,

FCC Report and Order, FCC-05-57A1, March 2005.

[2] S. Haykin, Cognitive radio: brain-empowered wireless com-

munications, IEEE JSAC, vol. 23(2), pp. 201-220, Feb. 2005.

[3] H. Urkowitz, Energy detection of unknown deterministic sig-

nals,Proc. of the IEEE, vol. 55(4), pp. 523 - 531, April 1967.

[4] A. Sahai, D. Cabric, Spectrum Sensing Fundamental Limitsand Practical Challenges, A tutorial presented at IEEE DyS-

pan Conference, Baltimore, Nov. 2005.

[5] P. Feng, and Y. Bresler, Spectrum-blind mimimum rate sam-

pling and reconstruction of multiband signals, Proc. IEEE

Intl. Conf. on ASSP, Atlanta, pp. 1688-1691, 1996.

[6] E. J. Candes, J. Romberg and T. Tao, Robust Uncertainty Prin-

ciples: Exact Signal Reconstruction from Highly Incomplete

Frequency Information, IEEE Trans. on Information Theory,

vol. 52, pp. 489-509, Feb. 2006.

[7] D. L. Donoho, Compressed Sensing, IEEE Trans. on Infor-

mation Theory, vol. 52, pp. 1289-1306, April 2006.

[8] C. La, and M. N. Do, Signal reconstruction using sparse tree

representation.Proc. SPIE Conf. on Wavelet Apllications in

Signal and Image Processing, San Diego, Aug. 2005.

[9] S. S. Chen, D. L. Donoho, and M. A. Saunders, Atomic de-

composition by basis pursuit, SIAM J. Sci. Comput., vol. 20,

no. 1, pp. 33-61, 1999.[10] Z. Tian, and G. B. Giannakis, A Wavelet Approach to Wide-

band Spectrum Sensing for Cognitive Radios, Proc. of Intl.

Conf. on CROWNCOM, Mykonos, Greece, June 2006.

[11] S. Mallat, W. Hwang, Singularity detection with wavelets,

IEEE Trans. Info. Theory, vol.38, pp. 617-643, 1992.

0 10 20 30 40 50 60 70 80 90 1000

20

40

0 10 20 30 40 50 60 70 80 90 1000

50

50%

0 10 20 30 40 50 60 70 80 90 1000

50

75%

0 10 20 30 40 50 60 70 80 90 1000

20

40

100%

f

B1 B

2B

3 B4B

5 B6

Fig.2 . signal frequency response: (top) noise-free |Xf|; (rest)

recovered | Xf| at compressing ratiosK/M= 50%, 75%,1.

50 60 70 80 90 100 110 120 130 140 1500

0.01

0.02

0.03

s=21

50 60 70 80 90 100 110 120 130 140 1500

1

2x 10

13

s=23

f

Fig. 3. wavelet modulus for edge detection at scaless= 2 j .

10 0 10 20 3010

3

10

2

101

100

SNR (dB)

RMSE

K/M = 33%

K/M = 50%

K/M = 75%

K/M = 100%

K/M=50%, 4 bits

K/M=50%, 8 bits

0.4 0.6 0.8 110

3

10

2

101

100

Compression ratio (K/M)

RMSE

SNR = 0 dB

SNR = 5 dB

SNR = 10 dB

SNR = 20 dB

Fig. 4. frequency band location estimation errors RMSE.

Strong compression ofK/M = 33% is demonstrated. Im-pact of quantization (bits/sample) is shown forK/M= 50%.

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