Channel Estimation by Using Short Training Sequences in CDMA Systems

Wireless Personal Communications (2005) 34: 359–371DOI: 10.1007/s11277-005-8262-8 C© Springer 2005

Channel Estimation by Using Short Training Sequencesin CDMA Systems

AHMET RIZANER1, HASAN AMCA1, KADRI HACIOGLU2 and ALI H. ULUSOY1

1Department of Information Technology, School of Computing and Technology, Eastern MediterraneanUniversity, Magosa, Kibris, Via Mersin 10, TurkeyE-mail: [email protected] of Colorado at Boulder, Center For Spoken Language Research, Boulder, CO 80309, U.S.A.

Abstract. Multiuser detection techniques are known to be effective to counter the presence of multiuser interferencein code division multiple access channels. Multiuser detectors can provide excellent performance only when thechannel impulse responses of all the users are precisely known. Hence, channel estimation becomes a challengingissue in mobile communication systems. In this paper, we address the problem of efficient maximum likelihoodmobile radio channel estimation at high channel efficiency that requires a short training sequence along with theknown spreading sequence. The proposed system can be employed in both the uplink and downlink of a heavilyloaded multiuser CDMA system. The extension of the approach with unknown users’ delays are also proposed.We present results that show the success of this method in recovering the transmitted bits with a relatively smallnumber of preamble bits.

Keywords: CDMA, channel estimation, mobile communication, multipath channel

1. Introduction

Direct sequence code division multiple access (DS-CDMA) has become one of the favoritesfor future mobile radio systems. Recently, there is a growing interest in multiuser detectors thatprovide excellent detection performance. A significant performance improvement of multiuserdetectors is due to the capability to suppress multiple access interference (MAI). Although,CDMA-based systems provide high power efficiency and moderate error rates, coherent mod-ulation does not provide reliable communication on fading channels if the impulse responsesof these channels are not known. Traditionally, channel estimation is achieved by sendingtraining sequences or using pilot channel. These approaches rely on periodic transmission oflong training sequences [1], making the identification of signature waveforms feasible sinceboth input and output signals are known during the transmission of these sequences. Generally,for better estimation accuracy, more training symbols or higher power for pilot channel shallbe required. Consequently, one must pay the price of using long training sequences with asignificant reduction of channel efficiency. Recently, some training based channel estimationalgorithms are proposed for CDMA systems [2, 3]. Although, the method proposed in [2] iscomputationally complex; it only requires the short training sequences of the desired users forthe estimation. Alternatively, subspace-based algorithms have been successfully developed fordifferent CDMA systems, which eliminate the use of training sequences [4–6]. The methodpresented in [4] is only applicable when the system is underloaded or when a few users are

360 A. Rizaner et al.

active. In [5], a subspace-based method, which can estimate the channel responses of over-loaded systems, is proposed. However, the computational complexity of this method is quitehigh due to complicated matrix manipulations.

This paper proposes a multiuser maximum-likelihood (ML) channel estimation methodthat uses a short training sequence and users’ spreading codes, and exploits the structure ofthe received multiuser data. The extension of the proposed algorithm without the knowledgeof the chip time delay index of each user is also developed. The algorithm is suitable for usein, for instance, slotted system where each user transmits a data burst with a short trainingsequence. The proposed system can be employed in both the uplink (mobile station to basestation) and downlink (base station to mobile station) of a heavily loaded multiuser CDMAsystem. The system can be heavily loaded since the multiuser interference can well be takencare of by the proposed channel estimator.

The rest of the paper is organized as follows. In the next section, the assumed communica-tion system model is presented. Section 3 describes the channel estimation method with shorttraining sequences. Finally, the simulation and analytical results, illustrating the performanceof the estimation method is presented together with some conclusions.

2. System Model

In a CDMA system, several users transmit simultaneously over a common communicationchannel. Figure 1 shows the equivalent baseband system model used in this paper. The receivedbaseband signal with a single receiver from P users can be represented as the superpositionof the active users with additive channel noise

y(t) =P∑

i=1

ri (t) + n(t) (1)

where subscript i denotes the user index, and n(t) is assumed to be white Gaussian noise withzero mean and a two-sided power spectral density of No/2.

The multipath-fading channel, which can be represented by the tapped delay line model asshown in Figure 2, can be implemented by a series of Dirac delta functions as [7]:

hi (t) =L∑

l=1

hi,lδ(t − τl) (2)

Figure 1. Asynchronous CDMA system.

