Design of low intersymbol interferencepartial response data transmission filters

A. Adama, M.Sc, and L.F. Lind, M.Sc., Ph.D., C.Eng.. M.I.E.E.

Indexing terms: Circuit theory and design, Filters and filtering

Abstract: The paper presents a method for the design of partial response data transmission filters. Two classesof filters are considered: class 1 and class 2. An analytical theory is developed for the design of timing-jitter-suppressed, low intersymbol interference class 1 filters. This is then extended to the generation of class 2 filters.The optimal poles and zeros of the filters are located by means of an iterative numerical search. Numericalresults show that the filters have optimal performance in both time and frequency domains. Filter responses to arectangular input pulse (NRZ) are given and the performance characteristics are discussed.

1 Introduction

The use of partial response systems in data transmission,in recent years, has greatly enhanced the transmission effi-ciency. In particular, using partial response techniqueswill permit transmission at higher bit rates for an availablebandwidth. The advantages to be derived from usingpartial response spectral shaping have been reported[1-3]. However, partial response systems using symbol-by-symbol detection possess reduced noise immunity, due tothe fact that the superposition of the signal waveformscauses the number of output levels to be more than thenumber of input levels. Nevertheless, partial responsesystems are widely employed for commerical systems[4-7]. The filters used in practical partial response systemsare conventional ones such as Chebychev or Butterworth.Although these filters have satisfactory frequency selecti-vity they are probably not optimal. They require groupdelay equalisers with their attendant cost and sensitivityproblems. The components due to the equaliser createadditional loss and tuning problems.

This paper is concerned with the design of realisablefilters that are tolerant to timing jitter, as well as havinglow intersymbol interference (ISI), for class 1 and class 2partial response systems. These types of filters are useful inbandwidth efficient digital radio systems, for high-speeddata transmission. The work by Nader and Lind [8] formsthe theoretical basis for the design of these filters.However, a novel extension of this work, to includetiming-jitter suppression, is presented. By flattening theimpulse response in the neighbourhood of the ideal sam-pling points, timing jitter suppression is achieved. Theresulting filters have optimal performance, both in timeand frequency domains. Every degree of freedom is used tooptimise their performance for partial response. Thisresults in putting the error criterion at a global minimum.The filters have good attenuation and group delay per-formance, and so no separate group delay equalisation isneeded.

2 Partial response system

A partial response system utilises a prescribed amount ofintersymbol interference to achieve transmission at theNyquist rate of 2J3 bits per second in a bandwidth B. Thedeliberate introduction of intersymbol interference resultsin more than two received levels for binary input. The

Paper 4691G (E10), first received as 2 papers 24th June and 10th July 1985, and as asingle paper 7th March 1986

The authors are with the Department of Electronic Systems Engineering, Universityof Essex, Wivenhoe Park, Colchester, Essex CO4 3SQ, United Kingdom

IEE PROCEEDINGS, Vol. 133, Pt. G, No. 4, AUGUST 1986

partial response signal is formed by filtering the binaryinput signal with a specially shaped lowpass filter, toproduce a waveform which takes one of three (class 1) orfive (class 2) possible levels at each bit sampling time.These lowpass filters are used to shape the overall spec-trum of the system as well as the input signal. This reducesthe bandwidth required for transmission to significantlyless than that of a 2-level signal at the same bit rate.

2.1 Class 1 filterThe impulse response of an ideal causal class 1 partialresponse filter is given in Fig. 1. It has two nonzero pulse

r\

m t t r-̂ t \t o -3 to -2 t o - l t 0 to . l to.2 to*3

Fig. 1 Impulse response of an ideal causal, class 1 partial response filtersymbol spacing = 1 s

samples within the main lobe. A major requirement is thatthese nonzero pulse samples, which are located at t0 andt0 — 1, should have equal amplitudes.

The condition for zero intersymbol interference can bestated as

g(t0 + kT) = 0; k = -M, - ( M - l ) , . . . , 1 , 2, . . . , oo

k±-l, orO (1)

where M is the integer part of [t0 — 1], t0 is the mainsample point and T is the symbol spacing in seconds. (Thiswill be assumed to be unity in the remainder of this paper).To achieve timing jitter suppression, the time response isflattened at t0 + k(k ± — 1 or 0) ideal sample points. Thiscan be done by imposing the zero-derivative condition forall t = t0 + k(k ± — l or 0). This type of design constraintintroduces a 'first-order' immunity to the timing jitter. Thiscondition can simply be stated as g'(t0 + k) = 0 for all kexcept the points k = — 1 or 0.

