A new realization of the whitened matched filter, incorporating easy symbol synchronization for high-speed transmission

A

New Realization of the Whitened Matched Filter Incorporating Easy Symbol

Synchronization for High-speed Transmission

Cheng-Kun Wang Li-Shan Lee Department of Electrical Engineering,

Rm

512

National Taiwan University Taipei, 10764, Taiwan

Abstract

The whitened matched filter (WMF) is the optimum receiver filter for pulse amplitude modulation (PAM) and quadrature amplitude modulation (QAM) transmis- sions in the presence of intersymbol interference (ISI) and additive white Gaussian noise (AWGN). The con- ventional realization of the WMF requires sophisticated algorithm for timing recovery such that it is not suitable for high-speed applications. In this paper a new realiia- tion of the WMF is proposed, which incorporates very easy timiig recovery for high-speed transmissions such

as optical fiber communications. The proposed realiza- tion scheme uses a specially designed filter to equalize the ISI, such that only relatively simple timing recovery circuitry for 1.51-free signals will be needed.

1

Introduction

Forney developed the whitened matched filter (WMF) [l] as the optimum receiver filter for pulse amplitude modulation (PAM) and quadrature amplitude modula- tion (QAM) transmissions in the presence of intersym- bo1 interference (ISI) and additive white Gaussian noise (AWGN). Additional signal processing can be applied to the WMF output to complete the optimum receiver design, e. g.

,

maximum-likelihood sequence estimation (MLSE), zero forcing (ZF), decision-feedback equaliza- tion (DFE), linear minimum mean-square-error estima- tion (MMSE), and mean;square decision feedback equal- ization (MSDFE) etc. However, such WMF is seldom used in practical communication systems. Presently, the WMF finds its way only in the low-speed voice-band modem and some medium-speed applications where the enormous amount of computation required in signal pro- cessing to realize the WMF is both technically achiev- able and economically feasible, and is not adopted when the complexity is considered too expensive, or when the

speed is simply too high such as in optical fiber commu- nications.

- 1 -

Figure 1: The whitened matched filter.

The WMF, as depicted in Figure 1, is composed of a matched filter, a symbol-rate sampler, and a transver- d filter whose transfer function is obtained from the spectral factorization

[I]

of the pulse autocorrelation function. The primary difficulties in realizing the the WMF lie partly in the synthesis of the matched filter, and mainly in the timing recovery for the symbol-rate sampler. While equalization [2] can be used to synthe- size the matched filter with ease, the complicated t i m ing recovery for ISI-corrupted signals [3,4,5] remains to be the major obstacle of the adoption of WMF in high- speed applications. In this paper a new realiiation of the WMF is proposed, which requires only relatively simple timing recovery circuits for ISI-free signals, therefore is especially suitable for high-speed applications. In the following, some background of the WMF will be briefly reviewed first for the development of this paper.

2

A

Brief

Review

of

the

WMF

2.1

The WMF

Since QAM is basically cornposed of two PAME in phase quadrature, it is sufficient to describe only the PAM. The block diagram of a baseband-equivalent PAM trans- mission system is depicted in Figure 2. Let {zk} be a sequence of integer symbols with finite length

N

such

that 0

5

z k

5

rn

-

1, 0

5

k

<_

N

-

1. The impulse response and transfer function of the transmitting filter are h ( t ) and H ( f ) , respectively. Then the transmitted signal s ( t ) is

N-1

s ( t )

=

zkh(t -

kT),

(1) k = O

where T is the symbol duration, and the received signal

r(t) is the additive-noise corrupted version of s ( t )

r(t)

=

s ( t )

+

n(t),

(2)

where n(t) is assumed to be a white Gaussian noise with two-sided power spectral density

(PSD)

N0/2. Let

D

denotes the delay operator such that @h(t)

=

h(t

-

kT),

then the symbol sequence { Z k } can be expressed

as z(D)

=

E:

:,

z k D k l and the transmitted signal aa s ( t )

=

z(D)h(t). (3)

At the receiver side a matched filter F ( f ) followed by a symbol-rate sampler 17, 81 provides a set of sample values sufficient for estimating z ( D ) .