Channel Estimation by Using Short Training Sequences in CDMA Systems 361

Figure 2. Tapped-delay line model of channel (hi (t)).

where L is the total number of propagation paths of the channel, hi,l is the complex gain of thel-th propagation path, and τl is the delay of the l-th propagation path. The channel coefficients,hi,l , are zero-mean complex Gaussian variables and are not changing within a symbol duration.

It is assumed that hi (t) has a finite support [0, LTc] where Tc is the chip period of spreadingcode waveforms. In addition, we may assume that the channel order is much less than the codelength (L � Lc) since the maximum delay spread of channel is usually insignificant relativeto the symbol period [8]. The received signal ri (t) for each user can be represented as:

ri (t) =N∑

n=1

Ai bi (n)L∑

l=1

hi,lci (t − ki − (l − 1)Tc − (n − 1)Ts) (3)

which can be viewed as a superposition of short signature waveforms ci (t) spaced by multiplesof symbol period Ts and linearly modulated by the transmitted information symbol sequencebi (n) with amplitudes Ai . In the case of asynchronous systems, each user may have a differentdelay ki where 0 ≤ k1 ≤ k2 ≤ · · · ≤ kp < Ts. Here, N is the length of the packet containingNe preamble bits.

3. Channel Estimation

We assume a training sequence of Ne symbols. By sampling the part of y(t) containing thepreamble symbols at chip rate (Tc), over the training period, the received signal vector Y oflength NeLc can be obtained as follows:

Y = A1G1h1 + A2G2h2 + · · · + ApGphp + N (4)

where, N is the noise vector, Gi is the spread training sequence matrix of the i-th user, hi =Ai [hi,1 hi,2 . . . hi,L ] is the channel coefficients vector of the i-th user in which the amplitudes(Ai ) are incorporated into the channel model. The spread training sequence matrix of eachuser can be represented as:

Gi = [zi (ki ) zi (ki + 1) . . . zi (ki + L − 1)] (5)

where

zi (m)def= [0 . . . 0︸︷︷︸

m

fi,1(Lc) fi,2(Lc), . . . fi,Ne (Lc − m)]T. (6)


Figure 3. Delayed transmitted signal of i-th user over the training period.

Here (·)T denotes the transpose operation and,

fi,n(x)def= bi (n)[ci (1) ci (2) . . . ci (x)]. (7)

The delayed transmitted signal of the i-th user (zi (m)) over the training period is illustrated inFigure 3. The vector Y given in (4) can be rearranged as follows:

Y = [G1 G2 . . . Gp]︸︷︷︸G

h1

h2...

hp

︸︷︷︸H

+ N. (8)

3.1. ESTIM ATION METHOD

The target function for channel estimation can be defined as:

f (H) = E[(Y − GH)H(Y − GH)]. (9)

The channel estimate H that minimize the target function is the maximum-likelihood estimatewhich satisfy the following equation:

H = arg minH

f (H). (10)

The unique minimum of f (H) can be determined by setting the partial derivative ∂ f (hi )/∂hHi

to zero and solving the resulting expression for H,

∂ f (H)

∂H= GHGH − GHY = 0. (11)

Denote G† = (GHG)−1GH as the pseudoinverse [9] of G, under assumption that G has fullcolumn rank and G†G = I, where I is the identity matrix. The maximum-likelihood estimateof the channel coefficients of all the users, which amounts to the multiplication of Y by the


pseudoinverse of G, is given by:

H = G†Y. (12)

H in (12) can be written as:

H = G†(Y) = G†(GH + N) = H + G†N. (13)

The expected value of G†N is E[G†N] = 0. Here, H is an unbiased linear estimate of H. H isalso the maximum-likelihood estimate of H from Y [10]. The error in the estimation is G†N.Thus, the mean square error (MSE) of the estimation can be given as:

ε = E[(H − H)(H − H)H]

= E[(G†N)(G†N)H] (14)

= E[NNH]E[G†(G†)H].

