Optimal Discretization of Analog Filters via
Sampled-Data H ∞ Control Theory
Masaaki Nagahara1 and Yutaka Yamamoto1
Abstract— In this article, we propose optimal discretizationof analog filters (or controllers) based on the theory of sampleddata H ∞ control. We formulate the discretization problem as
minimization of the H ∞ norm of the error system betweena (delayed) target analog filter and a digital system including
an ideal sampler, a zero-order hold, and a digital filter. The
problem is reduced to discrete-time H ∞ optimization via thefast sample/hold approximation method. We also extend the
proposed method to multirate systems. Feedback controller
discretization by the proposed method is discussed with respect
to stability. Numerical examples show the effectiveness of the
proposed method.

I. INTRODUCTION
Discretization of analog systems is a fundamental transformation in control and signal processing. Often an analog(or continuous-time) controller is designed based on standard
methods such as PID control [1], and then it is implemented
in digital devices after discretization. In signal processing,an analog filter is discretized to implement it on DSP(digital signal processor), an example of which is active noise
control [3] where the analog model of the secondary path isdiscretized and used for an adaptive filter algorithm realized
on DSP.
For discretization of analog filters, step-invariant transformation [2, Chap. 3] is conventionally and widely used. The
term “step-invariant” comes from the fact that the discretetime step response of the discretized filter is exactly thesame as the sampled step response of the original filter.
By this fact, step-invariant transformation is effective forsufficiently low-frequency signals. Another well-known discretization method is bilinear transformation, also known
as Tustin’s method, which is based on the trapezoidal rule
for approximating the definite integral. The following are
advantages of bilinear transformation:• stability and minimum-phase property are preserved,• there is no error at DC (direct current) between thefrequency responses of the original filter and the discretized one.
If a zero error is preferred at another frequency, one can use
a prewarping technique for bilinear transformation; see [2,Sec. 3.5] for details.
Although these methods are widely used, there may leadconsiderable discretization errors for unexpected signals suchas signals that contains high-frequency components. To solve
this, we apply the sampled-data H ∞ -optimal filter design
1 M. Nagahara and Y. Yamamoto are with Graduate School of Informatics,
Kyoto University, Sakyo-ku Yoshida-Honmachi, Kyoto 606-8501, JAPANnagahara@ieee.org, yy@i.kyoto-u.ac.jp[5], [7], [12], [13] to the discretization problem. The proposed design procedure is summarized as follows:1) give a signal generator model F (s) as an analog filter
for L2 input signals,
2) set a digital system K that contains a sampler, a hold,and a digital filter (see Fig. 1),
3) construct an error system E between K and a (delayed)target analog filter G(s) with signal model F (s) (seeFig. 2),4) find a digital filter in K that minimizes the L2 -induced
norm (or H ∞ norm) of the error system E.
Since the error system is composed of both analog and digitalsystems, the optimization is an infinite dimensional one. To
reduce this to a finite dimensional optimization, we introduce
the fast sample/hold approximation method [4], [11]. By thismethod, the optimal digital filter can be effectively obtainedby numerical computations.
The remainder of this article is organized as follows. In
Section II, we formulate our discretization problem as asampled-data H ∞ optimization. In Section III, we review
two conventional methods: step-invariant transformation andbilinear transformation. In Section IV, we give a design
formula to compute H ∞ -optimal filters. In Section V, we
extend the design method to multirate systems. In Section VI,
we discuss controller discretization. Section VII presents design examples to illustrate the effectiveness of the proposed
method. In Section VIII, we offer concluding remarks.
Notation
Throughout this article, we use the following notation.
We denote by L2 [0, ∞) the Lebesgue space consisting of
all square integrable real functions on [0, ∞). L2 [0, ∞) issometimes abbreviated to L2 . The L2 norm is denoted by∥ · ∥2 . The symbol t denotes the argument of time, s theargument of Laplace transform, and z the argument of Z
transform. These symbols are used to indicate whether asignal or a system is of continuous-time or discrete-time;for example, y(t) is a continuous-time signal, F (s) is acontinuous-time system, K(z) is a discrete-time system. The
operator e−ls with nonnegative integer l denotes continuoustime delay (or shift) operator: (e−ls y)(t) = y(t − l). Sh
and Hh denote the ideal sampler and the zero-order holdrespectively with sampling period h > 0. A transfer function
with state-space matrices A, B, C, D is denoted by{][C(sI − A)−1 B + D, (continuous-time)
A B
:=
C D
C(zI − A)−1 B + D, (discrete-time)√
We denote the imaginary number −1 by j.

