On MIMO Channel Shortening For Cyclic-prefixed Systems Toufiqul Islam

Md. Kamrul Hasan

Dept. of Electrical and Electronic Engineering, Bangladesh University of Engineering & Technology, Dhaka, Bangladesh. Email: [email protected]

Dept. of Electrical and Electronic Engineering, Bangladesh University of Engineering & Technology, Dept. of Electronics and Communication Engineering, East West University, Dhaka, Bangladesh.

Abstract—In this paper, we consider channel shortening for cyclic-prefixed block transmission system over multiple input multiple output (MIMO) channels. A time domain equalizer (TEQ) is necessary at the receiver front head to mitigate inter symbol interference (ISI). Melsa, Younce, and Rohrs proposed the most popular MSSNR channel shortening method for single input single output (SISO) channels based on minimizing the energy of the shortened impulse response (SIR) outside the target window while keeping the energy inside constant. Because of computational simplicity of the MSSNR method, we extended this method directly for MIMO channels unlike other MIMO TEQ design methods which perform shortening in multiple stages. We compared our scheme with other MIMO TEQ design techniques for equalization SNR, energy compaction ratio, signal to interference plus noise ratio and bit rate. The results show significant improvement over some reported techniques. Keywords- channel shortening, MIMO, TEQ, MSSNR, MMSE, cyclic prefix.

I.

INTRODUCTION

Cyclic-prefixed communication systems such as discrete multitone (DMT), orthogonal frequency division multiplexing (OFDM) are very robust to multipath, provided that the delay spread of the transmission channel is less than the length of the cyclic prefix (CP),ν , inserted between transmission blocks. If the channel is short, then equalization of the channel can be done tone-wise in the frequency domain by a bank of complex scalars. This is called frequency domain equalization (FEQ). However, if the channel is longer than the CP, additional equalization is required to minimize ISI. Typically, this takes the form of a TEQ which is designed to shorten the overall channel response to one sample more than the length of the CP used. This way TEQ restores orthogonality of the subcarriers and ensures scalar multiplication in the frequency domain. Since the channel encountered in a traditional DMT system is SISO channel, most, if not all, design methods for TEQs have been only for SISO channels. For DMT/OFDM modulation, several methods were proposed for the design of TEQs. In [1], Al-Dhahir and Cioffi proposed minimum mean squared error (MMSE) optimal decision feedback equalizer (DFE) training algorithm. Melsa, Younce, and Rohrs proposed MSSNR method which directly minimizes the part of the SIR that causes ISI [2]. This is a more effective method to reduce ISI than the methods based on the

MSE. In a MIMO system multiple channels need to be shortened simultaneously. Joint channel shortening can be combined with multiuser detection and precoding to mitigate crosstalk. MIMO channel shortening using a generalized MIMO-MMSE-DFE structure has been studied by Al-Dhahir [3]. Youming in [4] proposed a MIMO channel shortening algorithms which operate in multiple stages to jointly shorten the channels. In this paper, we directly extend the original MSSNR algorithm into the MIMO case. Based on this structure, we successfully develop MSSNR channel shortening method to jointly shorten MIMO ISI channels. The MIMO TEQ is formed with eigenvectors corresponding to some maximum eigenvalues of a particular matrix. This design is suitable for any arbitrary length TEQ [7] as well. We tested this scheme with some common figures of merits such as equalization SNR, energy compaction ratio, signal to interference plus noise ratio and obtained good shortening performances compared to [3]. We also proposed a heuristic bit rate formula for MIMO channel shortening systems and compared bit rates with [3-4] and achieved encouraging results. The structure of the paper is as follows. In Section II, we present the MIMO system model. MIMO MSSNR channel shortening algorithm is derived in Section III. Numerical results are given in Section IV and the paper is concluded in Section V. II.

