Projected Barzilai-Borwein Methods Applied to Distributed Compressive Spectrum Sensing

Projected Barzilai-Borwein Methods Applied to
Distributed Compressive Spectrum Sensing
Le Thanh Tan∗, Hyung Yun Kong∗, Vo Nguyen Quoc Bao∗
∗School of Electrical Engineering
University of Ulsan, San 29 of MuGeo Dong, Nam-Gu, Ulsan, Korea 680-749
Email: {tanlh,hkong,baovnq}@mail.ulsan.ac.kr
Abstract—Cognitive radio allows unlicensed (cognitive) users to
use licensed frequency bands by exploiting spectrum sensing tech-
niques to detect whether or not the licensed (primary) users are
present. In this paper, we present a compressed sensing applied
to spectrum-occupancy detection in wide-band applications. The
collected analog signals from each cognitive radio (CR) receiver
at a fusion center are transformed to discrete-time signals by
using analog-to-information converter (AIC) and then employed
to calculate the autocorrelation. For signal reconstruction, we
exploit a novel approach to solve the optimization problem
consisting of minimizing both a quadratic (l2) error term and an
l1-regularization term. Specifically, we propose the Basic gradient
projection (GP) and projected Barzilai-Borwein (PBB) algorithm
to offer a better performance in terms of the mean squared
error of the power spectrum density estimate and the detection
probability of licensed signal occupancy.
I. INTRODUCTION
The explosive growth of wireless communication has made
the problem of spectrum utilization more critical. On the one
hand, the increasing diversity and demand of high quality-
of-service applications are the main reasons of the crowded
nature of spectrum allocation. On the other hand, the spectrum
scarcity is not the result of heavy usage of spectrum, but
is due to the inefficiency of the static frequency allocation.
For example, the typical spectrum utilization of around five
percent or even less is reported in [1] . To solve this problem,
the Federal Communications Commission (FCC) proposed the
opening of licensed bands to unlicensed users and cognitive
radio (CR) was born to improve the utilization of spectrum
resource. In addition, the IEEE has established an IEEE 802.22
workgroup to build the standards of WRAN based on CR
techniques [2].
Spectrum sensing for wide-band CR applications associates
with considerable technical challenges. The radio terminals are
required to employ a bank of tunable narrowband bandpass
filters and search one narrow frequency band at a time. The
large bandwidth operation requires an unfavorably large num-
ber of RF components. Furthermore, high speed processing
units (DSPs or FPGAs) are needed to flexibly search over
multiple frequency bands concurrently. In particular, input
analog signals must be converted to discrete-time signals at
Nyquist sampling rate or higher by using analog-to-digital
converter (ADC). However, the extremely high sampling rates
of wide-band ADCs are a main challenge with this model.
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 sufficient
statistic when traditional linear signal reconstruction methods
are employed.
Compressed sensing (CS) [3], [4] builds on the surprising
revelation that a signal having a sparse representation in one
basis can be recovered from a small number of projections
onto a second basis that is incoherent with the first. Especially,
compressed sampling approach can get the sparse signal at
the rates lower than the Nyquist sampling method; signal
reconstruction which is a solution to a convex optimization
problem called min-l1 with equality constraints makes use
of the basic pursuit (BP) or some modified methods such as
orthogonal matching pursuit (OMP), tree-based OMP (TOMP)
[5], [6]. In [5], authors present a model of a single wide-band
CR that uses CS based spectrum sensing schemes.
This paper considers the problem of determining spectrum
occupancy of a wide-band system. Our proposed CR system
has a number of CRs and a centralized fusion center that
collects data from individual CRs with different signal-to-noise
ratios (SNRs) and decides these spectra to be available or not.
Specifically, the same wide-band analog signal is transformed
to discrete-time signal while preserving its salient information
by using a so-called analog-to-information converter (AIC).
