Comparison of Channel Shortening Equalizers for OFDM Systems Muhammad Danish Nisar, Student Member IEEE Institute for Circuit Theory and Signal Processing Munich University of Technology, 80290 Munich, Germany E-Mail: [email protected]

I. I NTRODUCTION Orthogonal Frequency Division Multiplexing (OFDM) has proven to be an efficient underlying technology for wireless communication. The major motivation for OFDM, or in general DMT systems comes from the fact that dividing the overall data among various low bit rate narrowband sub-carriers effectively renders the individual sub-channels to behave as frequency flat [1]. A prominent way to combat the leftover channel distortion due to ISI is the use of a Cyclic Prefix (CP) — prepending a block of N data symbols with its last ν symbols. A CP of reasonable length is all what is required to transform a channel with memory into a memoryless channel. The obvious problem with the CP approach however is the reduction of system efficiency by a factor of N/(N +ν). While this factor is not of much concern in low bitrate applications, but for applications like Digital Video Broadcast (DVB), the large required length of the CP causes the system efficiency to be considerably degraded. This motivates the design of Channel Shortening (CS) equalizers because reduction in effective Channel Impulse Response (CIR) length leads to smaller required length of the CP and hence increased system efficiency. II. C YCLIC P REFIX - U NDERLYING P RINCIPLE Transmission of a block of N samples over a frequency selective channel with impulse response length Q, can be intitutively expressed in the following matrix representation. From the convoluted channel output sequence y(n) with n = 0, 1, . . . , N +Q−1, we are interested in the first N samples which can be represented as, 2 3 y(0) h0 6 y(1) 7 6 h1 6 6 7 6 6 . 7 . . 6 6 .. 7 . 6 6 7 6 y(Q−1) 7 = 6h 6 6 Q−1 7 6 6 7 6 6 . 7 . . 4 4 .. 5 . y(N −1) 0 2

0 h0 .. . . . . h1 .. . ...

... 0 .. . h0 .. . hQ−1

... ... .. . 0... .. . . . . h1

32 3 0 s(0) 0 7 6 s(1) 7 76 7 7 . 76 . . 76 . 7 . 76 . 7 6 7 0 7 6 s(Q−1) 7 7 76 7 . 76 7 . . 54 . 5 . . h0 s(N −1)

(1)

Consider now the similar representation when a CP of length ν ≥ Q − 1, has been added at the beginning of the block i.e. s(−i) = s(N − i) for i = 1, 2, . . . , ν. For the case of ν = Q − 1, we get 2 y(−ν) 3 2 h 0 6 6 . 7 . 6 6 .. 7 . . 6 6 7 6 6 7 6 y(0) 7 = 6hQ−1 6 6 7 6 6 . 7 . . 4 4 .. 5 . y(N −1) 0

0 .. . . . . h1 .. . ...

... .. . h0 .. . hQ−1

... .. . 0... .. . . . . h1

0 3 2 s(−ν) 3 7 . 76 . 7 . 76 . . 76 . 7 76 7 0 7 6 s(0) 7 76 7 7 . 76 . . 54 . 5 . . h0 s(N −1)

(2)

Owing to the use of CP, above representation can be reformulated for y = [y(0), y(1), . . . , y(N − 1)]T and s = [s(0), s(1), . . . , s(N − 1)]T as, y = Hs

(3)

where the modified channel matrix, h 0 ... 0 h

Q−1

0

h1 . H=  .. 0 0

h0 .. . ... ...

0 .. . 0 0

... .. . hQ−1 0

0 .. . ... hQ−1

... hQ−1 .. . h1 ...

h2 ... .. . h0 h1

h1 h2  ..  . 0 h0



is interestingly a N × N Circulant matrix, and as such its Eigen Value Decomposition (EVD) can be given as [2], H = F ΛF H

(4)

where H • F : is the Fourier Matrix i.e X(f ) = F x(n), • Λ: is diag{ λ0 , λ1 , . . . λN−1 } with • λi as the channel frequency response at ith subcarrier. Note that for OFDM systems, with the IDFT at transmitter followed by DFT at receiver, the expression for recovered the data block dˆ boils down to dˆ = F H y = F H H s = ΛF H s = ΛF H F d = Λd

(5)

