Pre- and Post-Processing for Optimal Noise Reduction in Cyclic Prefix

Pre- and Post-Processing for Optimal Noise Reduction in Cyclic Preﬁx Based Channel Equalizers

Bojan Vrcelj and P. P. Vaidyanathan Dept. of Electrical Engineering 136-93 California Institute of Technology Pasadena, CA 91125-0001

Abstract— Cyclic prefix based equalizers are widely used for It is preceded (followed) by the optimal precoder (equalizer) high-speed data transmission over frequency selective channels. for the given input and noise statistics. These blocks are real- Their use in conjunction with DFT filterbanks is especially attrac- ized by constant matrix multiplication, so that the overall com- tive, given the low complexity of implementation. Some examples munications system remains of low complexity. include the DFT-based DMT systems. In this paper we consider In the following we first give a brief overview of the cyclic a general cyclic prefix based system for communication and show prefix system with DFT matrices used as the basic ISI can- that the equalization performance can be improved by simple pre- celer. Then, we introduce a way to deal with noise suppres- and post-processing aimed at reducing the noise at the receiver. This processing is done independently of the ISI cancellation per- sion separately and derive the optimal constrained pair pre- formed by the frequency domain equalizer.1 coder/equalizer for this purpose. The constraint is that in the absence of noise the overall system is still ISI-free. The performance of the proposed method is evaluated through com- I. INTRODUCTION puter simulations and a significant improvement with respect to There has been considerable interest in applying the eq– the original system without pre- and post-processing is demon- ualization techniques based on cyclic prefix to high speed data strated. transmission over frequency selective channels, such as twisted Throughout this paper we assume that the frequency selec-

pair channels in telephone cables [1], [2]. Cyclic preﬁx systems tive channel is a known FIR system of order Ä, i.e.

are invariably used in conjunction with the discrete Fourier Ä

transform (DFT) matrices, and one example is the DFT-based X

´Þ µ= c´k µÞ :

discrete multitone modulation (DMT) system [3]. Although it C =¼

has been shown by Kalet [9] that the use of ideal transmit and k Ì

receive ﬁlters in the multitone environment can lead to achiev- Ý A The notations A and represent the transpose and trans- ing signal to noise ratio (SNR) within 8-9 dB of the channel

pose conjugate of a matrix A. capacity (depending on the probability of error), in practice the DFT filterbank is most commonly used because of its low com- II. CYCLIC PREFIX SYSTEM WITH DFT MATRICES plexity. The development of these systems lead, for example, In this section we give a review of a cyclic prefix based sys- to asymmetric digital subscriber loops (ADSL) and high-speed tem for channel equalization that employs DFT filterbanks. Be- digital subscriber loops (HDSL) [2], [3]. fore we demonstrate the complete system, we first introduce the The initial goal of cyclic prefix based methods for equaliza- concept of cyclic prefix. This is a way of inserting redundancy tion is to cancel the intersymbol interference (ISI) induced by into the input data stream and it becomes useful in the process the channel. This task is achieved by introducing some redun- of channel equalization as will be explained later. Consider dancy in the form of a cyclic prefix (see Sec. II). Depending on the symbol stream from Fig. 1(a). It is divided into blocks of

the nulls of the channel in question, the noise can get severely Ä length Å . The last symbols from each block are copied and ampliﬁed by the equalizer alone. In this paper we consider a inserted at the beginning of that block (we assume hereafter general cyclic preﬁx based system with the aim of further re-

that the redundancy Ä< Å). This is achieved by “squeezing” ducing the noise at the receiver. This is achieved by perform- the samples as explained in Fig. 1(b)-(c). Obviously, this op- ing the ISI cancellation and the noise suppression separately, eration of inserting the redundancy into the input data stream in different modules. The main module for ISI cancellation is results in the bandwidth expansion. In this case the bandwidth

based on cyclic preﬁx with DFT ﬁlterbanks and is unchanged.

