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Sampled-data H ∞ Design for Digital Communication Systems,
Masaaki Nagahara, Yutaka Yamamoto

Graduate School of Informatics, Kyoto University
Abstract
A design procedure for digital communication systems is developed based on thesampled-data H ∞ control theory. The procedure provides transmitting/receiving filters, which
optimize an analog otimal performance. Furthermore, the trade-off between the attenuation of thereconstruction error and the energy of the transmitting signal is considered by the H ∞ design withan appropriate weighting function. Numerical examples are presented to illustrate the effectivenessof the proposed method.

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uz+-c Fc Sh KT -d Cd - d-KR - Hh -c Pc -c+6nd

1:1wc ∈ L2 [0, ∞)Fc (s)yc.

Fc (s)(PAM)(PCM)[5]

hSh

yc

ydyd

KT (z)Cd (z)

[4]

[5]

LTI

Cd (z)Cd (z)nd

1KR (z)

Hh∞0Pc (s)zc

KT (z)(FSFH)[3, 7]KR (z)

zc

[4]

zc

ycyc

§©¨tu~ xv3x
1 PCM

2ydnd

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DL+ ? ecwc
eFc yc−vuu+dddce- Sh yKTCdKR Hh Pc 6zc+?
6WzWnzdnd6?
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³ÝùlwHgxTy¤ERH¥NqC±#²}ç
(D y )(t) := y (t − L)âèk7¾#¿}RH\ ZTÆ ! Èzlx :Æ[ÒWàgZ)ì)p)B V F (s)^  Æ[ÒWì!
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ÇLpv J!W³ÝÕ ]ØF× Ny
TRFSFHS h TR H h
N

TR,N :=TT,N := S h TT H h (N = 1, 2, . . . )NNkTR,N k∞ → kTR k∞ ,G&H ­1nIZ7!n7ykTT,N k∞ → kTT k∞

4cnNTT(1)(N → ∞).

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Wn (z) := 1.
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8 ZNáHwNv ­(-ÝÆòÇw%x-¤)¥Ý³-µ)Ò}xTy
$Y)Zz1^ E¶WCeEf[§-¨E©!ª)ZE!wCnoxÝ\§ [<]^NB^ X~§ _@-1­ 7)nNyCff)·¥#Çzv`¥)ÇLv­Cùlw%xTyóf#f!· ­ 8­)Wnlx¶-fÈô!bdgA ­ }ÐH\TyëH@)ìzdEí!±-² ¥ 8 ®-¯ #Ý±#² Z 7%nÝy ëW@¤#¤C|^ R+. Pík ¥)Ç^ ¥ ­/ }ìTdN|)í#qN±-vW²w#Tx7%R)LZN)ów!fgÇ^ô-®bt¯dAÝC±³Wã-²W¥Ý|)RzEk f-Xf#~o·}R{É-A­ Ê]10WnNy­Lp}qÈrÿ&ó UEvE^{f#·W³ â mg|r  H|)qEvWwÝV%w#¤-¥L³)µÒoxTy2$354(687:9;<"$#±#Ë ²WçCèá[\10k  -# ­zÜh7^ Ã1&}RLë)@-ìdÝí}R{±®²)¯ ç{è¥TV%xT³C^
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22wc ∈L2 ,nd ∈l2 kwc kL2 + knd kl2
supKT (z)h=1
N
4

FSFHKR (z)L=2Cd (z)[2]
kTew k∞ = −26[dB]Tew

r = 0.21

2wc

ec

3.1

3[4]

[4](:

Step 12r = 0)

Wz

KTkTR k2∞ :=kec k2L222
wc ∈L2 ,nd ∈l2 kwc kL2 + knd kl2
supKR

Step 22Tew

4nd (k) = sin(3k)wc (t) = sin(0.1t)
zc (t)

KR

kec k2L2 + kzdk2l2
kwc k2L2
wc ∈L2kTT k2∞ := supKT

KT

3.2

4

1

Step 15()6(Step 2FSFH

Step 1[3, 7]

Step 2

DLFSFHH∞)

% &'

` bcd e fhgji gklmnkg`[b c
#$z{vw x ys tu!"[a [b c
a `
 

 
   

a `b c [\[(solid),_[`[[8 5: #+V$W : ëH@%ìzdí!±#²:r = 0(thick dash), r =0.21(thin dash))dot)(` bc(dash-d e fhgji gklmnkg(dot)).

