Multipath Channel Estimation Using Fast Least-Squares Algorithm

Xue Jiang∗, Wen-Jun Zeng†, En Cheng†, and Cong-Ren Lin†∗Sunplus mMobile Inc., Beijng Branch, Beijing 100085, China

†Key Laboratory of Underwater Acoustic Communication and Marine Information Technology of the Ministry of EducationXiamen University, Xiamen 361005, China

Abstract—Least-squares (LS) method is widely used formultipath channel estimation. The conventional algorithms forcomputing the LS solution involve matrix inversion, which iscomputationally demanding. In this paper, by exploiting thespecial structure of Toeplitz matrices, a fast and memory-saving numerical algorithm is developed for calculating the LSestimate of a multipath channel. The proposed fast channelestimation algorithm is easy to implement and has a lowcomputational complexity of O(N log N) with N the receivedsignal length. Simulation results are provided to verify theperformance and computational efficiency of the proposed fastLS algorithm.

Keywords-Channel estimation; least-squares (LS); Toeplitzmatrices; fast algorithm; conjugate gradient (CG) method.

I. INTRODUCTION

Multipath propagation often arises in such fields as wire-less communications [1], underwater acoustic communica-tions [2], sonar [3], and source localization [4], [5], [6].In a multipath environment, due to multiple reflections orscattering from the boundaries, the received signal can berepresented as the superposition of a number of time-delayedand amplitude-attenuated versions of the original transmittedwaveform [7], [8]. The performances of these systems aregreatly affected by the multipath channels. For example,the multipath causes time spread and the inter-symbol-interference (ISI) to communication systems.

In many applications, it is of interest to estimate theparameters of a multipath channel, including time-delaysand amplitudes, and remove the multipath effects. Multipathchannel estimation becomes an important problem of wire-less communications, underwater acoustic communications,GPS anti-multipath, and wireless localization.

The basic approach to channel estimation is the matchedfiltering, which is equivalent to the cross-correlation of thetransmitted signal and the received signal. This method hasbeen used in the GSM system. However, the performanceof the matched filtering is dominated by the property ofthe auto-correlation function of the transmitted sequence.To obtain a good estimate, it requires the auto-correlationfunction of the transmitted signal has a sharp main lobe andlow-level side lobes. Hence the transmitted training sequenceshould be carefully designed to possess such property.

The least-squares (LS) is a widely used channel esti-mation method. The conventional numerical algorithms for

LS problem need to compute the pseudoinverse of theToeplitz matrix formed by the transmitted sequence, whichis computationally expensive. When the signal length orchannel order is large, computation of the pseudoinverseof a large-size matrix is time-consuming. In addition, theconventional LS algorithms for need huge memory space tostore the large-size Toeplitz matrix.

In this paper, a fast LS algorithm for estimating theimpulse response of a multipath channel is developed byexploiting the special structure of the Toeplitz matrix. Theproposed method reduce the computational complexity fromO(NM2) to O(N log N), where N and M are the receivedsignal length and channel order.

II. PROBLEM FORMULATION

The received signal x(t) of a channel in response to theknown transmitted signal s(t) is

x(t) = h(t) ∗ s(t) + v(t) =∫

h(τ)s(t − τ)dτ + v(t) (1)

where ∗ denotes linear convolution, h(t) is the channelimpulse response, and v(t) is additive noise. Since a varietyof channels are prone to multipath propagation due torefraction, reflection, and scattering, the channel impulseresponse h(t) can be modeled as

h(t) =K∑

k=1

akδ(t − τk) (2)

where δ(·) denotes the delta function, K is the numberof paths, ak and τk are the amplitudes and time-delays ofpath k, respectively. The received signal x(t) then can beexpressed as the sum of K scaled and delayed replicas ofs(t):

x(t) =K∑

k=1

aks(t − τk) + v(t). (3)

In this paper, we consider the problem of identifying thechannel response h(t) from the known transmitted signals(t) and the received signal x(t).

