DESIGN FOR DIGITAL COMMUNICATIONSYSTEMS VIA SAMPLED-DATA H∞ CONTROL

M. Nagahara ∗,1 Y. Yamamoto ∗,2

∗ Department of Applied Analysis and Complex DynamicalSystems, Graduate School of Informatics, Kyoto University,

Kyoto 606-8501, JAPAN

Abstract: The design procedure for the equalization of digital communication channelsis developed based on the sampled-data H∞ control theory. The procedure providestransmitting/receiving filters so as to minimize the error between the original signaland the received signal with a time delay, and to reduce the noise added to thechannel. While the system has an ideal sampler, a zero-order hold and a time delay, thedesign problem can be reduced to a finite-dimensional discrete-time problem using theFSFH (fast-sample and fast-hold) approximation. Numerical examples are presentedto illustrate the effectiveness of the proposed method. Copyright c© 2001 IFAC

Keywords: Sampled-data systems; Digital communications; Digital filters; H-infinityoptimization; Time delay

1. INTRODUCTION

Nowadays the importance of digital communica-tions is increasing owing to the rapid growth of theInternet, the cellular phones, and so on(Proakis,1989). In digital communication, especially inpulse amplitude modulation (PAM) or in pulsecode modulation (PCM), the analog signal whichis to be transmitted is sampled and becomes adiscrete-time signal. In the conventional way, thecharacteristic of the analog signal is not consid-ered, and hence the whole system is regarded asa discrete-time system. For example, one usuallyassumes that the original analog signal is band-limited up to the Nyquist frequency. But in realityno signals are entirely band-limited.

This paper proposes a new design methodologybased on the sampled-data control theory(Chenand Francis, 1995b) that takes account of inter-sample behaviors or frequency components be-yond the Nyquist frequency in discrete-time. Re-

1 nagahara@acs.i.kyoto-u.ac.jp2 yy@i.kyoto-u.ac.jp; author to whom all correspon-

dence should be addressed.

cently, the sampled-data control theory is appliedto some digital signal processing systems(Chenand Francis, 1995a; Khargonekar and Yamamoto,1996; Yamamoto et al., 1997; Nagahara and Ya-mamoto, 2000). We propose a design for the dig-ital communication system via the sampled-datacontrol. In (Erdogan et al., 2000) a discrete-timeH∞ design for receiving filters or equalizers isintroduced, but no design for transmitting filteris mentioned. However, it is difficult to attenuateboth the signal reconstruction error and the ad-ditive noise by only an equalizing after the signalis received. Therefore we show the transmittingfilter design which is executed in the same wayas the receiving filter design. Moreover, we in-troduce the H∞ method which takes account ofa tradeoff between the signal reconstruction er-ror and the energy of transmitted signal with anappropriate weighting function. Design examplesare presented to illustrate the effectiveness of theproposed method.

Fc Sh KT Cd KR Hh Pc++uc

nd

zcycwc d- - -yd -vd -6- - - -

Fig. 1. Digital communication system.

2. DESIGN PROBLEM FORMULATION

The block diagram Figure 1 shows a digital com-munication system which is known as PAM orPCM. The incoming signal wc ∈ L2[0,∞) goesthrough an analog low-pass filter Fc and becomesyc which is nearly (but not entirely) band lim-ited. The filter Fc governs the frequency-domaincharacteristic of the analog signal yc

3 . The signalyc is then sampled by the sampler Sh to becomea discrete-time signal (or PAM signal) yd withsampling period h. Then the signal is shaped orenhanced by the transmitting digital filter KT tothe signal vd to be transmitted to a communica-tion channel.

The transmitted signal vd is corrupted by thecommunication channel Cd and the additive noisend. In PCM communication, nd is also consideredas the noise generated by quantizing and codingerror. The received signal goes through the receiv-ing digital filter KR which tries to attenuate thecorruption and the noise, then becomes an analogsignal uc by the hold device Hh with samplingperiod h and smoothed by an analog low-passfilter Pc and finally we have the output signal zc.

Our objective is to reconstruct the original analogsignal yc by the transmitting filter KT and thereceiving filter KR against the corruption causedby the channel Cd and the additive noise nd.Therefore consider the block diagram Figure 2which is the signal reconstruction error system forthe design. In the diagram the following points aretaken into account:

• The time delay e−Ls is introduced becausewe allow a certain amount of time delay forsignal reconstruction.

• The transmitted signal vd is estimated witha weighting function Wz because the energyor the amplitude of the transmitted signal vd

is usually limited.• The noise has a frequency characteristic Wn.