u

v

Sh

K(z)

ψ

Hh

ŷ
w

Fig. 1. Digital system K consisting of ideal sampler Sh , digital filter
K(z), and zero-order hold Hh with sampling period h.

F (s)K = Hh KSh ,shown in Fig. 1 that mimics the input/output behavior of theanalog filter G(s). The digital system in Fig. 1 includes anideal sampler Sh and a zero-order hold Hh synchronizedwith a fixed sampling period h > 0. The ideal sampler
Sh converts a continuous-time signal u(t) to a discrete-time
signal v[n] asv[n] = (Sh u)[n] = u(nh),n = 0, 1, 2, . . . .

The zero-order hold Hh produces a continuous-time signalŷ(t) from a discrete-time signal ψ[n] asŷ(t) =∞∑

ψ[n]ϕ(t − nh),t ∈ [0, ∞),

+ e
−

u
Sh

II. PROBLEM FORMULATIONIn this section, we formulate the problem of optimal
discretization.
Assume that a transfer function G(s) of an analog filter isgiven. We suppose that G(s) is a stable, real-rational, propertransfer function. Let g(t) denote the impulse response (orthe inverse Laplace transform) of G(s). Our objective is tofind a digital filter K(z) in a digital system

y

e−mhs

G(s)

Fig. 2.

Hh

K(z)

ŷ

Error system E.

Problem 1: Given target filter G(s), analog characteristic
F (s), sampling period h, and delay step m, find a digital
filter K(z) that minimizes( −mhs)eJ=
supG − Hh KSh F w 2w∈L2 , ∥w∥2 =1(1)()= e−mhs G − Hh KSh F ∞ .
The corresponding block diagram of the erros system()
E := e−mhs G − Hh KSh F(2)is shown in Fig. 2.
III. STEP-INVARIANT AND BILINEAR METHODS
Before solving Problem 1, we here briefly review two
conventional discretization methods, namely step-invariant
transformation and bilinear transformation [2].
A. Step-invariant transformation
The step-invariant transformation Kd of a continuous-timesystem G is defined by
Kd := Sh GHh .

n=0

where ϕ(t) is a box function defined by{
1, if t ∈ [0, h),ϕ(t) =
0, otherwise.

When the continuous-time input u applied to G is the step
function:{
1, if t ≥ 0,u(t) = 1(t) :=0, if t < 0,

The filter K(z) is designed to produce a continuous-time
signal ŷ after the zero-order hold Hh that approximates thedelayed output∫ t−ly(t) = (g ∗ u)(t − l) =g(τ )u(t − l − τ )dτthen the output y and the approximation ŷ processed by thedigital system shown in Fig. 1 with K = Kd are equal on
the sampling instants, t = 0, h, 2h, . . . , that is,0of an input u. A positive delay time l may improve the approximation performance when l is large enough as discussed
in e.g., [7], [12]. We here assume l is an integer multiple ofh, that is, l = mh, where m is a nonnegative integer. Then,
to avoid a trivial solution (i.e., K(z) = 0), we should assume
some a priori information for the inputs. As used in [7], [12],we adopt the following signal subspace of L2 [0, ∞) to which
the inputs belong:{}
F L2 := F w : w ∈ L2 [0, ∞) ,where F is a linear system with a stable, real-rational, strictly
proper transfer function F (s). This transfer function, F (s),defines the analog characteristic of the input signals in thefrequency domain.
In summary, our discretization problem is formulated asfollows:

y(nh) = ŷ(nh),n = 0, 1, 2, . . . .

