A novel and efficient mixed-signal compressed sensing for wide-band cognitive radio

IFOST 2010 Proceedings
1
A Novel and Efficient Mixed-Signal Compressed
Sensing for Wide-Band Cognitive Radio
Le Thanh Tan*, Hyung Yun Kong*
*School of Electrical Engineering
University of Ulsan, San 29 of MuGeo Dong, Nam-Gu, Ulsan, Korea 680-749
Email: {tanlh.hkong}@mail.ulsan.ac.kr
Abstract-In cognitive radio (CR) networks, unlicensed (cogni
tive) users can exploit the licensed frequency bands by using spec
trum sensing techniques to identify spectrum holes. This paper
proposes a distributed compressive spectrum sensing scheme, in
which the modulated wide-band converter can apply compressed
sensing (CS) directly to analog signals at the sub-Nyquist rate and
the central fusion receives signals from multiple CRs and exploits
the multiple-measurements-vectors (MMV) subspace pursuit (M
SP) algorithm to jointly reconstruct the spectral support of the
wide-band signal. This support is then used to detect whether the
licensed bands are occupy or not. Finally, extensive simulation
results show the advantages of the proposed scheme. Besides,
we also compare the performance of M-SP with M-orthogonal
matching pursuit (M-OMP) algorithms.
Index Terms-Wide-band spectrum sensing, distributed com
pressive spectrum sensing, MMV subspace pursuit algorithm,
cognitive radio, compressed sensing, modulated wide-band con
verter.
I. INTRODUCTION
Cognitive Radio (CR) provides a new paradigm to exploit
unoccupied wireless spectrum of the licensed users (LUs) [1],
[2]. Moreover, the popular wide-band applications are nowa
days increasingly interested since they pronounce dynamic
spectrum access opportunities for secondary users (SUs).
However, spectrum sensing in the wide-band regime also
associates with considerable technical challenges. Normally,
the RFfront-end can either do narrow-band sensing via a bank
of passband filters or use a wide-band RFfront-end followed
by DSP blocks to sense the whole bandwidth. Nonetheless, the
former has strict constraints of designing filters whereas the
latter needs a high-speed analog-to-digital converter (ADC).
Recent work in compressed sensing (CS) reveals that a
signal having a sparse representation in one basis can be
recoveredfrom a small number of projections onto a second
basis that is incoherent with the first [3], [4]. Especially, a
compressed sampling approach can get the sparse signal at
the rates lower than Nyquist sampling; signal reconstruction,
which is a solution to a convex optimization problem called
min-ll 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]. Besides,
authors in [5] present a model of a single wide-band CR that
uses CS based spectrum sensing schemes. However, this work
bases on digital approaches that generally require full-rate
sampling before spectrum estimation. Thus, when it comes
to practical implementation, there are several issues to be
considered. The first problem is that the random projections
in CS are done over a discrete-time signal that is obtained by
sampling the continuous-time signal at Nyquist rate. So we
must avoid to work at Nyquist rate by applying CS directly
to the analog signal. Moreover, a model must be appropriate
to a practical application by reducing a hardware complexity
at a receiver.
To cope with the above questions, authors in [7] propose
a parallel segmented compressed sensing structure that can
directly apply CS to analog signals. Specifically, a received
analog signal is time-windowed into segments. Then CS is
independently applied to each segment by the bank of mixers
and integrators. And OMP is used to jointly recover a signal.
Nonetheless, this model developsfrom random demodulator
(RD) [8] that has constraints on huge-scale CS matrices, even
though a finite-precision of matrix entries also lead to another
issue called undesired high correlations between the columns
of the CS matrix. Thus, the performances of CS algorithms
are dramatically degraded in comparison with an ideal infinite
precision. Overall, the bank of RD channels [7] duplicates
analog issues and a computational complexity is not much
improved.
