International Journal of Scientific & Engineering Research, The research paper published by IJSER journal is about Spectrum Sensing using Compressed Sensing Techniques for Sparse Multiband Signals 1
ISSN 2229-5518
Spectrum Sensing using Compressed Sensing
Techniques for Sparse Multiband Signals
Avinash P, Gandhiraj R, Soman K P
Abstract— Spectrum is scarce and the primary users (licensed users) do not use them always. There are free spaces called spectrum holes. Spectrum is not utilised efficiently in certain bands. A technique which scans the spectrum for the given bandwidth and finds the spectrum holes so that secondary users can use them, was proposed. But for high bandwidths the sampling rates are high such that practical Analog to Digital Converters cannot achieve. Compressed sensing techniques sample at rate less than the Nyquist rat e and still are able to reconstruct the original signal except that the signal should be sparse in some domain. So spectrum sensing using compressed sensing methods were proposed and found to be more efficient.
Index Terms— Blind spectrum sensing, Cognitive Radio, Compressed Sensing, Randomness, Sparse multiband signals, Spectrum
Sensing, Support.
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PECTRUM is a potentially scarce resource. Spectrum sensing is a process which ensures that Cognitive Radios will not interfere with Primary Users. They reveal the un-
used bands so that unlicensed users (secondary users) can establish communication in free bands. They are called spec- trum holes. As we go for higher frequency applications the sampling frequency will be higher and the practical ADCs impose bandwidth restriction on the signal thus resulting in loss of information.
In order to overcome this problem a method called Com- pressed Sensing [1] was introduced, which is capable of recon- structing the signal using lesser samples than Nyquist rate. The condition to be satisfied by the signal is that it should be sparse in any domain. The reconstruction method is consid- ered as an optimization problem and can be solved using dif- ferent algorithms like Orthogonal Matching Pursuit (OMP), Basis Pursuit (BP) etc.,
The available Spectrum Sensing techniques are listed in Sec- tion 2.Compressed Sensing based Spectrum sensing tech- niques [11] are described in Section 3. The Modulated Wide- band Converter technique is detailed in Section 4, and the re- construction algorithms along with results are explained in Section 55, the paper is concluded in Section 6 and Future scope is discussed in Section 7.
Spectrum sensing techniques perform sensing directly at
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Gandhiraj R is currently Asst. Prof. of Electronics and Communication
Engineering department in Amrita Vishwa Vidyapeetham, India, PH-
+919942223982. E-mail: r_gandhiraj@cb.amrita.edu
Nyquist rate. They manipulate the different properties of the signal in
1. Matched Filter Detection
2. Energy Detection
3. Cyclostationary Method
In high frequency applications the sampling rate becomes very high. The currently available ADCs are incapable of han- dling such high rates. So, we employ compressed sampling techniques to obtain the high frequency information at a rate lower than Nyquist rate. Three methods will be discussed be- low.
1. Random Demodulator
2. Multi- Coset sampling
3. Modulated Wideband Converter
Random demodulator is used to acquire sparse, band li- mited signals. The input signal is multiplied by a pseudo ran- dom sequence which spreads the tone across the entire spec- trum. Then a low pass filter is used to capture a part of mes- sage in the baseband which is later sampled at a lower rate. The obtained samples do not have linear relationship with the message.
Fig 1. Block diagram of Random Demodulator
IJSER © 2012
International Journal of Scientific & Engineering Research The research paper published by IJSER journal is about Spectrum Sensing using Compressed Sensing Techniques for Sparse Multiband Signals 2
ISSN 2229-5518
Convex optimization teniques are employed for efficient
We assume the input signal is a sparse multiband signal
reconstruction of the message. It was found that this method is
x(t)
which consists of N active bands each with
more efficient for multi tone signals.
bandwidth B . We also assume that the signal is sparse in frequency domain. And let the Nyquist frequency of the signal
Multi coset sampling method is used for compressive sampling of sparse multi band signals.
be
ith
f NYQ .The signal enters m channels simultaneously. In the
channel the signal is multiplied by a mixing function,
which has a period TP . After mixing, the signal spectrum is
1
truncated by a low-pass filter with cutoff
2TS
and the filtered
signal is sampled at rate TS . The sampling rate of each channel is low, so that existing commercial ADCs can be used.
