SIR-optimized DFT-bank transceivers for multipath fading channels

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SIR-OPTIMIZED DFT-BANK TRANSCEIVERS FOR MULTIPATH FADING CHANNELS

See-May Phoong, Yubing Chang, Chun-Yang Chen

Dept. of EE & Grad. Inst. of Comm. Engr.

National Taiwan Univ.

Taipei, Taiwan, ROC

Yuan-Pei Lin

Dept. Electrical and Control Engr.

National Chiao Tung Univ.

Hsinchu, Taiwan, ROC

ABSTRACT

In this paper, we study DFT-bank transceivers with ﬁlter length longer than the block size. We show how to optimize the DFT-bank transceiver so that both the intra-band and cross-band interferences are minimized for unknown multipath channels. No costly post processing technique is needed and the only channel dependent part is a set of one-tap equalizers at the receiver. The optimization problem is formulated as a Rayleigh-Ritz ratio whose solution is well-known. Experiments are carried out for transmission over random multipath channels and the results show that satisfactory SIR performance can be obtained.

1. INTRODUCTION

The OFDM (orthogonal frequency division multiplexing) and DMT (discrete multitone) techniques have found many applications in both wireless and wired transmissions [1]. These systems have low complexity and can effectively combat intersymbol interference (ISI). However the DFT ﬁlters suffer from very poor frequency re-sponses. The stopband attenuation is only 13dB and it decays at a rate of

only. In many applications, it is often desirable to have ﬁlters with better frequency responses.

Many solutions have been proposed for the design of transceivers with better transmitting and receiving filters. In particular, the fil-ter bank approach [2] [3] [4] has drawn many attentions recently. In [2], the authors propose the so-called discrete wavelet multitone (DWMT) system, in which perfect reconstruction (PR) filter bank is used as the transceiver. For multipath channels, there is intra-band as well as cross-intra-band interference in these transceivers. Un-like the OFDM system, there is no simple equalization technique for DWMT systems. Comparisons and performance evaluations of these filter bank transceivers have been conducted in [3]. The re-sults show that though the filter bank has near PR property, the ISI introduced by the channel can seriously degrade the system perfor-mance. To reduce the amount of ISI, intra- and cross-band equal-ization are performed on the receiver outputs in [2]. Filter bank transceivers that enjoy the ISI-free property for unknown multipath channels have been studied in [4]. By judiciously placing the zeros of the transmitting filters, the authors in [4] showed that the pro-posed AMOUR transceiver can obtain ISI-free transmission by us-ing a Vandermonde matrix at the receiver. However, the AMOUR transceiver belongs to the class of block transmission schemes, i.e. transmitting filter length restricted to the block size. Moreover

This work was supported in parts by National Science Council, Tai-wan, ROC, under NSC 92-2219-E-002-015 and 92-2213-E-009-022, Min-istry of Education, Taiwan, ROC, under Grant # 89E-FA06-2-4, and the Lee and MTI Center for Networking Research.

there is no simple method to design AMOUR transceivers with good frequency responses.

In this paper, we design DFT-bank transceivers for multipath fading channels. Although the proposed transceiver belongs to the class of overlapping-block transmission, the only channel de-pendent part is a set of one-tap equalizers at the receiver, like the OFDM system. For a set of good receiving filters, the trans-mitting prototype filter can be optimized so that SIR (signal-to-interference ratio) is maximized. Conversely, we can also design the receiving prototype filter to maximize SIR given transmitting filters. We show that such an optimization problem can be formu-lated as a Rayleigh-Ritz ratio, whose solution is well known [5]. Simulation results show that DFT-bank transceivers with satisfac-tory SIR value can be obtained.

2. ISI-FREE DFT-BANK TRANSCEIVERS Fig. 1 shows a ﬁlter bank transceiver. The number of bands is

whereas the up/downsampling factor is . We assume that

. The number of redundancy samples is

. We consider only DFT-bank transceiver where the transmitting ﬁlters

and the receiving ﬁlters

are DFT modulated versions of the prototypes

and

respectively. Their relation is given by ! # % # ) * + , -/ (1) for 0 2 0 5

Let6 7 and6 9 be respectively the orders

of

and

. In this paper, both6 7 and6 7 can be larger

than . The advance element :;

, where<

is an arbitrary integer, is added at the receiver for the convenience of discussion. We assume that the transmission channel does not vary rapidly so that it can be modeled as an LTI channel with transfer function=

, as shown in Fig. 1. Let> be the channel order so that =

@ A C D E 6 * C .

