Review Of Sparse Channel Estimation Basic focus on the development of the Sparse Channel Estimation using pilot signal &

comparison of different algorithms being employed.

Sanjeev Baghoriya Electronics & Communication Engineering department

Indraprastha Institute of Information Technology Delhi, India

sanjeev13161@iiitd.ac.in

Abstract—Sparsity occurs in various communication applications which can be infact a boon in estimating the channel to recover the signal transmitted through it. In addition to this , if the Channel State Information(CSI) are known at the receiver then the recovery becomes easier. Therefore , we have taken into account the pilot signal utilization technique for the channel estimation by equalisers. This paper reviews the various algorithms & methods that have been described over the years by many pioneers in their work. The later portion of the paper compares the algorthms to discuss the pros ,cons & possible future advancements .

Index Terms— Pilot signal, Sparsity

INTRODUCTION Wireless communication applications basically involves lots

of problems but the main problem is the successful reception & recovery of the transmitted signal through the channel . So , it is very much critical to understand the behaviour of the channel in the sense that how will the signal be affected .Whether it will degrade or completely lost or have multipath etc .So knowing the channel characteristics is basically termed as the process of channel estimation. With the improvements in the channel estimation techniques it is even possible to transmit & receive over the various range of frequencies over different fading phenomenas. We have divided our content in sub-sections where firstly we will be discussing about the fundamentals of our problem such as sparsity of a channel , equalisers

Sparsity: Exploiting the sparsity of channel with different methods has

been a hot topic for the researchers . It not only reduces the work of the receiver but also complexity & the cost . Sparsity[1] is basically a notion where the information or the data rate of a signal is smaller than depicted by its bandwidth or the number of degrees of freedom on which signal depends is smaller than its length[2]. Sparse channel is the channel impulse response where the zero components outnumber the non-zero components. So,it is not necessary to estimate all the enteries of the sparse signal rather only the knowledge of the non-zero enteries as well as the location of those enteries is required for the process of reconstruction. The condition can mathematically can be stated as the signal ‘x’ is sparse in the basis 훹. The signal can be represented using only a small number K<<M of elements from 훹.

|| 훹x||lo ≤ M .

Fig.1 : Sparse Channel Impulse Response Equalisers: [3] Since many practical channels are bandlimited & distort the transmit signal linearly which causes ISI [3]. Therefore to mitigate the affects of the channel an intelligent receiver component is employed which is known as the equaliser. The complexity of equalisers increases as the channel length increases ,thus the sparse channels are well suited for their easy functioning . Let us consider the following block diagram:

Fig.2 Basic Wireless communication block diagram [4]

Channel Estimation Techniques: Channel estimate allows the receiver to approximate the effects of the channel on the signal. It is of three types: 1) Pilot Assisted: - It is the most straight forward where the symbols are known to the receiver. 2)Blind(Without Pilots): - It is based on the channel statistics employment rather than the pilots. 3)Semi-Blind: - It is the combination of the above where the initial estimation is pilot based & next on channel tracking. For example , an ofdm fram contains preambles ,guard bands , data symbols,cyclic prefix & a pilot tone which contains the information of the mentioned parameters so that it becomes easy for the receiver to recover the signal & process accordingly. Basically the channel is observed as : Y = H*X + n where length of Y is << length of X to fulfill the sparsity condition.

OVERVIEW OF THE ALGORITHMS a) Classical Least squares

The classical Least Square channel[5] estimation ,popular in the 90s ,assisted by the pilots includes two steps in the frequency domain. Initially , the channel impulse response for the all the pilots is estimated separately. Then , the channel impulse response for the non-pilots is obtained through process of interpolation.

,where J is the cost function, where Xp is the pilot symbol , 퐻 channel response estimate of each pilot signal, Yp is the received signal.

then the estimator 퐻 becomes 퐻 = 푋 푋 -1푋 Yp . For LS , we can say that every tap in this channel estimate will have a non-zero value which will adversely affect the equalizer performance. So , to estimate a sparse channel reponse the non-zero taps are reduced by means of threshold which is known as thresholdedLS(ThLS). It provides very slow convergence & is expensive in comparison to other available methods but will perform efficiently for high SNR. b)Zero Tap Detection using Approximate Maximum Likelihood This[6] known as AMLE (Maximum likelihood estimation) is generally used in the detection problems & outperforms LS at low to moderate SNR but they have a disadvantage of having high computational costs than classical LS. Let us assume irrespective of the channel structure the first estimate as per LS is : h= [UTU]−1UTx . But here the assumption is based only on the fact that the priori is available only for the LS estimator after that the whole process depends on the channel statistical measurements , which is not the area of our interest but to explain the development & approaches over the years it is critical to undertstand the comparison.

from [6,section 3.1] we have ,

,where 휆 is the optimising parameter &p=q=1/2 is assumed for the approximate detection.

