DESIGN OF PILOT-SYMBOL ASSISTED POWER DELAY PROFILE ESTIMATION FOR MIMO-OFDM SYSTEMS

International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print),

ISSN 0976 – 6553(Online) Volume 4, Issue 5, September – October (2013), © IAEME

36

DESIGN OF PILOT-SYMBOL ASSISTED POWER DELAY PROFILE

ESTIMATION FOR MIMO-OFDM SYSTEMS

PRATHIMA BOPPANA Mtech( Wireless and Mobile Communication),

Vaagdevi College of Engineering, Bollikunta, Warangal, Andhra Pradesh, India

M. SHIVA PRASAD Assistant Professor, ECE Department,

Vaagdevi College of Engineering, Bollikunta, Warangal, Andhara Pradesh, India

ABSTRACT

This work proposes a power delay profile (PDP) estimation technique for linear minimum

mean square error (LMMSE) channel estimator of multiple-input multiple-output orthogonal

frequency division multiplexing (MIMO-OFDM) systems. For practical applications, only the pilot

symbols of all transmit antenna ports are used in estimating the PDP. The distortions caused by null

subcarriers and an insufficient number of samples for PDP estimation are also considered. This

technique effectively reduces the distortions for accurate PDP estimation. Simulation results show

that the performance of LMMSE channel estimation using the proposed PDP estimate approaches

that of Wiener filtering due to the mitigation of distortion effects.

Index Terms: Channel estimation, power delay profile, MIMO, OFDM, 3GPP-LTE.

INTRODUCTION

MULTIPLE-INPUT multiple-output orthogonal frequency division multiplexing (MIMO-

FDM) is one of the most promising techniques for wireless communication systems, including the

3rd Generation Partnership Project Long Term Evolution (3GPP LTE) and IEEE 802.16 (WiMAX).

MIMO-OFDM provides a considerable performance gain over broadband single-antenna systems by

obtaining the spatial diversity or multiplexing gain. Most receiver techniques of MIMO-OFDM

systems are designed with the assumption that channel state information (CSI) is available, in order

to achieve the maximum diversity or multiplexing gain. The performance gain depends heavily on

accurate channel estimation, which is crucial for the MIMO-OFDM systems.

INTERNATIONAL JOURNAL OF ELECTRICAL ENGINEERING &

TECHNOLOGY (IJEET)

ISSN 0976 – 6545(Print) ISSN 0976 – 6553(Online) Volume 4, Issue 5, September – October (2013), pp. 36-46 © IAEME: www.iaeme.com/ijeet.asp Journal Impact Factor (2013): 5.5028 (Calculated by GISI) www.jifactor.com

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The pilot-aided channel estimation, based on the linear minimum mean square error

(LMMSE) technique, is optimum in the sense of minimizing mean square error (MSE) when the

receiver knows the channel statistics .To obtain the frequency domain channel statistics at the

receiver, power delay profile (PDP) estimation schemes have been proposed . These schemes are

based on the maximum likelihood (ML) estimation by taking advantage of the cyclic prefix (CP)

segment of OFDM symbols. However, the ML PDP estimators require very high computational

complexity for obtaining an accurate PDP.

Another approach for improving the performance of LMMSE channel estimation employs an

approximated PDP (i.e., uniform or exponential model) with the estimation of second-order channel

statistics, which are mean delay and root-mean-square (RMS) delay spread. The channel delay

parameters are estimated using pilots with low computational complexity. Therefore, the LMMSE

channel estimator with the approximated PDP is appropriate for practical applications such as a

WiMAX system. However, the performance degradation is caused by both the correlation mismatch

and the estimation error of delay parameters.

To reduce the mismatch in the frequency domain, we propose a PDP estimation technique for

the LMMSE channel estimator of MIMO-OFDM systems. For practical applications, the proposed

technique uses only the pilot symbols of all transmit antenna ports to estimate the PDP with low

computational complexity. In addition, the proposed technique effectively mitigates the distortion

effects, incurred by null subcarriers and an insufficient number of estimated channel impulse

response (CIR) samples. Simulation results show that the performance of LMMSE channel

estimation with the proposed PDP estimate approaches that of Wiener filtering.

SYSTEM MODEL AND DEVELOPMENT

The system under consideration is a MIMO-OFDM system with P transmit and Q receive

antennas, and K total subcarriers. Suppose that the MIMO-OFDM system transmits Kd subcarriers at

the central spectrum assigned for data and pilots with K-Kd virtual subcarriers, in order to control

interferences with other systems. The CIRs corresponding to different transmit and receive antennas

in MIMO systems usually have the same PDP .