Then our design problem is as follows:

Problem 1. Given stable analog filters F (s) andP (s), digital filters (weighting functions) Wn(z)and Wz(z) and a channel model Cd(z), find digitalfilters KT (z) and KR(z) which minimizes

J2 := supwc∈L2,nd∈l2

‖ec‖2

L2 + ‖zd‖2

l2

‖wc‖2

L2 + ‖nd‖2

l2. (1)

3 In the conventional design Fc is considered as an ideal

filter which has a cut-off frequency up to the Nyquist

frequency.

yc

zc

nd

−

+

+wc

+ ece

e

6

6

--

--

6

?

---?

--

?

-Fc

Sh KT

e−Ls

Wz

Cd

Wn

KR Hh Pc

zd

yd vd ud uc

Fig. 2. Signal reconstruction error system.

yc

zc

nd

−

+

+wc

+ ece

e

6

6

--

--

6

?

--- -- -Fc

Sh KT

e−Ls

Cd

Wn

KR Hh Pcyd vd ud uc

Fig. 3. Error System TR for receiving filter design

3. DESIGN ALGORITHM

3.1 Decomposing Design Problems

Problem 1 is a simultaneous design problem of atransmitting filter and a receiving filter, and itis difficult to solve the problem directly. There-fore we introduce a decomposition of the designproblem into two steps, that is the design for thereceiving filter and that for the transmitting filter.

Obviously the transmitting filter KT cannot at-tenuate the additive noise nd, hence the receivingfilter KR has to play that role. Moreover KR

has to reconstruct the original signal from thecorrupted signal (if KR did not have to recon-struct, the optimal filter will be clearly KR = 0)Therefore we first design the receiving filter KR

in order to reconstruct the original signal and toattenuate the noise by the block diagram Figure2 with Wz = 0 and with KT = 1. Then design thetransmitting filter by the block diagram Figure 2with Wn = 0 and with KR which is obtained theprevious design, that is we consider the channelas KRCd.

The design procedure is as follows:

Step 1(Design for receiving filter) Find a re-ceiving filter KR which minimizes

‖TR‖2

∞ := supwc∈L2,nd∈l2

‖ec‖2

L2

‖wc‖2

L2 + ‖nd‖2

l2, (2)

in Figure 3 with fixed KT (the initial filter isKT = 1).

Step 2(Design for transmitting filter) Find atransmitting filter KT which minimizes

‖TT ‖2

∞ := supwc∈L2

‖ec‖2

L2 + ‖zd‖2

l2

‖wc‖2

L2

, (3)

in Figure 4 with KR which is obtained in theprevious step.

yc

zcuc

−udvd

wc+ ece

yd

zd?

--

--

6

?

---?-

Fc

Sh KT

e−Ls

Wz

Cd KR Hh Pc-

Fig. 4. Error System TT for transmitting filterdesign

Sh/N ���� Hh/NTecedN wc wdN

Fig. 5. fast sample/hold discretization

3.2 Fast Sample/Hold Approximation

The design problems (2) and (3) involve a con-tinuous time delay component e−Ls, and hencethey are infinite-dimensional sampled-data prob-lems. To avoid this difficulty, we employ the fastsample/hold approximation method (Keller andAnderson, 1992; Yamamoto et al., 1999). By themethod, our design problems (2) and (3) areapproximated to finite-dimensional discrete-timeproblems assuming that the delay time L to bemh where m is a positive integer:

Theorem 2. Assume that L = mh, m ∈ N. Then,

(1) for the error system TR in Step 1, there ex-ist finite-dimensional discrete-time systems{TR,N : N = 1, 2, . . .} such that

limN→∞

‖TR,N‖∞ = ‖TR‖∞.

(2) for the error system TT in Step 2, there ex-ist finite-dimensional discrete-time systems{TT,N : N = 1, 2, . . . } such that

limN→∞

‖TT,N‖∞ = ‖TT ‖∞.

PROOF. By the fast sample/hold method, weapproximate continuous-time inputs and outputsto discrete-time ones via the ideal sampler andthe zero-order hold that operate in the periodh/N (Figure 5). Then apply the discrete-timelifting(Yamamoto et al., 1997) LN to the dis-cretized input/output signal edN and wdN , we canget the lifted signals

edN := LN (edN ), wdN := LN (wdN ).

Then we can approximate the continuous signalas

‖ec‖L2 ≈

√h

N‖edN‖l2 , ‖wc‖L2 ≈

√h

N‖wdN‖l2 .