For that kind of sparse input signal, an AIC can be acquired
at sub-Nyquist rates (matching the information rate of the
signal), and it also offers the capability of extracting the feature
of data; therefore, the performance of the AIC system is
comparable to that of ADC system in term of SNR or effective
number of bits (ENOB) [7], [8], [9]. Then output signals of
AIC are used to generate autocorrelation vectors that will be
collected at the fusion. To obtain an estimate of the signal
spectrum, we propose a novel version of distributed CS algo-
rithm based on [10] by using a standard approach to minimize
an objective function includes a quadratic (squared l2) error
term combined with a sparseness-inducing (l1) regularization
term. It can be easily observed that basic gradient projec-
tion (GP) can perform the high quality reconstruction [11];
however, the time-consuming property makes this approach
not appropriate for spectrum sensing scheme which requires
time-constraint. Thus, the novel approach, which exploits the
projected Barzilai-Borwein (PBB) technique embedded in a
continuation heuristic to recover the efficient performance
[12], is introduced to improve both time-reduction and the
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE DySPAN 2010 proceedings
978-1-4244-5188-3/10/$26.00 ©2010 IEEE

performance of signal reconstruction.
We also compare the performance of the distributed com-
pressive spectrum sensing scheme with that of the scheme of
[5] for a single CR to show the gains accrued from spatial
diversity and exploiting the joint sparsity structure. We use
(i) the mean squared error (MSE) between the reconstructed
power spectrum density (PSD) estimate and the PSD based
on Nyquist rate sampling, and (ii) the probability of detecting
spectrum occupancy over the channels as performance mea-
sures.
The rest of this paper is organized as follows. Section II
contains the model of the wide-band analog signal compressed
sensing. A compressive spectrum sensing scheme for single
CR is presented in Section III. And an extention to collabo-
rative compressed spectrum sensing for multiple CR is shown
in Section IV. Section V demonstrates simulation results.
Finally, concluding remarks are given in Section VI.
II. SYSTEM MODEL OF WIDEBAND ANALOG
SIGNAL COMPSESSED SENSING
A. Signal model
We consider the frequency range of interest to be comprised
of maxI non-overlapping consecutive spectrum bands, a CR
network consisting of J CRs and a centralized fusion center.
Sensing is performed periodically at each CR and the results
are sent to the fusion center, where a decision is made on
whether or not there is a licensed signal present in each
channel.
B. Overview of compressed sensing
According to Donoho [3], in CS theories, an N ×1 vector of
discrete-time signal x = Ψs, where Ψ is the N × N sparsity
basis matrix and s is the N ×1 vector with K N non-zero
(and large enough) entries si, can be used to reconstruct the
signal from M measurements; especially, M depends on the
reconstruction algorithm and is usually much less than N. This
measurement can be done by projecting x on to an M × N
basis matrix Φ that is incoherent with Ψ [13]
y = Φx = ΦΨs. (1)
The reconstruction is done by solving the following l1-norm
optimization problem as
ˆs= arg min
s
s 1 s.t. y = ΦΨs. (2)
Linear programming techniques, e.g., basis pursuit [14], or
iterative greedy algorithms [15] can be used to solve (2).
C. Compressed sensing of analog signals
Because CS was proposed for discrete-time signal process-
ing, we must use ADC sampling at Nyquist rate to discreterize
the analog signal before applying the CS. After that, the
compressed sensed data are sent to DSP blocks for further
manipulation. While it is true that the data volume to be
processed by DSP blocks is reduced due to the CS, a high-
speed ADC sampling at Nyquist rate is still required when
the received signal is wideband. It is natural to think about
ways to avoid the high-speed ADC by applying CS to the
analog signal directly. A related idea was first described in
[8], where the analog signal was first demodulated with a
pseudo-random chipping sequence p(t), then passed through
an analog filter h(t), and the measurements were obtained in
serial by sampling the filtered signal at sub-Nyquist rate. The
serial sampling structure is appropriate for real-time process-
ing. However, to achieve a satisfactory signal reconstruction
quality, the order of the filter is usually higher than 10. In
addition, because the measurements are obtained by sampling
the output of the analog filter sequentially, they are no longer
independent due to the convolution in the filter, which brings
some redundancy in the measurements.
