Using Subspace Pursuit Algorithm to Improve Performance of the Distributed Compressive Wide-Band Spectrum Sensing

JOURNAL OF THE KOREAN INSTITUTE OF ELECTROMAGNETIC ENGINEERING AND SCIENCE, VOL. 11, NO. 4, DEC. 2011 JKIEES 2011-11-4-03
http://dx.doi.org/10.5515/JKIEES.2011.11.4.250
250
Using Subspace Pursuit Algorithm to Improve Performance of the
Distributed Compressive Wide-Band Spectrum Sensing
Le Thanh Tan․Hyung-Yun Kong
Abstract
This paper applies a compressed algorithm to improve the spectrum sensing performance of cognitive radio techno-
logy. At the fusion center, the recovery error in the analog to information converter (AIC) when reconstructing the
transmit signal from the received time-discrete signal causes degradation of the detection performance. Therefore, we
propose a subspace pursuit (SP) algorithm to reduce the recovery error and thereby enhance the detection performance.
In this study, we employ a wide-band, low SNR, distributed compressed sensing regime to analyze and evaluate the
proposed approach. Simulations are provided to demonstrate the performance of the proposed algorithm.
Key words: Wide-Band Spectrum Sensing, Subspace Pursuit Algorithm, Cognitive Radio, Compressed Sensing,
Power Spectrum Density Estimate.
Manuscript received July 19, 2011 ; revised October 25, 2011. (ID No. 20110719-022J)
School of Electrical Engineering, University of Ulsan, Ulsan, Korea.
Corresponding Author : Hyung-Yun Kong (e-mail : hkong@mail.ulsan.ac.kr)
Ⅰ. Introduction
Fueled by the dramatically increasing demand for high
quality of services, numerous novel wireless technologies
have been invented and are leading to a crowding of
spectrum allocation. This, in turn, raises the critical
problem that insufficient spectrum space is available for
new kinds of application. However, most of the licensed
bands are sporadically located and under-utilized, rather
than in actual shortage. In fact, less than 5 % of the total
licensed spectrum may be in use [1]. The Federal Co-
mmunications Commission (FCC) has therefore proposed
the idea of an open licensed frequency band, which un-
licensed users would be allowed to occupy opportu-
nistically. In addition, the IEEE 802.22 workgroup has
built the standards of WRAN based on cognitive radio
(CR) techniques [2]. CR is now considered as the most
competitive candidate for a secondary system that could
co-exist with the primary one.
Based on the ability to provide high data rates and
high quality of services, wide-band applications are re-
ceiving increasingly more attention recently. However,
wide-band applications in CR encounter considerable cha-
llenges in spectrum sensing. On the one hand, wide-band
sensing applications usually employ a large number of
RF devices to deal with the wide frequency range. On
the other hand, a trade-off exists between high-speed
processing units and detection performance due to the
sensing time constraints and insufficient samples.
In order to provide a reliable but low complexity
model, many studies have exploited a compressed sen-
sing (CS) framework for wide-band sensing. Initially, the
CS theory, which was innovated by Donoho [3], allowed
a highly sparse signal to be reconstructed from a small
number of measurements. In other words, this method is
able to compress the sparse signal at the sub-Nyquist rate
during sampling in the first stage. The reconstruction
stage requires state-of-the-art algorithms to solve the
convex optimization problem; for example, the basic
pursuit (BP) or orthogonal matching pursuit (OMP) [3],
[4]. Zhi et al. [5] next presented a single wide-band CR
model that uses CS based spectrum sensing schemes;
however, the input signal was still sampled by an ana-
log-to-digital convertor (ADC) operating at a Nyquist
rate. The authors in [6] improved compressive wide-band
spectrum sensing (CWSS) systems for single CR by em-
ploying an analog-to-information converter (AIC) [7]～
[9], which operates at a sub-Nyquist rate due to direct
application of CS to the analog signal. This group [10]
further extended their early work to multiple CRs in
order to design a distributed CWSS (DCWSS) based on
[11]. These studies simply applied wide-band spectrum
sensing to CS; hence, improvement of the model is still
required for greater robustness of the performance of
spectrum sensing.