In other words, we have
Kd Sh 1 = Sh G1.
The term “step-invariant” is derived from this property.
If G(s) has a state-space representation[]
A B
G(s) == C(sI − A)−1 B + D,
(3)
C D
then the step-invariant transformation Kd (z) has the following state-space representation [2, Theorem 3.1.1]:
[
]
Ad Bd
Kd (z) == C(zI − Ad )−1 Bd + D, (4)
C D
∫ h

where
Ad := eAh,

eAt Bdt.

Bd =0

We denote the above transformation by c2d(G, h).

IV. H ∞ -OPTIMAL DISCRETIZATION

B. Bilinear transformation
Another well-known discretization method is bilinear
transformation, also known as Tustin’s method. This is basedon the trapezoidal rule for approximating the definite integral.
The bilinear transformation Kbt (z) of a continuous-timetransfer function G(s) is given by()
2z−1
Kbt (z) = G.
hz+1
The mapping from s to z is given bys 7→ z =1 + (h/2)s,
1 − (h/2)sand this maps the open left half plane into the open unit
circle, and hence stability and minimum-phase property arepreserved under bilinear transformation.
A state-space representation of bilinear transformation
Kbt can be obtained as follows. Assume that G has a statespace representation given in (3), then Kbt has the following
state-space representation [2, Sec. 3.4]:[]
Abt Bbt
Kbt (z) == Cbt (zI − Abt )−1 Bbt + Dbt ,Cbt Dbt
where
(
)−1 (
)
h
h
I− A
I+ A ,
2
2
(
)−1
h
h
Bbt :=
I− A
B,
22
Cbt := C(I + Abt ),Abt :=(5)

Dbt := D + CBbt .

In this section, we give a design formula to numerically compute the H ∞ -optimal filter of Problem 1 via fastsample/hold approximation [4], [2], [11], [9]. This method
approximates a continuous-time signal of interest by a piecewise constant signal, which is obtained by a fast sampler
followed by a fast hold with period h/N for some positive
integer N . The convergence of the approximation is shownin [11], [9].
Before proceeding, we define discrete-time lifting LN and
its inverse L−1
N for a discrete-time signal as[]T
LN := (↓ N ) 1 z · · · z N −1
,
(7)
]
[
−1
−N +1
1
z
·
·
·
z
(↑
N
),
L−1
:=
N
where ↓ N and ↑ N are respectively a downsampler and an
upsampler [8] defined as{}∞}{↑ N : x[k] k=0 7→ x[0], 0, . . . , 0, x[1], 0, . . . ,| {z }
N −1

{
}∞
{
}
↓ N : x[k] k=0 7→ x[0], x[N ], x[2N ], . . . .
The discrete-time lifting for a discrete-time system is thendefined as
([
]
)
A B
lift
,N
C D
[
]
A B
:= LN
L−1
N
C D


AN −1 BAN −2 B . . . B
AN

C
D
0
... 0 



.. 
..
.
.
=
CA
CB
D
.




..
..
..
.
.

. 0 
.
.
.
CAN −1 CAN −2 B CAN −3 B . . . D

The frequency response of the bilinear transformation Kbt
at frequency ω (rad/sec) is given by
( jωh
)
(
)
(
)
2e
−1
h
ωh
Kbt ejωh = G
=
G
j
·
tan
.
h ejωh + 1
2
2