Specifically, suppose that we have an analog signal x(t)
which is K − sparse over some basis Ψ for t ∈ [0, T] as in
the following expression:
x(t) =
N
i=1
siψi(t), (3)
where x is the N × 1 vector x = Ψs, Ψ is the N × N sparsity
basis matrix Ψ = [ψ0(t), ψ1(t), . . . , ψN (t)] and s an N × 1
vector with K N non-zero elements si. It has been shown
that x can be recovered using M = KO(log N) non-adaptive
linear projection measurements on to an M × N basis matrix
Φ that is incoherent with Ψ [13]. The received signal y can
be viewed as the transmitted signal plus some additive noise
y = Φx + n = ΦΨs + n. (4)
There are several choices for the distribution of Φ such as
Gaussian, Bernoulli.
Reconstruction is achieved by solving the l1-norm opti-
mization problem as in (2). In this paper, the reconstruction
problem, that has been highly interested in solving the convex
unconstrained optimization problem, is a standard approach
consisting in minimizing an objective function which includes
a quadratic (squared l2-norm) error term combined with
a sparseness-inducing (l1-norm) regularization term. So the
problem can be given by
min
s
1
2
y − ΦΨs
2
2 + τ s 1. (5)
Basic GP is able to solve a sequence of problems (5) effi-
ciently for a sequence of values of τ. The gradient projection
algorithms for solving a quadratic programming reformulation
of a class of convex nonsmooth unconstrained optimization
problems are significantly faster (in some cases by orders
of magnitude), especially in large-scale settings. Instances of
poor performance have been observed when the regularization
parameter is small, but in such cases the gradient projection
methods can be embedded in a simple continuation heuristic
to recover their efficient practical performance. The new
algorithms are easy to implement, work well across a large
range of applications, and do not appear to require application-
specific tuning.

Analog filter
h(t)
Autocorrelation
x(t)
yk
ry
pc(t)
k/M
quantizer
AIC block
Fig. 1. CS acquisition at individual CR sensing receiver.
III. COMPRESSIVE SPECTRUM SENSING AT
SINGLE CR
We begin by describing the CS acquisition and recovery
scheme for a single CR (J = 1) case. Fig. 1 depicts the
acquisition at a single CR sensing receiver. The analog base-
band signal x(t) is sampled using an AIC. Following the
circuit implementation of the AIC system in series of previous
works [7], [8], [9], an AIC may be conceptually viewed as an
ADC operating at the Nyquist rate, followed by compressive
sampling. Denote the N × 1 stacked vector at the input of the
ADC by
xk = xkN xkN+1 ... xkN+N+1
T
k = 0, 1, 2 ...,
(6)
and the M × N compressive sampling matrix by ΦA. The
output of the AIC denoted by the M × 1 vector
yk = ykM ykM+1 ... ykM+M+1
T
k = 0, 1, 2 ...,
(7)
is given by
yk = ΦAxk. (8)
The respective N × N and M × M autocorrelation matrices
of the compressed signal and the input signal vectors in (6)
and (7) are related as follows:
Ry = E ykyH
k = ΦARxΦH
A , (9)
where subscript H denotes the Hermitian. The elements of the
matrices in (9) are given by: [Ry]ij = ry (i − j) = r∗
y (j − i),
[Rx]ij = rx (i − j) = r∗
x (j − i).