Figure 2: Baseband equivalent model of a PAM trans- mission system

Let h(t) be a finite impulse response of length

L

sym- bol durations, where

L

is an integer. The autocorrela- tion coefficients of

h(t)

are

(4)

and the pulse autocorrelation function of h(t) is defined

as

V

(5)

k = - v

where v

=

L

-

1 is called the span of h(t). The symbol- rate samples of the matched filter output are

A m

ak

=

r(t)h(t - k T ) d t , 0

5

k

5 N

-

1, ₍₆₎ and the sequence { a i ) can be expressed as

N-1 k=O Since 0)

=

-

- where ni

where nc(D) is a zero-mean colored Gaussian noise se- quence with autocorrelation function

%

Rhh(D). The pulse autocorrelation function &h(D) is real and sym- metric such that there exists the canonical factoriza- tion Rhh(D)

=

f ( D)f (D- ' )

[I]

where f(D) is a real polynomial of degree v that contains all the roots of &(D) outside the unit circle, and Equation (9) can be expressed as

a ( D )

=

z(D)f(D)f(D-')

+

n(D)f(D-'), (10)

where n ( D ) is a zero-mean white Gaussian noise se- quence with variance N0/2. Hence a transversal filter characterbed by l / f ( F 1 ) can be used to obtain the output sequence

z ( D )

=

a ( D ) / f ( D - ' ) = z ( D ) f ( D ) + n ( D ) (11) in which the noise is white. Thus the WMF is the cas- cade of the matched filter, a symbol-rate sampler with correct sampling instants, and a transversal filter char- acterised by l/f(D-l), as depicted in Figure 1.

2.2

Conventional Realization

It is practically very difficult to realize the WMF directly in the form as shown in Figure 1 because the correct sampling instants at the matched filter output is very difficult to determine. In fact almost no realiztion of the WMF using this approach can be found. Instead, be- cause of linearity, the traneversal filter and the symbol- rate sampler can be interchanged, and the matched filter and the transversal filter can be combined into one fil- ter whose frequency response is W ( f )

=

F ( f ) / F - l ( f ) , where

F-l(f)

is

the Fourier transformof f(D-') and

is

periodic with period 1/T. This filter W(f), also pro- posed by Forney [l] and in fact a direct consequence of the work of Ericson

[SI,

can be view as the alterna- tive form of the WMF. This form has the important interpretation that the receiving filter, usually realised by an equalizer, designed with the desired impulse re- sponse (DIR) set to f ( D ) , is a WMF [2]. Timing recov- ery can then be applied to this equalised output, which

is now simpler than that in the original form in Fig- ure 1 because the IS1 is now less severe and the noise is white, as was demonstrated by several standard al- gorithms [3, 4, 51. This realization scheme is plotted in Figure 3.

Figure 3: Alternative reahation of WMF However, the IS1 of the equaliied pulse characterized by f ( D ) can still be very severe such that all the algo- rithms [3, 4, 51 for timing recovery requires enormous amount of computation in signal processing. This is the major obstacle of applying WMF in high-speed trans- missions, such as optical fiber communications.

3

The

Proposed

Realization

Scheme

A new realization of the WMF featured by easy tim- ing recovery is proposed here as follows.

As

depicted in Figure 4, the proposed scheme consists of a matched filter in cascade with a filter C(f) t o be specified later, a symbol-rate sampler employing only simple ISI-free timing recovery circuitry, and a transversal filter char- acterized by f ( D ) . The filter C(f) must satisfy two constraints:

1. It must be information-losslees after symbol-rate sampling.

2. The pulse shape a t the filter output must satisfy the Nyquist zero-IS1 criterion for easy timing recovery.

is invertable and periodic with period 1/T. And by the Nyquist zero-IS1 criterion, constraint 2 states that

2

H ( f

-

n/T)a*(f

-

n / T ) c ( f - n/T)

=

constant.

n=-m

(12)

Define the folded spectrum a t the matched filter output

as 00 Shh(f) H ( f - n / T ) H * ( f - n / T ) n=-m m

=

IWf-

n / m (13) n=-m

which is the Fourier transform of the autocorrelation function Rhh(D)

Shh

(f)

=

Rhh ( D )

10

= a-ja=T f

.

(14)

By the fact that Shh(f) and C(f) are periodic with pe- riod 1/T, Equation (12) can be expressed as

Shh(f)C(f) = Constant, O

5

f

5

1/T. (15)

Hence we require that

c(f)

=

1/Shh(f)

#

0, 0

5 f

5

1/T.

As depicted in Figure 4, the matched filter H * ( f ) and the filter C(f)

=

l / S h h ( f ) can be combined into asingle filter

H*(f)/Shh(f).

The output of this filter is ISI-free for easy timing recovery, and the symbol-rate output are sent through a transversal filter characterized by f ( D )

to produce the final output z'(D). We shall prove that

this output is equivelent to that of the WMF output, that is, we shall show that

z'(D)

=

z ( D ) f ( D )

+

n'(D) (16)

where n'(D) should be a white Gaussian noise with vari- ance N0/2. Below is the proof.

Because of the ISI-free receiver filter H*(f)/Shh(f), the symbol-rate sampler output sequence is

N-1 ... , a'(D)

=

akDk (17) timing a: = 2 k

+

n p (18) "(f)C(f)Z"(f)67h

(9

i

r(t) k=O

(q-qpypt+

... where

reoovery _{and {nr} is a colored Gaussian noise sequence whose} PSD is

No

-

Figure 4: The proposed realization of the

WMF.