Using multirate identities, one can verify from Fig. 1 that the transfer function fromH I

6 toHK + 6 is an LTI system: L + I N I = :; + PR S (2) A T : D E < N I + :; * :PR S where NV PR S

denotes the -fold downsampling operation. From

the above equation, it is clear that the DFT-bank transceiver is ISI-free for unknown multipath channels if

L + I X I * C ; Y Z [ 5 (3)

,9

,(((

,&$663

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Figure 1: A ﬁlter bank transceiver.

Because<

is an arbitrary integer, we can assume6

without any loss of generality. In this paper, we set6

. From (2) and (3), one can immediately conclude that interchanging the roles

of transmitting and receiving ﬁlters does not change the ISI-free property. Deﬁne the sequences I

: 6 and I + : 6 as: I + :; * : R S I : @ C C D I : 6 * C [ Z @ C I + : 6 * C [ Z (4) for 0 [ Z 0 and 0 < 0 > . Note that I + : 6 is not deﬁned for[ Z

. For convenience, we deﬁne

II : 6 % [ < 6 5

Using these sequences, the ISI-free condition can be rewritten as

I : 6 I : Y 6 I + : 6 % [ Z < 6 5 Whenever we have I : 6 for6 or I + : 6 , then I : 6 and I + : 6

contribute respectively to the intra-band and the cross-band intersymbol interference.

Recall that the transmitting and receiving ﬁlters are DFT mod-ulated versions of prototype ﬁlters. Using this fact, one can show the following lemma (see [6] for a proof).

Lemma 1 For DFT modulated ﬁlter bank transceivers with ﬁlters

deﬁned in (1), the sequences I

: 6 and I + : 6 deﬁned in (4) satisfy I : 6 * I! :; * :" C # : 6 I + : 6 * I! :; * :" C # !! + * I ## / : 6 whereZ [ % representsZ [ m' ( ) <' .

Using Lemma 1, the ISI-free condition can be further simpliﬁed as: : 6 : Y 6 + : 6 % Z < 6 5 (5) When the transceiver achieves the ISI-free condition, any frequency selective channelE

6

with order0

> is converted into a set of

parallel frequency nonselective subchannels. The gain of the[th

subchannel is given by X I A T :D E < I : * I :; A T :D : I : E < 5 Note that : 6 and + : 6

are functions of the prototype ﬁlters, 6 and 6

. Using (4), one can write

-. . ./ 6 0 6 .. . A 6 12 2 2 3 4 6 -. . ./ .. . 6 7 12 2 2 3 -. . ./ + 6 + 0 6 .. . + A 6 1 2 2 2 3 5 + 6 -. . ./ .. . 6 7 1 2 2 2 3 where4 6 and5 + 6 are > by 6 7

matrices and their entries consist of the impulse response ,

6

. The exact forms of these matrices are given in the Appendix B of [6]. Deﬁne the vectors 6 7 -. . ./ .. . 6 7 12 2 2 3 9 7 -. . ./ 0 .. . A 12 2 2 3 5

Then the ISI-free conditions in (5) can be written as

4 6 6 9 Y 6 5 + 6 6 ; 5 (6) 3. DESIGN OF DFT-BANK TRANSCEIVERS In many applications, it is often desired to have transmitting filters or receiving filters with good frequency responses. Depending on applications, our design problem is either (1) given a good lowpass transmitting prototype filter

, design the receiving prototype ﬁlter

to achieve the ISI-free property or SIR maximization, or (2) given a good lowpass receiving prototype ﬁlter

, de-sign the transmitting prototype ﬁlter

to achieve the ISI-free property or SIR maximization. As interchanging the roles of the transmitting and receiving ﬁlters does not affect the ISI-free prop-erty, Problem 1 can be easily formulated into Problem 2, and vice verse. In this paper, we will study Problem 2 only. Hence in the rest of the paper,

is a predetermined good lowpass ﬁlter. 3.1. Least-Squares Solution

Recall the ISI-free condition given in (6). For a ﬁxed

, our goal is to design

so that (6) is satisﬁed. One way to solve this problem is to use the least-squares method. One can write the conditions in (6) as a single matrix equation:

= 6 > 9 ; @ 5

When the desired parameters9

are known, one can use the least-squares method to solve the above linear equations and obtain

6 A B = E = * 0 = E > 9 ; @ 5 (7)

,9

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In many applications, it is desired to have transceivers that maxi-mize SIR. In this least-squares approach, the optimal

:

that maximizes SIR is not known.