The respective 2N × 1 and 2M × 1 autocorrelation vectors
corresponding to (6) and (7) can be expressed as follows:
rx = 0 rx(−N + 1) ... rx(0) ... rx(N − 1)
T
,
(10)
ry = 0 ry(−M + 1) ... ry(0) ... ry(M − 1)
T
,
(11)
here the first zero values are artificially inserted. And these
above vectors represent the first column and row of the
respective autocorrelation matrices. To obtain the CS recovery
like the formula (5), we must to make the relation between
the autocorrelation vectors in (10) and (11). Using operations
in matrix algebra, we can derive as
ry = Φrx, (12)
note that
Φ =
¯ΦAΦ1
¯ΦAΦ2
ΦAΦ3 ΦAΦ4
. (13)
Denote that φ∗
i,j is the (i, j)-th element of ΦA, the M × N
matrix ¯ΦA has its (i, j)-th element given by
¯ΦA i,j
=
0
φM+2−i,j
i = 1, j = 1, ..., N,
i = 1, j = 1, ..., N,
(14)
and the N × N matrices Φ1, Φ2, Φ3, Φ4 are defined
as Φ1 = hankel [0N×1] , 0 φ∗
1,1 ... φ∗
1,N−1 ,
Φ2 = hankel φ∗
1,1 ... φ∗
1,N , φ∗
1,N 01×(N−1) ,
Φ3 = toeplitz [0N×1] , 0 φ1, N ... φ1, 2 ,
Φ4 = toeplitz φ1,1 ... φ1,N , φ1,1 01×(N−1) ,
where hankel(c, r) is a hankel matrix (i.e., symmetric and
constant across the anti-diagonals), note that c is the first
column and r is the last row of this matrix. toeplitz(c, r)
is a toeplitz matrix (i.e., symetric and constant across the
diagonals), note that c is the first column, and r is the first
row of this matrix. And 0a×b is the a × b zero matrix.
We also know that using the wavelet-based edge detection
in [16-17], the band boundaries (locations) can be recovered
from 2N-1 local maxima of the wavelet modulus zs and the
band number is determined by the number of local peaks; as
an experiment when N M in [5], zs can be recovered
under the sparseness constraint, and therefore there is a
linear transformation equality linking zs to the compressed
measurement vector ry. And rx has a sparse representation in
the edge spectrum domain [5], that is
rx = Gzs, (15)
where zs is the discrete 2N × 1 vector, and G = (ΓFW)
−1
.
The 2N × 2N matrices W and F represent respectively a
wavelet-based smoothing and a Fourier transform. The 2N ×
2N matrix Γ is a derivative operation given by
Γ =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 0 · · · 0
−1
... · · · 0
0
...
...
...
0 · · · −1 1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
Combining (12) and (15), we can formulate the CS recon-
struction of the edge spectrum as a convex unconstrained
optimization problem:
min
zs
1
2
ry − ΦGzs
2
2 + τ zs 1 (16)
To solve the above problem, we use the GP approach which
is described in the Section IV for an individual CR case.
The spectrum estimate can be evaluated as a cumulative sum
of terms ˆzs = ˆzs (1) ˆzs (2) · · · ˆzs (2N)
T
. The
discrete components of the PSD estimate are given by
ˆSx (n) =
n
k=1
ˆzs (k) (17)

1 k,1 y,1
J k,J y,J
.
Fig. 2. Distributed compressive spectrum sensing scheme for multiple CRs.
IV. COLLABORATIVE COMPRESSED SPECTRUM
SENSING
Let xj(t) be the wide-band analog baseband signal received
at the j-th CR sensing receiver. Each CR sensing receiver pro-
cesses the received signal to obtain an 2M ×1 autocorrelation
vector ry, j of the compressed signal, as in the CS acquisition
step described in Section III, these vectors are then sent to
the fusion center. The fusion center applies a GP algorithm
to jointly reconstruct J received PSDs ˆSx,j, j = 1, . . . , J and
then obtains an average PSD. The average PSD is then used
to determine the spectrum occupancy.