Let Cp[kp,np] be the pilot subcarrier for the pth

transmit antenna at the npth OFDM symbol,

which is a QPSK modulated signal from known sequences between the transmitter and receiver. We

assume that the pilot subcarriers are distributed over a time and frequency grid as in Fig. 1, to

preserve the orthogonality of pilots among different transmit antennas. ��∈ℱ� and �∈�� represent

the index sets for the pilot subcarriers of the �th antenna port in the frequency and time domains,

respectively. At the �th OFDM symbol, the number of pilot subcarriers is defined as ��=∣ℱ�∣. The

pilot inserted OFDM symbol is transmitted over the wireless channel after performing an inverse fast

Fourier transform (IFFT) and adding a CP. It is assumed that the length of CP, ��, is longer than the

channel maximum delay, ��ℎ, making the channel matrix circulant (��慜ℎ ≤ ��).

At the receiver, after perfect synchronization, the removal of CP, and FFT operation, the

received pilot symbol for the �th receive antenna can be represented as

y�[�] = ��(x�)Fph�,� + n�, (1)

where h�,�=[ℎ�,�[�, 0], ℎ�,�[�, 1], . . . , ℎ�,�[�, ��ℎ], 0, . . . , 0]� is an ��×1 CIR vector at

the �th transmit antenna and �th receive antenna. (⋅)� and (⋅)� represent the transpose operation, and

the transpose and conjugate operation of a vector or matrix, respectively. x� = [��[�1, �],

��[�2, �], . . . , ��[��, �]]� denotes a pilot vector at the � th OFDM symbol for ��∈ℱ� and

�=1, 2, . . .,��.



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P P

p

p

^ ^

Fig. 1. Pilot symbol arrangement in a physical resource block (PRB) of the LTE OFDM system

��(x�) is the ��×�� diagonal matrix whose entries are the �� elements of the vector x�. F� is a

��×�� matrix with the (�k, �)th entry [F�]��,� =1/√�� {-j2∏ikl/K} where ��∈ℱ� and �=0, 1, . . .,

��−1. n� is a complex additive white Gaussian noise (AWGN) vector at the �th receiver antenna

with each entry having a zero-mean and variance of !2.

PROPOSED METHOD FOR THE PDP ESTIMATION

A. Derivation of the PDP in MIMO-OFDM systems

From (1), the CIR at the (�, �)th

antenna port can be estimated approximately using the regularized

least squares (RLS) channel estimation with a fixed length of �� as

hhhh",�,� = (FFFF� FFFF�+#IIII��)−1 FFFF� ��(xxxx�)�yyyy�[�] ≜ WWWW"�%,� yyyy� [�], (2)

where #=0.001 is a small regularization parameter, and I�� is the ��×�� identity matrix. F� F� in (2)

is ill-conditioned due to the sparsity of pilot tones in the frequency domain and the presence of

virtual subcarriers . To derive the PDP from the estimated CIR in (2), the ensemble average of

h",�,�h�,�,� is given by



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p p

p

R,p,q

P,q ˜

P,q

H

RLS,p

H H

˜

˜

(3)

where Rℎℎ=&{h�,�hH,p,�} and W=(F� F� +#I�� )

−1F� F�.Note that the diagonal elements of the

channel covariance matrix, Rℎℎ, represent the PDP of multipath channel within the length of ��, and

all off-diagonal elements are zeros. Hence, the covariance matrix can be expressed as Rℎℎ=��

(pℎ), where pℎ=[�0, �1, . . . , ��ℎ, 0, . . . , 0]� and ��=&{∣ℎ�,�[�, �]∣2}. Unfortunately, Rℎℎ is

distorted by W,which is an ill-conditioned matrix due to the presence of F� F�. Thus, instead of

calculating W−1

, we investigate the method for eliminating the spectral leakage of W.

The covariance matrix of the estimated CIR is defined as Rhh=WRhhWH which can be

expressed as

Rhh = (plul)WH, (4)

Where ul is a unit vector with the lth entry being one and otherwise zeros.Let ph and tl be the

Lg ×1 vectors defined as ph=Dg(Rhh) and tl=D(Wdiag(ul)WH),respectively where Dg(A) is the

column vector containing all diagonal elements of A.Then, the relation in (4) is simplified as

Ph = p0t0+ p1tl +…..+PLg-1tLg-1≜TpTpTpTph, (5)

Where T=[t0,t1,….tLg-1] is defined as a distortion matrix by W. It is noted that the distortion

matrix is a strictly diagonally dominant matrix, satisfying ∣[TTTT]��∣>Σ+∕=� ∣[TTTT]�+ ∣ for all I,j, since the

non-diagonal elements of T are composed of the leakage powers of ui for all i. From the

Gershgorin circle theorem, a strictly diagonally dominant matrix is non-singular. In

addition, the distortion matrix is a well-conditioned matrix. Hence ,the distortion of WWWW can be

eliminated as (6)