Moreover define

‖TR,N‖2

∞ := supwdN ,nd∈l2

‖edN‖2

l2

‖wdN‖2

l2 +√

Nh ‖nd‖2

l2

,

‖TT,N‖∞ := supwdN∈l2

‖edN‖2

l2 +√

Nh ‖zd‖

2

l2

‖wdN‖l2,

where the systems TR,N and TT,N are approx-imated to finite-dimensional discrete-time sys-tems, then we can show ‖TR,N‖∞ → ‖TR‖∞ �

‖TT,N‖∞ → ‖TT ‖∞ as N → ∞ by using themethod as shown in (Yamamoto et al., 1999)under the assumption L = mh. 2

Once the problems have been reduced to discrete-time problems, they can be solved by a controldesign toolbox such as those given by MATLAB.The resulting discrete-time approximant is givenby the following:

Theorem 3. The approximated discrete-time sys-tems TR,N and TT,N are given as follows:

TR,N := Fl (GR,N , KR) ,

TT,N := Fl (GT,N , KT ) ,

GR,N :=

[ [z−mFdN , 0

], −PdN[

CdKT JFdN , Wn

], 0

],

GT,N :=

[z−mFdN

0

],

[−PdNKRCd

Wz

]

FdN , 0

,

J := [I, 0, . . . , 0],

FdN (z) :=

ANFd AN−1

Fd BFd, AN−2

Fd BFd, . . . ,BFd

CF 0, 0, . . . , 0CF AFd CF BFd, 0, . . . , 0

......

.... . .

...

CF AN−1

Fd CF AN−2

Fd BFd,CFdAN−3

Fd BFd,. . . , 0

,

PdN (z) :=

ANPd

N∑

k=1

AN−kPd BPd

CP DP

CP APd CP BPd + DP

......

CP AN−1

Pd

N∑

k=2

CP AN−kPd BPd + DP

,

AFd := eAFh

N , BFd :=

∫ h

N

0

eAF tBF dt,

APd := eAPh

N , BPd :=

∫ h

N

0

eAP tBP dt,

F (s) =:

[AF BF

CF 0

], P (s) =:

[AP BP

CP DP

].

where Fl(G, K) denotes the linear fractionaltransformation of plant G and filter K.

GR,N

KR

edN wdN

nd

Fig. 6. Discrete-time H∞ design problem for re-ceiving filter KR

GT,N

KT

edN wdN

zd

Fig. 7. Discrete-time H∞ design problem fortransmitting filter KT

Then our design problems (2) and (3) are reducedto finite-dimensional discrete-time H∞ problems,which are shown in Figure 6 and Figure 7.

4. DESIGN EXAMPLES

4.1 Design for Wz = 0

We present a design example for

F (s) :=1

10s + 1, P (s) := 1, Wn(z) := 1,

Cd(z) := 1 + 0.65z−1 − 0.52z−2 − 0.2975z−3,

with sampling period h = 1 and time delay L =mh = 2. An approximate design is executed herefor N = 8. Here we design without considering thetransmitting signal, that is Wz(z) = 0. For com-parison, the discrete-time H∞ design(Erdogan etal., 2000) is also done.

Figure 8 shows the gain responses of the filters,and Figure 9 shows the frequency response of Tew

which is the system from the input wc to theerror ec, and Figure 10 shows that of Tzn fromthe additive noise nd to the output zc. Comparedwith the discrete-time design, the sampled-dataone shows better frequency response both in Tew

and in Tzn. Moreover, we can say that only anequalizer is not able to attenuate the corruptioncaused by the channel and the additive noise,that is we need an appropriate transmitter fortransmission.

To explain this fact, we show a simulation ofthese communication systems. The input signalyc is the rectangular wave whose amplitude is1, and the noise nd is the discrete-time sinusoidnd[k] = sin(2k). Figure 11 shows the output zc

0 0.5 1 1.5 2 2.5 3 3.5−40

−30

−20

−10

0

10

20

30Frequency Response

Frequency [rad/sec]

Gai

n [d

B]

Fig. 8. Gain response of filters:sampled-data H∞

design (transmitting filter: solid, receivingfilter: dots) and discrete-time H∞ design(dash).

10−3

10−2

10−1

100

101

−35

−30

−25

−20

−15

−10

−5

0Frequency Response

Frequency [rad/sec]

Gai

n [d

B]

Fig. 9. Frequency response of Tew: sampled-dataH∞ design (solid) and discrete-time H∞ de-sign (dash).

with the receiving filter and the transmitting filterdesigned via sampled-data method, and Figure12 shows that with the receiving filter designedin discrete-time (and without any transmittingfilter). We see that the former shows much betterreconstruction against the noise than the latter.

4.2 Design for Wz(z) 6= 0

Then we consider the design with the estimationof the transmitting signal vd, that is Wz(z) 6= 0.

We observe from Figure 8 that the transmittingfilter shows high gain around the Nyquist fre-quency (i.e. ω = π), and hence we take

Wz(z) = r ·z − 1

z + 0.5

as the weighting function of the transmitting sig-nal, whose gain characteristic is shown in Figure

10−3

10−2

10−1

100

101

−35

−30

−25

−20

−15

−10

−5Frequency Response

Frequency [rad/sec]

Gai

n [d

B]

Fig. 10. Frequency response of Tzn: sampled-data H∞ design (solid) and discrete-timeH∞ design (dash).