In this paper, we design a distributed compressive spectrum
sensing model in wide-band environment. Firstly, to acquire
analog multiband signals at sub-Nyquist rate, a sampling stage
exploits the modulated wide-band converter (MWC), which
consists of a bank of modulators and lowpass filters. Then a
recovery stage which makes use of the blind-reconstruction
approach using the continuous-to-finite block [9], [10] to find
the support for signal recovery. It is interesting that this support
contains the band edges to determine whether the bands are
occupied or not instead of fully reconstructing original signals;
thus, we can save a lot of time. To guarantee the exact recovery
support S, the reconstruction problem is solved by using the
multiple-measurements-vectors (MMV) subspace pursuit (M
SP) algorithm, which is modifiedfrom the subspace pursuit
[11] to apply for the MMV problem [12]. Moreover, based on
[13], the novel version of the distributed CS algorithm using
M-SP is proposed to jointly and efficiently reconstruct the
spectral support at the central fusion. Finally, we compare the
performance of our proposed scheme with that of models using
the standard MMV orthogonal matching pursuit (standard M
OMP) and the regularized MMV orthogonal matching pursuit
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ret)
<1>0(1)
<1>" JI)
Fig. I. Block diagram of compressed sensing of analog signals.
T
Fig. 2. Demonstration of overlapping windows.
(regularized M-OMP) in order to demonstrate the gainsfrom
a spatial diversity and make use of the joint sparsity.
The rest of this paper is organized as follows. Section II
presents compressive spectrum sensing scheme based on RD
[7]. Compressive spectrum sensing with modulated wide-band
converter is presented in Section III. Section IV proposes the
distributed compressive spectrum sensing. Section V demon
strates simulation results and discussions. Finally, concluding
remarks are given in Section VI.
II. PRELIMINARIES
A. Compressive spectrum sensing scheme based on [7]
The compressive spectrum sensing approach presented in
[7] bases on the RD [8] which is restricted to discrete multitone
signals and has a high computational load. To make this
method to work in practice, authors in [7] consider parallel
segmented compressed sensing (PSCS) structure as depicted
in Fig. 1.
Specifically, the received analog signal x(t) is K- sparse
over some basis IJf as in (1) for t E [0,T].
P-I
X (t)= LanWn (t)= lJfa, (1)
s=o
where IJf = [wo (t),WI(t),..., Wp-I(t)] consists of P
basis elements, a = lao (t),al (t),..., ap-I(t)] has only
K « P non-zero elements. We also assume that channel state
information (CSI) needs to be made available at the receiver,
thus the received signal r(t) can be evaluated by
r(t)= x(t)+n(t), (2)
where n(t) is an additive white Gaussian noise (AWGN).
In this structure, the received signal r(t) is divided into M
periods rm(t)= r(t)Wm(t) I�� with a duration time Tc
by the window function Wm(t) as shown in Fig. 2.
At every duration T, L = MN samples are received by
applying a random projection to each segment through N
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ret)
p,(t)
Fig. 3. Modulated wide-band converter model.
parallel channels, i.e., the m-th sample at the n-th channel
is calculated by
mTm+Tc
YmN+n=(rm(t),mN+n(t))=Jr(t) ;;'N+n (t)dt (3)
mTm
where the distribution of ;;'N+n (t) can be Gaussian,
Bernoulli ensembles and others. It is clearly observed that
the PSCS model can reduce the number of parallel ADC
channels, i.e., the number of measurements, while the re
maining incomplete information called salient volume can
be jointly reconstructed to the original signal. Because the
window function damages heavily at the edges, overlapping
is proposed to diminish the loss by averaging the errorfrom
reduced data.
For signal reconstruction, all received measurements are
stacked into a M x N matrix Y and given as
[
-T -T -T ]Y= YO,YI" " ,YM-I (4)
where Ym = [YmN,YmN+I,...,YmN+N-I]T is the vector
of measurements of the m-th segmentfrom N channels. The
reconstruction matrix VI= {vf,j}L x P
is defined as
V�N+n,s= (ws,m(t),mN+n(t))=l;s,m(t) ;;'N+n (t)dto
(5)
where Ws,m(t) = Ws(t)Wm(t).And reconstruction is imple
mented by using OMP algorithm.