In summary the inclusion of the CP leads to a circulant channel matrix which as shown above effectively renders the channel to be memoryless. The reconstructed block is then just an amplified version of the desired block d, the amplification being determined by the channel frequency response at that sub-carrier. III. C HANNEL S HORTENING A PPROACHES While the problem of ISI can easily be circumvented in case of low bitrate applications, when it comes to higher data rates where even a small impulse response may correspond to a large number of data samples, we are faced with the stringent requirement of using a longer CP, translating thereby into a considerable reduction of system efficiency. A natural solution to this problem is to somehow force the channel to exhibit an impulse response of shorter length. It may be notified here that this so called Time Domain Equalization (TEQ) for OFDM systems is somewhat weaker than conventional equalization where the goal is to invert the channel completely. In contrast here we just need to make the CIR to be of shorter length.

The birth of Channel Shortening dates back to [3] where D. Falconer et al. presented the idea of Channel memory truncation for ML Sequence Estimation, so as to reduce its complexity. The first attempt of channel shortening for DMT systems, however, has been presented much later by N. Al-Dhahir and J. Cioffi [4]. Since then a number of approaches have been presented that tackle the problem of CS from different viewpoints. These include besides the original Minimum Mean Square Error (MMSE) idea, the Maximum Shortening SNR (MSSNR), Maximum Geometric SNR (MGSNR), Minimum InterSymbol Interference (Min-ISI) and the Maximum Bit Rate (MBR) methods. In recent past some blind adaptive algorithms have also been presented for channel shortening. This report presents a comprehensive literature review of the various methods proposed for CS equalizer design.

The MMSE approach although minimizes the MSE between TIR and TEQ, but indeed it cant be termed as the best criterion for the design of TEQ. In fact the error part besides addressing the actual goal of keeping effective CIR to length s also takes into account its deviation from the virtual TIR. V. MSSNR A PPROACH Melsa et al. in [5] came up with the Maximum Shortening SNR (MSSNR) approach for the design of channel shortening equalizer. The underlying idea behind their approach lies in the following matrix representation for effective CIR, i.e. the convolution of original channel with the TEQ of length t < Q, he (0) he (1) 6 6 6 . 6 . . 6 6 h (t−1) 6 e 6 6 . . 6 . 6 6 h (Q−1) 6 e 6 . 6 . 4 . 2

IV. MMSE A PPROACH Based on the idea presented in [3], N. Al-Dhahir et al. presented an optimal channel shortening scheme with respect to the MMSE criterion [4]. The system model used is shown in Fig 1.

2 h0 6 h1 7 6 7 6 . 7 6 .. 7 6 7 6h 7 6 t−1 7 6 7 7=6 . 6 .. 7 6 7 6h 7 6 Q−1 7 6 7 6 . 7 4 .. 5

0 h0 .. . ... .. . ... .. . ...

3

0

he (Q+t− 2)

... 0 .. . h1 .. . hQ−t−1 .. . 0

3 0 . . . 07 72 w(0) 3 .. 7 . 7 7 6 w(1) 7 7 6 h0 7 . 7 76 . 7 76 . 7 . 76 . 76 7 . 76 7 . . 5 4 hQ−t 7 . 7 7 w(t−1) .. 7 . 5

(9)

hQ−1

Now given the desired goal of reducing the effective CIR to length ν + 1, we can decompose the above representation into the desired ’window’ and the undesired ’wall’ portion:

hwin

2 3 hd he (d) 6 hd+1 6he (d + 1)7 6 7 6 =6 7=6 . . . 4 .. 5 4 . hd+ν he (d+ν)

hd−1 hd .. . hd+ν−1

hwall

2 3 h0 he (0) he (1) 6 h1 7 6 6 7 6 6 . 7 6 . 6 .. 7 6 . . 6 7 6 6 7 6 = 6 he (d−1) 7 = 6 hd−1 6h 6 h (d+ν +1) 7 6 d+ν+1 7 6 e 6 7 6 . 6 . 7 6 . 4 .. 5 4 . 0 he (Q+t− 2)

2

Fig. 1.