= ´Å · Äµ=Å expansion ratio is given by « , which can be 1 Work supported in part by the ONR grant N00014-99-1-1002, USA. made sufﬁciently small by making Å large. The purpose of T R A N S M I T T E R R E C E I V E R block of M q(n ) s(n ) q(n ) r(n ) s(n ) -1 channel and Γ -1 W prefix system W e n IDFT DFT DFT domain (a) e(n ) equalizer

Figure 3: Conventional cyclic preﬁx system with DFT matrices used for ISI cancellation. (b) n room for L samples

copy copy copy x(n )

thus we have

¾ ¿

C [¼] ¼ ¼ ::: ¼

6 7

¼ C [½] ¼ ::: ¼

6 7

= ;

6 7

e . . . (2) 4

Figure 1: (a) Input symbol stream, (b)-(c) explanation of . .. . 5

¼ ¼ ::: C [Å ½]

how cyclic preﬁx is inserted. ¼

j ¾kÒ=Å

C [Ò]= c´k µe

with Å . As we noted before, the

k =¼

L L ISI cancellation is achieved by inverting C in (1). Therefore,

q(n ) C(z ) discard

after moving the matrix Ï to the transmitter side, the com- channel M prefix plete DFT-based cyclic preﬁx system is shown in Fig. 3. The unblocking noise blocking M r(n )

M block labeled “channel and preﬁx system” is given in Fig. 2.

Matrix e is better known as the frequency domain equal- Figure 2: Channel with the system for cyclic preﬁx.

izer and contains on its diagonal the reciprocals of the DFT

´Òµ coefﬁcients of the channel. The input vector sequence × in

Fig. 3 is obtained by blocking the symbol stream into blocks of

e´Òµ

length Å . The noise vector is obtained by blocking the

· Ä physical noise process into blocks of length Å and then inserting the cyclic prefix is to allow the receiver to remove ISI discarding Ä samples out of each block (see Fig. 2). using some low-complexity operations. The main merit of the cyclic prefix system with DFT ma- As the first step in understanding how this is achieved, con- trices lies in its simplicity. From Fig. 3 we see that the trans- sider Fig. 2, ignoring the noise for the time being. The box la- mitter only needs to perform the inverse DFT operation fol- beled “blocking” is a serial to parallel converter, similarly “un- lowed by the cyclic prefix insertion. On the other hand, the blocking” performs the reverse operation and the block “prefix” receiver removes the cyclic prefix, performs the DFT operation

inserts the cyclic preﬁx as described in Fig. 1. Notice that the and frequency domain equalization, which amounts simply to

´Òµ Ö´Òµ

blocked sequences Õ and are related through a mul- multiplying each channel by a constant. Moreover, if is cho-

¢ Å

tiplication by the Å right circulant [5] channel matrix sen as a power of two, both IDFT and DFT operations can be

C Ä ·½

C. The ﬁrst column of consists of the coefﬁcients implemented using the fast radix-2 algorithm. The noise vec-

e´Òµ

´k µ Å Ä ½

of the channel impulse response c followed by tor process at the receiver passes through the DFT ma- ½

Ï Ï

c´¼µ 6= ¼ zeros. Since we can safely assume , it follows that trix and the frequency domain equalizer e . Since

C is nonsingular and the effect of the channel can be neutral- is a scaled unitary matrix, the main contribution to amplify-

ized by inverting it. Next, note that any circulant matrix can be ing the noise power at the receiver comes from the multipliers

½=C [Ò] C ´Þ µ e diagonalized by a DFT matrix [5]. Thus we can write [5] Å embedded in . Obviously, if the channel

has zeros near the unit circle, these multipliers can get large, ½

consequently boosting the noise at the receiver and degrading

C = Ï Ï ;

e (1) the performance. In the next section we propose a simple mod-

iﬁcation of the basic equalization structure from Fig. 3 that will

Å ¢ Å

where Ï is the DFT matrix and the diagonal matrix improve the performance signiﬁcantly.

e has the eigenvalues of on its diagonal. Those eigenvalues,

c´k µ in turn, are nothing but the Å -point DFT coefﬁcients of , III. MODIFIED SYSTEM FOR NOISE SUPPRESSION Consider the system shown in Fig. 4(a)-(b). It consists of

´Þ µ Otherwise we can shorten the impulse response of C . the same basic cyclic preﬁx transceiver (the middle portion) s(n ) x(n ) r(n ) -1 -1 channel and T W prefix system Our objective as mentioned before is to minimize the out- IDFT (a) put noise power (subject to the power constraint). This can be written as r(n ) x(n ) s(n )

-1

½ ½

Ý Ý

Γ Ý

ÑiÒ fÌ ÏÊ Ï ´ Ì g;

