Application of compressed sampling

International Journal of Computer Networks & Communications (IJCNC) Vol.7, No.5, September 2015
DOI : 10.5121/ijcnc.2015.7504 39
APPLICATION OF COMPRESSED SAMPLING
FOR SPECTRUM SENSING AND CHANNEL
ESTIMATION IN WIDEBAND COGNITIVE
RADIO NETWORKS
Naveen Kumar and Neetu Sood
Department of Electronics & Communication, Dr B R Ambedkar NIT Jalandhar, India
ABSTRACT
In the last few years Compressed Sampling (CS) has been well used in the area of signal processing and
image compression. Recently, CS has been earning a great interest in the area of wireless communication
networks. CS exploits the sparsity of the signal processed for digital acquisition to reduce the number of
measurement, which leads to reductions in the size, power consumption, processing time and processing
cost. This article presents application of CS for the spectrum sensing and channel estimation in Cognitive
Radio (CR) networks. Basic approach of CS is introduced first, and then scheme for spectrum sensing and
channel estimation for CR is discussed. First, fast and efficient compressed spectrum sensing (CSS)
scheme is proposed to detect wideband spectrum, where samples are taken at sub-Nyquist rate and signal
acquisition is terminated automatically once the samples are sufficient for the best spectral recovery and
then, after the spectrum sensing, in the second phase notion of multipath sparsity is formalized and a novel
approach based on Orthogonal Matching Pursuit (OMP) is discussed to estimate sparse multipath
channels for CR networks. The effectiveness of the proposed scheme is demonstrated through comparisons
with the existing conventional spectrum sensing and channel estimation methods.
KEYWORDS
Compressed Sampling; Cognitive Radio; Spectrum Sensing; Channel Estimation; Orthogonal Frequency
Division Multiplexing.
1.INTRODUCTION
According to Kotelnikov, Nyquist and Shannon [1-3], a continuous-time band-limited signal x(t)
with bandwidth B > 0, can be exactly recovered from twice as many samples per second as the
highest frequency present in the signal i.e. 2B, also known as the Nyquist rate. Around
2004, David Donoho, Terence Tao and Emmanuel Candès [4-6] proved that if the signal is sparse
in one domain, the signal may be reconstructed back with even fewer samples than required by
Nyquist’s sampling theorem. In this way the sampling of the signal at sub-Nyquist rate is the
basis of compressed sampling.
There are two primary conditions for the perfect recovery of the signal [7]. The first one
is sparsity i.e. the non-zero terms, of the signal. CS approach is based on the fact that the most of
the signals can be well-approximated as sparse signals by expanding them in a suitable
sparsifying basis. The second condition is incoherence, discussed in detain in Section 2, which is
applied through the restricted isometric property for sparse signals [8].