A. Overview of GP approach
We now describe the GP algorithm used for reconstruction
of J PSDs. We write A = ΦG in terms of its columns
A = a1 a2 · · · a2N . (18)
At the j-th CR, we introduce vectors uj and vj and make the
substitution zs,j = uj − vj, uj ≥ 0, vj ≥ 0, j = 1, · · · , J.
Here uj (i) = (zs,j (i))
+
= max {0, zs,j (i)} and vj (i) =
(−zs,j (i))
+
= max {0, −zs,j (i)} for all i = 1, . . . , 2N. Note
that (a)
+
= max {0, a}. Therefore, we have zs,j = 1T
2N uj +
1T
2N vj, where 12N = [1, 1, . . . , 1]
T
is the vector consisting
of 2N ones. The problem (16) can be modified as
min
u,v
1
2 ry,j −A (uj − vj)
2
2 +τ1T
2N uj +τ1T
2N vj
s.t. uj ≥ 0
vj ≥ 0
(19)
Problem (19) can be written in more standard bound-
constrained quadratic programming (BCQP) form as
min
p
cT
pj + 1
2 pT
j Bpj ≡ F pj ,
s.t. pj ≥ 0
(20)
where pj =
uj
vj
, c = τ1T
4N +
−b
b
, b = AT
ry,j
and B =
AT
A −AT
A
−AT
A AT
A
. The next step is solving the
problem (20) by using a GP technique. From iterate p
(k)
j to
iterate p
(k+1)
j , we must follow the below steps,
• Step 1. Choose the scalar parameters α
(k)
j > 0.
• Step 2. Set: w
(k)
j = p
(k)
j − α
(k)
j ∇F p
(k)
j
+
.
• Step 3. Choose the second scalar λ
(k)
j ∈ [0, 1].
• Step 4. Set: p
(k+1)
j = p
(k)
j + λ
(k)
j w
(k)
j − p
(k)
j .
The following subsections represent two algorithms to solve
the above problem coresponding to two different ways of
choosing α
(k)
j and λ
(k)
j .
B. Basic Gradient Projection:The GP - Basic Algorithm
In this algorithm, we search from each iterate p
(k)
j along the
negative gradient −∇F p
(k)
j , projecting onto the nonnega-
tive orthant, and performing a backtracking line search until
sufficient decrease is attained in F. We define the vector g(k)
as
g
(k)
j,i =
∇F pj
(k)
i
, if p
(k)
j,i > 0 or ∇F pj
(k)
i
< 0
0, otherwise.
(21)
where i = 1, . . . , 2N. The procedure of this algorithm is
described as follows:
1) Input:
a) An initial p(0)
= p
(0)
1 p
(0)
2 · · · p
(0)
J .
b) A 2M × J data matrix R =
ry,1 ry,2 · · · ry,J received from J
CR sensing receivers.
c) Choose parameters β ∈ (0, 1) and μ ∈ (0, 1/2).
d) Set k = 0.
2) Output: A 2N × J reconstruction matrix Zs =
zs,1 zs,2 · · · zs,J , the average of J PSD esti-
mate vectors ˆS
(J)
x .
3) Procedure:
a) Step 1. Compute α0,j as the following expression
[12]:
α0,j =
g
(k)
j
T
g
(k)
j
g
(k)
j
T
Bg
(k)
j
. (22)
Note that α0,j is solved from the expression:
α0,j = arg min
αj
F p
(k)
j − αjg
(k)
j . (23)
Then to guarantee that α0,j is not too small or too
large, we replace α0,j by mid (αmin, α0,j, αmax).