W e T µ e e Tr ee that is

DFT DFT domain (b) Ì

½ ½

( )

Ý Ý

e n equalizer Ý

ÑiÒ fÈ ÈÉg; É = ÏÊ Ï ´ µ

e e

Tr ee (6) È Figure 4: Cyclic preﬁx system with separated ISI cancellation Notice that neither the objective (6) nor the constraint (5) de-

and noise suppression. pend on the choice of the unitary matrix Í. This leads us to =

conclude that without loss of generality we can choose Í

Á Ì = È

Å ,or . Furthermore, it follows that we can achieve

nothing in terms of noise suppression if Ì itself is unitary [this

½ Ì surrounded by the constant matrices Ì and , so that the can also be concluded directly from Fig. 4(b)]. transmitter of the new system is shown in Fig. 4(a), while the In order to solve this problem of nonlinear optimization, receiver is in Fig. 4(b). Notice that as for the signal part, this namely minimizing (6) subject to (5), it is useful to consider system is completely equivalent to the one from Sec. II, i.e. the the unitary diagonalization of the positive deﬁnite matrices

signal still goes through an identity system. Therefore, the only

Ý Ý

È = Í £ Í ; É = Í £ Í :

Ô Ô Õ

Õ (7)

Ô Õ

purpose of the precoder Ì and the corresponding equalizer

Ì ×´Òµ

is to reduce the noise power at the receiver output .It 4

Î = Í Í

Deﬁning the unitary matrix Ô , the objective function is clear that without any additional constraints this task can be Õ can be written as

trivially accomplished by scaling the matrix Ì by a very small ¾

constant; without changing the received signal, in this way we Ý

ÑiÒ fÎ £ Î£ g:

Tr Õ (8)

Î ;£ could arbitrarily reduce the received noise power. However, Ô

this would in turn arbitrarily increase the power in the transmit-

Î £

Notice that the matrices and Ô are arbitrary, subject to be-

´Òµ ted signal Ü , which is of course unacceptable. The remedy, ing unitary and diagonal positive deﬁnite, respectively. More

therefore, is to ﬁnd the optimal precoder/equalizer pair subject £

importantly, optimal Ô can be chosen independently of opti- to the constraint on the transmitted power. This constraint can

mal Î . Let us denote

be written as

A = Î £ Î ;

Õ (9)

Ý ½ ½ Ý

fE [ÜÜ ]g = fÌ Ê ´Ì µ gÈ ; Ü

Tr Tr ×× (3)

A a ;a ;:::;a

¾ Å

and the diagonal elements of by ½ . Also de-

f¡g E [¡]

; ;:::;

where Tr and denote the trace and expectation operators £

Ô;½ Ô;¾ Ô;Å

note the diagonal elements of Ô by and

£ ; ;:::;

respectively and ×× denotes the autocorrelation matrix of the

Õ;½ Õ;¾ Õ;Å

the diagonal elements of Õ by . When

×´Òµ È

f g input vector process . Quantity Ü denotes the maximum

considered as a function of Ô;i , the minimization problem

´Òµ

power in the transmit signal Ü . From now on, we make the becomes

Å Å X

assumption that the input symbols are independent, identically X

ÑiÒ =½: a ;

i s.t. (10) Ô;i

distributed, coming from a predeﬁned constellation. Therefore, Ô;i

Ô;i

i=½ i=½

Ê = È Á Á

× Å Å

the autocorrelation matrix becomes ×× , with

¢ Å È Å The problem nicely reduces to scalar optimization and this can denoting the identity matrix, and × the power in the

input symbol stream. Thus, if the transmit power needs to be be readily solved using the method of Lagrange multipliers.

f g unchanged after employing the precoder, the power constraint The optimal solution for Ô;i as a function of the diagonal

becomes elements of A is given by

½=¾

½ ½ Ý ½ Ý ½

fÌ ´Ì µ g = f´Ì µ Ì g =½; a

ÓÔØµ

Tr Tr (4) ´

= :

È (11)

Ô;i

i=½

ABg = fBAg A; B using the identity Trf Tr , for square. According to the polar decomposition theorem [7] an arbi-