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In the field of wireless communication system, there are many cases, where a large amount of
information should be processed to obtain the interior characteristics, such as signal detection,
channel estimation, network monitoring and data fusion in Cognitive radio (CR) networks. These
problems share a common feature that the information being processed is huge but sparse in
nature. For such cases, CS can be used to lower all of the cost attributes and increase efficiency.
Fast and accurate spectrum sensing is extremely important in wide band CR networks. To
implement wideband spectrum sensing, CR needs fast Analog-to-Digital Converter (ADC) but
the achievable sampling rate of ADC is only 3.6 Gsps. Capitalizing on the wideband signal
spectrum sparseness, CS technique can be employed in spectrum sensing [9] in CR network.
In most scenarios in CR networks, the number of used channels is comparatively much lesser
than total channels; those are vacant at a particular time and space. Therefore, when dealing with
channel estimation problem in CR system where the channel band is really wide and dynamic and
occupation information of the channels is compressible, CS can be exploited, since CS does not
require any knowledge of the underlying multipath channel, based on the fact that a sparse
structure is exhibited by the physical multipath channels in angle delay Doppler spreading,
especially at large signal space dimensions, it is advantageous to utilize sparse channel estimation
method based on convex/linear programming, which can be proved to outperform the existing
least square based methods [10]. Such channels with significant angle delay Doppler spreading
can be handled by CS algorithms in the form of Basis Pursuit (BP) and Orthogonal Matching
Pursuit (OMP), when coupled with a channel equalizer mitigating inter-carrier interference, as
demonstrated in [11] using numerical simulation and experiments.
The remainder of this paper is structured as follows– Section 2 presents notations. Section 3
summarises the related works, In Section 4, the fundamentals of CS theory are introduced, and
the reconstruction methods of CS are summarized. Then for spectrum sensing and channel
estimation, CR system model is presented in section 5. Section 6 explains the theory of different
components of CS algorithms. CS based fast and efficient spectrum sensing scheme is presented,
after that in section 7 we formalize the notion of sparse multipath channels to apply the
compressed channel estimation based on various CS algorithms. Section 8 demonstrates and
summarizes the performance advantages of proposed CS based scheme over traditional energy
detection spectrum sensing and maximum likelihood ratio based channel estimation techniques.
Finally, we conclude the article in Section 9 by discussing some of the finer technical details
pertaining to the results presented in the paper and future scope.
2. NOTATIONS
x Discrete time signal
 Sparsifying basis
S Non zero coefficients
 Measurement matrix
m Number of measurements
Z Additive white Gaussian noise
Ri Training subset
Ti Testing subset
Vi Verification parameter
C Constant
K Signal level
Ts OFDM symbol duration
TG Guard interval

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3. RELATED WORK
To implement wideband spectrum sensing, CR systems need some essential components, i.e.
wideband antenna, wideband radio frequency front-end, and high speed Analog-to-Digital
Converter (ADC). [12] and [13] developed the wideband antenna and the wideband filter
efficiently but the development of ADC converter is relatively behind as the ADC currently have
limited sampling rates, of the order of 1 GHz, which is not sufficient for wideband spectrum.
To solve this problem, Z. Tian and Giannakis [14] applied Compressive spectrum sensing (CSS)
theory to CR systems for acquiring wideband signals at sub-Nyquist sampling rates. Capitalizing
on the sparseness of the signal spectrum in open-access networks, compressed sensing techniques
were developed for the coarse sensing task of unutilized spectrum i.e. spectrum holes’ (SHs)
identification.
Additionally, Malioutov et al. [15] described a stopping rule for a desired tolerance in
reconstruction for the near-sparse case. Simin Bokharaiee et al. [16] applied cyclic prefix
correlation coefficient (CPCC) based blind spectrum sensing for orthogonal frequency division
multiplexing based cognitive radio (OFDM-CR) systems. Furthermore, by considering multipath
correlation in the generalized likelihood ratio test (GLRT) algorithm and employing the inherent
structure of OFDM systems a low complexity algorithm was obtained. Furthermore, M. Farrag et
al. [17], proposed Distributed Sensing Matrix (DSM) based algorithm to reduce complexity at
each sensing node and improve the unlicensed users i.e. primary users’ (PUs) detection
performance.
After spectrum sensing, deactivated subcarriers in non contiguous OFDM complicate the design
of efficient pilot symbols for channel estimation [18-21]. In [22], the pilot design is formulated as
an optimization problem minimizing an upper bound related to the Mean Square Error (MSE),
where the pilot indexes are obtained by solving a series of 1-dimentional low complexity
problems. To reduce the computational complexity for the optimal selection of pilot subcarriers,
cross entropy optimization was utilized in [24] to determine the position of pilot subcarriers.
Manasseh et al. [23], proposed a cross-entropy optimization based pilot design scheme using
convex optimization together to minimize the MSE. Min Jia et al. [25] presented channel
estimation algorithm for OFDM-CR, based on OMP and applied sparsity adaptive matching
pursuit (SAMP) algorithm for the first time for channel estimation in NC-OFDM systems.
Moreover, for the reconstruction time-consuming of SAMP algorithm was too large, modified
adaptive matching pursuit (MAMP) algorithm was introduced as an improved SAMP algorithm.
Chenhao Qi et al. [26] introduced sparse channel estimation (SCE) scheme in OFDM-CR, where
pilot design was formulated as an optimal column selection problem and constrained cross
entropy optimization based scheme was proposed to obtain an optimized pilot pattern.
4. COMPRESSED SAMPLING THEORY
According to the theory of CS, information about certain signals can be acquired directly and
recovered back with high probability from far fewer measurements than traditional methods,
provided that the signal is sparse, by solving an optimization problem with the small amount of
projections. Consider a discrete time signal x ϵ N
, which can be expressed as Х(n) where n = 1,
2, …, N. The claim of compressive sensing is that from m (m<<n) measurements, we can
reconstruct the original signal x with non-adaptive linear measurements. This does not violate the
Shannon-Nyquist sampling theorem as reconstruction of only sparse signals is possible.