2.2 Class 2 filterThe ideal impulse and amplitude responses of a class 2partial response filter are shown in Fig. 2. Here hid(t) = 0for all integer t except for t = 0, ± 1. At these pointshid(-1) = hJX) = 1 and hJQ) = 2hid(l) = 2. This impulseresponse can be described by the expression

hid(t) = 2 sine {t) + sine (t - 1) + sine (t + 1) (2)

195

which translates to the frequency domain as

00HJJ) = 4 cos2 -+J0 (3)

-3 -2 -1 u t

4.0

0 f = 0.5Hz f

b

Fig. 2 Responses for an ideal class 2 partial response filter

a Impulse response, signalling rate = 1 symbol/sb Amplitude response

This filter has an amplitude response similar to the fullraised cosine characteristic but requires half the band-width. It is therefore sampled at twice the usual rate whilestill retaining all the advantages of the full raised cosinefilter.

Ideally both class 1 and class 2 filters have frequencycharacteristics that fall to zero at the Nyquist frequency.Nevertheless, the class 2 system has less signal power inthe vicinity of the Nyquist frequency when compared withclass 1. This is due to its cosine2 spectrum (the class 1system has a cosine spectrum). Element timing informationcan, therefore, be easily transmitted using a 1/2T Hz sinewave, as the separation of the data from the timing signalsis now more easily accomplished at the receiver. Also theclass 2 system can often replace a quaternary signal with araised cosine spectrum, but with a relatively low toleranceto additive white noise. In a situation requiring high adjac-ent channel rejection, the class 2 system would have abetter performance than the class 1.

The ideal conditions stated in eqns. 1-3 cannot beachieved with finite-element filters. However, they can beapproximated in some optimal manner. The approxi-mation can either be done in the frequency domain or thetime domain. If frequency domain methods are employed,we are faced with the difficult problem of joint opti-misation of the amplitude characteristic and a flat groupdelay. Even if this problem is solved, it does not guaranteegood performance at the zero crossings of the impulseresponse. As a good time response is desired, it wasthought best to do the optimisation directly in the timedomain by an analytic procedure.

3 Design theory

If the ideal response in Fig. 2a is made causal with themain sample point t0 placed at time t = 1, then t0 — 1 andt0 — 2 correspond to the times t = 0 and t = — 1, respec-

tively. Under this condition, the postcursor sample pointsof the resulting impulse response become identical to thoseof the class 1 system given in Fig. 1. The design of bothfilters can therefore be handled simultaneously.

Previous work [8] using the analytic design procedureconsidered the design of minimum ISI Nyquist /* filters.This method allows the postcursor squared (normalised)ISI error to be fixed to a preset value. The results of thisdevelopment are now extended to include sensitivity totiming jitter.

The transfer function of a realisable filter can be writtenas:

G(s) = £ (4)

The inverse Laplace of this gives the impulse response as:N

(5)

0 otherwise

The total squared postcursor ISI (normalised with respectto the squared value of the response at the main samplepoint t0) is given by

I 92(h + k)A i^i

92(t0)(6)

In considering sensitivity to timing jitter, we define thetotal squared postcursor slope error (normalised withrespect to the squared response at the main sample point)as

+c AA if

g\t0)(7)

where the prime indicates first derivative.Now we form an error criterion ET defined as the sum

of the total intersymbol interference error and the totalslope error.

ET = E + yS

where y is the slope importance factor.From the definitions of E and S,

k) \.9'(t0

ET =

Now,

fc=l

92(to)

n = 1 j = 1

(8)

where xn = exp( — pn).Applying the geometrical series summation to the term

in square brackets gives

I 92(t0 + k) = £ £ rHrj(xmx^V{xHXj - 1)

Similarly, for the total slope error

t

(9)

(10)

• Nyquist's first criterion: equally spaced axis crossings in the impulse responseexcept for a peak at the centre of the main lobe.

196 IEE PROCEEDINGS, Vol. 133, Pt. G, No. 4, AUGUST 1986

Substituting eqns. 9 and 10 into eqn. 8 gives

rnrj\xnxj) n~1

J

Z I(11)

The quantity ET is to be minimised. Then, to obtain aglobal optimum, we force the condition

Using the formula for differentiating a quotient,

dET " , ._..