It is important to point out that this method does not improve
too much to efficiently implement in practical application
due to remaining the highly computational burden in digital
domain.
III. COMPRESSIVE SPECTRUM SENSING WIT H MODULATED
WIDE-BAND CONVERTER
In this section, we examine the MWC model which applies
CS algorithms to the traditionalfrequency-domain [9,10,14].
A. Infinite-measurement-vectors (IMV) modelfor Compressed
sampling
Fig. 3 shows that the MWC has m channels and in the i-th
channel, the input signal r(t) is multiplied by the periodic
2

3
Zo (ll) Them*M
ret) mixing
matrix
C
ZM I (Il)
exp(J27r(M-1)1 IT)
Fig. 4. The modified model of modulated wide-band converter.
o , lfN� IlvI ' , f�
Zo (II) Z'_l (II) Z, (II) Z'+l (II) ZM_l (11)
Fig. 5. The relation between the Fourier transform of the input multiband
signal r(f) and the lowrate sequences Zt (n).
waveform Pi(t), lowpass filtered, and then sampled at the
rate liT, where the sampling interval T equals the period
of waveforms. For simplicity, we choose the rate liT � B.
Fig. 4 depicts the equivalent model to acquire the multi
band signals. M is chosen as the smallest integer such
that M � TfNyq and stands for the compression ratio. In
each channel, r(t) is modulated and then filtered to create
the sequence Zl(n) as in Fig. 5. For each time-point, the
vector Z(n) = [zo(n),. . . , ZM-I(n)]T is compressed into
the output vector Y(n)= [YI(n),...,Ym(n)]T as follows:
y(n)= Cz (n) (6)
The equivalence to Fig. 3 holds due to the periodicity of
waveforms Pi(t). Since Pi(t)= Pi(t + T) for all t E R, we
have the Fourier expansion
+00 ( 2 )pdt)= L Cilexp j;It
l=- =
(7)
where Cil= -f; 1:[Pi(t)exp (-j'2,;It)dt.Choosing the matrix
C of Fig. 4 such that its il-th entry is equal to the Fourier
coefficient Cil results in the desired equivalence
Ydn)= (r(t)pdt)) * h(t)It=nT
=
Mfil ((r(t)exp (-j�It))* h(t)It=nT) (8)
l=O
Conceptually, MWC shifts the mixing matrix C = {Cil} into
the analog domain, such that each channel realizes a single
row of C in analog hardware.
B. Signal recovery
For signal reconstruction, [9], [10], [14] and references
therein suggest an efficient method based on the fact that the
bands occupy continuous spectrum intervals, i.e., the vectors
z (n) for different time instances share a common spectral
Y(II
Lowrate reconctruction
Frameconstruction
Q= Ly(n)yT(n)
SolveQ= v"v
V Reconstruction joint supportS
SolveV=CU
s=Usupp(u, ),
Fig. 6. The model of signal reconstruction.
support called a nonzero location set. Thus, instead of solving
separately for every n, we construct a basis Vfrom a set of
consecutive sample sets y (n). It is shown in [10] that any
suchframe has the same support as the joint sparsity of z(n).
The Continuous-to-Finite block (CTF), which is depicted in
Fig. 6, implements this principle.
The first task in CTF constructs aframe for y (n), which
involves computing
Q = LY(n)yT(n),
n
(9)
In the presence of noise, (9) is modified as Q' = Q + (T2I and
slight rate increase is required.
Since Q is positive semi-definite, it can be decomposed as
(10)
And the joint support is inferredfrom a MMV CS system
V= cu, (11)
where the sparse matrices have a few nonzero rows and
the support S is determined by merging the supports of all
columns iTi of the sparsest matrix iT. Once the support Sis
found, the pseudo-inverse c1 is computed and then used to
recover zs (n).The notation Cs means the column subset of
C indicated by S, and similarly zs (n) is the relevant vector
subset.
The last computation in CTF is solving the MMV system
(11) for the sparsest matrix iT which is presented in Section
IV.