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4.1 SPARSE REPRESENTATION OF THE SIGNAL
Based on the CS theory, efficiency of signal acquisition depends on the sparse representation of
the signal. Signal Xϵ N
can be expressed in a sparsifying basis  = [1,2,…N]as follows
[27] :
X = = (1)
where is the vector of inner products, N = <X, N>. When only S (S<<N) of the N
coefficients are nonzero, it is called S-sparse.
4.2 MEASUREMENT MATRIX
By using non-adaptive linear projections the useful information in the sparse signals can be
acquired, via directly compressing signals into a smaller data [27]. Measurement of the signal x is
done by sampling it with respect to a measurement matrix  ϵ M×N
. Substituting X = into
Equation (1):
Y =X =  Θ  (2)
where Y and  are M × 1 column vector and M ×N measurement matrix respectively. Based on
the CS theory signal recovery can actually be made possible when the matrix Θ =  obeys the
rule of Restricted Isometric Property (RIP). For each integer n = 1, 2, . . .N isometry constant δn >
0 of a matrix  can be defined as the smallest number and | · | is defined as the amplitude of a
complex number, For signal reconstruction to be successful, must satisfy [28]:
(1-δn) ≤ Θ ≤ (1+δn) (3)
From equation (3) we can say that a matrix  has the RIP of orders if δn is not too close to 1. The
restricted isometry constants give a measure of how much they can change the length of an S-
sparse vector.
4.3 Signal Reconstruction
As , X ϵ N
and Y ϵ RM
for m << n there are infinitely many solutions to equation (2). If  is
incoherent to  the original CS theory proposed l1 norm minimization for signal reconstruction
when dealing with noise free measurements:
min s. t. Y =  (4)
Convex optimization techniques can be used for the sparse signal recovery with high probability.
CS relied on the assumption that equation (4) provides the correct solution and is computationally
not complex. However work has been done to find alternative algorithms those can be easily
implemented, faster and give better reconstruction performance, such as Matching Pursuit (MP)
[29] and OMP [30] algorithms, which can be fast and easily implemented. There also exist a
variety of different kinds of iterative algorithms such as Compressed sampling Matching Pursuit
(CoSaMP), iterative hard thresholding (IHT), and various similar thresholding algorithms for
improved compressed signal reconstruction [31-37]. These algorithms are based on similar
structure can be easily understood by identifying the locations of the nonzero coefficients of 
and then estimating the values of those coefficients.