IH(f

-

n/T)/Shh(f

-

n/T)l'

=

NO/ZShh(f).

n=-m

By the theorem of reversibility [7] and the Nyquist

Hence we have

a'(D) = z ( D )

+

nco(D) (20) where the autocorrelation function of nco(D) is

No/2Rhh(D) by the convolution theorem of the Fourier transform. The final output sequence is then

z ' ( D )

=

a ' ( D ) f ( D )

= z(D)f(D)

+

nCO(D)f(D)

=

z ( D ) f ( D )

+

d ( D ) (21) where n'(D) is a Gaussian noise with autocorrelation function

ISI-free situations the optimal Data Transition Track- ing Loop (DTTL) and Early-Late Gate Tracking Loop (ELGTL) [lo] can be simply approximated by a phase- locked loop (PLL) [ll, 12, 131, which has already been implemented in giga-bit systems.

After the timing recovery, the transversal filter can be implemented by a analog-to-digital (A/D) converter and digital arithematic unit, which is not a problem even in giga-hertz range.

5

Conclusion

A new realization of the WMF incorporating easy sym- bol synchronization has been demonstrated in this pa- per. The proposed realisation scheme is easy t o imple- ment, and is especially suitable for high-speed applics tions such ae optical fiber communications.

and is thus white with variance N0/2.

References

4

Practical Consider at ions

In low- and medium-speed applications where sophis- ticated digital signal processing can be used, the fil- ter H * ( f ) / S h h ( f ) can be implemented by a zero-forcing

(ZF) fractionally spaced equalizer (FSE) [2]. In these situations the tap spacing T must be chosen such that

1/2r is larger than the cut-off frequency of the filter, i.e.,

H*(f)/Shh(f)

=

0,

I f 1

2

1/27. The channels in these applications are usually slowly time-varying to jus- tify the use of adaptive algorithm [2] t o update the tap weights. The discrete transfer function f ( D ) will then also vary slowly with time, and should be estimated from the tap weights.

In high-speed applications where digital signal pro- cessing is not feasible, such as in optical fiber com- munications, analog equalization can be used to real- ize H*(f)/&h(f) if both amplitude and phase equal- izations are employed [9]. In analog equalizations, a ZF equalizer is usually expressed as

(23) where R c o s ~ ( f ) is the raised cosine shaping filter with roll-off factor

p.

The roll-off factor is carefully cho- sen such that its cut-off frequency (1

+

P)/2T is close to the the cut-off frequency of H(f) t o avoid exces- sive noise enhancement. Comparing Hep(f) with the proposed filter H*(f)/Shh(f), its can be seen that we only have t o replace the raised cosine filter Rcosp(f) by

lH(f)I'/Shh(f), which is also an ISI-free filter. In these

[l] G. D. Forney, Jr. , "Maximum-likelihood sequence estimation of digital sequence in the presence of in- tersymbol interference," IEEE Trans. Inform. The- ory, vol. IT-18, pp. 363-378, May 1972.

[2]

S.

U. H. Qureshi, "Adaptive equalization," Proc. IEEE, vol. 73, pp. 1349-1387, Sept. 1985.

[3]

R.

D. Gitlin and J. sals, "Timing recovery in PAM systems," Bell Syst. Tech., vol. 50, pp. 1645-1669, May, 1971.

[4]

S.

U. H. Qureshi, "Timing recovery for equalized partial response systems," IEEE D a m . commun.,

vol. COM-24, pp. 1326-1330, Dec. 1976.

[5] L.

E.

Frank, "Carrier and bit synchronization in data communication," IEEE tram. commun., vol.

COM-28, pp. 1107-1120, Aug. 1980.

[e]

T. Ericson, "Structure of Optimum Receiving Fil- ters in Data Transmission Systems," IEEE "mm.

Inform. Theory, vol. IT-17, pp. 352-353, May 1971.

[7] J. M. Wosencraft and I. M. Jacobs, Principles of Communication Engineering. New York: Wiley,

1965.

[8] H. L. Van Trees, Detection, Estimation, and

Mod-

ulation Theory, Part I. New York: Wiley, 1968.

[9] G. Cancellieri and F. Frosini, UDiscuaion on equal- ization in optical fiber transmission system," J .

[lo]

W.

C. Lindsey and M. K. Simon, Telecommuni cation System Communication, Englewood Cliffs,

N.J.: Prentice-Hall, 1973.

[ll] F. M. Gardner, Phuelocked Techniques, New York:

Wiley, 1979.

[12] C. R. Hogge, 'Self correcting clock recovery cir- cuit," IEEE J . Lightwave Tech., vol. LT-3, Dec.

1985.

[13]

D.

Shin, M. Park, and M. Lee, "Self-correcting clock recovery circuit with improved jitter performance,"