3.2. SIR-Optimized Solution

In the following derivation, we will assume that the input signals

H I 6 satisfy H I 6 H + Y [ Z Y 6 (8) where

is the symbol energy. This mild assumption can be satis-ﬁed by properly interleaving the input data. We will consider two cases: (i) known channel and (ii) multipath fading channel. Known Channels: In this case, the impulse responseE

6

is known. One can write the output of theZ

th subchannel as K H + 6 A T : D + : E < H + 6 A T : D E < > + : 6 + : Y 6 @ H + 6 % * 0 T I D A T : D E < I + : 6 H I 6

where denotes convolution. The three summation terms on the

right hand side are respectively the desired signal, the intra-band and cross-band intersymbol interferences. Using (8), one can ex-press the signal and interference powers at theZ

th subchannel as I Z A T : D + : E < ; ! + # (9) I I Z T C C D A T : D + : 6 E < ! + # T I C A T : D I + : 6 E < , ! + # 5 (10) The SIR expression is given by

! " # $ @ % * 0 + D & Z @ % * 0 + D & 0 Z @ % * 0 + D & Z

Using the results in Lemma 1, we can write

% * 0 T + D & Z % * 0 T + D A T : D + : : E < ( ( ( * A " 0 , $ 9 ( ( (

where. / . denotes the 2-norm of/ . The diagonal matrix, $ ( [1 3 N E E 5 5 5 E > P and * A " 0 is the by > ma-trix formed by the ﬁrst

>

columns of the DFT matrix

*

. Substituting9 4

6

into the above equation, we get

% * 0 T + D & Z 6 E 4 E , E$ * E A " 0 * A " 0 , $ 4 5 ; 6 5

Similarly it can be veriﬁed that we can write

% * 0 T + D & 0 Z 6 E 6 0 6 % * 0 T + D & Z 6 E 6 6 where 6 0 T C C D 4 E 6 , E$ * E A " 0 * A " 0 , $ 4 6 6 T C % * 0 T + D 0 5 E + 6 , E$ * E A " 0 * A " 0 , $ 5 + 6 5

Using these results, we can write SIR as

! " # $ 6 E 6 6 6 E 6 0 6 6 5 Since6

I is positive deﬁnite (except for some rare cases), the above

SIR expression is a Rayleigh-Ritz ratio [5]. The optimal

6

can be found by solving an eigen problem.

Multipath Fading Channels: In many applications, E

6

might not be available and we may have only the statistics of the trans-mission channels. Consider multipath fading channels with

> tapsE < for 0 < 0 > . Assume that E <

are complex random variables that satisfy

E < 8 E < E < 2 : ; : Y 2 for 0 < 0

> . In this case, we consider average powers.

Tak-ing the expectation of (9) and (10) with respect toE

<

and using Lemma 1 to simplify the results, we get

= I Z A T : D > : > ; : @ ; ! + # = I I Z T C C D A T : D > : 6 > ; : @ ! + # T I C A T : D > I : 6 > ; : @ , ! + # 5

The average SIR is therefore given by

! " # @ % * 0 + D B Z @ % * 0 + D B 0 Z @ % * 0 + D B Z 5

Similar to the previous case, one can verify that

% * 0 T + D B Z 6 E 4 E , D 4 6 % * 0 T + D B 0 Z 6 E T C C D 4 E 6 , D 4 6 6 % * 0 T + D B Z 6 E T C % * 0 T I D 0 5 E I 6 , D 5 I 6 6

where, D is the diagonal matrix( [1 3 N; ; 0 5 5 5 ; A P . From the above expressions, one can clearly see that the average SIR can also be formulated into a Rayleigh-Ritz ratio. The optimal

6

can be obtained by solving a corresponding eigen problem. When no channel information is available, the channel is often modeled as iid channels. In this case, the results can be obtained by setting

;

I

and the transceiver becomes channel independent.