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5. SYSTEM MODEL AND PROBLEM STATEMENT
System model considered in our work is same as that in [38], which assumes that CR with
frequency range 0–W Hertz, as shown in Figure 1. Figures 1(a) and 1(b) corresponds to
frequency frame and time frame. In time frame 0–t (second) is required for spectrum sensing and
remaining t–T (second) is used for transmitting data. In out work, all the received signals from
different PUs and SUs are treated as one received signal, s(t). Continuous signal received at the
RF front-end of SU contains signal si and noise zi. Considered noise is a sequence of independent
and identically distributed, additive zero-mean, circularly symmetric, complex Gaussian noise
samples with variance E{|ni|2
} = σn
2
for all i. The power spectrum density (PSD) over each
spectrum sub-band is independent for two neighboring subbands. In a sensing period t–T, the
locations and the number of active subbands keep unchanged. By using sampling rate fN over t,
we obtain [n] = i(n/fN), n = 1, 2, · · · ,N.
PU
PSD PU PU SH W
(a)
F
0 1 2 3 4 .... b .... B
Sensing time Transmission time
(b)
0 t T Time
Figure 1. Wideband CR spectrum sensing model (a) Frequency Frame (b) Time Frame
By using sub-Nyquist sampling rate fS (fS < 2W), the compressed samples y (y ∈ Mx1
, M = τ fS
<< N) can be written as:
y = φ  (5)
where φ is an M × N sensing matrix and y is the measurement vector of m measurements. Based
on the CS theory, efficiency of signal acquisition depends on the sparse representation of the
signal. Signal  ϵ N
can be expressed in a sparsifying basis  = [1,2,…N follows:
 = (6)
where x is the vector of inner products, xi = <, i>. Let μ (φ, ψ) be the coherence between φ and
ψ, C is constant, we can choose minimum number of measurements required for spectrum
reconstruction:
m ≥ C0 · μ2
(φ, ψ) · S · log N (7)
From (7), it can be observed that m depends on the sparsifying basis. In this article, we compare
the cases if the sparsifying basis is the Discrete Fourier Transform (DFT) or Discrete Cosine
Transform (DCT) or Empirical Wavelet Transform (EWT). An estimate of sparse vector can be
obtained by solving the CS reconstruction problem:
= arg min||x||1 s. t. y = φ ψ x (8)
Above problem can be solved efficiently using Greedy algorithms, Tony Cai and Lie Wang at al.
[39] explains OMP algorithm, which is suitable to deal with noisy signal.

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After spectrum reconstruction SUs sense the channel in order to find the spectrum holes (SHs),
which can be described as the hypothesis testing problem, mathematically:
0 : i = zi , i = 1, 2, . . .,N (9)
1 : i = si + zi , i = 1, 2, . . .,N (10)
For large values of N, it can be approximated:
 =  (11)
as a Gaussian random variable by using the Central Limit Theorem, It can be verified that the
mean and variance of  are given by, respectively:
E{} = (12)
Var{} = (13)
An energy detector can be used to accumulate the energy E = ∈ received in sub-
band b. The SU decides the spectrum availability by:
(b) =


(14)
The threshold λ is a design parameter for the CR receiver system. The choice of λ provides a
trade-off between maximizing SU throughput and minimizing interference to the PU. Two
probabilities of interest are defined as follows
Probability of false alarm: Pf = Pr { ≥ λ| 0} (15)
Probability of detection: Pd = Pr { < λ| 1} (16)
After spectrum sensing, CR adopts NC-OFDM technique that decomposes wideband into
orthogonal sub-channels. In NC-OFDM based CR systems, as shown in Figure 2, the sub-
channels are activated when the spectrum is idle and when it is not available corresponding sub-
channels are deactivated (set to zero). The data stream is only assigned to active sub-carriers and
pilot design is performed accordingly.
In wide-band wireless communication systems, the actual bandwidth of the system is usually
larger than the coherence bandwidth and the channel is usually frequency selective fading
channel. The vector is S-sparse, Discrete Fourier Transform (DFT) size is N, active sub-carriers
are Q and pilot sub carriers (cp) are P (P  Q). The cyclic prefix (CP) length is greater than the
maximum possible path delay. OFDM symbol data X(n) contains mapping signals and pilot
signals. After removing CP, DFT is applied to the received signal yn for n ϵ [0, N−1] to obtain k ϵ
[0, N − 1]. The received signal is N×1 sample vector. The discrete-time channel model is:
h(n) = (17)
where the impulse response vector of the channel h = [h0, h1,…, hl-1]T
remains unchanged in
multiple OFDM symbol period of time reflects the slow time variation of the channel.