`; < =>?=@AB%< =CDE@C=[b c2 43z{vw x ys tu

2 4:
2 49[a [b c2 48QRa `OP 2 47MNKL2 46a `b c[2 45\[][^[_[`[[doe fhghp kgqr

8 6: #V5W : ®-¯ #7±#²23 )23 (^[doe fhghp kgqr

8 3: #óEfgô-btdlANR7ü?!@ V$W : ë!@!ì{dNí#±-² (ófgôEb d[A ®!Efl¯ ôb d[A^ %E±!² ó#fsô%bdÈA^ófÈô%bdlA][()+* ,()+* -()+* .
; < =>?=@ABF < GHI C=AJ()0/()1.

R#ºo¼½ VNW®-¯ ëÈ@gìd}íl±²^ #Ý±#²²}kÝ^L¡lZ¶~ !6 I7~E%^z-HV}ÝRE}f#·}RF-§#¨©%ª!Z 5 ¥úä°Hgx{y84: Tew
0.21(solid), r = 0(dot)):(r =(dash).

TSUWVYXYZ

controllers, IEEE Trans. on Automatic Control,
AC-37, pp. 214–223 (1992) ëH@%ìzdí)Â-îZ%\-xL`lb c ATd-ef§)¨©Hª!R+ )±%²§)@ c ª[4] |~} , :Theory, 

23  Dynamical System, pp. 91–94 (2000)[5] J. G. Proakis: Digital Communications, McGrawHill (1989)[1] T. Chen and B. A. Francis: Optimal SampledData Control Systems, Springer (1995)[2] A. T. Erdogan, B. Hassibi and T. Kailath: On
linear H ∞ equalization of communication cahnnels, IEEE Trans. on Signal Processing, SP-48,
pp. 3227–3232 (2000)[3] J. P. Keller and B. D. O. Anderson: A new approach to the discretization of continuous-time[6] Y. Yamamoto, P. P. Khargonekar: Frequency response of sampled-data systems, IEEE Trans. on
Automatic Control, AC-41, pp. 166–176 (1996)[7] Y. Yamamoto, A. G. Madievski and B. D. O. Anderson: Approximation of frequency response forsampled-data control systems, Automatica, vol.
35, pp. 729-734 (1999) :  1  mLÙpH R è[­7)n7y#¤¤z|E^E§¨N©#ª G RNM{ôL©HbA@!#Z%\-xº-¼)½(V5W ­ | }µn7y¥  R RZ%wWÇLv&J!W³ÝÕ ] ×FNy[6]G(ω)

1ω ∈ [−π/h, π/h]2{w ∈ L[0,h] : kwk2 ≤ 1}w ∈ U1 :=C$EsÇLv^ N ≥ N V
1³TÕÖ]¦×Cyz¤¤T|\[]  RkH h S h w0 − w0 k2 ≤ εk(H h S h − I)DNN¥  RDmh (ω) = e−jωmh[−π/h, π/h]Z%wWÇLv

(5)¥Táòrú¥¶^

(5)
(3)–

³Es#^-¾-Ú\[] é#ê ¥+*gZWÇLvkH h S h T H h S h w0 − T w0 k2 ≤ (1 + M )ε(ω)Fc (ω)wk2 → 0 (N → ∞).