In terms of digital processing, the sequences Zl(n) are first
zero padded:
{ Zl(n) n= nM,n E Z
Zl (n) =
0 elsewhere (12)
Then, Zl (n) is interpolated to the Nyquist rate, using an
ideal (digital) filter. Finally, the interpolated sequences are
modulated in time and summed:
x (n) = x (nT)= L (Zl(n) * hI(n))exp (21rlfpnT). (13)
lES
However, in this work, we just consider spectrum sensing so
we need to find the support S and Olnit the last step of finding
x (n).
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Yes
Quit
ilerat'ions
No
,
S goodetlough?
Sl ;;;;;;SI-l U {Kil1diceswith
Sf = {Kindiceswith the
largestpro).coefficients }
Fig. 7. Iterations in the proposed M-SP algorithm for K-sparse signals.
IV. DISTRIBUTED COMPRESSED SPECTRUM SENSING
Let rj(t) be the wide-band analog baseband signal re
ceived at the j-th CR sensing receiver. Each CR sensing
receiver processes the received signal to obtain an Y
j =
[Yj,l (n),...,Yj,m (n)] of the compressed signal and con
structs theframe Vj , as in the CS acquisition step described
in Section III, theseframes are then sent to the fusion center.
The fusion center applies M-SP to jointly reconstruct the
spectral support S, which is then used to determine spectrum
occupancy.
A. Overview of MMV SP (M-SP) approach
SP [11] has a provable reconstruction capability comparable
to that of linear programming (LP) methods and exhibits the
low reconstruction complexity of matching pursuit techniques
for sparse signals.
We now describe the modified SP algorithm called M-SP to
be used for reconstruction of a model with J CRs. We write
C in terms of its columns as
C= [ C
1 C2 ... CM ] (14)
The next step is solving the problem (11) to find the spectral
support S by using M-SP. We set the truncation for the
subspace Cs of the m x M matrix C as in Definition 1
and the space spanned by the columns of Cs is denoted by
span(Cs).
D£ifinition 1 (Truncation): Let CE Rmx M and UE RM xr.
For an index set S C {I, ...,M}, the matrix Cs consists of
the columns of C with indices iE S and the matrix Usdenotes
the matrix formed by entries with row indices i E S.
Besides, the m x M matrix C also satisfies the RIP as in
Definition 2.
D£ifinition 2 (RIP): [15], [16] The matrix C E nmx M
satisfY the RIP with the parameters (K,8) for K :::; m,0 :::;
8 :::; 1, if for all index sets S C {I"" ,M} and for all
qE nisI,
(1 - 15K) IIqll� :::; IICsqll� :::; (1+15K) IIqll� (15)
and (1 - 15K) :::; Amin (cICs) :::; Amax (cICs) <
(1+8K).
As in [4], most known families of matrices satisfYing the
RIP property are random. If the random matrices whose
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entries follow the Gaussian distribution with zero mean and
variance 1/M, the RIP for a randomly chosen matrixfrom
such ensembles holds with overwhelming probability when
ever K :::; 1Jog(';}jm) , where 1J is a function of the RIP
constant. However, when the reconstruction principle holds
for the Fourier ensemble, the random matrices satisfY the RIP
with overwhelming probability, given that K :::; 1J[Og�)J6'
There also exists a relation between LP reconstruction and
the RIP property. The h-LP approach can reconstruct all K
sparse signals if the RIP must be satisfied with constants 8K,
82K and 83K, which have a condition 15K+82K+83K < 1.
However, in [4], the authors improved the above condition
to 82K < v'2 - 1. The projection of V onto the subspace
span(Cs) is denoted by Vp and can be computed by
Vp= proj(V,Cs):= CsC1V. (16)
Note that C1= (cICs)-1cI is the pseudo-inverse of the
matrix Cs and the subscript T denotes matrix transposition.
Corresponding to the projection matrix, the projection residue
matrix Vr is defined as
Vr= resid(V,Cs):= V -Vp. (17)
Fig. 7 illustrates the schematic diagram of iterations in M-SP.