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Data
Data
Figure 2. NC-OFDM based CR system
where the impulse response vector of the channel h = [h0, h1,…, hl-1]T
remains unchanged in
multiple OFDM symbol period of time reflects the slow time variation of the channel. The
relation between the transmitted pilots and received pilots can be written as:
= · + (18)
where Z is additive White Gaussian Noise (AWGN) and is FK×L is a DFT sub matrix given by:
FK×L = (19)
where wnl
= . Let A= XFK×L. Then (18) can be written as:
y = Ah + Z (20)
R×N matrix S selects the location of P pilot from the N subcarriers. N×N matrix S selects R rows
corresponds to the pilot position from the unit matrix. The pilot signal received is:
yp = Xp Hp + Zp = Xp Wp h + Zp (21)
where R × 1 matrix yp = Sy , R × R matrix Xp = SXS’ , R × L matrix Wp = SW , R × 1 matrix nP =
S. In (21) yp, Xp and Wp are known. We can re-obtain system channel relation obtained in (20)
by:
Hlk= (22)
Pilot
Design
S/P Subcarrier Assignment
CP
Removal
IFFT
CP
Insertion
S/P
Channel
P/S
P/S Detection Channel Estimation
Spectrum
Sensing

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where
F[m,i]=Xh(m,i) (m,i/Nr) (23)
with the cross ambiguity function [40] Aγ,g(m,ε)= . The purpose of
channel estimation is to find the channel frequency response H. Since the channel delay spread is
much larger than sampling period, particularly for OFDM systems with over sampling, most
components of h are either zero or very close to zero, which implies that h is sparse. With this a
priori condition, CS algorithm can be applied to estimate h.
6. COMPRESSED SPECTRUM SENSING
Consistent with the theory of CS discussed in Section 4, an N point signal x, or its S sparse
representations can be reconstructed from only m measurements. According to equation (1) and
(2):
y = φ  = φ ψs x = φ ψs F-1
X (24)
where F-1
X is the inverse Fourier transform of x. From (7), the minimum number of
measurements depends on the sparsifying basis ψs. We apply method, empirical wavelet
transform scheme to find sparse representations of the spectrum in the context of compressive
sensing. The discrete signal x is not known prior to sensing so first we calculate an estimated
version of the signal using an appropriate conventional sparsifying matrix. Next, we apply
empirical wavelet scheme. Jerome Gilles [43] proposed a method is to build a family of wavelets
adapted to the processed signal. The process is equivalent to building a set of band pass filters.
Filters’ support depends on where information in the spectrum of analyzed signal is located.
Consider a spectrum [0, ] as shown in Figure 2, which is segmented into I segments; ωi is the
limit between each segment Λi. i is the transition phase of width 2τi.
2τ1 22τ2 2τi 2τi+1 τI
0 ω1 ω2 ωi ωi+1
Figure 3. Spectrum segmentation
The empirical wavelets are defined as band pass filters on each segment. Empirical scaling
function and empirical wavelets are defined by expressions, (25) and (26) respectively
Φi (ω) = (25)

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Ψi (ω) = (26)
The function is an arbitrary Ck
([0,1]) function [44] such that
β(x) = (27)
Using equation (24) problem reconstruction of can be converted into the problem of
reconstruction of :
= arg min|| ||0 s.t. y = (φ F-1
) (28)
It can be seen that (28) is a non-convex problem as there can be many optimal solutions. Equation
(28) has a unique solution when:
= arg min|| ||1 s.t. y = (φ F-1
) (29)
To deal with the signals with noise components, some variants of LASSO algorithm can be
developed by minimizing the usual sum of squared errors [41, 42]:
= arg min|| ||0 s.t. || φ F-1
– y ||2 ≤ ε (30)
where ε is recovery error threshold. The EWT-CSS problem can be solved with a two-step
scheme: First, use compressed measurements y to estimate the sparse sequence and second,
reconstruct signal according to ψs.
ℓ2 norm approach is used to terminate the signal acquisition automatically at CR receiver. As
discussed in Section 3 time frame is divided into sensing interval (0 – t) and transmitting interval
(t - T), if we decrease the sensing time, we can increase the time interval to transmit data. First,
we divide complete sensing time interval t into I mini time slots. Ri is training subset, Ti is testing
subset and Vi is the verification parameter, for best spectrum recovery:
Vi = – ψi F-1
i (31)
Using equation (2) and (10), Ri and Ti can be written as correspondingly
Ri = φi xi + n = φi F-1
i + ni (32)
Ti = ψi xi + n = ψi F-1
i + ni (33)
where φi is a ri × N measurement matrix, ψi is a ti × N testing matrix, and n denotes the noise in
measurements.