Where is defined as the received sample

vector for estimating PDP at the (p,q)th antenna port on the npth OFDM symbol, and

W =

B. PDP Estimation in Practical MIMO-OFDM Systems

The received sample vector in (6) can be expressed as

gp,q[np]=Dg(hp,qhH ) + np,q + ep,q, (7)

where np,q = T-1

Dg(WRLS,pnqnqWH

) and ep,q = 2Re T-1

Dg(WRLS,pnqnqWRLS,p)

Here,Re a denotes the real part of a.We assume that np,q is an effective noise by AWGN. Then



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z init

The sample average of gp,q[np] is given by

(8)

Where N ≜ |Tp|PQ respected the total number of sample for PDP estimation |Tp| is the

number pilot symols at the kpth subcarrier in atime slot.When N is sufficiently large,the PDP can be

perfectly estimated.since

However,it is difficult for a receiver of practical MIMO-OFDM systems to obtain such a

large number of samples.With an insufficient number of samples, the PDP can be approximated as

Ph≈

To improve the accuracy of PDP extimation with insufficient samples, we mitigate the

effective noise as follows

(9)

Where ZN ≜ is a defined as a residual noise vector, in which

each entry has a zero-mean. Then, the error of PDP estimation with N samples can be

calculated as

(10)

Since[ph]I ≥0 for all i,the PDP can initially be estimated as

(11)

Where sp,q[np] is the sample vector of proposed PDP estimator with the lth entry

(12)

where ��,�[��]= [g�,�[�]]� and ˜N�=[˜w]�. To mitigate the detrimental effect of residual

noise zO, the proposed scheme estimates the average of residual noise at the zero-taps of pℎ.At the

�th entry of ˆp��P, the zero-tap can be detected as

P� = 1 if ˆ��< QPℎ,

0 otherwise , (13)



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p

n

p n

p n

p p p p

^

z

^ ^ ^ ^ ^

^

^

n p

whereQPℎ = plinit is defined as a threshold value for the zero-tap detection. Then, the average of

residual noise at the zero-taps can be estimated as

nR,avg = linit

tlz’ (14)

where OR=l represents the total number of detected zero-taps. With the mitigation of residual

noise, the �th tap of the PDP estimate, pℎ, can be expressed as

�l = ��P − ",�S� if ��

��P > ",�S�

0 otherwise . (15)

Then, the estimated PDP in (15) can be used to obtain the frequency-domain channel correlation in

the LMMSE channel estimator.

PERFORMANCE AND COMPLEXITY ANALYSIS

The LMMSE channel estimator with the imperfect PDP in (15) is given by

(16)

where F� is the �d× �g matrix obtained by taking the first �� columns of the DFT matrix. pℎ = pℎ

+ e�� is expressed as the estimated PDP, where the �th element of e�� is defined as

(17)

From the matrix inversion lemma, (F�T�(ˆpℎ)F�+ !2I�� )

−1 in (22) is converted as

(18)

where A A A A ≜ (FFFF�T� (ppppℎ)FFFFHHHH + !2I��) and B B B B ≜T�(eeee��) (I��+F

HA−1FFFF�T�(eeee��))−1. Then, the

coefficient matrix for LMMSE channel estimation with ˆppppℎ can be rewritten as

WWWWU,� =WWWWV�P,� +WWWW�WW,�, (19)

where WV�P,� ≜ F�T�(pℎ)F� (F�T�(pℎ)F�+ !2I��)

−1 is the coefficient matrix for Wiener filtering,

and W�WW,� is given by W�WW,� = −F�T�(pℎ)F�A

−1F�BF�A

−1 + F�T�(e��)F�(F�T�(ˆpℎ)F�+ !

2I)

−1 (20)



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g g

^ ^

n p f,p’

The error covariance matrix of LMMSE channel estimation with the imperfect PDP can be obtained

as

E� = & (F�h�,�−WU,�h�%,�,�)(F�h�,�−WU,�h�%,�,�)�

= (F� −WU,�F�)T�(pℎ) (F� −WU,�F�)� + !2WU,�F�F�W� (21)

Fig.2. Performance of LMMSE technique using the estimated PDP over ETU channel.

where ˆh�%,�,� ≜ ��(x�)�y�[�]. Using the error covariance matrix, the frequency-domain MSE of

the proposed scheme is given by

(22)

where �W(E�) denotes the trace operation of E�. With a sufficiently large number of samples, e�� →

0. Thus, the MSE of the proposed scheme achieves that of Wiener filtering because WU,� →WV�P,�.

The additional complexity by the proposed PDP estimation technique is X(�牡3+ ��2

+

∣�p∣YZ��),

which mainly comes from computing (2) and (6). When the pilot spacing is fixed in the frequency

domain, all entries of F� and T are constant. Thus, can be computed only

once, and their values can be stored. The additional complexity is then reduced to X(�2+ ∣��∣YZ��).