By using the discrete-time lifting, we obtain the fastsample/hold approximation EN of the sampled-data error
system E given in (2) as()
EN (z) = z −mN GN (z) − HN K(z)SN FN (z), (8)It follows that at ω = 0 (rad/sec), the frequency responses
Kbt (ejωh ) and G(jω) coincide, and hence there is no error
at DC (direct current). If a zero error is preferred insteadat another frequency, say ω = ω0 (rad/sec), frequency
prewarping may be used for this purpose. The bilinear
transformation with frequency prewarping is given by
)
(
z−1
,
Kbt,ω0 (z) = G c(ω0 ) ·
z+1
(6)
(
)−1
ω0 h
c(ω0 ) := ω0 tan
.
2

where

It is easily proved that Kbt,ω0 (ejω0 h ) = G(jω0 ), that is, there
is no error at ω = ω0 (rad/sec). A state-space representationof Kbt,ω0 can be obtained by using the same formula (5)with 1/c(ω0 ) instead of h/2.

()
GN (z) = lift c2d(G(s), h/N , N ),()
FN (z) = lift c2d(F (s), h/N ), N ,HN = [1, . . . , 1]⊤ ,| {z }
N

SN = [1, 0, . . . , 0].
| {z }(9)N −1

The optimal discretization problem formulated in Problem
1 is then approximated by a standard discrete-time H ∞ optimization problem: we find the optimal K(z) that minimizes
the discrete-time H ∞ norm of EN (z), that is,()
min ∥EN ∥∞ = min z −mN GN − HN KSN ∞ .
K

KThe minimizer can be effectively computed by a standard
numerical computation software such as MATLAB.
Moreover, a convergence result is obtained as follows.

u

↑L

Sh

K(z)

ŷ

Hh/L

F (s)

Fig. 3. Multirate system Kmr consisting of ideal sampler Sh , upsampler↑ L, digital filter K(z), and fast hold Hh/L .

G(s)
w

F (s)

y

e−mhs

Fig. 5.

+e−uShFig. 4.

↑L

K(z)

G(s)

Hh/L

ŷ

Multirate error system Emr .

Feedback control system.

We use the method of fast sample/hold approximation tocompute the optimal filter K(z). Assume that N = Ll forsome positive integer p, then the approximated discrete-timesystem for Emr is given by()
e N K(z)Se
Emr,N (z) = GN (z)z −mN − H
N FN (z), (11)where GN (z), FN (z) and SN are given in (9), andTheorem 1: For each fixed K(z) and for each ω ∈[0, 2π/h), the frequency response satisfiesLe jωh )∥, as N → ∞,∥EN (ejωh )∥ → ∥E(ewhere Ee is the lifted system of E. The convergence is uniformwith respect to ω ∈ [0, 2π/h) and with K in a compact setof stable filters.
Proof. This is a direct consequence of results in [11], [9]. □The theorem guarantees that for sufficiently large N theerror becomes small enough with a filter K(z) obtained by
the fast sample/hold approximation.
Remark 1: If an FIR (finite impulse response) filter ispreferred for filter K(z), it can be obtained via LMI (linear
matrix inequality) optimization. See [10], [6] for details.
V. EXTENSION TO MULTIRATE SYSTEMSIn this section, we extend the result in the previous section
to multirate systems. If one uses a fast hold device withperiod h/L (L is an integer greater than or equal to 2) insteadof Hh , the performance may be further improved. Thisobservation suggests us to use a multirate signal processing
systemKmr := Hh/L K(z)(↑ L)Sh ,shown in Fig. 3. The objective here is to find a digital
filter K(z) that minimizes the H ∞ norm of the multirate
error system Emr shown in Fig. 4 between the delayedtarget system G(s)e−mhs and the multirate system Kmr withanalog characteristic F (s). More precisely, we solve thefollowing problem.
Problem 2: Given target filter G(s), analog characteristic
F (s), sampling period h, delay step m, and upsampling ratioL, find a digital filter K(z) that minimizes( −mhs
)
Jmr =
sup
e
G − Hh/L K(↑ L)Sh F w 2w∈L2 , ∥w∥2 =1=( −mhs)e
G − Hh/L K(↑ L)Sh F ∞ .

e N = blkdiag{1p , 1p , . . . , 1p }, 1p = [1, . . . , 1]⊤ ,H| {z }
{z
}
|

(10)

See the corresponding block diagram of the error system Emr
shown in Fig. 4.