The superscript (·)H is used to denote the conjugate transpose operation. Since, E[NNH] = NoI,the MSE can be written as,

ε = No E[G†(G†)H]. (15)

Hence, the corresponding MSE of the i-th user is,

εi = No

i L∑

l=(i−1)L+1

gl(gl)H (16)

where gl is the l-th row of G†.One of the important properties of this algorithm is that, channel coefficient of the desired

user can be estimated without having to estimate all other users. That is, the correspondinguser rows of G†, which is used to estimate the channel response of user i, is given by,

gi = [g(i−1)L+1 g(i−1)L+2 . . . gi L ]. (17)

Then, the channel estimation of the desired user can be obtained as,

hi = g†i Y. (18)

3.2. INITIAL SYNCHRONISATION

In (8) the users’ delays are assumed to be known at the receiver or estimated. In the uplinksignal processing (i.e. base station), the spreading code sequences of the users are knownbut their delays are unknown. In the case of unknown delays, it is not possible to constructGi matrices. Hence, the procedure for establishing initial synchronization is critical to thealgorithm performance. Therefore, we develop a novel multi-user channel estimation algorithmwhich does not require any timing information. We suggest incorporating the delays into thechannel models. Assuming that the maximum possible delay is Lc − 1, the order of the


channels is overestimated and hi is taken as if it has Lc + L − 1 elements. Then, G isreconstructed in accordance with this modification. Hence, the size of the resulting G matrixis now Ne Lc × (Lc + L − 1)P instead of Ne Lc × L P . The Gi ’s can be constructed as:

Gi = [zi (ki ) zi (ki + 1) . . . zi (ki + Lc − L − 1)]. (19)

In the absence of noise, the channel estimate of the i-th user turns out to be

hi =

0 · · · 0︸︷︷︸ki

hTi 0 · · · 0︸︷︷︸

Lc−ki −1

T

. (20)

By investigating the structure of (20), in the presence of additive noise, we suggest a simplethresholding to get the actual channel coefficients from the estimate. We can detect ki zerosby checking how many smaller values of hi are below a predetermined threshold level.

3.3. CHANNEL ESTIM ATION COMPLEXITIES

We have evaluated the computational complexities of the estimation algorithms by countingthe number of floating-point operations (flops) [9]. Multiplications, divisions, additions andsubtractions were counted as one flop. Maximum-likelihood channel estimation with delayknowledge involves matrix pseudoinverse at the end of the preamble symbols. The GHG matrixof dimension L P × L P should be inverted that requires L3 P3 floating point operations foreach packet transmission. In the subspace-based estimation method, eigenvalue decompositionshould be applied to Lc × Lc data matrix to obtain the necessary subspaces [2]. Both ofthese estimation approaches delay the start of detection beyond the length of the preamblesequence until the estimate has been computed. Moreover, pseudoinverse of G ((GHG)−1GH)or eigenvalue decomposition should be calculated periodically for each frame at the end of thepreamble sequence, which makes the problems even more severe.

Although the suggested channel estimation approach without users’ delay information con-siderably increases the size of matrix (G) that should be inverted, the computational complexityof the proposed approach is not high. The main complexity of the proposed channel estimationmethod without chip synchronization is the inversion of the (Lc + L − 1) × (Lc + L − 1)matrix GHG. However, G matrix is now constant for each transmission even the users’ delaysmay vary since the users’ delays are incorporated into the channel model. In this approach,the inverse of GHG is pre-computed offline and only once and then each user stores only thenecessary part (gi ) of this inverted matrix for channel estimation. Although, there is a mem-ory requirement to store the necessary matrices, the channel estimates can be carried out byonly simple matrix multiplications. An example with a representative system setup is givenin Table 1. In this example, the required number of floating point operations to transmit asingle frame is calculated. It is seen that, subspace method that requires approximately threetimes more cumulative number of floating point operations than exact ML estimation methodfor channel estimation is the most complex algorithm [13]. The modified ML method with-out synchronization is computationally simple. It approximately requires 100 times less flopsthan ML method with known delays and 300 times less flops than subspace-based estimationapproach.


Table 1. Computational complexities of the necessary operations for channel estimation

ML (known ML (unknown Subspace methodOperation delays) (Mflops) delays) (Mflops) (Mflops) [13]

Inverse of G (Online) 1.603 – –

Inverse of G (Offline) – 91.458 –

Eigenvalue decomposition – – 4.759

Estimation of hi 1.619 0.016 5.036

Here P = 7, L = 4, Lc = 31, N = 400, Ne = 16.

4. Numerical Results

The effectiveness of the proposed estimation method is illustrated by analytical results andextensive computer simulations. In all of the following examples, an asynchronous CDMAsystem with P = 7 users was analyzed by using N = 400 data symbols in each packet.The channel length (L) was preselected to be four and the channel response of each user wasgenerated Gaussian randomly based on (2). The desired and interfering users employed Goldsequences [11] of length Lc = 31 as short spreading codes. The preamble bits and data bitswere generated randomly for each user as antipodal signaling.