|}ÞxEÒÈL^  RZ%wWÇLv

NN2 := max{N0 , N1 }N ≥ N2NmhN(2)ω ∈w ∈ U1NNN[7]³&7zgxTyl¤Tg\[]kTN (ω)k ≥ρ − (M + 2)ε1+ε³NNÕ ] ×A[¤%¥N³)µ%ÒxLyom#Né%ê ¥ *Z}Ç7v-^³Õ>]a×N#y ³  ÆÒ&:HàZHìW BW|}ÞlxÒò R OQ !
k Ú}RU:* C½l¥CVÈ] RW³
Õ ] ×F7yãúÕ¹]_×T^ ER !P%|Ýº¼½ V$WW³ %nox¤-¥L³µ>$? %  G H k7^ Ò[ mÝ|%R§-¨E©!ªEá Ò}xy%!7Z Qok U*H|!ÞWxE¤#¥-ã 7lTlxyÝ¾EÚ\\0 Ò mE|%R-§#¨©!ª%Z%wNv)k lÇ{^ ] Æ!ï ³ÝÕ ZE]&á F× U[HAx ¤#5l¥ÝÃ ³-µ) Òo®)xT¯ y !pgR R)N¬ S%#V S!
Ò mE|%R-§#¨©%ª)­zäg°%xTy)n%V-µH
­Ý¾#¿Z7%nÝyl¤#¤Ý| ^ á[\0 RE¥7Tá¹UWr¦VHy 7Ó m^­Wì S@ G %¥ }µ¥;n7ô%y bdgA R!ô%bÝB )YXpF ? Z-l­¥7áWq­7}n7yl¤)¤7||!ÞHxTy
GH RLõ-ö}kLé-ê ¥  *H|!ÞWxy%mCÙ ­ ÆÇa"^ ! ZÝw)ÇvLºE¼E½ V(W%³ #-noxN¤¥CA­ 7nNy`aoR-Ð ­ Ç­ %¥ $ r¦y¥Cá r yl¤ERH¥Nq & R Z%w
ÇTvN^Þ%x<³ CEòÇzv
³Õ ]Ø× Cy}¤¤C&| UVj(­ '%µÙ kL Ý½¥aÄ#Æ
|%qHxTy è á[\ 0Eé#ê R\o]Ø^}Þ
x @PB#½ +³ CEsÇLv^V lim inf kTN (ω)k ≥ kT (ω)kk(H h S h − I)Dmh (ω)Fc (ω)wk2
NN →∞

N

[7]= k(H h S h − I)Fc (ω)wk2
NNFc[0, h](Fc w)(θ)

lim inf kTN (ω)k ≤ kT (ω)k(2)

N →∞

ωwc

1ndec

zdec

wc

(1)

1

T := (Dmh− Pc Hh KSh )Fc ,kTN k∞ → kT k∞

P (s)
F (s)
(AF , BF , CF , 0) (AP , BP , CP , DP )GKFl (G, K)

TN := S h T H h
NN(N → ∞)TR,N := Fl (GR,N , KR ) , TT ,N := Fl (GT ,N , KT ) , −mz FdN , 0 , −PdN
GR,N :=,Cd KT JFdN , Wn ,0   −m−PdN KR Cdz FdN,,Wz
0GT ,N := FdN ,0K := KR Cd KT[7]ωωωsupw∈U1 kT wk2

G(ω)

Gw0 ∈ U1

ρ := kT k =ε>0ρ − ε < kT w0 k2 ≤ ρw0

1[7]
Lemma 4.2
N ≥ N0N0kH h S h T w − T wk2 ≤ k(H h S h − I)Dmh Fc wk2
NNNN³  RZWwWÇLvEÕ ] ×F yl%NZ k
Ú[RL-J@ ]C!ù$H|%ÞHxNR|^}ÞHx³
C$EsÇLv+ k(H h S h − I)Pc Hh KSh Fc wk2 ≤ ε (3)NN2w ∈ U1

L2[0,h]TM ≥0FdN (z) :=−1−2ANAN. . . , BF dF d BF d ,F d BF d ,0,0,... , 0 CF BF d ,0,... , 0 ,..
..
.. ..
.
.
.
. N −2N −3
CF AF d BF d , CF d AF d BF d , . . . , 0NPN −kANAPBPdPddk=1CDPP CP A P dCB+DPPPdPdN (z) := ,
.
.
..
..

NPN −kN −1
CP AP d BP d + DP
CP A P dANFd
CF

CF
..
.
−1
CF A NFd

k=2³, ) Rv, w ∈ U1

[0, h]

2

NZ !w%ÇzvÝÕ¹]Ø×F7y!m k+*ÚoRzgÝ½W|!Þ%xCÒ[¶^!ÞWx(@ B½ ³kT (v − w)k2 ≤ M kv − wk2(4)w0N1¤¥ t=<\u w0> £ y .¨ -0z@/2?\1.z 3 ¡5476 z(@{0AC8#BE9 DG"FI:0H; vIJ#9L K740M 6 ¨(D12 = 0)

Pc (s)[7]

D12 6= 0

AF d := eh
AF Nh,AP d := eAP N ,BF d :=
BP d :=

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

ZZhNeAF t BF dt,0h
N0eAP t BP dt,