Specifically, an estimate is recalculated and updated during
each iteration. An index, which is considered reliable in some
iteration but shown to be wrong at a later iteration, can be
added to or removedfrom the estimated support set at any
stage of the recovery process. The expectation is that the
recursive refinements of the estimate of the support set will
lead to subspaces with strictly decreasing distancefrom the
measurement matrix . The following subsection represents the
algorithm to solve the above problem.
B. The jointly recovery M-SP algorithm
The main difference between M-SP and M-OMP is the
method to generate S, that is the estimate of the correct support
set SaRI.
Note that in this algorithm, to update the spectral support,
we use the following method:
Each j-th CR (j = 1,00', J), at the l-th iteration, we find
the nonzero rows in the matrix Aj. Firstly, we use 12-norm to
evaluate each row of Aj:
i'J-1= IIRowsof [Aj] II= [ t;J.1 tl-1j,2
r (18)
and then calculate
i'1-1=average{i'J-1}= [ ti-1 tl-12 ... r (19)
J
where t�-l = J 2: tt/. Finally, the nonzero rows are
J=l
specified as
K indicescoresponding }tothelargestmagnitude
entries in thevectori'1-1
(20)
4

5
Algorithm 1 The jointly recovery M-SP algorithm
Input: A common dictionary m x M matrix C. A m x J
data matrix V = [ VI (n) V2 (n) ... VJ (n) 1 re
ceivedfrom J CR signalsYj = [Yjdn),...,Yjm (n)] ,
j = 1,. . . , J.
Output: The M x J estimated signal Zs
lZs,dn) Zs,2(n) ... zs,J (n) ], the estimate of
t e correct support set S.
Procedure:
1) Initialization: For each j-th CR 0 = 1,..., J), we have
the correlation matrix Aj = CTVj then apply (18),
(19), (20) to calculate TjO , TO, So, respectively.
The residue vector of the projection for the j-th CR
is �,j= resid(Vj,Cso)
2) Iteration: At the I - th iteration, we perform the
following steps.
Step 1: For each j-th CR 0 = 1,..., J), we have the
correlation matrix Aj= CTV��:/ then apply (18), (19),
(20) to calculate fj-I, f1-1, 51-1, respectively.
And then get an updated set 51 = SI-1 U 51-1.
Step 2: For each j-th CR, we set the projection coeffi
cients Up,j= C�,Vj , then apply (18) withAj = Up,j,
(19), (20) to calculate TJ, Tl, Sl, respectively.
The residue vector of the projection for the j-th CR
is V�,j= resid(Vj,CsI).
3) Termination test: The M-SP iteration is terminated
when satisfYing min 11;�,jII�_� min 11.v��:/1.12' � =
1, 2,. . . , J. Then let S = S and qUIt the IteratIOn.
If the limit is not reached then increased I and return
to iteration.
4) Store the results: The estimated signal Zs,j, satisfYing
Zs,j Irows{{I,... ,M}-SI} = 0 and Zs,j lSI = C1,yj·
The estimate of the correct support set S = Sl.
So the algorithm is summarized as Algorithm 1.
It is noted that we can improve the performance by a
backsolve, i.e., when reaching the termination, we can check
the neighbor locations of the entries in the spectral support set
S and perform the some iterations to get the exact result.
V. SIMULATION RESULTS AND DISCUSSIONS
To evaluate the performance of MWC model for compres
sive wide-band spectrum sensing, we simulate the system on
signals contaminated by white Gaussian noise. The signal
consists of N = 8 bands, each of width B = 50 MHz, and
is given as
N
X (t) =
L VEiBsinc(B(t - 7i))exp (21fJit) (21)
i=1
where sinc(x) = sin(x)/x. The energy coefficients are
equal to Ei 1 and the time offsets are 7i
{0.4,0.7,0.2,0.4,0.2,0.3,0.5, 0.3}f-Lsecs. The carriers fi
0.8
1;' 0.6"
5
�
� 04
"
-if;
0.2
10 20 30
-T- Standard M-OMP, 1CR
____ Regularized M-OMP, 1CR
,-,-, M-SP 2CRs
- - - M-SP 5CRs
40 50 60 70
Sampling channels (m)
80
Fig. 8. Recovery rate with different in at SNR � 10 dB in the case of 1
CR, and varying SNRs from 5dB to l5dB in the case of multiple CRs.