= l J =

yv N

T'° ~1^ ) = 0

J

=l XnXj ~

Zj=\ XnXj

(12a)

(126)

(12c)

The right-hand side of eqn. 12c is a constant (independentof pn). Using the method of Lagrange multipliers and thepolynomial theory [8], a closed form result for the residuescan be obtained from eqn. 12c as:

(1 + yp2n) sinh I f p

" sinh (pn + p,.)/2

""•/M sinh (p. - p,)/2

n= 1, 2, . . . ,

(13)

We also know that an all-pole filter has residues defined by

N

;= In N

i = 1 , i ^ n

(14)

Our optimum filter must satisfy both residue equations,thus eqns. 13 and 14 can be equated to give

(i+ypa)e-'-'°(i + ypi)

, sinh (p. + p,)/2

> sinh (p. - p,

sinhi=\

where

Q =g'(t0)

g(t0)(15)

Substituting eqn. 13 into eqn. 11, the total error ET can befound to be

ypnQ)/["exp ( -2 (16)

Eqns. 15 are an implicit expression in terms of the poles.Putting y = 0 (corresponding to ISI error criterion only)these equations reduce to the results obtained by Naderand Lind. When pole values are found that satisfy eqns. 15we have an optimal filter. The resulting filter is optimalover the infinite postcursor points. Precursor equalisationis achieved by introducing transmission zeros [8].

Eqns. 15 are used in two distinct ways. First, they canbe simplified to give an approximate solution in closedform. These solutions are used as starting poles for a sub-sequent iterative procedure. Secondly, they are usedexactly in a computer program to iterate to the globaloptimum. The remainder of the computer program is con-cerned with iterative adjustment of f0 and the transmissionzeros which are used for precursor equalisation.

The solution output consists of poles, zeros, ISI, slopeerror and t0. The inputs to the program are N, the degreeof the filter; ET, the total squared postcursor error; y, theslope importance factor, even or mixed numerator transferfunction (corresponding to passive or active filterrealisation) and choice of input waveform. The computerprogram can be described by the flowchart given in Fig. 3.

Although the theoretical development above was forimpulsive inputs, the design theory is capable of extensionto handle nonimpulsive inputs. The case of the rectangularinputs: N =numberof poles

Ej = total errorYj - slope importance factoreven or mixed numeratorchoice of input waveformfilter type(class 1 or class2)

find starting valuesfor poles and for t0

solve equation set(15) for the poles

class 1 design class 2 design

introduce transmissionzero for precursor ISIelimination

find transmissionzero such thatg( t o -1) = 2g(t0)

introduce moretransmission zerosfor precursor ISIelimination

g(t0).

outputs: poles and zero locationsISI and slope errort0 = main sample point

Fig. 3 Flowchart for computer program

1EE PROCEEDINGS, Vol. 133, Pt. G, No. 4, AUGUST 1986 197

input pulse (NRZ) was solved and the following resultswere obtained.

4 Numerical results

Several filters were designed on a PDP10 computer usingthe procedure of this paper. Both even and mixed numer-ator transfer functions were considered, corresponding topassive and active realisations, respectively.

4.1 Class I designBy varying ET and y, it is possible to examine the ISI as afunction of the slope error. In fact, it was observed that fora fixed value of ET, the slope error can be reduced byincreasing y. However, this has the effect of increasing E;therefore some compromise is desired. For a fixed value ofy (say y = 0.007), E was found to be always 10 ~4 irrespec-tive of the value of ET or the degree of the filter, N. Other yvalues also gave similar constant E results; for example, toobtain an E of 10"3, it was necessary to assign the value0.0125 to y for all ET = 10"5 to 10"10. It was thereforepossible to keep E constant while changing S. This gaverise to several designs each having a different slope error.

As an example [11], two 8-degree filters each havingE = 10 " 4 but with different slope errors, are considered.These designs are for the even numerator transfer functioncase and the input waveform is a full-width rectangularpulse. This type of input pulse has been chosen because ofits wide application in practice.

The resulting poles and zeros are given in Table 1.The time response of these filters is given in Figs. 4a and bwith their corresponding eye diagrams given in Figs. 5aand b, respectively. These results show that filter A has ahigher total slope error (precursor plus postcursor) thanfilter B. In fact, an 88% reduction has been achieved in thetotal slope error. This change in slope error can be seenfrom the location of the poles of filter B with respect to A.