After sampling with sampling period Ts, we obtain thediscrete-time version of (1)

xxx = SSShhh + vvv (4)

2011 Third International Conference on Communications and Mobile Computing

978-0-7695-4357-4/11 $26.00 © 2011 IEEE

DOI 10.1109/CMC.2011.12

433

where the vectors xxx = [x(0), · · · , x(N − 1)]T , hhh =[h(0), · · · , h(M − 1)]T and vvv = [v(0), · · · , v(N − 1)]T areformed from the samples of x(t), h(t) and v(t), respectively.The Toeplitz matrix SSS ∈ C

N×M is written as

SSS =

⎡⎢⎢⎢⎣

s(0) s(−1) · · · s(−M + 1)s(1) s(0) · · · s(−M + 2)

......

. . . · · ·s(N − 1) s(N − 2) · · · s(N − M)

⎤⎥⎥⎥⎦ (5)

with the first column being denoted sss = [s(0), · · · , s(N −1)]T .

The length of the transmitted training symbols s(n) isdenoted by Ns. Hence s(n) = 0 if n /∈ [0, Ns − 1]. Thegoal of channel identification is to estimate the channelresponse hhh from the transmitted signal s(n) (matrix SSS) andthe received signal xxx, i.e., solving the linear system in (4).

III. FAST LEAST-SQUARES ALGORITHM FOR CHANNELESTIMATION

In this section, we elaborate the fast LS algorithm forchannel estimation.

A. Principle of the Least-Squares Method

The LS method minimizes the following squared-errorfunction

minhhh

‖xxx −SSShhh‖2 (6)

which gives the LS solution

hhh = SSS†xxx = (SSSHSSS)−1SSSHxxx (7)

The superscript (·)† represents the pseudoinverse of a matrix.Although (7) gives the closed-form of the LS solution, it stillhas the following two drawbacks

1) It requires to construct and store the N × M matrixSSS, which needs a large amount of memory space whenthe signal length or the channel order is large.

2) It involves computing matrix multiplication and inver-sion of SSSHSSS, which requires a high computationalcomplexity of O(NM2).

In the next subsection, we will propose a fast andmemory-saving implementation of the LS method with alow complexity of O(N log N).

B. Fast LS Algorithm with Low Complexity

The leading computational cost of the LS method is tocompute the matrix multiplication and inversion. In addition,it needs to construct and store the N ×M matrix SSS, whichwill cost tremendous storage space when the length of thereceived signal N is large. Herein, we aim at reducingthe computational complexities of (7) and storage space byharnessing the structure of SSS. Our implementation of thechannel estimation method only requires to compute thematrix-vector products of the forms SSSHxxx and SSSHSSSbbb with avector bbb ∈ C

M . By considering the special structure of SSS

shown in (5), the matrix-vector products of SSSHxxx and SSSHSSSbbbcan be efficiently performed.

Define a circulant matrix CCC ∈ CN×N whose first column

is sss = [s(0), · · · , s(N − 1)]T . The relation between CCC andthe Toeplitz matrix SSS ∈ C

N×M in (5) is then can beexpressed as

SSS = CCCEEE (8)

where EEE = [eee1, · · · , eeeM ] ∈ CN×M with {eeei}M

i=1 ∈ CN

the basic vector whose ith coordinate is unit and othercoordinates are zeros. We will exploit the circulant propertyof CCC to fast compute the two matrix-vector products. Theeigenvalue decomposition of the circulant matrix CCC andCCCH is given by

CCC = FFFHΛΛΛsFFF (9)

andCCCH = FFFHΛΛΛ∗

sFFF (10)

HenceCCCHCCC = FFFH |ΛΛΛs|2 FFF (11)

where the superscripts (·)∗ and (·)H are the complex con-jugate and the Hermitian transpose, and FFF is the N × NFourier matrix whose (k, l)th element is FFF kl = 1√