The next step is to ﬁnd the optimal set of diagonal elements

Ì = ÍÈ

trary square matrix Ì can be written as a product ,

A fa g

of , namely i that minimize the objective (10). After sub- È

where Í is unitary and a positive semideﬁnite matrix (in

f g

stituting the solution for Ô;i , the problem becomes È our case, since Ì is obviously desired to be nonsingular, is

positive deﬁnite). Therefore, the power constraint (4) becomes Å

X X

a · ÑiÒ a a ]: [

i j

i (12)

Ý ½ Ý ½ ¾

Í Í´È µ È g = fÈ g =½:

Tr f Tr (5)

i=½

i6=j p1 p2 p3 IV. EXPERIMENTAL RESULTS We now consider an experimental example designed to com- pare the equalization in a traditional cyclic preﬁx system versus

the modiﬁed system with optimal precoder for noise suppres-

´Þ µ Ä =8 generated by Ω sion. The channel C was taken to be of order , with coefﬁcients 1.0, -0.5974, 0.2346, -0.0305, 0.8519, 0.5680, -

0.0975, -1.0479, 0.6939. The magnitudes of the four complex

´Þ µ Figure 5: Convex polytope deﬁned by the doubly stochastic pairs of zeros of C are given by

matrix ª; see text.

¼:68¿9; ¼:9656; ½:¼79¿; ½:½687:

´Þ µ

We see that four complex zeros of C are very close to the È

Å unit circle. As a consequence, several DFT coefﬁcients of

a = f£ g = É Õ

Notice from (9) that i Tr const. (since

i=½

C [Ò]

Å are very low in magnitude, so that the frequency domain is ﬁxed), so that the objective given by (12) becomes that of

equalizer at those frequencies ampliﬁes the noise severely. This

f ´aµ = a a a j

minimizing i , with the vector deﬁned as

6=j

i resulted in very high probabilities of error at signal to noise ra- 4

Ì tios that are moderate to high, when the traditional cyclic preﬁx

a =[a ;a ;:::;a ] f ´¡µ

¾ Å ½ . It is important here to notice that

system is used. This can be seen from Fig. 8. The quantity Å is a concave function of a, since it is a positive linear combina-

Ô was chosen to be 128, so that the FFT algorithm can be used.

a a j tion of concave functions of the form i . If we denote by

The input constellation was chosen to be 64-QAM. The sig-

Ú ´i; j µ

i;j the th element of V deﬁned previously, then from (9) we have nal to noise ratio used in the experiments was calculated at the

input of the receiver [see Fig. 4(b)], i.e.

a = jÚ j ;

i k;i Õ;k

jjÖjj

=¾¼ ÐÓg : =½

k SNR

½¼

jjejj ¾ or in other words In Figs. 6-7 we show the results of the channel equaliza-

Ì tion in the traditional system and the modiﬁed system with

a = ª ¢ [ ::: ] :

½ Õ;¾ Õ;Å Õ; (13) precoder, respectively. The plots are obtained at the signal to

noise ratio of 24 dB. The average probability of error corre-

´i; j µ Here ª is a doubly stochastic matrix [7], with the th el-

sponding to the traditional system from Fig. 6 is slightly more

= jÚ j ª

j;i i;j

¢ ½¼

ement given by . It follows [8] that (13) de- than 8 , while the error probability in the system with

¢ ½¼ ﬁnes a convex polytope in the ﬁrst quadrant of the real - precoder is less than ¾ . In Fig. 8 we show the aver- dimensional vector space. This is shown in Fig. 5. The corners age symbol error probability as a function of signal to noise

of that polytope are given by the permutations of the vector ratio for the two systems (traditional and with pre- and post-

[ ; ;:::; ] Ô

½ Õ;¾ Õ;Å i

Õ; and are denoted by in Fig. 5. Since processing). We see that in this example even at the moderate

´¡µ f is a concave function deﬁned over a convex polytope its probabilities of error the improvement is signiﬁcant (more than absolute minimum is reached at one of these corners. That is 6-8 dB).

to say, the optimal matrix Î is a permutation matrix. But from

´aµ

the form of the objective function f we conclude that with- V. C ONCLUDING REMARKS