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Algorithm 1 Fast and Efficient Compressive Sensing
Initialize: Recovery threshold criterion ε ≥ Vi / Ti –
While the spectrum recovery threshold criterion is not true and i ≤ I,
Step 1. Samples at sub Nyquist rate for time slot i to get compressed samples’ set yi.
Step 2. Devide yi into two complementary subsets Ri and Ti.
Step 3. Estimate the spectrum by recovery equation (30), leading to spectral estimate
Xi.
Step 4. By using (31) and testing subset, calculate Vi.
If the recovery threshold criterion is satisfied
End
Else i = i + 1
repeat Step 1
Repeating this procedure, a sequence of spectrum estimates, i.e., , will be obtained,
and by increasing measurements mi, we can get a a better estimate of the spectrum with
minimum spectrum recovery error . Terminating the signal acquisition automatically
and by using the remaining slots for transmission, leads to fast and efficient spectrum sensing and
improved system throughput.
7. SPARSE CHANNEL ESTIMATION
In the CS model discussed in Section 3, the signal x is not directly measured, but projected onto
the observation matrix ξ = [ξ1, ξ2,…ξM], and then the sampling vector y is obtained. The matrix
form is
y = ξx (34)
where x is an N × 1 vector, ξ is an M × N matrix, and y is an M × 1 sampling vector. If signal x is
sparse in the transform basis ψT
= [ ψ1, ψ2,.. ψN] , it can be expressed as follows:
x = = ψα (35)
where α is an N ×1 vector, ψ is an N × N sparse vector. (8) is substituted into (7):
y = ξx = ξψα = Θα (36)
where M × N matrix ξ=ψΘ. The observation dimension M is far less than the signal dimension N,
so the signal x can’t be solved with M sampling values of y. As α is sparse, with the sparse
decomposition algorithm based on the theory of sparse decomposition, α can be solved by the
inverse problem of solving (36), and be substituted into (35) to obtain the signal x. Consider the
baseband channel model of (4).
Due to the channel fading and noise, to determine the length of training sequence Θ is important
in terms of both spectrum efficiency and estimation robustness. Therefore, the length N of Θ must
satisfy
N ≥ C· S· (log L)4
(37)
where C is a constant and μx= maxi,j|Xi,j| which is known as the maximum coherence between
the ith column and jth column of Θ.

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OMP algorithms suggests the reconstruction under the conditions of a given iteration number, as
the iterative process is forced to stop, OMP algorithm needs a lot of linear measurement to ensure
accurate reconstruction. The basic idea of the OMP algorithm is to select the columns of Θ with
greedy iterative algorithm, make sure the correlative value between the columns selected in each
iteration and the current redundant vector is maximum, and then subtract the correlative value
from the sampling vector and repeat iteration until the number of iterations achieves the sparse
degree S. Here, Θ is sensing matrix, y is sampling vector, S is sparse degree, r is residual and Λ is
the index set . Output of the OMP algorithm is the S-sparse approximation of α.
Algorithm 2 OMP-CS
Input: Θ, y, S;
Output:
Initialization: r0=y , Λ0=N=1;
Step 1. find the maximum value of the inner product of r and the column of
sensing matrix yj the corresponding foot mark is λ, λt = argj=1..N max | |
Step 2. renew the index set Λt = Λt-1 U{λt} , the sensing matrix Θt = [ Θt-1, ]
Step 3. solve t = arg min || y- Θt, ||2 by least-square method;
Step 4. renew the residual rt= y- Θt t t= t+1
Step 5. if N > S stop the iteration,
else repeat step 1.
OMP algorithm selects an atom in each iteration to update the atom collection, which will
certainly pay a large time for reconstruction. The number of iterations is closely related to sparse
degree S and the number of samples M, with their increase, time consumption will also increase
significantly.
Problem with Algorithm 2 is that it is not adaptive, pre-estimate of the sparse degree of the sparse
signal is needed and the reconstruction accuracy is not satisfactory. In reality, the sparse degree of
the sparse channel is usually unknown. Nam Naguyen et al. [30] proposed extended OMP-CS
algorithm in order to improve the accuracy of reconstruction, and make the algorithm adaptive. In
the ExtOMP-CS algorithm, one key issue is how to choose the step size. Unlike the OMP-CS
algorithm, the iteration times of ExtOMP-CS algorithm is not certain and is related to step size
and computational complexity and computational time are higher in the ExtOMP-CS algorithm
than OMP-CS algorithm.
An extension to OMP algorithms is the Compressed Sampling Matching Pursuit (CoSaMP)
algorithm [45]. The basis of the algorithm is OMP but CoSaMP can be shown to have tighter
bounds on its convergence and performance.
Algorithm 3 Compressed Sampling Matching Pursuit Algorithm
Input: S, y, 
1. x(0) 0
2. v y
3. k0
4. while Halting condition false do
5. kk + 1
6. z T
v : signal proxy
7. Ωz2S
: find the largest 2S components of the signal proxy (Identification)

