SIViP (2012) 6:171–178DOI 10.1007/s11760-011-0213-0

ORIGINAL PAPER

Time delays and angles of arrival estimation using known signals

Pushpendra Singh · Pradip Sircar

Received: 8 January 2009 / Revised: 20 January 2011 / Accepted: 20 January 2011 / Published online: 11 February 2011© Springer-Verlag London Limited 2011

Abstract Channel estimation in a multipath mobile com-munication system is addressed in this paper, and a novelapproach based on the linear prediction in frequency domainand the singular value decomposition technique is presentedfor joint estimation of the angles of arrival and the time delaysof multiple reflections of a known signal. Simulation resultsillustrating the performance of the proposed algorithm areincluded, and the results show that the proposed method isclose in accuracy when compared to the iterative maximum-likelihood method. However, when the two methods are com-pared in computational complexity, it is demonstrated thatthe proposed method reduces the complexity to nearly halfof that of the maximum-likelihood method. The Cramer–Raobounds are computed for comparison.

Keywords Wireless communication ·Channel estimation · Array processing ·Source localization · Joint angle and delay estimation

1 Introduction

In a mobile communication system, channel estimation bythe base station involves the estimation of the angles of arrival(AOAs) and the time delays of multipath rays of a known sig-nal transmitted by a mobile unit [1–3]. The angle and delayestimates in a multipath channel are useful for equalizationand transmit diversity [4].

In this paper, we present a novel approach for estimat-ing the AOAs and time delays of multiple reflections of a

P. Singh · P. Sircar (B)Department of Electrical Engineering, Indian Institute of TechnologyKanpur, Kanpur, UP 208016, Indiae-mail: sircar@iitk.ac.in

known signal. The direct sequence spread spectrum (DS/SS)technique is used to generate the source signal, which bringsthe desirable features of spread spectrum communications inour system [5]. The proposed method utilizes the geometricalproperties of the signal in the frequency domain alternatelyfor the time delay and the AOA variables. We introduce thediscrete-frequency variables as ratios of the frequency sam-ples of the sensor output signal and the frequency samples ofthe known transmitted signal for this purpose. The frequencyvariables are then fitted in linear prediction models for theestimation of parameters.

Several papers are available in the literature [6–9], whichdemonstrate that the estimation of angle and delay param-eters of the multipath multiple-access wireless systems cangreatly improve the receiver performance. The methods usedfor the joint estimation of angle and delay are mainly someversions of two-dimensional MUSIC (Multiple Signal Clas-sification) and ESPRIT (Estimation of Sinusoidal Parametersby Rotational Invariance Technique). An iterative algorithmto implement a two-dimensional (2-D) maximum-likelihood(ML) estimation is presented in [1], where the joint estima-tion problem is solved by transforming the 2-D ML crite-rion into two sets of 1-D maximization problems in eachiteration.

The well-known SAGE (Space Alternating General-ized Expectation) maximization algorithm can estimate therelative delay, the azimuth and elevation angles, and the com-plex amplitude of a known received signal by using spher-ical arrays [10]. However, the convergence of the iterativemethod depends to a large extent on the initialization of theparameter vector. Hence, for mobile personnel communica-tion systems where the time delay and the azimuth angleof arrival parameters can characterize the channel effec-tively, the SAGE algorithm may not be a desirable solutionprocedure.

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2 Data model

We consider an array of M sensors receiving L reflections ofa known signal s(t). Let τl be the time delay, θl be the angleof arrival (AOA), and βl be the complex reflection coeffi-cient corresponding to the lth propagation path; then, usingthe complex envelope representation, the signal received bythe array is given by the following M × 1 vector,

x(t) =L∑

l=1

a(θl)βl s(t − τl) + n(t) (1)

where the vector a(θl) is the response of the array along thedirectional angle θl , and n(t) is the complex noise vectorassumed to be zero-mean, white, and Gaussian [4].