The poles of filter B are more spread out along the im-

0.8

0.5

0.4

0.3

0 2

0.10.0

Table 1: Poles, zeros and total slope error for 8th-degreefilters with E = 10~4 (even numerator)

Pi-PaP2.P7Pa.PeP4.P5z,,z2z3.zAz5.z6Total 1slope >error J

Filter A

-0.40836 ±y2.77274-0.57244 ±y2.02382-0.64645 ±y1.22018-0.67535 ±/0.40781-2.44106 ±y1.00744

2.44016 ±y1.00744±12.18077

0.236

Filter B

-0.72490 ±y'4.04982-1.02553 ±/2.72310-y'1.34415 ±/1.57551-1.50752 ±y0.52828± 36.58857± 4.66688

—

0.028

Fig. 5 Eye diagrams of 8th-degree filters of example

a Filter A b Filter B

0.7

0.6

0.5

^ 0 . 4

9, 0.30.2

0.1

0.03t .s

Fig. 4 Time response of 8th-degree filters of example

a Filter A b Filter B

aginary axis than those of filter A. They also have higher(in the negative sense) real parts. The implications of thisbehaviour are a faster decay and higher damping of thepulse tails. Fig. 6 shows the poles plotted in the s-plane.

The eye diagrams show that the filters have practicallyno ISI at the zero crossings. This results in a very goodwaveshape for timing recovery. The eye diagram of filter B,on the other hand, has a wider eye, resulting in more toler-ance to timing jitter.

In the frequency domain, it is expected that the cost ofthis improvement is an increase in the minimum band-width. Fig. 7 shows the amplitude and group delay charac-teristics of the filters of this example.

These results show that filter B requires more band-width than filter A to achieve this improved timing-jitter

198 IEE PROCEEDINGS, Vol. 133, Pt. G, No. 4, AUGUST 1986

performance. The selectivity of both filters is good withfilter B outperforming filter A in the far stopband. Thegroup delay is reasonably flat over the passband. Theattenuations at fN, 2fN, 3/N and 4/N are given in Table 2,{fN is the Nyquist cut-off frequency given by 1/2 T, whereT = symbol spacing).

J77

2

a

a

-200-1.75-1.50 -1.25-1.00 -0.75-0.50-0.25 0I V

aI

a

I a

1

Fig. 6 Pole locations

2 filter A• filter B

0 -

- 2

- j T T

2.5 3.0

-100-

0

3

2

2

o"o 1

1

0

0

.3

.0

.5

.0

.5

.0

.5

.0

/w/V

-

'•—T\i

- \V0.0 0.5 1.0 1.5 2.0 2.5 3.0

f .Hz

b

Fig. 7 Characteristicsa Amplitude characteristicsb Group delay characteristics

Table 2: Comparison of attenuations for 8th-degree filters ofexample

A, dB Filter A Filter B

Af

A I,A3tlA

18.8858.2573.1281.86

9.3149.2875.1391.49

Fig. la can be used to define a suitable point at whichthe bandwidths can be compared. The — 20 dB point hasbeen chosen in this work. The percentage increase in band-width in going from filter A to B is 33.3%. This is thepenalty paid for the 88% improvement achieved in theslope error. This can be regarded as a 'fair deal'. In general,for a particular value of slope error, the bandwidth wasfound to decrease with N, the degree of the filler.

Fig. 8 gives the bandwidth as a function of slope error

1.0r

0.9

N 0.8i

0.7

0.8

0.5

0.A

Nyquist'sminimumbandwidth

0.05 0.10 0.15 0.20 0.2 5 0.30normalised squared slope error

Fig. 8 20 dB bandwidth as a function of slope error for class I partialresponse filters, E = 10 ~4

N = 9

for these filters with N varying from 5 to 9. These resultsare obtained with E = 10 ~4 and the even-numerator trans-fer function case.

These curves can be used to see the trade-offs betweenbandwidth, slope error and the degree of filter. It can beseen that the higher the degree of the filter, the lower thebandwidth required for a given slope error. For highdegree (N = 9) filters, small slope errors can be achievedwith a little increase in bandwidth.

4.2 Probability of error boundsTo compare the probability-of-error bounds for thesefilters, we define an 'equivalent noise bandwidth'. Thenoise power at the output of the filter is given by

oz = 'oi \G(f)\2dfjo

(17)

where No = Gaussian noise power spectral density andG(f) = transfer function of the filter.

Therefore the integral term above can be regarded asbeing equivalent to a 'bandwidth'. This is termed the'equivalent noise bandwidth', Bn, that is

(18)= \G(f)\2df

IEE PROCEEDINGS, Vol. 133, Pt. G, No. 4, AUGUST 1986 199

Using eqn. 18, the equivalent noise bandwidth of each ofthe filters in the example was computed.