Ne−j 2π

N kl.Premultiplying matrix FFF is equivalent to performing discreteFourier transform (DFT) and premultiplying FFFH meanscarrying out inverse discrete Fourier transform (IDFT). Inthis paper, the notation “˜” is used to represent the DFT ofa vector, e.g., sss = DFT [sss]. ΛΛΛs is a diagonal matrix whosediagonal elements are the DFT of sss, i.e.,

ΛΛΛs = diag {sss} . (12)

For an N -dimensional vector xxx = [x1, · · · , xN ]T , we use xxx↓

to represent the operation that selects the first M elementsof xxx, i.e.,

xxx↓ = EEETxxx = [x1, · · · , xM ]T ∈ CM .

For an M -dimensional vector bbb = [b1, · · · , bM ]T , we define

bbb↑ = EEEbbb = [bbbT ,000T ]T ∈ CN

i.e., the operation bbb↑ denotes padding N − M zeros to bbb.Using these notations, SSSHxxx in (7)can be calculated as

SSSHxxx = EEETCCCHxxx = EEETFFFHΛΛΛ∗sFFFxxx

= EEET IDFT[sss∗ � xxx

]= EEETcccsx = ccc↓sx

(13)

with cccsxdef= IDFT

[sss∗ � xxx

]the circulant cross-correlation of

sss and xxx.Now, we consider how to compute (SSSHSSS)−1 in (7) with

a low complexity. In fact, explicit matrix inversion is notrequired. Pre-multiplying SSSHSSS to (7), we have

SSSHSSShhh = SSSHxxx (14)

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Let AAA = SSSHSSS and bbb = SSSHxxx. We obtain the followingequation

AAAhhh = bbb (15)

where bbb has been computed in (13).All we need is to solvethis linear equation. Hence the conjugate gradient (CG)method [9] can be applied to solve (15). The CG method forsolving AAAhhh = bbb is listed as Algorithm 1, where the M × 1vector ppp(i) and the scalar α(i) denote the search directionand the step size in the ith iteration of the CG. εcg andNcg are the tolerance and the maximum iteration numberfor terminating the iteration of the CG, respectively.

Algorithm 1 The CG method for solving AAAhhh = bbb

Given tolerance εcg > 0 and the iteration maximum Ncg;Initialize hhh(0) = 0, ppp(0) = ddd(0) = bbb;for i = 0, 1, 2, · · · do

Stopping criterion: quit if∥∥AAAhhh(i) − bbb

∥∥ / ‖bbb‖ < εcg ori > Ncg

qqq = AAAppp(i)

α(i) =(ddd(i)

)Tddd(i)/

(ppp(i)

)Tqqq

hhh(i+1) = hhh(i) + α(i)ppp(i)

ddd(i+1) = ddd(i) − α(i)qqq

β(i+1) =(ddd(i+1)

)Tddd(i+1)/

(ddd(i)

)Tddd(i)

ppp(i+1) = ddd(i+1) + β(i+1)ppp(i)

end for

The main computational cost of the CG is calculating qqq =AAAppp(i). Since AAA = SSSHSSS, AAAppp(i) can be computed as

AAAppp(i) = SSSHSSSppp(i) = EEETCCCHCCCEEEppp(i)

= EEETFFFH |ΛΛΛs|2 FFFEEEppp(i) = EEET IDFT[|sss|2 � ppp

↑]

= EEETcccsp = ccc↓sp(16)

with cccspdef= IDFT

[|sss|2 � ppp

↑].

Therefore the CG method for solving (15) only needsO (N log N) operations per iteration. Generally speaking,the iteration number required by the CG is from 30 to 40for a given accuracy εcg = 10−9.

Now it is clear that the complexity of the proposed fastLS algorithm for channel estimation is O (N log N) periteration. It is also worthy pointing out that the proposedfast algorithm does not require to construct and store theN × M matrix SSS, which will save huge memory space.