The first experiment is conducted to show that presented ML channel estimation methodis near–far resistant. Figure 4 presents the root mean square error (RMSE) behavior of the

Figure 4. RMSE vs. NFR for user 1 (SNR = 10 dB, N = 400, P = 7, L = 4, Lc = 31).


proposed channel estimator for several preamble sizes at 10 dB signal to noise ratio (SNR)with respect to near–far ratio (NFR). The average RMSE of the estimation is defined by,

RMSEi =√√√√ 1

M

M∑

j=1

‖h ji − h j

i ‖2 (21)

where M is the number of iteration in each Monte-Carlo trials, and h ji is the estimation of the

channel vector from the j-th iteration. The RMSE of the channel parameter estimate has beencalculated by averaging 200 Monte-Carlo runs. In each run different channel parameters wereused. The near–far ratio is defined as the ratio of the average received power (A2

av) of eachinterferer to that of the desired user by A2

av/A2i , where the amplitude of the transmission for

the interfering user k is Ak = 1 for all k �= i . The power level of the desired user was adjustedwhile the signal strength for all the other interfering users were kept constant. As evident fromthe simulation results for Ne = 4 and Ne = 16, the estimation algorithm is near–far resistant.With increasing the NFR the performance is not changing. For the rest of the simulations,perfect power control was assumed (i.e. all users have equal powers, A2

k = 1).The RMSE performance of the estimation method is presented in Figure 5, for both known

and unknown delays, with respect to SNR. In the former, the curves were obtained both analyt-ically and by simulations for different preamble lengths using (16). The corresponding resultsfor unknown delays were obtained through simulations. There is an excellent agreement be-tween analytical and simulated results. The performance improvement by increasing the num-ber of preamble bits used is obvious. The channel estimates without using the delay knowledgeis close to the ones obtained using the delay information. The performance of a subspace-basedestimator [2] is also shown for comparison by using the actual delay knowledge.

Figure 5. RMSE vs. SNR for user 1 (NFR = 0 dB, N = 400, P = 7, L = 4, Lc = 31).


Figure 6. RMSE vs. Ne for user 1 (SNR = 10 dB, NFR = 0 dB, N = 400, P = 7, L = 4, Lc = 31).

Figure 7. BER vs. Ne for user 1 (SNR = 10 dB, NFR = 0 dB, N = 400, P = 7, L = 4, Lc = 31).

The effect of the preamble length on estimation is shown in Figure 6. In this example the es-timation performance was investigated at 10 dB SNR by increasing the length of the preamble(Ne = 1, 2, 4, 8, 16, . . .). It is seen from this example that increasing the number of preamblesymbols used does not improve the estimation performance significantly after a large numberof preamble sizes (300). The subspace-based estimator [2], which exhibits slightly lower es-timation performance especially at high preamble lengths, is also shown. Figure 7 illustratesthe effect of preamble size on bit error rate (BER) at 10 dB SNR. In these simulations after


the channel coefficients were estimated by using different number of preamble sizes, a simpledecorrelating filter [12] was constructed to recover the original signal for each user. It is worthmentioning that the BER performance levels off at a bit length much smaller than that of theMSE performance. Since the ultimate performance indicator is the BER, we conclude that, asmall number of bits (10–20) are sufficient for successful training. Signal constellation plots arepresented in Figure 8. This is another way of visualizing the success of the estimation approach.

Figure 8. Signal constellations for user 1 (SNR = 10 dB, NFR = 0 dB, N = 400, P = 7, L = 4, Lc = 31).

Figure 9. RMSE vs. number of users in the system (P) for user 1 (SNR = 10 dB, NFR = 0 dB, N = 400, P =7, L = 4, Lc = 31).


As can be easily seen; the clusters by estimated channel responses are separated as well as theactual channel knowledge case. The proposed method can successfully estimate the channelresponses and by using these estimates the original signal of each user can be reconstructedsuccessfully.

We claim that, the system can be heavily loaded since the multi-user interference is welltaken care of by the proposed channel estimator. To demonstrate this, under 10 dB SNR, theroot mean square error estimation performance of a particular user by gradually increasing thenumber of total users in the system was examined. The results are presented in Figure 9. Forboth 4 and 16 training symbols, it is shown that the channel estimator is robust to MAI. Thatis, the performance of the proposed estimator is not degraded for a heavily loaded system.