SYSTEM MODEL

We consider a MIMO communication system with nt transmit and nr receive antennas. The baseband system model can be written as nr

p −1

yk( j ) = ∑∑ hm(i , j ) xk(i−) m + uk( j )

(1)

i =1 m = 0

( j)

where yk (i , j ) m

h

is the received signal at the jth antenna for time k,

is the mth channel tap for the channel between ith ( j)

transmit antenna and jth receive antenna, uk is the noise (modeled as white noise) affecting the received signal of the jth receive antenna at time k, and p is the maximum length of all the nt nr channels. Let at time k,

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y k = [ yk(1) , yk(2) , K , yk( nr ) ]T

(2)

x k = [ xk(1) , xk(2) , K , xk( nt ) ]T

(3)

u k = [u , u , K , u

(4)

(1) k

(2) k

( nr ) T k

]

where y k , x k and u k are received signal vector, transmitted signal vector and noise vector respectively. Then in vector form, (1) can be written as p −1

y k = ∑ H m xk −m + uk

where H m is the MIMO channel matrix coefficient of size nr × nt . By stacking q samples of the received signal vector, (5) can be written as

H1 H0 O L

L H1

H p −1 L

0

H0

matrix of size N x N and 0M denotes M x M zero matrix. Then the desired and remainder portion of the channel matrix H can be defined, respectively, as

and

(5)

m=0

y k ⎤ ⎡H0 ⎡ ⎢y ⎥ ⎢ 0 ⎢ k -1 ⎥ = ⎢ ⎢ M ⎥ ⎢ M ⎢ ⎥ ⎢ ⎣⎢ y k -q +1 ⎦⎥ ⎣⎢ 0

where ∆ is the transmission delay parameter which lies within the range (0 ≤ ∆ ≤ p + q − d − 1) , Ι N denotes the identity

L 0 0 ⎤ M ⎥⎥ H p −1 O O 0 ⎥ ⎥ H1 L H p −1 ⎦⎥

⎡ xk ⎤ ⎡ uk ⎤ ⎢ x ⎥ ⎢u ⎥ k −1 ⎥ k -1 ⎥ ⎢ × +⎢ ⎢ M ⎥ ⎢ M ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ x k − p − q +1 ⎥⎦ ⎢⎣u k -q +1 ⎥⎦

(7)

y and u are nr q × 1 vectors and x is a nt ( p + q − 1) × 1 vector. III. MIMO MSSNR CHANNEL SHORTENING A. Problem formulation Given the MIMO channel matrix H which has p matrix taps, our objective is to design a MIMO TEQ, W = [ W0 , W1 ,K, Wq−1 ]T of q matrix taps, each of size nr × nt , to

equalize H to maximize the energy in a window of d = ν +1 matrix taps of the effective channel, Ceff = HT W (of size

Let us define two window matrices, each of size nt ( p + q − 1) × nt ( p + q − 1) :

Ι dnt 0

G ∆ = Ι nt ( p + q −1) − G ∆

power

at

equalizer

output

is

Now, MSSNR problem in MIMO case can be formulated (12)

Wopt = arg max W trace( WT CW) subject to WT AW = Ι nt

(13)

where C = A + B = HH T

(14)

Note that (12) and (13) will lead to the same TEQ, except that they are different by a amplitude scale factor [5-6]. But (13) will lead to faster TEQ computation as matrix C is the same for all delays, whereas matrices A and B have to be recomputed for each delay. Apart from computational saving, (13) works for any value of q as well [7]. B. Optimum MIMO TEQ design Assuming A is positive definite and has full rank of nr q such that ( A ) −1 exists, A can be decomposed using Cholesky decomposition into

0

(8)

(9)

A = A ( A )T

(15)

Y = ( A )T W

(16)

Let us define Then using (13), we obtain YT Y = WT A ( A )T W = WT AW = Ι nt

(17)

Hence from (13) and (16), it follows that T

nt ( p + q − 1) × nt ) keeping the energy in the remainder of the effective channel fixed.

⎤ ⎥ 0 ⎥ 0( p + q − d −∆−1) nt ⎥⎦

(11)

or equivalently

where H is a block Toeplitz matrix of size nr q × nt ( p + q − 1) ,

0

H wall = HG ∆

Wopt = arg max W trace( WT BW) subject to WT AW = Ι nt

In compact form, (6) can be written as

⎡ 0 ∆nt ⎢ G∆ = ⎢ 0 ⎢ 0 ⎣

signal

(10)

σ x2 trace( W T BW ) and residue signal power at equalizer output is σ x2 trace( W T AW ) , where σ x2 is the input symbol power, B = H win H Twin and A = H wall HTwall . as