For a uniform linear array (ULA) with element spacing d,the response vector a(θl) is given by

a(θl) =[

1, exp

{j2π

λd sin θl

},

. . . , exp

{j (M − 1)

2π

λd sin θl

}]T

where λ is the wavelength of propagation.In the frequency domain, (1) becomes

X(ω) =L∑

l=1

a(θl)βl S(ω) exp(− jωτl) + N(ω) (2)

where X(ω), S(ω), and N(ω) are the Fourier transforms ofx(t), s(t), and n(t), respectively [11]. Note that it is sym-bolic to write the Fourier transform of (1) as (2). In practice,the K samples of the signal {xm(tk)} sampled at t = tk = kTs

at the output of the mth sensor are converted into K samples{Xm(ωk)} in the frequency domain by the discrete Fouriertransform (DFT). The circular frequencies ωk are expressedas ωk = 2πk

K Ts, where Ts is the uniform sampling interval.

3 Parameter estimation

For the characterization of the multipath channel, we areinterested in estimating the parameter sets {τl} and {θl},which are unknown constants. We propose a method basedon the geometrical properties of the signal in the frequencydomain alternately for the time delay and the AOA variables.The solutions of the polynomial equations arising from thegeometrical properties of the signal [12], in turn, provide thetime delay and the AOA estimates.

The complex coefficients {βl} are random variables forfading channels. In simulation study, we use {βl} as complexconstants over the duration of observation. We also carry outsimulation using {βl} as complex Gaussian random variables,

which shows that the proposed model is applicable for Gauss-ian fading channels [3].

3.1 Time delay estimation

Using the frequency samples of the sensor output signal{Xm(ωk)} and the frequency samples of the known trans-mitted signal {S(ωk)}, we compute the discrete-frequencyvariables,

pm[k] = Xm(ωk)

S(ωk); m = 1, . . . , M; k = 1, . . . , K .

which will be of finite value at each point with S(ωk) non-zero over the entire frequency range for the DS/SS signal.

At this point, note that when noise is not present, (2) canbe rewritten in the following form,⎛

⎜⎜⎜⎝

p1[k]p2[k]

...

pM [k]

⎞

⎟⎟⎟⎠ =

⎛

⎜⎜⎜⎝

γ11 γ12 · · · γ1L

γ21 γ22 · · · γ2L...

γM1 γM2 · · · γM L

⎞

⎟⎟⎟⎠

⎛

⎜⎜⎜⎝

zk1

zk2...

zkL

⎞

⎟⎟⎟⎠ (3)

where we use the notations

γml = am(θl)βl and zl = exp

(− j

2πτl

K Ts

)

Assume that z1, z2, . . . , zL are the roots of the polynomialequation [12]

D(z) = zL + bL−1zL−1 + · · · + b1z + b0 = 0 (4)

Then, using the relation

pm[k] = γm1zk1 + γm2zk

2 + · · · + γmL zkL ,

one can derive the following identity for the mth sensor,

pm[k] + bL−1 pm[k − 1]+ · · · + b1 pm[k − L − 1] + b0 pm[k − L] = 0 (5)

From the above identity and knowing that the set {pm[k] ;k = 1, . . . , K } is available for processing, we can write thefollowing matrix equation for the mth sensor,

Pmb = pm (6)

where

Pm =

⎛

⎜⎜⎜⎝

pm [K − 1] pm [K − 2] · · · pm [K − L]pm [K − 2] pm [K − 3] pm [K − L − 1]

......

...

pm [K − J ] pm [K − J − 1)] · · · pm [1]

⎞

⎟⎟⎟⎠ ,

b = [bL−1, bL−2, . . . , b1, b0]T ,

pm = [−pm [K ],−pm [K − 1], . . . , −pm [K − J + 1)]]T .

Note that the matrix Pm has the dimension J × L; thus, whenthe integer J = K − L is chosen such that J ≥ L , assuming

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SIViP (2012) 6:171–178 173

K ≥ 2L , then (6) can be solved for the coefficient vector b.For J = L and K = 2L , Pm becomes a square matrix.

For a better approach, we combine equations similar to(6) for M sensors to obtain the matrix expression⎛

⎜⎜⎜⎝

P1

P2...

PM

⎞

⎟⎟⎟⎠ b =

⎛

⎜⎜⎜⎝

p1

p2...

pM

⎞

⎟⎟⎟⎠

or

Qb = q (7)

which can be solved for the coefficients vector b as follows:

b = (QH Q)−1QH q (8)

Once we have the solution for the coefficients {bL−1, . . . ,

b0}, the polynomial equation D(z) = 0 can be formedand roots extracted. The time delays {τl} can be determineddirectly from the roots {zl}. Note that we can estimate the timedelays by utilizing data from one of the sensors. However,by utilizing data from all sensors, we get a robust estimate inthe sense of best fit.