0 20 40 60 80 100−1.5

−1

−0.5

0

0.5

1

1.5Time Response

Fig. 11. Time response with sampled-data design.

0 20 40 60 80 100−1.5

−1

−0.5

0

0.5

1

1.5Time Response

Fig. 12. Time response with discrete-time design.

13 where the parameter r = 0.21. The otherdesign parameters are the same as the exampleabove.

Figure 14 shows the H∞ norm of Tew and Tvw

which is the system from wc to vd in Figure 2which varies with r ∈ [0, 5]. We can take account

� ��� � � ��� � � ��� � � ��� ��

��� �

��� �

��� �

���

��� �

���

��� �

��� �

��� ��� �����������������������������

��� �������������

!" #$%&'

Fig. 13. Gain characteristic of the weighting filterWz(z).

0 1 2 3 4 5−30

−25

−20

−15

−10

−5

0

5

10

r

H∞ n

orm

[dB

]

Fig. 14. Relation between r and ‖Tew‖∞ (solid),‖Tvw‖∞ (dash).

0 0.5 1 1.5 2 2.5 3 3.50

5

10

15

20

25

30

Frequency [rad/sec]

Gai

n [d

B]

Fequency Response

Fig. 15. Gain response of transmitting filters de-signed for r = 0.21 (solid) and r = 0 (dash)

of a trade-off between the error attenuation leveland the amount of the transmitting signal withFigure 14. For example, we choose r = 0.21 inorder to attenuate the error less than −26dB.

10−3

10−2

10−1

100

101

−30.5

−30

−29.5

−29

−28.5

−28

−27.5

−27

−26.5

−26

−25.5Frequency Response

Frequency [rad/sec]

Gai

n [d

B]

Fig. 16. Frequency response of Tew designed forr = 0.21 (solid) and r = 0 (dash).

10−3

10−2

10−1

100

101

−40

−30

−20

−10

0

10

20Frequency Response

Frequency [rad/sec]

Gai

n [d

B]

Fig. 17. Frequency response of Tvw designed forr = 0.21 (solid) and r = 0 (dash)

Figure 15 shows the gain response of transmittingfilters designed for r = 0 and r = 0.21. We can seethat the new filter shows better attenuation thanthe filter designed for r = 0 at high frequency.

Figure 16 shows the frequency response of the er-ror system Tew. We see that the attenuation levelof Tew designed for r = 0.21 is less than −26dB.Figure 17 shows the frequency response of Tvw.We can see that the amount of the transmittingsignal is attenuated at high frequency.

5. CONCLUDING REMARKS

We have presented a new method of designingtransmitting/receiving filter in digital communi-cation. An advantage here is that an analog otimalperformance can be obtained, and this can be ad-vantageous in audio/speech signal transmission.Another advantage is that the trade-off betweenthe attenuation of the reconstruction error andthe energy of the transmitting signal is consideredby the H∞ design with an appropriate weighting

function. By the fast sample/hold method, thedesign is reduced to a finite dimensional discretetime design, which can be easily implemented toCAD (e.g. MATLAB).

6. REFERENCES

Chen, T. and B. A. Francis (1995a). Designof multirate filter banks by H∞ optimiza-tion. IEEE Trans. on Signal Processing SP-43, 2822–2830.

Chen, T. and B. A. Francis (1995b). OptimalSampled-Data Control Systems. Springer.

Erdogan, A. T., B. Hassibi and T. Kailath (2000).On linear H∞ equalization of communicationcahnnels. IEEE Trans. on Signal ProcessingSP-48(11), 3227–3232.

Keller, J. P. and B. D. O. Anderson (1992). A newapproach to the discretization of continuous-time controllers. IEEE Trans. on AutomaticControl AC-37(2), 214–223.

Khargonekar, P. P. and Y. Yamamoto (1996).Delayed signal reconstruction using sampled-data control. Proc. of 35th Conf. on Decisionand Control pp. 1259–1263.

Nagahara, M. and Y. Yamamoto (2000). A newdesign for sample-rate converters. Proc. of39th Conf. on Decision and Control pp. 4296–4301.

Proakis, J. G. (1989). Digital Communications.McGraw Hill.

Yamamoto, Y., A. G. Madievski and B. D. O. An-derson (1999). Approximation of frequencyresponse for sampled-data control systems.Automatica 35, 729–734.

Yamamoto, Y., H. Fujioka and P. P. Khargonekar(1997). Signal reconstruction via sampled-data control with multirate filter banks.Proc. of 36th Conf. on Decision and Controlpp. 3395–3400.