In this work, we adopt CWSS and DCWSS schemes
for single and multiple CRs, respectively. In addition, we
propose the use of the subspace pursuit (SP) method [12]
in the reconstruction stage. The novel SP method pro-
vides the robustness to cope with inaccurate measure-

TAN and KONG : USING SUBSPACE PURSUIT ALGORITHM TO IMPROVE PERFORMANCE OF THE DISTRIBUTED COMPRESSIVE…
251
ment (due to noisy environment) as well as efficiency
and low complexity, owing to its restricted isometry pro-
perty (RIP) [13]. In an undesirable condition of low SNR
licensed signals, we also propose a DCWSS based SP
(DCWSSSP) method that jointly reconstructs the sparse
signal from multi-CR received signals. Finally, we com-
pare the performance of the DCWSSSP algorithm with
those of the schemes using compressive sampling mat-
ching pursuit (Cosamp) [12] and OMP [6], [10] to illu-
strate the accuracy of the proposed method.
The rest of this paper is organized as follows: section
Ⅱ presents the relevant concepts and terminology of the
CS reconstruction, while a compressive spectrum-sensing
scheme for single CR is presented in Section Ⅲ. An
extension to the collaborative compressed spectrum sens-
ing for multiple CR is shown in Section Ⅳ, while Sec-
tion V demonstrates the corroborating simulation results
to illustrate the effectiveness of the novel approach in
detecting the spectrum holes. Finally, concluding rema-
rks are given in Section Ⅳ.
Ⅱ. Preliminaries
2-1 Signal Model
We assume that the frequency range of the signal
consists of max I channels with equal bandwidth. The
spectrum-sensing model presented here includes a fusion
center that collects data from J CR nodes. An AIC is
used to sample the received signal at each CR node.
Finally, the determination of which bands are occupied
by licensed users (LUs) at the fusion center.
2-2 Compressed Sensing of Analog Signals and the
Restricted Isometric Property x(t), [0, ]t TÎ
We present an analog signal x(t), in a discrete format
as a finite weighted sum of the basic elements as [7]～
[9]:
1
( ) ( )
N
i i
i
x t s ty
=
= å (1)
where x is an 1N ´ vector x=Ψs , which is represented
in the sparse form of an 1N ´ vector s with K N<<
non-zero elements si via the N N´ matrix Ψ . CS de-
monstrates that x can be recovered using M N<<
measurements [13]. The measurements y are expressed
as:
y = Φx+ n = ΦΨs + n (2)
Several choices are available for the distribution of Φ,
such as the Gaussian, Bernoulli, or Fourier ensembles.
The reconstruction stage is performed by solving the
following standard approach to an objective function as
according to:
1s
min . .s y = ΦΨsst (3)
The problem (3) can be efficiently solved using BP or
some types of constructive algorithms such as matching
pursuit (MP) and OMP [3], [4].
In order to ensure the accuracy of each reconstruction
algorithm, the projected matrix Φ must satisfy the RIP
[14], which is presented as follows:
Definition 1 (Truncation): Let M N´
Î RF M N´
Î RF
M N´
Î RF and N
Îx R . The matrix TF with { }1, ,T NÌ L
has an i-th column ( )i TÎ in F and Tx is calculated
through TF .
Definition 2 (RIP): The matrix M N´
Î RF satisfies the
RIP with ( ), KK s for ,0 1KK M s£ £ £ , if
( ) ( )
2 2 2
2 2 2
1 1q q qK T Ks s- £ £ +F (4)
for all { }1, ,T NÌ L and for all T
qÎR , and ( )1 K T Ts l- £
( ) ( ) ( )min max1 1H H
K T T T T Ks l l s- £ £ £ +F F F F , where ( )min
H
T Tl F F
and ( )max
H
T Tl F F represent the minimal and maximal
eigenvalues of H
T TF F , respectively.
Ⅲ. Compressive Spectrum Sensing at a
Single CR [6], [10]
In this section, we summarize the procedures for the
receiving and reconstruction at each CR node.
Fig. 1 shows that the analog input x(t) is [ ]x
T
k kN ix += ,
and the output of AIC is y
T
k kM jy +
é ù= ë û where 0,1,2...k = ,
0, , 1i N= +K , and 0, , 1j M= +K . The AIC is modeled
by the M N´ projected matrix AF as
k A ky = Φ x (5)
Using some mathematical operations [6], [10], we ha-
ve
y xr = Φr (6)
where [ ]0 ( )xr
T
xr i= and 0 ( )yr
T
yr ié ù= ë û , 1, , 1i N N= - + +K
1, , 1i N N= - + +K are 2 1N ´ and 2 1M ´ autocorrelation vectors, res-
pectively.