3.2 Estimation with noisy data

When the signal in (1) or (2) is embedded in noise, we usean extended model order to accommodate L signal and − L noise components [13]. Extra dimensions of the sys-tem matrix generate the noise subspace, which can be sepa-rated from the signal subspace by utilizing the singular valuedecomposition (SVD) technique for extracting the principalcomponents of a matrix, and in the process, we get the desiredsolution with high accuracy, which is unperturbed by addi-tive noise [13, pp. 115–146]. With extended model order, (3)is rewritten as⎛

⎜⎜⎜⎝

p1[k]p2[k]

...

pM [k]

⎞

⎟⎟⎟⎠ =

⎛

⎜⎜⎜⎝

γ11 γ12 · · · γ1

γ21 γ22 · · · γ2

...

γM1 γM2 · · · γM

⎞

⎟⎟⎟⎠

⎛

⎜⎜⎜⎝

zk1

zk2...

zk

⎞

⎟⎟⎟⎠ (9)

In this case, we find the roots of the equation

E(z) = z + b−1z−1 + . . . + b1z + b0 = 0 (10)

and the coefficient vector

be = [b−1, b−2, . . . , b1, b0]T

is determined by solving the matrix equation for the mthsensor,

Pembe = pe

m (11)

where Pem =

⎛

⎜⎜⎜⎝

pm[K − 1] pm[K − 2] · · · pm[K − ]pm[K − 2] pm[K − 3] pm[K − − 1]

......

...

pm[K − Je] pm[K − Je − 1] · · · pm[1]

⎞

⎟⎟⎟⎠

and pem = [−pm[K ],−pm[K −1], . . . ,−pm[K − Je +1]]T .

The integer Je = K − is chosen such that Je ≥ , assum-ing K ≥ 2.

Rewriting (7) with (11) we get,

⎛

⎜⎜⎜⎝

Pe1

Pe2...

PeM

⎞

⎟⎟⎟⎠ be =

⎛

⎜⎜⎜⎝

pe1

pe2...

peM

⎞

⎟⎟⎟⎠

or

Qebe = qe (12)

which can be solved for be using the SVD technique [13],

be = Q+e qe (13)

where Q+e = V−1

e UH is the rank-constrained generalizedinverse of the matrix Qe, Qe = UVH , = diag(σ1, σ2,

. . . , σL , σL+1, . . . , σM ) and −1e = diag(1/σ1, 1/σ2,

. . . , 1/σL , 0, . . . , 0). Note that the effective rank of Qe isL . It is to be pointed out that if the number of multipaths Lin (1) is not known a priori, then L can be determined by theSVD technique.

The performance of an SVD-based parameter estimationalgorithm is often far superior to that of a classical method;however, since the SVD is a nonlinear operation, it is diffi-cult to get a precise characterization of this performance [13,p. 116]. For instance, it is beyond the scope of this paper tocharacterize how is to be chosen as a function of the sig-nal-to-noise ratio (SNR) and of L . In simulation study, wetake an empirical approach and choose which provides thebest performance at the lowest SNR. It is advisable to main-tain the similar ratio of and L in estimation of a practicalchannel.

3.3 AOA estimation

Using the values of {pm[k]; m = 1, . . . , M; k = 1, . . . , K }computed by the expression given earlier, we can write thefollowing matrix equation from (3),

P = ZG (14)

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174 SIViP (2012) 6:171–178

where

P =

⎛

⎜⎜⎜⎝

p1[1] p2[1] · · · pM [1]p1[2] p2[2] pM [2]

......

...

p1[K ] p2[K ] · · · pM [K ]

⎞

⎟⎟⎟⎠

Z =

⎛

⎜⎜⎜⎝

z1 z2 · · · zL

z21 z2

2 z2L

......

...

zK1 zK

2 · · · zKL

⎞

⎟⎟⎟⎠

and

G =

⎛

⎜⎜⎜⎝

γ11 γ21 · · · γM1

γ12 γ22 · · · γM2...