V. SIMULATION RESULTS

We consider a MIMO-OFDM system with the physical layer parameters for the downlink of

3GPP LTE . The system bandwidth is 5 MHz with 301 subcarriers for transmitting data information

and pilots including a DC subcarrier at 2-[�R carrier frequency. The width of each subcarrier is 15

kHz with an FFT size of 512. The MIMO-OFDM system utilizes four transmit and two receive

antennas(P=4,Q=2). We assume that the pilots of the four transmit antenna ports are distributed as

the time and frequency grid of the LTE system in Fig. 1. The length of CP is 40 (��=40). For all



43

simulations, the channel estimator is based on a cascaded 2×1D LMMSE technique during 14

OFDM symbols (∣�1∣ = ∣�2∣ = 2, ∣�3∣ = ∣�4∣ = 1), as shown in Fig. 1, where the filtering in frequency

domain is followed by the filtering in time domain over slowly fading channels with the Doppler

frequency of 5 Hz.

Figure 2 shows the MSE performance of the 2×1D LMMSE technique using the estimated

PDP. All underlying links are modeled as extended typical urban (ETU) channels . The performance

of the 2×1D Wiener filter with exact PDP is included as a lower bound. For performance

comparisons, we plot the performance of frequency domain regularized LS channel

estimation in which the PDP information is not required. The performance of the 2×1D LMMSE

technique using the approximated PDP, which is uniform or exponential model with the channel

delay parameter estimation in [11], is also plotted. Note that the LMMSE technique using the

estimated PDP outperforms the conventional methods, since the correlation mismatch is reduced by

the proposed PDP.

Fig.3. Performance of LMMSE technique using the estimated PDP over 6-ray exponential channel

maximum delays(Pilot SNR=30dB).

Fig.4. Performance of LMMSE technique using the estimated ETU channel with different mobile

equipment speeds (Pilot SNR=30 dB and Doppler frequency = 9.26-203.7Hz



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5

estimation. We also observe from Fig. 3 that the proposed method has a performance loss within

only a 2.4-dB gap,compared with 2×1D Wiener filtering.

In Fig. 3, we investigate the MSE performance of the proposed scheme over the

exponentially power decaying six-path Rayleigh fading channel model, where the channel maximum

delay, ��ℎ, is variable. The PDP of the channel model is defined as for �=0,∆\, . . . , 5∆\ and ∆\=��ℎ . Here, is the normalization factor (%ℎ=Σ� �−�/\W]^)

.

The performance of the proposed scheme is better than that of the conventional methods, and

approaches that of Wiener filtering in various channel environments.

Figure 4 shows the MSE performance of the 2×1D LMMSE technique using the estimated

PDP for different mobile equipment speeds at 30-�_ SNR. All underlying links are modeled as ETU

channels. In Fig. 4, it can be seen that the MSE of LMMSE technique using the estimated PDP

achieves that of Wiener filtering even at high Doppler frequencies.

Figure 5 shows simulation and analysis results of the frequency-domain LMMSE channel estimation

with various samples for obtaining the PDP at 20-�_ SNR (O = ∣��∣YZ).We assume that 2×2

MIMO-OFDM system over ETU channels with 70-�R Doppler frequency. The simulation results

correspond to the channel estimation performance at the first OFDM symbol of antenna port 1 shown

in Fig. 1. We obtain the analytic results in (22) by using the coefficient matrix for LMMSE channel

estimation with the perfect or imperfect PDP at the antenna port. In Fig. 5, it is observed that the

MSE of the proposed scheme improves the MSE performance with an increase in the number of

samples for PDP estimation.

Fig.5 Simulation and analysis result of LMMSE channel estimation over ETU channel with various

number of samples for the PDP estimation (Pilot SNR=20dB)



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VI. CONCLUSIONS

We proposed a PDP estimation technique for the LMMSE channel estimator in MIMO-

OFDM systems. The CIR estimates at each path of the MIMO channels were used to obtain the PDP.

For accurate PDP estimation, we considered the spectral leakage effect from virtual subcarriers, and

the residual noise caused by the insufficient number of estimated CIR samples. The proposed

technique effectively mitigates both the spectrum leakage and residual noise. Simulation results

show that the performance of LMMSE channel estimation using the proposed PDP estimate

approaches that of Wiener filtering.

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DESIGN OF PILOT-SYMBOL ASSISTED POWER DELAY PROFILE ESTIMATION FOR MIMO-OFDM SYSTEMS

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