Probability-of-error upper bounds have also been com-puted for these filters. The error bounds are due toLuganani [5] and Saltzberg [6]. These computations havebeen done for a signal-to-noise ratio in the absence of ISIof 8.0 dB. It has been assumed, for simplicity, that all theISI terms contribute to the noise power.

The error bounds are given in Fig. 9. It can be seen

or

-2

-A

-6

-8

-100.0 0.1 0.2 0.3 0. A 0.5

peak deviation from sampling timeFig. 9 Probability-of-error bounds as a function of peak deviation fromoptimum sampling time, N = 8, E = 10~A

filter A, Bn = 0.572filter B, Bn = 0.837

from these error bounds that the Pe for filter A is notaffected significantly if the sampling time is within 8% ofbit duration from the optimum sampling time. Filter B, onthe other hand, can tolerate deviations of up to about 20%of bit duration before the Pe starts to deteriorate adversely.

4.3 Class 2 designHere we present, as examples, the results of two designs.We consider 9-pole passive and active realisations, eachhaving a total post-cursor error E = 10" 8. The design pro-cedure then gives the poles, zeros and the main samplepoint as follows:

(a) For passive realisation:

Pl,P2 = -0.695955850 ±j2.914115459

P 3 , P4 = -0.967618108 ±/2.256880492

P5, P6 = -1.126846731 +J1.518542454

/>7, P8 = -1.202836037 ±j0.759157278

P9 =-1.223827451

Zt,Z2= ±y3.141001701

Z3,Z*= ±27.989299774

Z5,Z6= ±3.170467287

tm =4.1143

(b) For active realisation:

P l 5 P 2 = -0.670303240+./2.924511760

P 3 , P 4 = -0.949857 = 93 ±7*2.287992209

P5, P6 = —1.128761500 ±7*1.557789072

P 7 , P 8 = -1.227039918 ±7*0.787276633

P9 =-1.258416831

Zt,Z2= ±73.141524076

Z 3 = 7.794664502

Z 4 = 2.989967942

tm = 4.4804

The locations of the zeros associated with g(tQ — 1) provedto be very interesting. The design procedure gives zerosthat are located on the imaginary axis. Their location isindependent of the type of realisation (active or passive)and the value of E. This behaviour has been seen toimprove immensely the selectivity of the filters.

In both these results, Zx and Z2 are located near ±771.The implication of this is that the amplitude characteristicswill have a high attenuation near fN, the Nyquist fre-quency.

The passband amplitude response of these filters followsvery closely the theoretical raised-cosine shape. Table 3gives the stopband attenuations a t / N , l.5fN, 2fN, 2.5fN and3.5/N. Both filters have high attentuation at the Nyquistfrequency ensuring very good selectivity in the passband.In the stopband, the active filter has a slightly better per-formance. The amplitude characteristics and group delayof these filters are shown in Fig. 10. The group delay isreasonably flat over the passband.

Table 3: Comparison of attenuations (in dB) for 9-pole filter.

Filter

Passive

/

89 .77

Attenuation (dB) at

1.5f* 2fN 2.5fN

50.50 63.60 73.70

3

81.75

Active 108.88 52.78 67.14 78.22 87.06

As hid(t) has its maximum at t = 0 and g(t) at t = f0 — 1,we compare Hid(f) in the frequency domain with

H(f) = G(f) exp [j2nf(t0 - 1)] (19)

which has both real and imaginary parts. The results forthe 9-pole passive filter are shown in Fig. 11.

The worst errors for the real and imaginary parts are0.77 x 10~2 and 0.58 x 10~2, respectively. The actualimpulse responses of the filters in the above examples havebeen computed and are shown in Fig. 12. The overshootsare 2.332% and 2.383% for the passive and active filters,respectively, as compared with 2.647% for eqn. 2. The eyediagram is shown in Fig. 13. This has been computed on adigital computer using a pseudorandom sequence of 127binary bits. It is evident from this Figure that the ISI ispractically nonexistent at the zero crossings and that a5-level output results [2].