푏 ,hence this problem can be solved using the Viterbi algorithm . They have a problem of unstable output at the high SNR wjere the LS has upperhand. Below is the BER comparison of both for a single user bound.

Fig.3 LS vs AMLE BER comparison.. [6] The advancements in AMLE will be discussed in the inferences section of all the algorithms. b) Matching Pursuit Algorithm [3] Let the stationary channel impulse impulse response be denoted as C(m), m=0,….,M-1 is sent through the channel & the received entries are r(m), m=0,…,M-1& for sparse channel C(m) ≠0 for very few values of m. The response is given as the . As per [3],

where b denotes the residual matrix & A the column matrix which when has maximum correlation with b provides orthogonal projection & the iterations go on till the solution is reached.

where K represents the tap value position & correspondingly the tap coefficients. It has the faster convergence rate advantage over the classical least squares . But it has the problem of basis re-selection as it sequentially selects the column matrix from the dictionary which makes it of less use for equaliser in various applications.

c) Orthogonal Matching pursuit More accurate estimation of the channel is derived in [8] which completely eliminates the problem of the basis-reselection. The work in [8] explains that orthogonal matching pursuit at the equaliser employs an intelligent method that selects the base vector such that it has maximum correlation with the residual vector. This makes the job of the decision feedback equaliser a lot easier in finding the approximate number of tap. d) Block Orthogonal Matching Pursuit[9]

Fig.4 Cluster Sparse channel response In various channel environments there exist large obstacles like mountains , hills , buildings which tends to give sparse cluster-structure in the channels which was not considered in the application of the MP as well as the OMP algorithms. They loose their importance when large multipath interference arises due to the long duration pulse shapes caused by these cluster sparse channels. The channel vector h is , ||h||Lo=K<<L where it has the K-sparsity. Lo norm counts the non-zero taps in a vector. Also , Y=HX+n where :

problem :

which depicts that the cluster is counted only if it is exceeding the noise floor otherwise the calculated & estimated values are considered as a part of the noise matrix ,n. All the cluster components are assigned same tape values as they exhibit same characteristics thereby reducing the complexity & calculations required by the equaliser. BOMP works in the same manner as the OMP except that it consider the correlation of the block of sparse clusters channel matrix. e) OMP using Sensing Measurement matrix[10],[11] Linear Inverse problems are encountered in the channel estimation applications especially for the sparse channels ,where the accurate estimation by using the shorter designed

training/pilot sequence is a major problem to solve. It should be desgned to satisfy the restricted isometric property [12]. Conventionally it has been a problem due to coherence interference of columns in training signal which is defined as :

where 휇 is the mutual incoherence . Input: Observation signal vector 휙 Output: Sparse signal Vector 훽SMM Step1: Initialise the residual r0= 휙 & set the selected variable X=0/ . Iteration=1 count start. Step2: Find variable Xi that solves maximisation problem

where W is SMM

& uses . Add the variable Xi to the seet of selected variables & update X. Step 3: Update ri Step 4: If the stopping condition is attained stop algo else move to step 2.

where signal length is N=48 & 1000 monte-carlo runs are used. f) Compress Sensing Technique[12][13][14] Considering the conventional linear model for the measurement : y= 휙x= 휙 훹 휃 where 훹 휃 denoted the effective matrix for the estimation of the k sparse vector 휃 .

Fig 6. Compressed sensing mathematical representation. Methods: RIP(Restricted Isometric Property

It provides the basis to not to deteriorate the compressed samples but it doesn’t signy how to recover the sparse vector.

Lo norm It estimates the sparse solution accurately but it is not feasible.

L1 norm Gives correct estimation but includes mild oversampling

L2 norm Solution is never sparse approximately.

CONCLUSION & INFERENCES >> Orthogonal matching pursuit algorithms are bandwidth

efficient , good convergence rate with shorter training sequences but they don’t have that much stable output as far as the procedure moves on with the number of iterations in the time varying channels, Thus , it creates little difficulty for the receiver equalisers in estimation of the channel.

>> Whereas the least square classical approach works well for high SNR signals with pretty good convergence rate when modified by providing threshold to all the non-zero taps to estimate the correct number. They are not as much bandwidth efficient as OMP based equalisers are.

>> Cluster –sparse estimation estimates smaller number of channel freedom of degree & shorter training sequences. They are spectrally efficient in wideband applications.

>>Approximate maximum likelihood estimation has been in advancement now a days as it is modified with an iterative approach by utilizing an initial estimate . widely used in the ofdm systems.[15]

>>Compressive sensing is a very hot topic as it provides the shortest training sequence along with minimum number of samples for the sparse channels analysis. Since the correct estimation using the Lo norm is not feasible to efficient optimization thus showing a way for the convex optimization techniques.Basis pursuit * lasso’s algorithm are known to work well & are still under research to give more correct estimate.

ACKNOWLEDGEMENT I wish to thank Dr. Angshul Majumdar to provide me this

opportunity to increase my bank of knowledge with such a new field of signal processing.

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