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 channels
is developed based on the sampled-data H ∞ control theory. The procedure provides
transmitting/receiving filters so as to minimize the error between the original signal
and 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 presented
to 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 communications 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 pulse
code 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 considered, and hence the whole system is regarded asa discrete-time system. For example, one usuallyassumes that the original analog signal is bandlimited up to the Nyquist frequency. But in reality
no signals are entirely band-limited.
This paper proposes a new design methodology
based on the sampled-data control theory(Chenand Francis, 1995b) that takes account of intersample behaviors or frequency components beyond the Nyquist frequency in discrete-time. Re1

nagahara@acs.i.kyoto-u.ac.jp
yy@i.kyoto-u.ac.jp; author to whom all correspondence should be addressed.

2

cently, the sampled-data control theory is applied
to some digital signal processing systems(Chenand Francis, 1995a; Khargonekar and Yamamoto,
1996; Yamamoto et al., 1997; Nagahara and Yamamoto, 2000). We propose a design for the digital communication system via the sampled-datacontrol. In (Erdogan et al., 2000) a discrete-time
H ∞ design for receiving filters or equalizers isintroduced, but no design for transmitting filter
is mentioned. However, it is difficult to attenuate
both the signal reconstruction error and the additive noise by only an equalizing after the signal
is received. Therefore we show the transmitting
filter design which is executed in the same way
as the receiving filter design. Moreover, we introduce the H ∞ method which takes account ofa tradeoff between the signal reconstruction error and the energy of transmitted signal with anappropriate weighting function. Design examples
are presented to illustrate the effectiveness of theproposed method.

yd v
wyc
uc -zc-c Fc d- - Sh KT d Cd ++ KR Hh Pc
6ndw
-c Fc

Fig. 1. Digital communication system.
2. DESIGN PROBLEM FORMULATIONThe block diagram Figure 1 shows a digital communication system which is known as PAM orPCM. The incoming signal wc ∈ L2 [0, ∞) goes
through an analog low-pass filter F c and becomes
yc which is nearly (but not entirely) band limited. The filter Fc governs the frequency-domaincharacteristic of the analog signal y c 3 . The signal
yc is then sampled by the sampler Sh to becomea discrete-time signal (or PAM signal) y d withsampling period h. Then the signal is shaped orenhanced by the transmitting digital filter K T to
the signal vd to be transmitted to a communication 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 coding
error. The received signal goes through the receiving digital filter KR which tries to attenuate thecorruption and the noise, then becomes an analog
signal uc by the hold device Hh with sampling
period h and smoothed by an analog low-passfilter Pc and finally we have the output signal z c .
Our objective is to reconstruct the original analog
signal yc by the transmitting filter KT and thereceiving filter K R against the corruption causedby the channel Cd and the additive noise nd .
Therefore consider the block diagram Figure 2
which is the signal reconstruction error system forthe design. In the diagram the following points aretaken into account:• The time delay e−Ls is introduced because
we allow a certain amount of time delay forsignal reconstruction.
• The transmitted signal vd is estimated witha weighting function Wz because the energy
or the amplitude of the transmitted signal vd
is usually limited.
• The noise has a frequency characteristic W n .
Then our design problem is as follows:

kec k2L2 + kzd k2l2
22 .
wc ∈L2 ,nd ∈l2 kwc kL2 + knd kl2
supFig. 2. Signal reconstruction error system.

w-c Fc- e−Ls+ ?ec
yce−+yv
uuddce- Sh -dKT zc
CdKR Hh Pc 6+6Wnnd6Fig. 3. Error System T R 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 it
is difficult to solve the problem directly. Therefore we introduce a decomposition of the design
problem into two steps, that is the design for thereceiving filter and that for the transmitting filter.
Obviously the transmitting filter K T cannot attenuate the additive noise nd , hence the receiving
filter KR has to play that role. Moreover K R
has to reconstruct the original signal from thecorrupted signal (if K R did not have to reconstruct, the optimal filter will be clearly K R = 0)
Therefore we first design the receiving filter K R
in order to reconstruct the original signal and toattenuate the noise by the block diagram Figure
2 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 channel
as KR Cd .
The design procedure is as follows:
Step 1(Design for receiving filter) Find a receiving filter KR which minimizes
kTR k2∞ :=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 digital
filters KT (z) and KR (z) which minimizes
J 2 :=- e−Ls+ ?ecyc
e−+eydvduu- Sh zc
KT Cd - KR -dHh -c Pc 6+?
6WzWnndzd6?