Block Diagram of MMSE model

2

As can be noted it creates a virtual Target Impulse Response (TIR) of length s < ν, and attempts then to minimize the MSE between the output of TEQ and TIR. The solution for TEQ obviously depends upon the TIR selection, and as such a joint optimization is performed to look for the optimal TIR and TEQ that lead to minimum MSE, i.e. [wMMSE , bMMSE ] = argmin E[|rk − zk |2 ] [w,b]

This optimization problem as described in [4] leads finally to the solution bMMSE as eigenvector of R∆ corresponding to its minimum eigenvalue where R∆ = D

„

1 It+ν + H H R−1 η H Sx

«−1

DT

(6)

where Rη is the t-dimensional noise covariance matrix, Sx is the average energy of input symbols and the matrix D is defined below Is+1

0(s+1)×(t+ν−∆−s−1)

˜

wMMSE = 0(1×∆)

bMMSE

˜

0(1×s) H

H

„

... 0 .. . ... ... .. . 0

0 ...0 .. . hd−t

3

7 72 3 w(0) 7 7 7 6 w(1) 7 7 76 7 . 76 . 5 4 hd−t+ν+27 . 7 7 w(t−1) 7 .. . 5 hQ−1 (11)

SSNR =

H ||hwin ||2 hH hwin w H Hwin Hwin w = Hwin = H H 2 ||hwall || hwall hwall w Hwall Hwall w

(12)

where Hwin and Hwall are respectively the channel matrices in (10) and (11). Intitutively the MSSNR problem can also be posed as H wMSSNR = argmin w H Hwall Hwall w w H w H Hwin Hwin w = 1

(7)

The optimal solution of wMMSE can be expressed as [4], ˆ

3 32 w(0) hd−t+1 hd−t+2 7 6 w(1) 7 7 76 7 76 . .. . 5 54 . . w(t−1) hd−t+ν+1 (10)

Defining the Shortening ’SNR’ as the ratio of energy contained in the desired to that of undesired effective CIR, we obtain [5]

s.t. ˆ D = 0(s+1)×∆

0 h0 .. . hd−2 hd+ν .. . ...

... ... .. . ...

«−1 1 H Rη + H H Sx (8)

H H Defining A = Hwall Hwall and B = Hwin Hwin , we proceed by Cholesky decomposing B as

B = QΛQH = QΛ1/2 ΛH /2 QH = (QΛ1/2 )(QΛ1/2 )H = B 1/2 B H /2

(13)

Defining y = B H /2 w the MSSNR problem reads as yMSSNR = argmin y H Cy

yHy = 1

s.t.

y

with C = B −1/2 AB − H /2 . The solution to this well known optimization is the eigenvector of C corresponding to minimum eigenvalue, λmin , of the matrix C. The optimal TEQ filter w and the MSSNR can be respectively expressed as wMSSNR = B − H /2 yMSSNR

(14)

wH BwMSSNR yH yMSSNR 1 MSSNR = MSSNR = HMSSNR = H λmin wMSSNR AwMSSNR yMSSNR CyMSSNR (15)

Interesting point to note about the MSSNR solution is that in its definition of SNR it ignores the effect of noise and only considers the interference. This drawback of MSSNR will be addressed in the next section. VI. M IN -ISI A PPROACH

0

0 SNR2 .. . ...

... ... .. . 0

0 3 0 7 H .. 7 5F . SNRN

(16)

H ˜ wall ˜ wall w wMin-ISI = argmin w H H SH w H ˜ ˜ win wHH Hwin w = 1

˜ wall and H ˜ win are respectively related to the where H matrices Hwall and Hwin except for the fact that instead of being of reduced dimensions, both are N × N matrices with ineffective rows being replaced by zeros. Note that this optimization problem can be solved in a manner similar to the approach presented above in Section V to yield a similar solution. VII. MGSNR A PPROACH The Max-Geometric SNR approach presented by N. AlDahir et al. [8], attacked the problem of channel shortening from the most direct perspective i.e. the maximization of channel capacity. The motivation for MGSNR can be developed from the following derivation for the total capacity, C, of the multicarrier system:

N X

(17)

where Γi defined in [9] is a function of allowed error probability. Now formulating the above expression as below, we define the effective SNR, SNRe . 2

Γ 6 6 6 C = N log2 61 + 4

N „ Y

i=1

SNRi 1+ Γi

«1/N

Γ

„ « SNRe = N log2 1 + Γ

!3

−1 7 7 7 7 5

(18)

The effective SNRe can be approximated as geometric mean of SNRi s, i.e. N Y

SNRi

i=1

!1/N

= GSNR

(19)

The maximization of system capacity under this approximation therefore translates into the maximization of so called Geometric SNR. Pursuing their MMSE TEQ design approach, the authors argue that SNRi for the equalized system can be written as SNRi =

Psi |Bi |2 Pni |Wi |2

(20)

where Bi and Wi are the ith FFT coefficients of the TIR and the TEQ respectively. Trying to find the optimal TIR, b, the authors assume its independence from the optimal TEQ and as such arrive at

where F is the Fourier Matrix, the Min-ISI Approach can be written as,

s.t.