Here mid (α1, α2, α3) is defined to be the middle
value of three scalar values.
b) Step 2. Backtracking line search: choose
α
(k)
j to be the first number in the sequence
α0,j, βα0,j, β2
α0,j, . . . and satisfy the following
inequality
F p
(k)
j −α
(k)
j ∇F p
(k)
j
+
≤ F p
(k)
j −
μ∇F p
(k)
j
T
p
(k)
j − p
(k)
j −α
(k)
j ∇F p
(k)
j
+
(24)

and update the new set of values
p
(k+1)
j = p
(k)
j − α
(k)
j ∇F p
(k)
j
+
.
c) Step 3. Termination test:
• Condition: We now convert p
(k)
j =
u
(k)
j
T
, v
(k)
j
T T
to an approximate
solution z
(k)
s,j,GP = u
(k)
j − v
(k)
j . And scanning
through the entire J CR to check the termination
condition, i.e. the convergence gradient (CG)
iteration is terminated when satisfying
ry,j − Azs,j
2
2 ≤ εD ry,j − Azs,j,GP
2
2 ,
(25)
where εD is a small positive parameter. How-
ever, the termination iteration is also performed
when the number of CG steps reaches to
maxiterD.
• We perform the convergence test and terminate
with approximation solution p
(k+1)
j if it is satis-
ﬁed the above conditions; otherwise we increase
k to k + 1 and go back to step 1.
d) Step 4. Store the results: Store pj =
(uj)
T
, (vj)
T
T
, and calculate the reconstruction
vector ˆzs,j = uj − vj. The j-th PSD estimate
vector is ˆSx,j (n) =
n
k=1
ˆzs,j (k). And the average
of J PSD estimate vectors is
ˆS(J)
x =
1
J
J
j=1
ˆSx,j. (26)
C. Projected Barzilai-Borwein Reconstruction Algorithm
The improvement of this algorithm is updating the step by
the following formula [18]
γ(k)
= −H−1
k ∇F p(k)
, (27)
where Hk is an approximation to the Hessian of F p(k)
.
The procedure of this algorithm is similar to the basic GP
algorithm except the following steps:
1) Step 1. Compute step γ
(k)
j for the j-th CR as the
following expression:
γ
(k)
j = p
(k)
j − α
(k)
j ∇F p
(k)
j
+
− p
(k)
j . (28)
2) Step 2. Line search: The scalar λ
(k)
j , (λ
(k)
j ∈
[0, 1]) will be found to minimize F p
(k)
j + λ
(k)
j γ
(k)
j
and update the new set of values p
(k+1)
j =
p
(k)
j − α
(k)
j ∇F p
(k)
j
+
. Because F is quadratic, the
line search parameter λ
(k)
j can be evaluated by the
following closed-form expression:
λ
(k)
j = mid
⎧
⎪⎨
⎪⎩
0,
γ
(k)
j
T
∇F p
(k)
j
γ
(k)
j
T
Bγ
(k)
j
, 1
⎫
⎪⎬
⎪⎭
.
Note that if γ
(k)
j
T
Bγ
(k)
j = 0, we choose λ
(k)
j = 1.
3) Step 3. Update α
(k)
j : Denote
ξ
(k)
j = γ
(k)
j
T
Bγ
(k)
j . (29)
If ξ
(k)
j = 0, let α
(k+1)
j = αmax, otherwise
α
(k+1)
j = mid
⎧
⎪⎨
⎪⎩
αmin,
γ
(k)
j
2
2
ξ
(k)
j
, αmax
⎫
⎪⎬
⎪⎭
.
D. Performances
1) MSE Performance: The normalized MSE of estimated
PSD is computed by
MSE(J)
= E
⎧
⎪⎨
⎪⎩
ˆS
(J)
x − Sx
(J)
2
2
Sx
(J)
2
2
⎫
⎪⎬
⎪⎭
, (30)
where ˆS
(J)
x and Sx
(J)
denote the average of the J PSD
estimate vectors based on our compressed sensing approach
and the periodogram using the signals sampled at the Nyquist
rate, respectively.
2) Detection performances: We evaluate the probability
of detection Pd based on the averaged PSD estimate ˆS
(J)
x .