Fig. 1. CS acquisition at an individual CR sensing receiver.

JOURNAL OF THE KOREAN INSTITUTE OF ELECTROMAGNETIC ENGINEERING AND SCIENCE, VOL. 11, NO. 4, DEC. 2011
252
After acquiring the output vector of the autocorrelation
operation, we use the wavelet-based approach as pre-
sented in [15], [16] to detect the band edge locations.
For an experiment when N M<< in [5], the edge
spectrum sz can be determination from measurements
and the relation between sz and xr is:
x sr = Gz (7)
where sz is the discrete 2 1N ´ vector and ( )
1
FW
-
=G G .
The 2 2N N´ matrices G , W, and F represent a first
derivative operation, Wavelet, and Fourier transforms,
respectively.
Combining (6) and (7), the optimization problem for
an edge spectrum reconstruction is given by:
1
min . .s y sz r = ΦGzs t (8)
To solve this problem, we use the CWSS based SP
(CWSSSP) that is presented in the next section. The
spectrum estimate can now be evaluated as a cumulative
sum of elements in vector ( )s s
ˆ ˆz =
T
z ié ùë û , 1, , 2i N= K .
Therefore, the estimated values of PSD are given by
( ) ( )x s
1
ˆ ˆ=
n
k
S n z k
=
å
(9)
Ⅳ. Collaborative Compressed Spectrum Sensing
In Fig. 2, ( )jx t is the input of AIC at the j-th CR
node. The output of AIC is processed to give the 2 1M ´
autocorrelation vector ,yr j . The fusion center collects the
autocorrelation vectors and applies the DCWSSSP app-
roach to reconstruct the J received PSD ,
ˆ
x jS ; 1, ,j J= K
and then obtains an average PSD. Finally, the center
determines whether the frequency ranges are occupied,
based on the average PSD.
4-1 Overview of the SP Approach [17]
The SP algorithm [17] is less complicated but it re-
sults in a comparable recovery performance to LP te-
Fig. 2. DCWSS for multiple CR nodes.
Fig. 3. Subspace pursuit algorithm applied to reconstruc-
tion.
chniques.
First, the matrix A = ΦG can be expressed in a row
of its columns as:
[ ]1 2 2A a a a N= L (10)
The next step is solving the problem (8) by using an
SP technique. We set the truncation for subspace AS of
the matrix 2 2M N´ A as in Definition 1, Section III
and ASspan( ) is represented to the space span of AS .
In addition, the matrix 2 2M N´ A also satisfies the
RIP, as in (4), Definition 2, Section III, by replacing M
by 2M, N by 2N, each TF by AS , and each F by A .
The l1-linear program approach can successfully
reconstruct a K-sparse signal if the RIP must be satisfied
with constants Ks , 2Ks and 3Ks , which have a con-
dition 2 3 1K K Ks s s+ + < [18]. However, in [14], the
authors improved the above condition to 2 2 1Ks < - .
For any given vector
2M
y Îr R , the projection of ry onto
the subspace ASspan( ) is denoted by ,ry p and can be
computed as:
( ) †
, , :r r A A A ry p y S S S yproj= =
(11)
Note that ( )
1†
A A A AH H
S S S S
-
= is the pseudo-inverse of
the matrix AS ,where subscript
H
denotes the conjugate
transposition. Corresponding to the projection vector, the
projection residue vector ,ry r is defined as
( ), ,, :r r A r ry r y S y y presid= = - (12)
Fig. 3 illustrates the schematic diagram of iterations in
the SP algorithm [17], demonstrating that the subspace is
updated during each iteration; i.e., elements can be added
to or deleted from the subspace.
The following subsection represents the algorithm to
solve the above problem.
4-2 The Jointly Recovery SP Algorithm

253
The advantage of the SP algorithm comparing with the
OMP method is the way to generate l
S , that is the
estimate of the correct support set S .
We describe the procedure of this algorithm as
follows:
1. Input:
․A 2 2M N´ matrix A.
․A 2M J´ input matrix ,1 ,2 ,y y yR r r r Jé ù= ë ûL recei-
ved from J CR sensing receivers.
2. Output: The 2N J´ estimated signal s s s sZ z z z=
,1 , 2 ,s s s sZ z z z Jé ù= ë ûL , the average of j PSD es-
timate vectors ( )
x
ˆS J
.