(1)

3 In the conventional design F is considered as an ideal
cfilter which has a cut-off frequency up to the Nyquist
frequency.

kec k2L222 ,wc ∈L2 ,nd ∈l2 kwc kL2 + knd kl2
sup(2)

in Figure 3 with fixed KT (the initial filter isKT = 1).
Step 2(Design for transmitting filter) Find atransmitting filter KT which minimizes
kec k2L2 + kzdk2l2,kwc k2L2
wc ∈L2kTT k2∞ := sup(3)in Figure 4 with KR which is obtained in theprevious step.

ke
edN k2l2q,2w
edN ,nd ∈l2 kw
edN k2l2 + Nknk2dhlqNke
edN k2l2 + h kzd k2l2kTT,N k∞ := sup,kw
edN kl2w
edN ∈l2- e−Ls

kTR,N k2∞ :=+ ?ecw
e-c Fc yc−ydvduu- Sh zc
KT Cd -KR -dHh -cPc 6?
Wzzd?
Fig. 4. Error System T T for transmitting filter
designedNec
 Sh/N wc
wdNT  Hh/N Fig. 5. fast sample/hold discretization
3.2 Fast Sample/Hold Approximation
The design problems (2) and (3) involve a continuous time delay component e−Ls , and hence
they are infinite-dimensional sampled-data problems. 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-time
problems assuming that the delay time L to be
mh where m is a positive integer:

where the systems TR,N and TT,N are approximated to finite-dimensional discrete-time systems, then we can show kTR,N k∞ → kTR k∞
kTT,N k∞ → kTT k∞ 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 discretetime problems, they can be solved by a control
design toolbox such as those given by MATLAB.
The resulting discrete-time approximant is givenby the following:
Theorem 3. The approximated discrete-time systems TR,N and TT,N are given as follows:

Theorem 2. Assume that L = mh, m ∈ N. Then,(1) for the error system T R in Step 1, there exist finite-dimensional discrete-time systems{TR,N : N = 1, 2, . . . } such that
lim kTR,N k∞ = kTR k∞ .

N →∞(2) for the error system T T in Step 2, there exist finite-dimensional discrete-time systems{TT,N : N = 1, 2, . . . } such that
lim kTT,N k∞ = kTT k∞ .

N →∞PROOF. By the fast sample/hold method, weapproximate continuous-time inputs and outputs
to 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 discretized input/output signal edN and wdN , we canget the lifted signals

supTR,N := Fl (GR,N , KR ) ,TT,N := Fl (GT,N , KT ) ,  −mz FdN , 0 ,  −PdN
GR,N :=,
Cd KT JFdN , Wn , 0  −m z FdN−PdN KR Cd,
,0Wz
GT,N := FdN ,0J := [I, 0, . . . , 0],
FdN (z) :=−2ANF d BF d , . . . ,BF d0,... , 0 
0,... , 0 ,..
. . .. . . .
N −1
N −2
N −3
CF AF d CF AF d BF d ,CF d AF d BF d ,. . . , 0
NXN −kN
AP d BP d AP d


k=1


 CP

DP

 CP AP dCP BP d + DP
PdN (z) := ,

..
..
.
.


NX

N −k
N −1CAB+DCP APPPdPPdd

ANFd CF CF AF d

..
.

N −1
AFd BF d ,0,
CF BF d ,..
.

k=2he
edN := LN (edN ),w
edN := LN (wdN ).

Then we can approximate the continuous signal
asrrhhkeedN kl2 , kwc kL2 ≈kw
edN kl2 .
kec kL2 ≈N
N
Moreover define

AF d := eAF N ,BF d :=Z NheAF t BF dt,0h
AP N

Z Nh

AP d := e, BP d :=eAP t BP dt,0



AF BF
AP BP,
P (s) =:
F (s) =:.
CF 0CP DP
where Fl (G, K) denotes the linear fractionaltransformation of plant G and filter K.

eedNGR,Nnd

Frequency Responsew
edN

3020KR
Fig. 6. Discrete-time H ∞ design problem for receiving filter KReedN
zdGT,N

Gain [dB]

10Fig. 7. Discrete-time H ∞ design problem fortransmitting filter KT−10−20w
edN

KT

0−30

−40
00.5

1

1.5
2
Frequency [rad/sec]

2.5

3

3.5

Fig. 8. Gain response of filters:sampled-data H ∞design (transmitting filter: solid, receiving
filter: dots) and discrete-time H ∞ design(dash).
Frequency Response

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

0−5

4. DESIGN EXAMPLES
4.1 Design for Wz = 0We present a design example for
1, P (s) := 1, Wn (z) := 1,10s + 1
Cd (z) := 1 + 0.65z −1 − 0.52z −2 − 0.2975z −3,F (s) :=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 comparison, 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 T ew
which is the system from the input wc to theerror e c , and Figure 10 shows that of T zn fromthe additive noise nd to the output zc . Comparedwith the discrete-time design, the sampled-data
one shows better frequency response both in T ew
and in Tzn . Moreover, we can say that only anequalizer is not able to attenuate the corruption
caused 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 signal
yc is the rectangular wave whose amplitude is1, and the noise nd is the discrete-time sinusoidnd [k] = sin(2k). Figure 11 shows the output z cGain [dB]−10−15−20−25−30−35 −310−210−110Frequency [rad/sec]