The detection analysis to follow, strictly speaking, holds only
for samples collected at Nyquist rate. We however use this
as a simple way to analyze the detection performance in
the compressive sampling case as well. The decision of the
presence of licensed transmission signals in the certain channel
is made by an energy detector using the estimated frequency
response over that channel, i.e., the test statistic is
E
(J)
I =
IK
i=(I−1)K+1
ˆS(J)
x (i), I = 1, 2, . . . , maxI, (31)
where I is the channel index, maxI is the number of channels,
and K is the number of PSD samples of each channel. The
PSD estimate of the j-th CR node can be evaluated as
ˆSx,j(i) =
1
H
H
h=1
|Xh,j(i)|
2
, (32)
whereXh,j(i) is the Fourier transform of the h-th block of the
received signal xh,j(n), j representing the CR node index,
n representing the time sample index, each block containing
2N time samples, and H denoting the number of blocks.
Substituting (26) and (32) to (31), the test static can be
obtained by
E
(J)
I =
1
JH
IK
i=(I−1)K+1
J
j=1
H
h=1
|Xh,j(i)|
2
. (33)
The decision rule is chosen as
E
(J)
I
H1
>
<
H0
μ, I = 1, 2, . . . , maxI, (34)

0.1 0.2 0.3 0.4 0.5
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Compression rate [M/N]
Reconstructionerror
Basic GP, 5CRs
PBB, 5 CRs
Basic GP, 1 CR
PBB, 1 CR
Ref. [5], 1 CR
Fig. 3. Reconstruction error (MSE) for Basic GP and PBB approaches versus
compression rate M/N for various number of collaborating CRs (SNRs of
active channels varying from 8dB to 10dB).
0.1 0.2 0.3 0.4 0.5
0
0.05
0.1
0.4
0.5
0.6
0.7
0.8
0.9
1
Compression rate [M/N]
Performance
Pd
, 5 CRs
Pd
, 1 CR
P
d
, 1CR, Ref. [5]
Pfa
, 5 CRs
Pfa
, 1 CR
Pfa
, 1 CR, Ref. [5]
Fig. 4. Probability of detection Pd and probability of a false alarm Pfa for
PBB versus compression rate M/N for various number of collaborating CRs
(SNRs of active channels varying from 8dB to 10dB).
where H0, H1 represent the hypotheses of the absence and
presence of primary signals, respectively, and μ is the decision
threshold. Under H0, E
(J)
I / σ2
n/ (JH) ∼ χ2
2JKH has a
central χ2
distribution with 2JKH degrees of freedom. The
probability of a false alarm P
(J)
fa can be obtained by
P
(J)
fa = 1 −
Γ JH, μ
JH
Γ (JH)
, (35)
where Γ(., .) is the upper incomplete gamma function [19, Sec.
(8.350)], Γ(.) is the gamma function [19, Sec. (13.10)]. Under
H1, the probability of detection P
(J)
d is evaluated by
P
(J)
d =
1
a
Ia
I=I1
Pr E
(J)
I > μ, (36)
where Ii, i = 1, . . . , a denote the indices of a active channels.
Parameters 2k mode
Elementary period T 7/64µs
Number of carriers K 1,705
Value of carrier number Kmin 0
Value of carrier number Kmax 1,704
Duration of symbol part TU 2,048× T 224µs
Carrier spacing 1/TU 4,464 Hz
Spacing between carriers Kmin and Kmax 7.61 MHz
TABLE I
THE OFDM PARAMETERS FOR THE 2K MODE.
V. SIMULATION RESULTS
The model for simulation can be briefly described in this
section. We consider at baseband, a wide frequency band of
interest ranging from -38.05 to 38.05 MHz, containing maxI
= 10 non-overlapping channels of equal bandwidth of 7.61
MHz. Our simulations will focus in the 2k mode of the DVB-T
standard. This particular mode is intended for mobile reception
of standard definition DTV. The structure of signal is followed
an OFDM frame. Each frame has a duration of TF , and
consists of 68 OFDM symbols. Four frames constitute one
super-frame. Each symbol is constituted by a set of C = 1,705
carriers in the 2k mode and transmitted with a duration TS.