3. Procedure:
1) Initialization:
․For each j-th CR (j=1, …, J), we have
0
,
0 0 0
,1 ,2 ,
j
y j
j j j K
K indiceswith respecttothelargest
elementsin
S
u u u
ì üï ï
= í ý
ï ïî þ
é ù= ë û
H
A r
L
and then calculate
{ }0 0
0 0
1 2
j
o
K
S average S
u u u
=
é ù= ë ûL
,
Where ,
1
1 J
o o
k k j
jJ
u u
=
= å .
․The projection residue vector for the j-th CR is
( )0
0
ˆ, , ,r r Ar j y j S
r e s id= .
2) Iteration: The following steps will be performed at
every l th- iteration.
․For each j-th CR (j=1, …, J), we evaluate
{ }1 1
1 1 1
1 2
늿l l
j
l l l
K
S average S
u u u
- -
- - -
=
é ù= ë ûL
1
H 1
,
ˆ
A r
l
j l
r j
K indices respect tothelargest
elementsin
S -
-
ì üï ï
= í ý
ï ïî þ ,
1ˆl
jS -
(13)
where
1 1
,
1
1
, 1, 2, ,
J
l l
k k j
j
k K
J
u u- -
=
= =å K ,
(14)
and then 1 1ˆl l l
S S S- -
=% U .
․For each j-th CR, we set the projection coefficients:
†
, , ,z A rls p j y jS
= % .
․
,
l
j
s j
K indiceswith respecttothelargest
elementsin
S
ì ü
= í ý
î þz
; And we cal-
culate l
S using (13), (14) and replacing (l—1) with l.
․The residue vector of the projection for the j-th CR
is ( ), , ,r r A l
l
r j y j S
resid= .
3) Termination test: The SP iteration is terminated
when 1
, ,2 2
m in m inr rl l
r j r j
-
> , 1,2, ,j J= K . Then
let 1l l
S S -
= and quit the iteration. If the limit is
not reached, increase l and return to the iteration.
4) Store the results:
The estimated signal ˆzs, j satisfies { }1, ,
ˆz 0ls, j N T-
=L and
†
,
ˆz A rl ls, j y jS S
= . The j-th PSD estimate vector is
( ) ( )x , s ,
1
ˆ ˆS =
n
j j
k
n z k
=
å .
The average of J PSD estimate vectors is
(15)
4-3 Performances:
4-3-1 MSE Performance
The MSE of PSD from our approach is calculated as:
( )
( ) ( )
( )
2
2
2
2
ˆ
MSE = E
x x
x
S S
S
J J
J
J
ì ü-ï ï
í ý
ï ï
î þ (16)
where ( )ˆ
xS J
and ( )
xS J
denote the average PSD output and
the PSD, respectively for in case of signals sampled at
the Nyquist rate.
4-3-2 Probability of Detection
To compute the detection probability Pd, we apply the
energy detection method, where the test static is cal-
culated from the averaged PSD estimate ( )ˆ
xS J
[19]. As in
[10], we identify the static test as:
( )
( )
2
,
1 1 1 1
1
= ( )
IL J H
J
I h j
i I L j h
E X i
JH = - + = =
å åå (17)
where L is the total samples from each channel, I maxI=
1,2, ,I maxI= K , H is the total number of blocks, X repre-
sents the Fourier transform from x.
Using the Neyman-Pearson hypothesis test, we deter-
mine the decision threshold m [19]:
( )
( )
,
1
J
f
JH
JH
P
JH
mæ ö
Gç ÷
è ø= -
G (18)
, ,

254
where (.,.)G is upper incomplete gamma function [20],
Sec. (8.350)], Γ(.) is the gamma function [20], Sec.
(13.10)]. Hence, the probability of detection ( )J
dP is eva-
luated as
( ) ( )
{ }
1
1
Pr
aI
J J
d I
I I
P E
a
m
=
= >å
(19)
where , 1, ,iI i a= K denote the indices of a active cha-
nnels.
Ⅴ. Simulation Results
In this section, the simulation results are shown to
evaluate the proposed approach. In this simulation mo-
del, the frequency band ranges from —38.05 to 38.05
MHz, which is the same as that described in [21], and
the number of channels is maxI=10 with 7.61 MHz
bandwidth. The OFDM frame length TF includes 68
symbols, and each of these super-frames contains four
frames. The number of carriers per symbol is C=1,705
with a duration TS, composed of a useful part TU and a
guard interval ∆ (set to 0 in this simulation).