IV. SIMULATION RESULTS

In this section, we conduct several numerical simulationsto illustrate the performance of the proposed fast LS al-gorithm for channel estimation. The parameters of the CGmethod are taken to Ncg = 30 and εcg = 10−7. The signal-to-noise ratio SNR = 10 log10

(Ps/σ2

v

)= 10 dB, where Ps

and σ2v are the variances of the transmitted signal and the

noise.

A. An Underwater Acoustic Channel Estimation Example

In the first numerical example, we consider an underwateracoustic communication system in a multipath propagationenvironment. The acoustic source is positioned at 5 m depth,and the receiver is located at 10 m depth with 30 m rangeof the source, as shown in Fig. 1. The depth of the shallowocean is 20 m. The sound velocity is 1500 m/s. The multipleray paths between the source and the receiver are displayedin Fig. 1. The discrete equivalent complex baseband impulseresponse of the multipath acoustic channel is

h(nTs) =K∑

k=1

ake−j2πfcτkδ(nTs − τk)

where Ts is the sampling interval, fc is the carrier frequency,τk and ak are the time-delay and amplitude of path k. Thetransmitted sequence s(n) is a QPSK modulated signal. Thesampling rate is equal to the symbol rate. We set the carrierfrequency fc = 20 kHz, and the symbol duration Ts =2 × 10−4 s corresponding the symbol rate 5 k baud.

The first 10 dominant rays are taken into account and theamplitude of the direct ray is normalized to unit. Note thatthe multipath time-delays are not the integer multiples of thesampling period Ts; therefore the leakage effect will appear,i.e., the number of nonzero elements of the channel responsewill be larger than the number of paths. The channel orderis about M = 380 (corresponding the maximum time-delay79 ms).

Fig. 2 illustrates the true channel response with theestimate result of the proposed fast LS algorithm with thelength of the transmitted training sequence being Ns = 64.Then the length of the received signal, i.e., signal processinglength is N = Ns + M − 1.

Figure 1. An underwater multipath propagation environment.

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Figure 2. Channel response and its estimate by fast LS algorithm.

B. Running Time Comparison

In the second experiment, we compare the running timeof the proposed fast LS algorithm with the conventional LSby directly computing (7). We run the two algorithms inMATLAB on a CPU of Intel L7100 and a 1.2 GB RAM. InMATLAB, the command h = S\x is taken as the numericalalgorithm of the conventional LS to compute the solution ofxxx = SSShhh.

The experimental setup is the same as the one in SectionIV-A with different length of the transmitted signal Ns.Hence the length of the received signal N = Ns + M − 1is also varying. The values of N are taken to N =1024, 2048, 4096. Table I reports average CPU times (inseconds) over 50 runs of the two methods under differentvalues of N .

Table ICPU TIMES (AVERAGE OVER 50 RUNS) OF THE FAST LS ALGORITHM

AND THE CONVENTIONAL LS.

Value of N 1024 2048 4096Fast LS 0.0156 0.0262 0.0488

Conventional LS 0.6616 1.4461 3.0961

The proposed fast LS algorithm is much faster thanthe conventional numerical algorithms for LS. Conventionalalgorithms has a high complexity of O (

NM2)

since it doesnot take into account the special structure of the Toeplitzmatrix SSS. In addition, it requires storing the huge matrix SSS,which will also slow down the speed. Hence the proposedmethod is very efficient for solving large-scale channelestimation problems.

V. CONCLUSION

A fast and memory-saving LS algorithm is developedfor multipath channel estimation. The proposed method

reduce the computational complexity from O(NM2) toO(N log N), where N and M are the received signal lengthand channel order. Numerical simulations verify the accu-racy and computational efficiency of the proposed algorithm.

ACKNOWLEDGEMENT

This work was supported by the National Natural Sci-ence Foundation of China under Grant 61071150, and theNatural Science Foundation of Fujian Province under Grant2010J01344.

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