010

110Fig. 9. Frequency response of T ew : sampled-data
H ∞ design (solid) and discrete-time H ∞ design (dash).
with the receiving filter and the transmitting filterdesigned via sampled-data method, and Figure
12 shows that with the receiving filter designed
in discrete-time (and without any transmitting
filter). 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 transmitting
filter shows high gain around the Nyquist frequency (i.e. ω = π), and hence we takeWz (z) = r ·z−1z + 0.5

as the weighting function of the transmitting signal, whose gain characteristic is shown in Figure

Frequency Response 

−5
−10

Gain [dB]−15&'  $%−20" # !
 −25 −30 −35 −3
10−2

10

−1

10
Frequency [rad/sec]0 11010Fig. 10. Frequency response of T zn : sampleddata H ∞ design (solid) and discrete-time
H ∞ design (dash).

    Fig. 13. Gain characteristic of the weighting filterWz (z).
10Time Response
1.550

H∞ norm [dB]10.5

0−5−10−15−20−0.5−25−1−300−1.5012345r
20406080100Fig. 11. Time response with sampled-data design.

Fig. 14. Relation between r and kT ew k∞ (solid),kTvw k∞ (dash).

Time Response

Fequency Response

1.530
1250.5Gain [dB]20015−0.5

10−15−1.50204060

80

100000.5

11.5
2Frequency [rad/sec]

2.5

33.5

Fig. 12. Time response with discrete-time design.
13 where the parameter r = 0.21. The otherdesign parameters are the same as the example
above.
Figure 14 shows the H ∞ norm of Tew and Tvw
which is the system from wc to vd in Figure 2
which varies with r ∈ [0, 5]. We can take accountFig. 15. Gain response of transmitting filters designed for r = 0.21 (solid) and r = 0 (dash)of a trade-off between the error attenuation level
and the amount of the transmitting signal with
Figure 14. For example, we choose r = 0.21 inorder to attenuate the error less than −26dB.

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

Frequency Response−25.5−26−26.5

Gain [dB]−27−27.5

6. REFERENCES−28−28.5−29−29.5−30−30.5 −310−210−110Frequency [rad/sec]

010110Fig. 16. Frequency response of T ew designed forr = 0.21 (solid) and r = 0 (dash).
Frequency Response
2010Gain [dB]0−10−20−30−40 −310−210−110Frequency [rad/sec]

010110Fig. 17. Frequency response of T vw designed for
r = 0.21 (solid) and r = 0 (dash)
Figure 15 shows the gain response of transmitting
filters designed for r = 0 and r = 0.21. We can seethat the new filter shows better attenuation than
the filter designed for r = 0 at high frequency.
Figure 16 shows the frequency response of the error 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 T vw .
We can see that the amount of the transmitting
signal is attenuated at high frequency.

5. CONCLUDING REMARKS
We have presented a new method of designingtransmitting/receiving filter in digital communication. An advantage here is that an analog otimal
performance can be obtained, and this can be advantageous in audio/speech signal transmission.
Another advantage is that the trade-off between
the attenuation of the reconstruction error andthe energy of the transmitting signal is consideredby the H ∞ design with an appropriate weighting

Chen, T. and B. A. Francis (1995a). Designof multirate filter banks by H∞ optimization. IEEE Trans. on Signal Processing SP43, 2822–2830.
Chen, T. and B. A. Francis (1995b). Optimal
Sampled-Data Control Systems. Springer.
Erdogan, A. T., B. Hassibi and T. Kailath (2000).
On linear H ∞ equalization of communication
cahnnels. IEEE Trans. on Signal Processing
SP-48(11), 3227–3232.
Keller, J. P. and B. D. O. Anderson (1992). A new
approach to the discretization of continuoustime controllers. IEEE Trans. on AutomaticControl AC-37(2), 214–223.
Khargonekar, P. P. and Y. Yamamoto (1996).
Delayed signal reconstruction using sampleddata control. Proc. of 35th Conf. on Decisionand Control pp. 1259–1263.
Nagahara, M. and Y. Yamamoto (2000). A new
design 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. Anderson (1999). Approximation of frequency
response for sampled-data control systems.
Automatica 35, 729–734.
Yamamoto, Y., H. Fujioka and P. P. Khargonekar(1997). Signal reconstruction via sampleddata control with multirate filter banks.
Proc. of 36th Conf. on Decision and Control
pp. 3395–3400.