For this scheme, the over-sampling factor is 02, and
only 50 % of the channels are active. The SNRs of the
active channels are assumed to be in the range [—10 dB,
—8 dB] and the AGWN variance is
2
1ns = . The smoo-
thing signal scheme is performed using a Gaussian
wavelet. The length of input signal is 2N=512 and the
compressed rate is varying from 5 % to 100 %, and
H=160, and ΦA has a zero-mean Gaussian ensemble wi-
th variance 1/M. The number of PSD samples of each
channel is L=25.
Fig. 4 illustrates the MSE performances for the SP,
OMP, iteratively reweighted least squares with regula-
rization (IRLS) [21] and Cosamp algorithms [22]. The
results show that better performance is achieved for SP
than for these other approaches, while all versions take
Table 1. Parameters for the simulations.
Parameter 2 k mode
Elementary period T 7 / 64 sm
Number of carriers C 1,705
Value of carrier number minC 0
Value of carrier number maxC 1,704
Duration of symbol part UT
2, 048
224
T
sm
´
Carrier spacing 1/ UT 4,464 Hz
Spacing between carriers minC and
( )max 1 / UC C T-
7.61 MHz
Fig. 4. MSE for SP, IRLS, OMP and Cosamp approaches
versus compression rate M/N for various numbers of
collaborating CRs (SNR=[—10 dB, —8 dB]).
the same time to reach convergence. The OMP algorithm
has the worst performance in the conditions used for this
simulation, with a low sampling factor corresponding to
low sparsity and the noisy environment. The SP algo-
rithm is robust in this case because it adds the good basis
candidates and it also removes the bad candidates. This
figure also shows the signal recovery quality, where
MSE decreases when the compression rate M/N increa-
ses. However, to complement this degradation, we take
advantage of the multi-CR scheme, where MSE can be
significantly reduced. Therefore, we reduce the cost of
high speed by using the CS method and we also improve
the MSE by exploiting the multiple CRs.
The detection performance is shown in Fig. 5, which
Fig. 5. Probabilities of detection Pd for SP and Cosamp ap-
proaches versus compression rate M/N for various num-
bers of collaborating CRs (SNR=[—10 dB, —8 dB]).

255
depicts the probability of detection ( )J
dP vs. the compre-
ssion ratio M/N when the number of CRs is J=1 and 5
at a given ( )J
fP of 0.01. This figure demonstrates that the
detection probability was high at a low compression ratio
M/N. The use of multiple CRs significantly improves the
detection probability. In addition, the probability of
detection is improved by the SP approach compared with
the IRLS and Cosamp algorithms.
Ⅵ. Conclusion
In this paper, we used the DCWSS and SP algorithm
to reduce the recovery error in the reconstruction stage in
the CRN. A new iterative algorithm, termed the DCW-
SSSP approach, is exploited for joint compressive spec-
trum sensing.
This research was supported by Basic Science
Research Program through the National Research
Foundation of Korea(NRF) funded by the Ministry of
Education, Science and Technology(No. 2010-0004-
865)
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256
Le Thanh Tan Hyung-Yun Kong
received the B.S. degrees in Telecommu-
nication Engineering from Poly-technique
University of Danang, Vietnam in 2005.
In 2008, he got the degree of Master
from Hochiminh University of Technol-
ogy, Vietnam in major of Electrical and
Electronics Engineering. Since 2010, he
has been studying Ph.D. program at
University of Ulsan, Korea. His major researches are Cogni-
tive Radio Network, Cooperative Communication.
received the M.E. and Ph.D. degrees in
electrical engineering from Polytechnic
University, Brooklyn, New York, USA,
in 1991 and 1996, respectively, He re-
ceived a B.E. in electrical engineering fr-
om New York Institute of Technology,
New York, in 1989. Since 1996, he has
been with LG electronics Co., Ltd., in
the multimedia research lab developing PCS mobile phone
systems, and from 1997 the LG chairman's office planning
future satellite communication systems. Currently he is a pro-
fessor in electrical engineering at the University of Ulsan,
Korea. His research area includes channel coding, detection and
estimation, cooperative communications, cognitive radio and
sensor networks. He is a member of IEEK, KICS, KIPS,
IEEE, and IEICE.

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