A useful part with duration TU and a guard interval with a
duration Δ (choosen to 0) compose TS. The over-sampling
factor is 2. The occupancy ratio of the total 76.1 MHz band
is 50%. The received signal is damaged by additive white
Gaussian noise (AWGN) with a variance of σ2
n = 1. The
received SNRs on the a = 5 active channels are randomly
varying from 8dB to 10dB. A Gaussian wavelet function is
used for smoothing. For compressive sampling, 2N is 4096,
the compressed rate M/N is varying from 5% to 50% and H =
160 is the number of blocks. The compressive sampling matrix
ΦA has a Gaussian distributed function with zero mean and
variance 1/M. The number of PSD samples of each channel
is K = 25. We set αmin = 10−30
, αmax = 1030
for PBB
algorithm, and use β = 0.5, μ = 0.1, and τ = 0.1 AT
ry,i
∞
for both Basic and PBB algorithms.
Fig. 3 illustrates MSE performance for Basic GP and PBB
algorithms and compares with the performance result in [5].
In comparison with [5], our proposed approach in case of
1 CR slightly decreases the MSE performance because of
the reduced mutual incoherent of Φ in (12), however, our
approach can reduce the hardware cost due to AIC acquisition
at the lower sampling rate. The results show that in comparison
with Basic GP version, the PBB algorithm achieves the same
performances while the Basic GP version takes a lot of time
to get convergence [12]. So the following results are imple-
mented by using the novel PBB algorithm. This figure also
shows the performances of signal recovery quality in which
MSE decreases when the value of compression rate M/N
increases. However, as considering the effects of multiple CRs
in spectrum sensing scheme, it is easily to observe that MSE
also decreases as the number of CRs J increases; therefore,
we can obtain the lower compression rate but not degrade the

recovery performance by using more CRs in networks. For
example, to receive MSE at 0.37, we must compress wideband
signals at the rate 0.4 in the 1 CR case, while networks with
5 CRs can be implemented at the lower compression rate 0.3.
For detection performance, Fig. 4 depicts the probability
of detection P
(J)
d with respect to both the compression ratio
M/N and the number of CRs J =1, 5, under a ﬁxed P
(J)
fa
of 0.01. This ﬁgure demonstrates that in order to obtain
reliable performances, the joint collaboration and compression
is necessary. Especially, collaboration among CRs can avoid
the hardware cost of each CR by reducing the compression rate
M/N while remaining the high detection performance. For
instance, the probability of detection in case of 1 CR is ≈ 1
at the compression rate M/N over 0.2, while the collaboration
among 5 CRs requires the compression rate M/N from the
lower value 0.15.
Especially, analyzing the results reveals the interesting con-
clusion, i.e., in Fig. 4, the detection performances under both
our method and the approach in [5] over the examined range
of compression rates are similar while in Fig.3, the MSE
performances of these approaches have a bit differences.
VI. CONCLUSION
In this paper, we presented a distributed compressive spec-
trum sensing scheme for CR networks. To avoid the high speed
ADC systems, the alternative converters called AICs are ex-
ploited to acquire the salient information of received signals at
sub-Nyquist rates. Moreover, the GP approach is used for joint
cooperation and compressive sensing. The major barrier of GP
method, which takes a lot of time to reach the convergence,
can be solved by modifying the backtracking line search for
updating parameters. Among new fast CS techniques, PBB
algorithm, which is used to update the step of the iterations
in the recovery stage, demonstrates its outperformance, i.e.,
it not only achieves high quality of signal recovery but also
increases the speed to quickly reach convergence.
ACKNOWLEDGMENT
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. 2009-0073895)
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