Î = Á out loss of generality we can take Å . Summarizing, we We have considered a cyclic preﬁx system with DFT matri- have shown the following. ces, that is commonly used in several equalization methods, Theorem. Optimal precoder/equalizer. Consider the sys- such as the DFT-based DMT. Note that apart from the fre- tem for digital communication shown in Fig. 4. The optimal quency domain equalizer introduced in Sec. II, in practice precoder/equalizer matrix Ì in the sense of minimizing the out- there would typically be a time domain equalizer for channel

put noise power for the ﬁxed transmitted power constrained as shortening as well [6], but this was not considered here. Our

Ì = Í £ Í Í

Ô Õ in (3) is given by Õ . Unitary matrix is ob- Õ approach was to treat the problem of noise suppression sepa-

tained as in (7), given the deﬁnition of É in (6). The diagonal rately from the ISI cancellation and implement it as a separate £ elements of Ô are given by module. With this in mind we have constructed the optimal

constant (precoder, equalizer) pair for noise reduction, subject

½=¾

to the constraint that there is no ISI in the absence of noise. Ob-

Õ;i

´ÓÔØµ

: = Ô

È viously, a more general solution would involve matrices with

Ô;i

Õ;i =½ i memory and/or substitute the restraint on the ISI-free solution 8 by the minimum mean-squared error objective. A similar ap- 6 proach was taken in [10], although the authors there do not 4 consider a cyclic preﬁx system, and unfortunately the solution 2 in that case involves ideal (unrealizable) ﬁltering. Another gen-

0 eralization of our approach would involve a “tall” instead of a square matrix precoder, and thus allow for some additional re- QUADRATURE −2 dundancy in the system. Moreover, a similar approach could be −4 applied to modiﬁed DFT-based systems and generalized perfect −6 DMT systems [4]. Finally, note that although our initial goal

−8 was not to allocate power in subbands, but perform a more gen- −5 0 5 IN PHASE eral precoding and equalization for noise reduction, some sort of power allocation is performed here as well. In particular, in Figure 6: Equalization results in a traditional system without the case when the noise autocorrelation matrix is a scaled iden-

optimal pre- and post-processing. tity, one can easily show that the optimal matrix Ì becomes indeed diagonal. Thus, in this case the optimal strategy is to allocate as much signal power in the frequency bands where the channel magnitude is low, so that the equalizer at those fre- 8 quencies does not amplify the noise.

6 REFERENCES 4

2 [1] J. A. C. Bingham, “Multicarrier modulation for data transmission: an idea whose time has come,” IEEE 0 Comm. Magazine, pp. 5-14, May 1990.

QUADRATURE −2 [2] T. Starr, J. M. Cioffi, P. J. Silverman, Understanding DSL −4 technology. Prentice Hall, Inc., 1999. −6 [3] P. S. Chow, J. C. Tu and J. M. Cioffi, “Performance evalu- −8 −5 0 5 ation of a multichannel transceiver system for ADSL and IN PHASE VHDSL services,” IEEE J. Select. Areas Comm., vol. 9, pp. 909-919, Aug. 1991. Figure 7: Equalization results using a modified system with optimal precoder/equalizer. [4] Y.-P. Lin and S.-M. Phoong, “Perfect discrete multitone modulation with optimal transceivers,” IEEE Trans. Sig- nal Proc., vol. 48(6), pp. 1702-1711, Jun. 2000. [5] A. Papoulis, Signal Analysis. McGraw Hill, 1977. 0 10 [6] J. G. Proakis, Digital Communications, McGraw Hill, Inc., 1995.

WITHOUT PRECODER T [7] R. A. Horn and C. R. Johnson, Matrix Analysis. Cam- −2 10 bridge Univ. Press, Cambridge, UK, 1985. [8] P. P. Vaidyanathan and S. Akkarakaran, “A review of the theory and applications of optimal subband and transform

−4 WITH PRECODER T 10 coders,” Appl. and Computat. Harm. Anal., vol. 10, pp. 254–289, 2001.

PROBABILITY OF ERROR [9] I. Kalet, “The multitone channel,” IEEE Trans. Commu-

−6 nications, vol. 37, pp. 119-124, Feb. 1989. 10 16 18 20 22 24 26 [10] J. Yang and S. Roy, “On joint transmitter and receiver optimization for multiple-input-multiple-output (MIMO) Figure 8: Probability of error vs. SNR: without precoder transmission systems,” IEEE Trans. Comm., vol. 42(12), (dashed line) and with precoder (solid line). pp. 3221-3231, Dec. 1994.