20
15
10
iii' 5
::s
-5
-10
-15
20 30 40 50 60
Sampling chalmels (orn)
70 80
Fig. 9. Regularized M-OMP recovery rate vs both sampling channels and
SNRs in the case of 1 CR.
20
15
5
o
-5
10 20
Lower bounds of
perfect reconstJuction range
30 40 50 60
Sampling channels (m)
70
0.8
0.6
0.4
0.2
80
Fig. 10. M-SP recovery rate vs both sampling channels and SNRs in the
case of 1 CR.
are chosen randomly in the range [0,fNyq] with fNyq =
lOGHz and the sampling stage is fs = fp = fNyq/195 =
51.3MHz. The number of channels is set to m = 80 (m �
2N), where each mixing function Pi(t) alternates sign at
most M= Mmin = 195 times. The continuous signals are
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represented by a grid of 50001 points in the interval 0.5p,secs,
the time resolution is T15. Gaussian noise is added and scaled
so that the test signal has the desired signal-to-noise ratio
(SNR), where the SNR is defined to be 10log (1111:11
11:)•
The support of the input signal is reconstructed from
in :::; m channels. To reduce IMV system to MMV system, we
follow the steps in Fig. 6. Then, theframe V is constructed
and the MMV is solved using the M-SP and compared with the
standard M-OMP or the regularized M-OMP [12], [17], [18],
[19]. It is noted that the reconstruction is successful only if the
estimate support S contains the true support SaRI, i.e., the
estimate support is allowed to have some additional entries.
Fig. 8 illustrates the recovery rate for various numbers in
of channels at SNR = 10 dB in the case of 1CR, and at
varying SNRsfrom 5dB to 15dB in the case of multiple
CRs. The results show that the standard M-OMP has low
reconstruction rate especially in low SNR regime (the figure
in low SNR regime is omitted due to the limited number of
pages). Although we apply some well-known techniques to
improve M-OMP, the performance of the regularized M-OMP
is lower than that of M-SP. So M-SP is appropriate to apply
for jointly reconstructing signal in the case of CR networks.
Especially, in the low SNR regime, the correct recovery
corresponding to 1 CR case is accomplished when using
in ;::: 40 for a stable recovery. However, the performances
can be significantly improved by reducing the number of
sampling channels when applying M-SP to jointly reconstruct
the spectral supportfrom multiple CRs. Furthermore, Fig. 9
and Fig. 10 illustrate the perfect recovery rates vs the number
of sampling channels and SNRs. It is easily observed that the
perfect-recovery-rate area of M-SP is larger than that of the
regularized M-OMP.
In comparison with model [7], in this wide-band scenario
with N = 8 bands of width B = 50 MHz and fNyq = 10 GHz,
we can get K= NB = 400.106tones.In this setting W =
1010, .pI has about [8] R � 1.7Klog(WjK+ 1) = 2.215·
109 rows, resulting in a huge scale CS system. Therefore,
the bank of RD channels [7] can't significantly diminish the
computational complexity.
VI. CONCLUSION
In this paper, we presented a distributed compressive spec
trum sensing scheme using MWC for wide-bandCR networks,
in which all operations such as sampling, processing the
information contents and recovering the input involve only
low-rate computations. Especially, M-SP is exploited to jointly
reconstruct the spectral support which is used to detect whether
the spectral bands occupy or not. The simulation results prove
the prominent advantages of the MWC model using M-SP.
ACKNOWLEDGMENT
This research was supported by Basic Science Research
Program through the National Research Foundation of Ko
rea(NRF) funded by the Ministry of Education, Science and
Technology(No. 2010-0004865).
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