(
)
e
K(z)
= lift K(z), L [1, 0, . . . , 0]⊤ .
| {z }p

L−1

The optimal discretization problem with a multirate system
formulated in Problem 2 is then approximated by a standardediscrete-time H ∞ optimization. The minimizer K(z)can benumerically computed by e.g. MATLAB. Then once the filtereK(z)is obtained, the filter K(z) in Fig. 3 is given by thefollowing formula:[]e L ).
K(z) = 1, z −1 , . . . , z −L+1 K(zWe have the following convergence theorem:e
Theorem 2: For each fixed K(z)and for each ω ∈[0, 2π/h), the frequency response satisfies∥Emr,N (ejωh )∥ → ∥Eemr (ejωh )∥, as N → ∞,where Eemr is the lifted system of Emr . The convergence is
e in auniform with respect to ω ∈ [0, 2π/h) and with K
compact set of stable filters.
Proof. The proof is almost the same as in [12, Theorem 1].
□
See Section VII for a simulation result, which shows thatas upsampling ratio L increases, the H ∞ norm of the error
system E decreases.
VI. CONTROLLER DISCRETIZATION
We here consider feedback controller discretization. Let
us consider the feedback system consisting of two linear
time-invariant systems, F (s) and G(s), shown in Fig. 5.
We assume that F (s) is a plant model that is a stable,
real-rational, strictly proper transfer function, and G(s) isa stable, real-rational, proper controller. We also assume thatthey satisfy∥GF ∥∞ < 1.
Then, by the small-gain theorem [14], the feedback system isstable. Then, to implement continuous-time controller G(s)in a digital system, we discretize it by using the H ∞ -optimal

Frequency Response

F (s)

10

(A)

0
−10

Fig. 6.

K(z)

Sh

−20
magnitude (dB)HhSampled-data feedback control system.

−30

−50
−60

discretization discussed above. Suppose that we obtain an
H ∞ -optimal digital filter K(z) with no delay (m = 0). Then
we have∥Hh KSh F ∥∞ = ∥(G − Hh KSh )F − GF ∥∞

−70
−80
−90

target analog filter
H−inf optimalbilinearstep invariant−100 −210≤ ∥E∥∞ + ∥GF ∥∞ .
It follows that by the small-gain theorem, the sampled-data
feedback control system shown in Fig. 6 is stable if ∥E∥∞ +∥GF ∥∞ < 1, or equivalently∥E∥∞ < 1 − ∥GF ∥∞ .

(B)

−40

(12)

−1

10

0

1

10
frequency (rad/sec)

10

Fig. 7. Frequency response of filters: target analog filter G(s) (solid black),
sampled-data H ∞ -optimal Ksd (z) (solid red), bilinear transformation
Kbt,ω0 (z) with frequency prewarping at ω0 = 1 (rad/sec) (dashed blue),and step-invariant transformation (dash-dotted magenta).

magnitude (dB)(A)

In summary, if we can sufficiently decrease the H ∞norm of the sampled-data error system E via the H ∞ optimal discretization to satisfy (12), then the stability of
the feedback control system is preserved under discretization.
The multirate system proposed in Section V can be also used
for stability-preserving controller discretization.

G(s) =

0.0031623(s2 + 1.33)(s2 + 1.899)(s2 + 0.3705s + 0.1681)(s2 + 0.1596s + 0.7062)(s2 + 10.31)
× 2.
(s + 0.03557s + 0.9805)