5. Conclusion

In this paper, we have considered a maximum-likelihood channel estimation method for asyn-chronous CDMA systems, which estimates the channel responses by using short trainingsequences together with the known spreading sequences.

It is shown that increasing the number of preamble bits used does not improve the estimationperformance significantly after a large number of preamble bits. The effect of preamble size ismuch less effective on BER than RMSE performance. The RMSE performance of the proposedmethod without using delay information is very close to the results obtained with perfect delayknowledge. Although the proposed estimation method is computationally simple, there is aneed to store some matrices for the estimation. But the memory requirement to store thesematrices is not important if the length of preamble sequences is kept moderately short. As aresult, we can conclude that it is possible to successfully estimate the channel responses bythe proposed method by using short training sequences.

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13. 2 A. Rizaner, K. Hacıoglu, H. Amca, and A.H. Ulusoy, “Signature Waveform Estimation by Using ShortTraining Sequences,” in Proceedings of Third International Conference on Information, Communications &Signal Processing, ICICS2001, Singapore, October 2002.

Ahmet Rizaner was born in Larnaca, Cyprus, on January 31, 1974. He received the B.S.and M.S. degrees in Electrical and Electronics Engineering from the Eastern MediterraneanUniversity, Famagusta, North Cyprus, in 1996 and 1998, respectively. He completed his PhD.degree in Electrical and Electronic Engineering in Eastern Mediterranean University andjoined Eastern Mediterranean University as a lecturer in 2004. He is lecturing in the Schoolof Computing and Technology. His main research interests include CDMA communications,adaptive channel estimation, and multiuser detection techniques.

Hasan Amca was born in 1961 in Nicosia-Cyprus. He graduated from the Higher Technologi-cal Institute in Magosa-Cyprus (which is renamed later as Eastern Mediterranean University).He joined EMU in 1985 after receiving a M.Sc. (Digital Signal Processing) degree from theUniversity of Essex in England (1985). He took his Ph.D. (Mobile Communications) from theUniversity of Bradford where he was on a Commonwealth scholarship. He has been teachingin the Electrical and Electronic Engineering Department of Eastern Mediterranean Universitysince 1993 where he also served as the vice chairman from Spring 1998 to Spring 2000. Hehas been appointed as the Director of the School of Computing and Technology of the EMUsince Spring 2000. His research interests include Multi User Detection of CDMA signals,Adaptive Equalisation, Multi Carrier Systems, Mobile Radio Systems and Networks, Internetand Information Technology Applications in Education.


Kadri Hacıoglu was born in Nicosia, Cyprus. He received the B.Sc., M.Sc., and Ph.D. degreesin electrical and electronic engineering from the Middle East Technical University, Ankara,Turkey, in 1980, 1984, and 1990, respectively. After his two-year military service, in 1992, hejoined the faculty of Eastern Mediterranean University, Magosa, North Cyprus, as an AssistantProfessor, and became an Associate Professor in 1997. While there, he taught several classeson electronics, digital communications, speech processing and neural networks. During thistime, he conducted research on applying fuzzy logic, neural networks, and genetic algorithmsto signal processing and communications problems. From 1998 to 2000, he was a VisitingProfessor in the Department of Computer Science, University of Colorado, Boulder. Here,he taught classes on neural networks and continued his research. Since 2000, he has been aResearch Associate at the Center for Spoken Language Research, University of Colorado. Hehas authored or coauthored numerous papers and supervised a dozen M.Sc./Ph.D. theses. Hiscurrent research interests are concept-based language modeling, speech understanding, naturallanguage generation, and search methods in speech recognition/understanding. He also doesresearch on multiuser detection and equalization in CDMA systems.

Ali Hakan Ulusoy was born in Eskisehir, Turkey, on June 3, 1974. He graduated from thedouble major program of the department of Electrical and Electronic Engineering and depart-ment of Physics in Eastern Mediterranean University as the first rank student of Faculty ofEngineering in 1996. He received his M.S. degree in Electrical and Electronic Engineering inEastern Mediterranean University in 1998. He completed his PhD. degree in Electrical andElectronic Engineering in Eastern Mediterranean University and joined Eastern MediterraneanUniversity as a lecturer in 2004. He is lecturing in the School of Computing and Technology.His current research interests include receiver design, multi-user detection techniques, blindand trained channel estimation in Code Division Multiple Access (CDMA).

Channel Estimation by Using Short Training Sequences in CDMA Systems

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