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8. Ω supp (x(k-1)
) : merge the support of the signal proxy with the support of the
solution from the previous iteration (Support Merge)
9. arg min||(x-y)||2 : estimate a solution via least squares with the constraint that the
solution lies on a particular support (Estimation)
10. xk
xs
: takes the solution estimate and compresses it to the required support (Pruning)
11. vy-x : update the sample, namely the residual in F-space (Sample Update)
12. end
13. while x(k)
Output: : such that it is S-sparse and y = x
If N-length training sequence X satisfies the RIP and δ2S≤ -1, for any 2S-sparse channel vector
h, it produces the channel estimator that satisfies
||h-ĥ||2 ≤ C max{ε, 1/ ||h-ĥ2S||1 + ||z||2} (38)
8. PERFORMANCE ANALYSIS AND SIMULATION RESULT
In NC-OFDM the power spectral density (PSD) of Pth
subcarrier signal is characterized of the
form
k(f) = K · sinc2
((f – fP)TS) (39)
where K is the signal level, fk is the sub-carrier center frequency, and TS is the OFDM symbol
duration and TG is guard interval. Assuming independent symbols in different subcarriers, the
PSD of an NC-OFDM signal is obtained as
(f) =  (40)
where index P is the number of active subcarriers. In this paper we are considering a spectrum (0-
500MHz) with four sub-bands located at 30MHz – 70MHz, 120MHz – 180MHz, 300MHz –
340MHz, 420MHz – 460MHz and their corresponding power spectral density (PSD) levels are in
the range of 0.0277 - 0.1126, 0.0157 - 0.0988, 0.0588 - 0.1294, and 0.0381 - 0.1201, as shown in
Fig. 3. The sub-band b is in the frequency range [fb – Bb/2, fb + Bb/2], where Bb = 10–30 MHz
and fb is in [Bb/2, W – Bb/2]. Each time frame has length T = 10 μs and t is 5 μs, which is divided
into I = 20 mini time slots. Instead of using the Nyquist sampling rate fN = 1 GHz, we are
sampling at the sub-Nyquist rate fS = 250MHz. The number of compressed samples in
conventional CS system is M = fs t = 1250, whereas N = fN t = 10,000. Calculating and Ti after
each time slot we calculate the verification parameter using (31). Vi becomes very close to 2σ2
at
time slot L = 9, so using the presented algorithm we can recover the spectrum at lesser time slots.
All the simulations are carried out in the MATLAB 2014.
8.1 RECOVERED PSD
As can be observed from Figure 3, the wideband signal contains the PSD. Using algorithm 1 for
fast and efficient compressed sensing, wideband spectrum can be successfully recovered, as
shown in Figure 4 and terminate the signal acquisition at comparatively lesser time slots leading
to a fast efficient spectrum sensing.