γ1L γ2L · · · γM L

⎞

⎟⎟⎟⎠

Furthermore, by allowing a decomposition of the matrix G,we find the expression

P = ZBAT (15)

where B = diag{β1, β2, . . . , βL} and A = [a(θ1), a(θ2),

. . . , a(θL)].By introducing a change of variable at this point, (15) can

be rewritten as follows

P = ZBY (16)

where

Y =

⎛

⎜⎜⎜⎝

1 y1 · · · yM−11

1 y2 yM−12

......

...

1 yL · · · yM−1L

⎞

⎟⎟⎟⎠

with

yl = exp

{j2π

λd sin θl

}

Assume that {y1, y2, . . . , yL} are the roots of the polyno-mial equation [12]

F(y) = 1 + c1 y + c2 y2 + · · · + cM−1 yM−1 = 0,

M − 1 ≥ L (17)

Then, post-multiplying both sides of (16) by the coefficientsvector

c = [1, c1, c2, . . . , cM−1]T ,

we get the following equation,

Pc = ZBYc = 0 (18)

or rewritten,

P̄c1 = p̄ (19)

where

P̄ =

⎛

⎜⎜⎜⎝

p2[1] · · · pM [1]p2[2] pM [2]

......

...

p2[K ] · · · pM [K ]

⎞

⎟⎟⎟⎠

c1 = [c1, c2, . . . , cM−1]T

and

p̄ = [−p1[1],−p1[2], . . . ,−p1[K ]]T

When K ≥ M − 1, (19) can be solved for c1. Once weget the solution for the coefficients {c1, c2, . . . , cM−1}, thepolynomial equation F(y) = 0 can be formed and rootscomputed. Among the M − 1 roots, L roots are identified assignal roots yl , and the corresponding AOA θl are estimatedfrom the roots. Note that by using M = L + 1, we can getthe rough estimates of the signal AOAs. These estimates canidentify the signal AOAs when the number of sensors M islarge compared to the number of multipaths L . More pre-cisely, assuming that the value of L is known from (13), weget the rough estimates of the L AOAs θl by first using thesampled data from L + 1 sensors. The condition, M > L , isnot too restrictive because what we consider here are signalscoming from distant scatterers/ reflectors, which are usuallylimited in number [14].

4 Simulation results

Simulation study is carried out to illustrate the performanceof the proposed algorithm. A uniform linear array (ULA)with elements spaced at half wavelength is used. The sourcesignal generated is a coherent BPSK signal with a fixedsequence of bits. The DS/SS technique is utilized to gener-ate the source signal, which in turn ensures that the elementspm[k] of the matrices Pm (6, 11) and P̄ (19) are of finite value,and the matrices are well-conditioned for inverse operation.The DS/SS technique guarantees that the frequency sam-ples of the transmitted signal are non-zero at each point overthe entire frequency range, which is a necessary conditionfor pm[k] to be of finite value. Furthermore, it should benoted that the proposed method developed with the sourcesignal generated by the DS/SS technique is readily applicablefor wireless communications using code-division multipleaccess (CDMA) [5].

The following specifications are used for generating theBPSK source signal:Carrier frequency fc = 100 MHz, symbol data rate fd =50 MHz, sampling rate fs = 600 MHz. The binary datasequence used is [1001110]. The values of fc, fd and fs

are chosen such that fs > 2( fc + fd), and fs/( fc + fd) isan integer.

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SIViP (2012) 6:171–178 175

For simulation, we set various parameter values as fol-lows:Number of multipaths L = 3, Complex reflection coeffi-cients: β1 = 1

3 + j4 , β2 = 1

4 + j3 , β3 = 1

3 + j3 , AOAs:

θ1 = sin−1(51/180) = 16.459◦, θ2 = sin−1(101/180) =34.132◦, θ3 = sin−1(151/180) = 57.023◦, Time delays:τ1 = 4.5Ts, τ2 = 8.6Ts, τ3 = 12.4Ts, Ts being the sam-pling interval, Number of sensors M = 15, Number of sam-ples of each sensor output K = 70, Extended model order = 8 used for noisy data.

The complex zero-mean white Gaussian noise is added tothe signal setting the SNR at various levels. The Monte Carlosimulation is carried out with 500 trials. The computed biasesand variances of estimates of the time delays and the AOAsversus SNR are plotted in Figs. 1, 2, 3, and 4. In the samefigures, we plot the respective bias and variance of estimatesof the time delays and the AOAs as computed by the iterativeML method for comparison purpose.