Ci =

SNRe ≈

In their papers G. Arsaln et al. [6], [7] attempted to generalize the MSSNR solution and included the effect of noise on each subcahnnel. The underlying idea behind their approach is to attach more significance to the subchannels with high SNR. ISI reduction there, is going to have a pronounced effect on the capacity of that subchannel. The channels already with low SNR, even if equalized well, will not influence the overall capacity much. Defining a matrix S as 2SNR 1 6 0 S=F6 4 .. .

N X

« „ SNRi log2 1 + Γi i=1 i=1 "N „ # « 1/N Y SNRi 1+ = N log2 Γi i=1

C=

bMGSNR = argmax b

s.t.

N X

ln|Bi |2

i=1

bH R∆ b ≤ MSEmax , ||b||2 = 1

where R∆ has been defined in (6). The solution needs to be found numerically, and then the optimal TEQ can be found using (8). The MGSNR approach although has been the first attempt to design TEQ with the goal of directly maximizing the system capacity, but it includes some approximations and assumptions which make the solution less appealing. VIII. MBR A PPROACH In [7] G. Arsalan et al. extended the idea behind MGSNR Approach to the optimal Maximum Bit Rate (MBR) equalizer design. In fact their approach can also be considered to be an extension to the MSSNR approach, in a sense that they generalized the definition of the subchannel SNR, which is given as, SNRi =

Psi |Hisignal |2 Pni |Hinoise |2 + Psi |HiISI |2

(21)

signal

where Hi , HiISI and Hinoise are the ith FFT coefficients of the corresponding portions of the impulse responses, so that ˜ win w Hisignal = fiH H ISI H ˜ Hi = fi Hwall w

(22)

Hinoise = fiH Qw

where fi is the DFT vector, i.e. ˆ fiT = 1

ej2πi/N

ej2π2i/N . . .

ej2π(N−1)i/N

the matrix Q is defined as Q=

»

It×t

0(N−t)×t

˜

–

(23)

(24)

and the remaining matrices/vectors follow from earlier sections. Substituting definitions from above equations into (21), we get w H Ai w SNRi = H w Bi w

(25)

H ˜ win ˜ win Ai = H fi Psi fiH H

(26)

H ˜ wall ˜ wall fi Psi fiH H Bi = QH fi Pni fiH Q + H

(27)

A blind approach to the design of channel shortening equalizer can be based upon the fact that data samples in general being indeterministic would lead to y(2) 6= y(10) except for the case when the effective channel coefficients h2 , h3 , h4 , . . . are all zero. As such if somehow we are able to attain y(2) = y(10) for successive frames, it is almost guaranteed that the effective CIR length is reduced to length ≤ ν. Note that the approach is blind in the sense that it does not require any knowledge of channel coefficients although it needs sufficient amount of data to converge to a solution. A stochastic gradient based algorithm for MERRY implementation can be outlined as under [11]: For symbol blocks k = 0, 1, 2, . . . r˜(k) = r(M k + ν) − r(M k + ν + N ) ˜ e(k) = w H r(k) ˆ + 1) = w(k) − µe(k)˜ w(k r ∗ (k) ˆ + 1) w(k w(k + 1) = ˆ + 1)||2 ||w(k

(29)

with

so that the capacity of overall system can be expressed as C=

N X

Ci =

i=1

N X

„ « 1 w H Ai w log2 1 + Γi w H B i w i=1

(28)

The authors propose in [7] the use of numerical methods such as conjugate gradient method [10] to arrive at the optimal (MBR) channel shortening equalizer i.e. directly by maximizing the overall capacity of the system. IX. MERRY B LIND APPROACH In [11] Martin et al. presented for the first time a blind algorithm for channel shortening equalizers for OFDM systems. Another prominent blind approach to TEQ design has been proposed under the name of Sum-squared Autocorrelation Minimization (SAM) in [12]. The underlying idea behind MERRY (Multicarrier Equalization by Restoration of RedundancY) can be understood from the following illustration [13] which uses the block of size N = 8 and a CP of length ν = 2.