51
Figure 4. Wideband Spectrum
Figure 5. Reconstructed Spectrum
8.2 PROBABILITY OF DETECTION PERFORMANCE
The energy detection scheme using (15) and (16) is used to detect spectrum hole by using the
reconstructed spectrum. The performance of different spectrum-sensing algorithms based on DF,
EWT and DCT, is compared by MATLAB simulation via Pd for constant Pf = 0.05.
Figure 6. Performance comparison of DFT CSS, DCT CSS and proposed EWT CSS spectrum-sensing
algorithms
Frequency (MHz)
Frequency (MHz)
PSD
PSD

52
8.3 ESTIMATION ERROR
The MSE performance of the proposed estimation method will be evaluated by simulations and
compared with the MSE performance of LS. It is obvious that smaller MSE means more accurate
channel estimation and vice versa. MSE is defined as
MSE = 1/M (41)
Figure 7 depicts the MSE versus SNR. It can be seen from the figure, that the CS-based scheme
significantly outperforms the least-squares method. Compared with the least square scheme with
1024 pilots, CS based scheme with 511 pilots shows only a small performance degradation.
Figure 7. Performance of the channel estimate MSE for different SNRs
8.4 BER PERFORMANCE
We are considering an OFDM-based CR system with M = 1024 subcarriers, after spectrum
sensing without any false alarm or missing detection and deactivating those subcarriers occupied
by PUs, we assume that there are 512 remaining OFDM subcarriers for SUs, including three non-
contiguous subcarrier blocks, i.e., {1, 2, . . . , 256}, {513, 514, . . . , 640} and {897, 898, . . . ,
1024}, with the number of subcarriers in each block being 256, 128, and 128, respectively. A
sparse multipath channel h is considered with L=60 taps where 5 non-zero taps are placed
randomly. We now evaluate the channel estimation performance using the designed pilot patterns.
Figure 8. Comparison of BER Performances
SNR(dB)
MSE

53
Figure 8 shows BER performance of two schemes- proposed CS channel estimation and channel
estimation scheme based on LS. Improved BER performance of the proposed CS based scheme
can be seen from the above figure, over the conventional LS based scheme.
9. CONCLUSION
CS is a very promising technique in wireless communication networks. However, the studies on
the applications of CS are just in fewer areas. Even in these areas, a lot of problems are still not
been fully settled, limiting the performance of CS. In CR network systems, if the number of
channels, is not large enough, the requirement of sparsity cannot be guaranteed, which limits the
advantages of CS. In this article, we demonstrated the application of CS in CR networks and
based on the advantages of the proposed scheme, the problem of designing a high-performance
CR receiver indicates that the approach should work both for spectrum sensing and channel
estimation. On future work, we will consider an adaptive CS algorithm which senses the
sparseness of the signal, random noise and other unexpected interferences and can differentiate
between primary user and secondary user signals.
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Authors
Naveen K. Yadav received his B. Tech degree from Gautam Buddh Technical University
in 2010 and M. Tech degree from National Institute of Technology, Jalandhar, India in
2015. He has worked as Assistant Professor in Electronics and Communication
Department, J. S. Institute of Technology, Uttar Pradesh from year 2010 to 2013.
Currently, he is working as Assistant Professor in Department of ECE., S.S. Institute of
Technology, Aligarh. His research interests include cognitive radio, wireless sensor
networks, massive MIMO and smart grid.
Neetu Sood received the B. Tech degree (with Honour) in Electronics and
Communication from Sant Longowal Institute of Engineering and Technology, India
2000. In year Jan. 2002, she completed her M.Eng. Degree (With Honour) in Electronics
and Communication from Thapar University, Patiala, India and PhD in Electronics and
Communication in year 2013. From year 2002 to 2007, she worked with DAV Institute of
Engineering and Technology, Jalandhar and Guru Nanak Dev University (Regional
Campus), Jalandhar. Currently, she is working as Assistant Professor at Dr. B Ambedkar National Institute
of Technology, Jalanadhar. Her current research includes the simulation of wireless systems based on
OFDM and simulation of fading channels.

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