5 Conclusions

An analytic design theory for the design of optimal realis-able class 1 and class 2 partial response filters having verylow intersymbol interference has been presented. The pole

200 IEE PROCEEDINGS, Vol. 133, Pt. G, No. 4, AUGUST 1986

and zero locations of the filters are located by using anefficient iterative computer program. In the class 1 case,

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40

-20

filters having controlled sensitivity to timing-jitter weredesigned. The resulting ISI and timing-jitter performance

- 40

~ -60o

- 8 0

-100

3.5

3.0

2.5

2.0

1.0

0.5

0.00.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

f .Hzb

Fig. 10 Frequency responses for the 9-pole class 2 filters, E = 10'a Amplitude responseb Group delay response

active filter

0.0 0.2 0.4 0.6 0.8 1.0

0.2 0.3 / 0.4 0.5f Hz f

0.4

0.3

0.2

0.1

0.0(

-

-

-

-

/ \

/ \

/ \• i , . . J \«=—=_i 1

2 3 4 5 f ''

Fig. 12 Time responses for the 9-pole class 2 filters T = 1, E = IQ~

Fig. 13 Eye diagram for the 9-pole passive class 2 filter

are at a global minimum, resulting in zero 'first order' dif-ferential sensitivity to any drifts in component values. Ithas been shown that to achieve any improvement in thetiming-jitter performance of the filter, more bandwidth isneeded. In fact, the lower the sensitivity to timing jitter, thewider the bandwidth required.

A comparative study of the performance of the class 1filters based on error probabilities has been presented. It isobserved that for large jitters, lower error probabilities arepossible by allowing a bit more bandwidth.

In the class 2 case, filters having very good frequencyselectivity in the passband were designed. These filterswere shown to possess good frequency domain error per-formance, and they do not require additional phase equal-isation.

Finally, graphs showing the trade-off that can beobtained between bandwidths, slope error and degree offilter have been presented. It is conclusive from thesegraphs that higher degree filters (iV > 7) are preferable forhigh speed data communication.

Fig. 11 Error performance of the 9-pole passive class 2 filtera real part of H(f) against frequencyb Imaginary part of H(f) against frequencyc Deviation of real part of H(f) from the ideal characteristics

IEE PROCEEDINGS, Vol. 133, PL G, No. 4, AUGUST 1986 201

6 Acknowledgments

One of the authors, Abu Adama, would like to acknowl-edge the financial support of the Federal Polytechnic,Idah, Benue State, Nigeria.

7 References

1 LENDER, A.: 'The duobinary technique for high speed data trans-mission', IEEE Trans., 1963, CE-82, pp. 214-218

2 K.RETZMER, E.R.: 'Generalization of a technique for binary datacommunication', ibid., 1966, COM-14, (2), pp. 67-68

3 HUANG, J.C.Y., and FEHER, K.: 'Performance of partial responsedigital radio systems in linear and non-linear bandlimited channels',ibid., 1979, COM-27, (11), pp. 1720-1725

4 SWARTZ, T.L.: 'Performance analysis of a three-level modified duo-binary digital FM microwave radio system'. IEEE International Con-ference on Communications, Minneapolis, MN, USA, 17th—19th June1974, pp. 5D-1-5D-4

5 BECHER, F.K., KRETZMER, E.R., and SHEEHAN, J.R.: 'A new

signal format for efficient data transmission', Bell Syst. Tech. J., 1966,45, pp. 755-758

6 ANDERSON, C.W., and BARBER, S.G. 'Modulation Considerationsfor a 91 Mbit/s digital radio', IEEE Trans., 1978, COM-26, (5),pp. 523-527

7 KUREMATSHU, K., OGAWA, K, and KATOH, T.: 'The QAM2G-1OR digital radio equipment using a partial response system',Fujitsu Sci. & Tech. J., June 1977, pp. 27-48

8 NADER, S.E., and LIND, L.F.: 'Optimal data transmission filters',IEEE Trans., 1979, CAS-26, (1), pp. 36-45

9 LIND, L.F., and NADER, S.E.: 'Realizable filters that minimise inter-symbol interference', Radio & Electron. Eng., 1978, 48, (12), pp. 612-618

10 LUCKY, R.W., SALZ, J., and WELDON, E.J.: 'Principles of datacommunication' (McGraw-Hill, 1968) pp. 83-92

11 ADAMA, A. and LIND, L.F.: 'High speed, low bandwidth data trans-mission filters that are tolerant to timing-jitter'. IEE Colloquium onElectronic Filters, IEE Colloquium Digest 1985/6, pp. 5.1-4

12 LUGANANI, R.: 'Intersymbol interference and probability of error indigital systems', IEEE Trans., 1969, IT-15, pp. 682-688

13 SALTZBERG, B.R.: 'Intersymbol interference error bounds withapplication to ideal bandlimited signalling', ibid., 1968, IT-14,pp. 563-568

202 IEE PROCEEDINGS, Vol. 133, Pt. G, No. 4, AUGUST 1986