Therefore the design parameters are number of channels ‘ m ’,
1
the time period TP
, and the sampling rate of the ADC
TS
Fig 2. Block diagram of Multi Coset Sampler
and the mixing function number of channels.
Pi (t) for1 i m , where m is the
The signal is fed into m parallel shifters which are also
In practice
Pi (t) can be any periodic function but here we
called cosets maintained at different shifts. These cosets pick
up at least one sample from each active band. The effective
m
take it as piecewise constant function alternating between the levels 1 for each of M intervals of the time.
sampling rate becomes
MT
which is a rate lower than
Pi (t) ik
, k t
k 1 Tp
, 0 k M 1
Nyquist rate. So, we obtain different versions of the same M M
signal [10]. ADCs introduce a bandwidth limitation which
causes distortion and maintaining time shifts of the order of
Nyquist rate is difficult.
Where
ik {1, 1}
and
Pi (t nTP ) Pi (t)
shows that Pi (t) is TP periodic.
In this method same signal
x(t) is passed parallely through
If X ( f )
is the spectrum of the signal and Yi [n]
are the
m channels. Each channel has a mixing signal Pi (t) that alias the input signal to baseband. The primary aim of this method
is to recover a sparse signal using relatively lesser number of
DFTS of each channel, then the relation between them can be shown by the following mathematical equation,
samples. An analog mixer aliases the spectrum, such that a spectrum portion from each band appears in baseband. The system consists of several channels, implementing different
Y ( f ) AZ ( f )
(1)
mixtures, so that, in principle, a sufficiently large number of
where Y ( f )
is a vector of length m with ith element being
mixtures allow to recover a relatively sparse multiband signal.
Yi ( f ) and Z ( f ) is the unknown vector to be calculated. This
Z ( f ) gives the spectral support of the input signal. A is an
m L Matrix and is called the sensing matrix
The necessary conditions for perfect reconstruction of the
signal are
fP B, fS fP
and
m 2N
The sufficient conditions for perfect reconstruction of the
signal is
fS fP B, M M min where
M ( f NYQ fS )
, and
m 2N
for blind
min 2
1
2 fP
Fig 3. Block diagram of Modulated Wideband Converter
reconstruction. The relation (1) is the compressed sensing framework and the reconstruction involves complex algorithms for perfect reconstruction of the original [1]. The
IJSER © 2012
International Journal of Scientific & Engineering Research The research paper published by IJSER journal is about Spectrum Sensing using Compressed Sensing Techniques for Sparse Multiband Signals 3
ISSN 2229-5518
above relation (1) is called the Infinite Measurement Vector problem.
In the simulation following specifications were used :
1.Initialize a vector A ' {} .
2.Find the column of A that has maximum correlation with Y .
3.Add that column to A ' .
4.Perform reconstruction using A ' .
5.Calculate mean square error E .
Mmin 267, L 267, B 6Mhz, N 6,
6.If
E ES
then
A ' is the reduced matrix of A and it
m 100, fS fP 7.5Mhz .
The main aim is to recover the unknown vector set x from the known measurements. It is called an Infinite measurement Vector problem if the set is continuous in time or it consists of large number of samples. The number of unknowns is large in number so that the computation load increases. So another model is proposed.
This model is similar to that of the IMV model except that the cardinality of is finite value ‘ l ’. It is proved that for every
IMV model there exists a MMV model such that the support is recovered. In this model we can see that the number of un- knowns have been decreased to some extent. But still this is a convex optimization problem so the computation load is high.
This model is obtained by further reducing the dimension of MMV model. Since we assume that the input bandsparse sig- nal has joint sparsity prior we will be able to breakdown the MMV model into multiple SMV models. As we are dealing with only one vector at a time the computational load decreas- es significantly.
The reconstruction of the signal from the obtained compressed samples is further split up into two sub problems. The first
is enough to reconstruct X from Y .