(6)

y = Hx + u

Desired

H win = HG ∆

WT CW = Y T ( A ) −1 C( A ) −1 Y

= YT ZY

where

(18) (19)

T

Z = ( A ) −1 C( A ) −1 If we define eigendecomposition of Z as

(20)

(21) Z = U ∑ UT = Udiag (γ 0 , γ 1 , K , γ nr q −1 ) UT where γ 0 ≥ γ 1 ≥ L ≥ γ n q −1 . Then optimal shortening can be r considered as choosing Y to maximize YT ZY constraining YT Y = Ι n . The solution to this problem occurs for t

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Yopt = U[e 0 e1 L e nt −1 ] (ei denotes the ith unit vector) which contains nt eigenvectors corresponding to maximum nt eigenvalues of Z. Finally, it follows that T

(22)

Wopt = ( A ) −1 Yopt

delay parameter ∆ (0 ≤ ∆ ≤ p + q − d − 1) present inside A is optimized to maximize trace( D) , where

D = diag (γ 0 , γ 1 , K , γ nt −1 ) IV.

(23)

SIMULATION RESULTS

This section presents experimental results of the proposed algorithm. The input and noise processes are assumed to be uncorrelated. Input signal to noise ratio (σ x2 / σ u2 ) is chosen to be 20 dB. Channels are generated as zero mean uncorrelated Gaussian random variables. To test our proposed channel shortening method, we have decided to compare it with MMSE method of [3] for the following figures of merit: T Equalization SNR = trace( W BW) T

(24)

trace( W AW) T

Energy compaction ratio, ρ = trace( W BW) trace( WT CW)

(25)

σ x2 trace( WT BW ) σ trace( WT AW ) + σ u2 trace( WT W)

(26)

2 x

V. CONCLUSION In this paper, we discussed the problem of MIMO channel shortening and derived MIMO MSSNR channel shortening algorithm. This extension is motivated by the simplicity and good channel shortening performance of MSSNR technique. MIMO MSSNR method achieves best channel shortening performance than other MIMO TEQ design methods. Even MIMO MSSNR algorithm gives SINR and bit rates comparable to MMSE method but at much less computational complexity. The design is adaptable to arbitrary length TEQ as well. Experimental results verified superior performance of the algorithm. ACKNOWLEDGMENT

Overall Signal to (Interference + Noise) Ratio, SINR

=

are compared for SINR as function of TEQ length. It is expected that MSSNR will face some performance loss as its output noise power is not considered in the cost function (12). Still for some TEQ lengths, SINRs are comparable. For Figs. 2, 3 and 4, the parameters chosen are p = 512, ν =15, nt =2 and nr =2. In Figs. 5 and 6, joint channel shortening for a 2 x 2 system is shown with the derived algorithm and MMSE method respectively. Finally, in Table I we compared our scheme with [3-4] for achievable bit rate using (28) and obtained performances comparable to that of MMSE method but with less computational complexity. When compared to [4], our scheme achieves higher bit rates.

We are thankful to the Department of EEE, BUET for all kinds of support for carrying out this research as a part of M.Sc. Engineering thesis work. One of the authors (Toufiqul Islam) would also like to thank R.K. Martin of AFIT, Dayton, OH, USA for his helpful discussion on the topic of this paper.

We propose a heuristic bit rate formula for MIMO channel shortening system as

=

SINR ) bits/symbol Γ

Fs SINR log 2 (1 + ) bps Γ ( N + v)

MMSE MSSNR

20

(27) (28)

where, Fs is the sampling frequency chosen to be 2.208 MHz, N is the IFFT size. ( N + v ) constitutes the total length of the DMT symbol including CP. Γ is the SNR gap required to achieve a target bit error rate. SINR is computed using (26). In Fig. 1, performances of MSSNR and MMSE schemes are compared for equalization SNR as a function of delay. It is clear from Fig. 1 that MSSNR can achieve higher SNR at the optimum delay than that of MMSE method. In Fig. 2, two methods are compared for equalization SNR as function of TEQ length. It shows that MSSNR method achieves good shortening performances for small number of TEQ taps. In Fig. 3, two methods are compared for energy compaction ratio, ρ as function of TEQ length. ρ is a very important metric as it gives a measure of how much of the effective channel energy is contained within the desired window of interest. Clearly, 0 ≤ ρ ≤ 1 and for efficient channel shortening, a larger ρ is required. Again, we can see MSSNR methods gives better result for small number of TEQ taps. In Fig. 4, two methods