We use the following analog characteristic
F (s) =

1.
(s + 1)3

We set sampling period h = 1, delay step m = 4, andthe ratio for fast sample/hold approximation N = 12.
With these parameters, we design the H ∞ -optimal filter,
denoted by Ksd (z) that minimizes the cost function J in
(1). We also design the step-invariant transformation Kd (z)given in (4) and the bilinear transformation Kbt,ω0 (z) withfrequency prewarping at ω0 = 1 (rad/sec) given in (6). Fig. 7
shows the frequency response of these filters. Enlarged plotsof the frequency response in passband and stopband arealso shown in Fig. 8 These plots show that the proposed
sampled-data H ∞ -optimal discretization Ksd (z) shows thebest approximation among the filters. The step-invariant−2−4−6 −0.3
10

−0.2

−0.1

10

0

10

10

(B)

VII. NUMERICAL EXAMPLES−45
magnitude (dB)

Here we present numerical examples to illustrate theeffectiveness of the proposed method. The target analog filter
is given as a 6-th order elliptic filter with 3 (dB) passband
peak-to-peak ripple, −50 (dB) stopband attenuation, and 1(rad/sec) cut-off frequency. The filter is computed with MATLAB command ellip(6,3,50,1,’s’) whose transfer
function is given by

0

−50
−55
−60
−65

0.1

10

Fig. 8.

0.2

0.3

10
10
frequency (rad/sec)

0.4

10

Enlarged plots of Fig. 7 in passband (top) and stopband (bottom).

transformation Kd (z) is almost the same as Ksd (z) in
passband, while its stopband response is quite different fromthat of the target filter G(s). The bilinear transformation
Kbt,ω0 (z) shows the best performance around the cut-off
frequency ω0 = 1 (rad/sec) at the cost of deteriorationof performance at the other frequencies. In summary, thesampled-data H ∞ discretization outperforms conventional
discretization methods in particular in high-frequency range.
Then we consider an advantage of multirate systems.
We compare the sampled-data H ∞ -optimal multirate system with the single-rate one designed above. We set theupsampling ratio L = 4. We simulate time response ofreconstructing a rectangular input signal u(t) shown in
Fig. 9. Fig. 10 shows the time response of the single-rate
and multirate systems. The multirate system approximatesthe ideal response more precisely with a slow-rate samples{u(0), u(1), . . . }. This is an advantage of the multirate

Input u(t)Upsampling ratio vs error
0.12

1
0.8

0.1

0.6
0.08

0.2
||E||

amplitude

0.4

0

0.06

−0.2
−0.4

0.04

−0.6
−0.8

0.02

−1
0

5

10

15
20
time (sec)

25

30

2

4

6

8

10

12

14

16

L

Input signal u(t).

Fig. 9.

0

Fig. 11.

Upsampling ratio L versus approximation error ∥E∥∞ .

Time response
1
0.8

ACKNOWLEDGMENT

ideal
H−inf single−rate
H−inf multirate

This research is supported in part by the JSPS Grantin-Aid for Scientific Research (B) No. 24360163 and (C)No. 24560543, and Grant-in-Aid for Exploratory Research
No. 22656095.

0.6

amplitude

0.4
0.2

R EFERENCES

0
−0.2
−0.4
−0.6
−0.8
−1
0

5

10

15
20
time (sec)25

30

Fig. 10. Time response: delayed ideal response y = (Gu)(· − mh)(dashed), response of H ∞ -optimal single-rate system (blue), and that of
H ∞ -optimal multirate system (red).

system. In fact, the approximation error becomes smallerwhen we use larger L. To show this, we take upsampling
ratio as L = 1, 2, 4, 8, 16, and compute the H ∞ norm of
the sampled-data error system E. Fig. 11 shows the result.
The figure shows the performance gets better when the
upsampling ratio is increased.
VIII. CONCLUSIONSIn this article, we have proposed a discretization method
for analog filters via sampled-data H ∞ control theory. The
design is formulated as a sampled-data H ∞ optimization
problem, which is approximately reduced to a standard
discrete-time H ∞ optimization. We have also proposed
discretization with multirate systems, which may improve
the approximation performance. We have also discussed
feedback controller discretization with respect to stability.
Design examples show the effectiveness of the proposed
method compared with conventional bilinear transformationand step-invariant transformation.

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