7.Else remove the maximum correlated column from A
and repeat from step 2.
8.Stop.
Fig. 4 Original Signal
one is to recover the support
S I (x()) from the equation
and the second one is to reconstruct the X with the know- ledge of S and Y [4].
This can be done in two ways:
1. Recovery of support directly from MMV system
using OMP algorithm.
2. Convert the MMV to SMV system and then recov-
ery of support using OMP algorithm.
The OMP algorithm is used to recover the support in this model.
Parameter: Acceptable error ES . Algorithm for the equation Y AX :
Fig. 5 Reconstructed signal
A sparse multiband signal with six active bands (including negative bands) was simulated in bandwidth range of 800Mhz . This is shown in Fig. 4.
This signal was sampled using MWC model. It was reconstructed using OMP algorithm directly from MMV. This is shown in Fig. 5. We can see from Fig. 4 and Fig. 5 that the
IJSER © 2012
International Journal of Scientific & Engineering Research The research paper published by IJSER journal is about Spectrum Sensing using Compressed Sensing Techniques for Sparse Multiband Signals 4
ISSN 2229-5518
active bands have been recovered. By changing the number of channels in MWC model the recovery rate varies. This is shown in Fig. 6 and Fig. 7
allowed iterations, error E , P the probability distribution and
S the optimization technique. The algorithm is as follows:
1. Take a vector ‘ a ’ of length ‘ l ’ that follows some
probability distribution P.
2. Calculate y Ya . This transforms Y into single
dimensional vector ‘y’. Solve y Ax
technique.
3. Let support be S I (x) .
using SMV
4. If S K
(1).
5. If S K
then find X by inverting the relation in
then discard this S and go to step 2.
Fig. 6 Recovery Rate for SNR=10dB
Fig. 7 Recovery Rate for SNR=25Db
This algorithm deals with reduction of a MMV model into many SMV models. Consider a MMV model
6. Perform this iterative process till the number of iterations is equal to iter.
By the end of this algorithm the spectral support S is found which may be directly used to find out the holes in the spectrum or the time domain signal can be reconstructed.
Fig. 8 Original Signal
Y AX
Where A is m n rectangular matrix, Y is m l
(2)
matrix and
X is n l
matrix. The matrix Y is converted into a single
column matrix y by using a matrix a which follows some
absolutely continuous distribution. The dimensions of a is l 1 . Then, a SMV problem is solved to obtain the support. Once the support is known the signal X can be reconstructed. The Reduce MMV and Boost (ReMBo) algorithm is an iterative process. Input to this algorithm is K which denotes the sparsity required, iter which denotes the maximum number of
Fig. 9 Reconstructed signal for 200 iterations
A sparse multiband signal with six active bands (including negative bands) was simulated in bandwidth range of
800Mhz . This is shown in Fig 8.
IJSER © 2012
International Journal of Scientific & Engineering Research The research paper published by IJSER journal is about Spectrum Sensing using Compressed Sensing Techniques for Sparse Multiband Signals 5
ISSN 2229-5518
This signal was sampled using MWC model. The MMV system is converted into SMV system using ReMBO algorithm [4] and then reconstructed using OMP algorithm for 200 iterations as shown in Fig. 9
Both methods discussed above show similar results except that the direct recovery from MMV involves less number of iterations than SMV. For the specifications mentioned in sec- tion IV, in MMV system the number of unknowns per itera- tions is 3204. In the SMV technique the number of unknowns per iteration is 267.
The MWC method recovers the required signal by sampling at a rate lesser than Nyquist rate and provides better results compared to Random demodulator and MC techniques for sparse multiband signals. Direct recovery from MMV system uses lesser number of channels to recover than Recovery from SMV system, but the number of unknown per iteration is higher in the former.
The matrix A used in this method is randomly generated to ensure perfect reconstruction. But currently studies are going on to use deterministic matrix instead of random matrix. This method finds wide applications in Software Defined Radio (SDR).
We would like to thank Ram R, Niranjan Manoharan, Archana C K, Manjari Das C and Krishna Teja V for their support dur- ing various stages of this work.
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