We compute the Cramer–Rao bound (CRB) of estimate ofeach parameter [1]. The computed lower bound of variance is

Fig. 1 Plot of computedtime-delay variance vs. SNR(dB); time delay = 4.5, 8.6,12.4, normalized by samplinginterval (top to bottom)

Fig. 2 Plot of computed AOAvariance vs. SNR (dB); AOA =16.459, 34.132, 57.023 degrees(top to bottom)

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176 SIViP (2012) 6:171–178

Fig. 3 Plot of computedtime-delay bias vs. SNR (dB);time delay = 4.5, 8.6, 12.4,normalized by sampling interval(top to bottom)

Fig. 4 Plot of computed AOAbias vs. SNR (dB); AOA =16.459, 34.132, 57.023 degrees(top to bottom)

plotted together with the variance of estimate in Figs. 5 and 6for comparison. We use 500 trials with reflection coefficients{βl} chosen as complex Gaussian random variables for com-puting the variances of estimates in Figs. 5 and 6 by theproposed method. In the same figures, we plot the variancesof estimates of the time delays and the AOAs as computed bythe iterative ML method [1] for comparison purpose. Notethat we are mainly interested to check the accuracy of esti-mation of the proposed method, and an ML method provides

a benchmark in achievable accuracy. It is to be pointed outthat since the proposed method is a direct method, it will nothave run-time difficulties like slow convergence, and con-vergence to local optimum, which are typical of an iterativeapproach.

It is apparent from the simulation results that the perfor-mance of the proposed method in accuracy is close to theperformance of the iterative ML method. However, the pro-posed method takes less than half of the time of computation

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Fig. 5 Plot of CRB andcomputed time-delaylog-variance vs. SNR (dB); timedelay = 4.5, 8.6, 12.4,normalized by sampling interval(top to bottom)

Fig. 6 Plot of CRB andcomputed AOA log-variance vs.SNR (dB); AOA = 16.459,34.132, 57.023 degrees (top tobottom)

what is needed in the iterative ML method. In a personalcomputer with the Intel Celeron processor (1.3 GHz clock,1 GB random access memory), the iterative ML method takes4.67 s for calculation of the parameters for 6 settings of SNR,i.e., 0.778 s per setting, whereas our method requires 2.25 s

for 6 settings of SNR, i.e., 0.375 s per setting. Note that theiterative ML method first estimates the AOAs ignoring thetime delays, and then the method estimates the time delaysby maximizing the likelihood function conditioned on theknown values of the AOAs. In the iteration, the AOAs are

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estimated using the estimated values of the time delays, andso on. Intuitively, when the problem is decoupled in theproposed method, the time taken to estimate two sets ofparameters is twice of the time needed to estimate one set ofparameters. However, for the coupled problem in the iterativeML method, we need to do a two-dimensional search, and thecomputational complexity becomes at least four times thatof a one-dimensional search for a single set of parameters.

5 Conclusion

We propose a new method employing the geometrical prop-erties of the radio signals received by an array of antennasfor estimating the time delays and the angles of arrival ofmultipath rays in a mobile communication system. An accu-rate solution of the joint estimation problem is importantfor incorporating various advantageous features of a mobilecommunication system equipped with smart antennas [15].The proposed method based on the polynomial factorizationand the SVD technique applied on the frequency samples ofthe radio signals provides a solution of the joint estimationproblem with high accuracy and efficiency.

Simulation results show that the performance of the pro-posed method in accuracy is close to the performance of theiterative ML method. This feature of the proposed methodcan be attributed to the fact that it is based on the SVDtechnique which makes a geometrical method perform likethe ML method [16]. The main advantage of the proposedmethod is that it takes less than half of the time of computa-tion compared to the iterative ML method.

An application of joint angle and delay estimation tech-nique is found in code acquisition tasks for DS/SS receivers[17]. The performance of these receivers is often evaluatedin terms of the mean acquisition time. This metric dependson computational complexity and achievable accuracy ofthe code acquisition strategy. The proposed method of jointangle and delay estimation reduces the time of computa-tion drastically while keeping the level of accuracy slightlyinferior. It will be interesting to know the overall effect onthe performance of the DS/SS receivers employing the pro-posed technique, and further research will be needed in thisdirection.

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