Fig. 2.

where r(i) = [r(i), r(i − 1), . . . r(i − t)]T . The normalization in last step is required to avoid the trivial solution of w = 0. To ensure faster convergence the authors in [11] H recommend for initialization the eigenvector of Hwall Hwall corresponding to its minimum eigenvalue. A variant of MERRY, has been proposed under the name of FRODO (Forced Redundancy with Optional Data Omission) in [14]. This algorithm has been shown to show faster convergence at the expense of over-shortening the channel. The idea has also been extended for the case of MIMO CS equalizers [14]. X. C OMPARITIVE A NALYSIS In this section we present a comparitive analysis of the channel shortening equalizers discussed in this paper. We use achievable bit rate after equalization and implementation complexity as the two major performance measures. Given in Fig 3 is the chart that compares the bit rates after equalization by various CS schemes described here.

Illustration of MERRY principle

Note that if y(.) denotes the equalized samples and hi the effective CIR coefficients, we can represent y(2) and y(10) as: y(2) = h0 s(2) + h1 s(1) + h2 s(0) + h3 s(−1) + h4 s(−2) + . . . = h0 s(10) + h1 s(9) + h2 s(0) + h3 s(−1) + h4 s(−2) + . . . y(10) = h0 s(10) + h1 s(9) + h2 s(8) + h3 s(7) + h4 s(6) + . . .

Fig. 3. Achievable bit rates by various CS schemes, A Delay optimized TEQ of length 17 taps has been designed for each algorithm [15]

Signal to be equalized is one of the ADSL measured signals provided by AST. As can be noted, the MGSNR and MBR methods that directly aim at bit rate enhancement outclass other algorithms such as MMSE, MSSNR and Min-ISI, by a significant margin. The recently introduced MERRY algorithm also shows reasonable performance eventhough its a blind algorithm.

Fig. 4.

Implementation Complexity of various CS schemes

The next chart in Fig 4 compares the various schemes from the implementation point of view. We note here that both the MGSNR and MBR schemes are nonimplementable in realtime, while the others can be implemented in realtime with reasonable complexity. This explains the fact that algorithms like MMSE and MSSNR still win the performance-complexity tradeoff in many practical applications. XI. ACKNOWLEDGEMENTS The author would like to thank Dr. Ing. Micahal Joham, seminar supervisor, at NWS for his useful discussions and continued guidance through out the literature survey. Timely assistance from Dr. Ing. E. Steinbach (LKN) and Dr. Ing. Michel Ivrlac (NWS) regarding report compilation using LATEX is also deeply appreciated. R EFERENCES [1] J. M. Cioffi, “A multicarrier primer,” Online, http://www.stanford.edu/group/cioffi/pdf/multicarrier.pdf, 1991. [2] Robert M. Gray, “Toeplitz and circulant matrices: A review,” Online, http://ee.stanford.edu/ gray/toeplitz.pdf, 1971. [3] D. D. Falconer and F. R. Magee, “Adaptive channel memory truncation for maximum likelihood sequence estimation,” in Bell Syst. Tech. Journal, Nov 1973, vol. 52, pp. 1541–1562. [4] N. Al-Dhahir and J. M. Cioffi, “Optimum finite-length equalization for multicarrier transceivers,” in Proc. IEEE Global Comm. Conf., San Francisco, Nov 1994, pp. 1884–1888. [5] P. J. W. Melsa, R. C. Younce, and C. E. Rohrs, “Impulse response shortening for discrete multitone transceivers,” in IEEE Trans. Commun., Dec 1996, vol. 44, pp. 1662–672. [6] G. Arslan, B. L. Evans, and S. Kiaei, “Optimum channel shortening for discrete multitone transceivers,” in Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Proc., 2000, vol. 5, pp. 2965–2968. [7] G. Arslan, B. L. Evans, and S. Kiaei, “Equalization for discrete multitone receivers to maximize bit rate,” in IEEE Trans. on Signal Processing, Dec 2001, vol. 49, pp. 3123–3135. [8] N. Al-Dhahir and J. M. Cioffi, “Optimum finite-length equalization for multicarrier transceivers,” in IEEE Trans. Commun., Jan 1996, vol. 44, pp. 56–63.

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