Equalization SNR (dB)

R = log 2 (1 +

22

18 16 14 12 10 8

0

5

10

15

Delay

Figure 1. Variation of the equalization SNR of the MIMO TEQ vs delay over 100 channel realizations (p = 7, q =12, ν = 2, nt =2, nr =2). TABLE I. ACHIEVABLE BIT RATE COMPARISON ( Γ = 9.8 dB , p = 512, q =13 , ν = 32, N = 512, nt =2, nr =2 ) Method

Bit rate (Mbps) nt =1 , nr =1

nt =2 , nr =2

Proposed MSSNR

5.58

8.67

MMSE[3]

5.61

8.68

Youming’s MSSNR[4]

4.43

7.53

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40

35

35 30

MMSE MSSNR

30 25

SINR (dB)

Equalization SNR (dB)

40

20 15

20 15

5

5

0 0

0

5

10 length of TEQ

15

5

10

20

15 20 25 length of TEQ

0.4

MMSE 0.999

h12

0

-0.2

-0.2

MSSNR

-0.4

0.9985

0

100

200

300

-0.4

0.998

0.4

0.9975

0.2

0.997

0

0

0.9965

-0.2

-0.2

0

5

10 length of TEQ

15

-0.4

20

0.6

h11

[1]

h12

0.2

0

0

-0.2

-0.2

-0.4

[2]

-0.4 100

200

300

0

0.6

0.6

0.4

0.4

h21

0.2

0

-0.2

-0.2

-0.4

-0.4 200

100

200

300

300

[3] [4]

h22

0.2

0

100

200

300

h21

0

100

200

0.2

300

-0.4

h22

0

100

200

300

REFERENCES

0.4

0.2

100

0.4

0.6

0.4

0

Figure 6. Original and shortened channels(p = 300, q =13, ν = 15, nt =2, nr =2) (MMSE method).

Figure 3. Energy Compaction ratio as a function of TEQ length.

0

40

0.2

0

0.9995

0

35

0.4

h11

0.2

1

0.996

30

Figure 4. Variation of the SINR as a function of TEQ length.

Figure 2. Variation of the equalization SNR as a function of TEQ length.

Energy compaction ratio

25

10

10

0

MMSE MSSNR

[5]

[6] 0

100

200

300

[7] Figure 5. Original and shortened channels(p = 300, q =13, ν = 15, nt =2, nr =2) (MSSNR method).

N. Al-Dhahir and J. M. Cioffi, “Efficiently-computed reduced-parameter input-aided MMSE equalizers for ML detection: A unified approach,” IEEE Trans. on Info. Theory, vol. 42, pp. 903-915, May 1996. P.J. W. Melsa, R.C. Younce, and C. E. Rohrs, “Impulse response shortening for discrete multitone transceivers,” IEEE Trans. on Comm., vol. 44, pp. 1662-1672, Dec. 1996. N. Al-Dhahir, “FIR channel-shortening equalizers for MIMO ISI channels,” IEEE Trans. on Comm., vol. 49, pp. 213-218, Feb. 2001. L. Youming, “Maximum shortening SNR design for MIMO channels,” IEEE Int. Symp. Microwave, Antenna, Propagation and EMC Tech. for Wireless Comm., vol. 2, pp. 1488-1491, Beijing, Aug. 2005. R. K. Martin, J. M. Walsh, and C. R. Johnson, Jr., “Low complexity MIMO blind, adaptive channel shortening,” IEEE Trans. on Signal Process., vol. 53, no. 4, pp. 1324-1334, Apr. 2005. R. K. Martin, M. Ding, B. L. Evans and C.R. Johnson, Jr., “Efficient channel shortening equalizer design,” EURASIP J. App. Signal Process., 2003:13, pp. 1279-1290. C. Yin and G. Yue, “Optimal impulse response shortening for discrete multitone transceivers,” IEE Electronics Letters, vol. 34, pp. 35-36, Jan. 1998.

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