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The Choice of :; : Note that 4 6 and 5 I 6 depend on the choice of the integer<

. One has to search for the best<

to maxi-mize either! " # $

or! " #

. In our simulations, we ﬁnd that the best

<

always falls within the range of 0 < 6 7 6 9 . 4. SIMULATIONS In the examples, the receiving prototype ﬁlter,

6

is a unit norm lowpass filter designed using the eigenfilter method. The coef-ficients,

6

are designed to minimize > ) + > ( 5 The channelsE 6

are multipath fading channels. The coefﬁcientsE

6

are independent circular complex Gaussian random variables with variances;

C . We have used 10,000 random channels in the

ex-periments. All the SIR values shown in the plots are computed by taking the average value over 10,000 random channels.

Example 1. In this example, we design transceiver with

and

. The receiving prototype ﬁlter has6 9 , 5

and a stopband attenuation of more than 61 dB. The random channels are iid channels with

>

taps. The SIR curves for>

are shown in Fig. 2. We can obtain a moderate SIR value even when the receiving ﬁlter has a 61 dB stopband attenuation. Example 2. We take , and > . The re-ceiving ﬁlter prototype

has6 9 , 5 and a stopband attenuation of 61 dB. The random channels have an ex-ponentially decay delay proﬁle and;

C 0 * C . We consider 3 scenarios: (i) the channel impulse responseE

6 is known; (ii) only;

C is known; and (iii) no channel information is available

and we assume iid channels. In Case (i), we design an optimal transceiver for each of the 10,000 random channels, whereas in Cases (ii) and (iii), we design only one optimal transceiver. The cost of designing optimal transceivers for known channels is sig-niﬁcantly higher. The results are shown in Fig. 3. As we might expect, ifE

6

is known, the transceiver will have the best SIR performance. Comparing Case (i) and Case (ii), the improvement is not significant. If we compare Case (ii) and (iii), we can obtain a moderate gain. Hence incorporating the channel profile in the design can significantly increase the SIR performance.

5. CONCLUSIONS

In this paper, we study DFT-bank transceivers with filter length longer than the block size. Given a fixed receiving (or transmit-ting) prototype filter, we have shown that the problem of finding the best transmitting (or correspondingly receiving) prototype fil-ter that maximizes the SIR can be formulated as a Rayleigh-Ritz ratio. Though our discussion is restricted to the case of DFT lated filters, the results can be generalized to the case of non modu-lated filters. Moreover, in some cases there is no loss of generality in assuming that the optimal transmitting filters are DFT modu-lated version of a prototype filter [6].

6. REFERENCES

[1] R. van Nee and R. Prasad, OFDM for wireless multimedia

communications, Artech House, Boston, 2000.

120 140 160 180 200 220 240 260 280 300 320 14 16 18 20 22 24 26 28 30 32 n_f SIR (dB) L = 4 L = 8 L = 16

Figure 2: SIR optimized 64-band transceivers.

20 25 30 35 40 45 50 55 60 65 14 15 16 17 18 19 20 21 22 23 n_f SIR (dB) c(n) known σ2_known iid channels

Figure 3: Performance of transceivers designed using different de-grees of channel proﬁle.

[2] S. D. Sandberg and M. A. Tzannes, “Overlapped Discrete Multitone Modulation for High Speed Copper Wire Commu-nications,” IEEE J. SAC, Dec. 95.

[3] A. D. Rizos, J. G. Proakis, and T. Q. Nguyen, “Compari-son of DFT and Cosine Modulated Filter Banks in Multicar-rier Modulation,” IEEE Global Telecommunications Confer-ence, pp. 687-691, 1994.

[4] G. B. Giannakis, Z. Wang, A. Scaglione, and S. Barbarossa, “AMOUR - Generalized Multicarrier Transceivers for Blind CDMA Regardless of Multipath,” IEEE Trans. Comm., vol. 48, no. 12, Dec. 2000.

[5] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1985.

[6] S. Phoong, Y. Chang, C. Chen and Y. Lin, “DFT modulated ﬁlter bank transceivers for multipath fading channels,” submit-ted to IEEE Trans. SP, Aug. 2003.

SIR-optimized DFT-bank transceivers for multipath fading channels