Multirate Digital Signal Processing
via Sampled-Data H ∞ Optimization

Dissertation
Submitted in partial fulfillment of
the requirements for the degree ofDoctor of Informatics

Masaaki Nagahara

Department of Applied Analysis
and Complex Dynamical Systems
Graduate School of Informatics
Kyoto University

Abstract
In this thesis, we present a new method for designing multirate signal processing anddigital communication systems via sampled-data H ∞ control theory. The difference between our method and conventional ones is in the signal spaces. Conventional designs
are executed in the discrete-time domain, while our design takes account of both thediscrete-time and the continuous-time signals. Namely, our method can take account of
the characteristic of the original analog signal and the influence of the A/D and D/Aconversion. While the conventional method often indicates that an ideal digital low-passfilter is preferred, we show that the optimal solution need not be an ideal low-pass when
the original analog signal is not completely band-limited. This fact can not be recognized
only in the discrete-time domain. Moreover, we consider quantization effects. We discuss
the stability and the performance of quantized sampled-data control systems. We justify
H ∞ control to reduce distortion caused by the quantizer. Then we apply it to differential
pulse code modulation. While the conventional ∆ modulator is not optimal and besides
not stable, our modulator is stable and optimal with respect to the H ∞ -norm. We also
give an LMI (Linear Matrix Inequality) solution to the optimal H ∞ approximation of IIR(Infinite Impulse Response) filters via FIR (Finite Impulse Response) filters. A comparison with the Nehari shuffle is made with a numerical example, and it is observed that
the LMI solution generally performs better. Another numerical study also indicates that
there is a trade-off between the pass-band and stop-band approximation characteristics.

Acknowledgments
I would like to express my sincere gratitude to everyone who helped me and contributedin various ways toward completion of this work.
First of all, I am most grateful to my supervisor Professor Yutaka Yamamoto. Since
I entered the Graduate School of Kyoto University, he has constantly guided me in every
respect of science and technology. Without this knowledge he has endowed me, it isunthinkable that I could proceed this far toward the completion of this thesis. He alsotaught me fundamental knowledge on sampled-data control and signal processing on which
this thesis is based. His strong leadership and ideas have been a constant source of
encouragement and have had a definite influence on this work. Without his supervision,
this thesis would not exist today.
I would also like to thank Dr. Hisaya Fujioka for his helpful discussions and support
in my research, in particular, those on sampled-data control. His warm encouragement
was also great help to completing this thesis.
My gratitude should be extended to Dr. Yuji Wakasa for his valuable advice. In
particular, his advice on convex optimization was very valuable to my research.
I would thank Mr. Kenji Kashima, Mr. Shinichiro Ashida, and current and past members of the intelligent and control systems laboratory for their academic stimulation anddairy friendship.
Special thanks are addressed toward my parents for their deep understanding andencouragement.
This research was partially supported by Japan Society for the Promotion of Science(JSPS).

Contents
1 Introduction

1

2 Sampled-Data Control Theory
5
2.1 Sampled-data control systems . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Frequency response and H ∞ optimization . . . . . . . . . . . . . . . . . . 8
2.4 Fast discretization of multirate sampled-data systems . . . . . . . . . . . . 10
2.4.1 Discrete-time lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4.2 Approximating multirate sampled-data systems . . . . . . . . . . . 11
3 Multirate Signal Processing
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Interpolators and decimators . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Design of interpolators . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Reduction to a finite-dimensional problem . . . . . . . . . . . . . .
3.4 Design of decimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Reduction to a finite-dimensional problem . . . . . . . . . . . . . .
3.5 Design of sampling rate converters . . . . . . . . . . . . . . . . . . . . . . .
3.6 Design examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.1 Design of interpolators . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.2 Design of decimators . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.3 Design of sampling rate converters . . . . . . . . . . . . . . . . . .
3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15
15
16
21
21
22
24
24
26
28
30
30
33
36
39

4 Application to Communication Systems
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Digital communication systems . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Design problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Design algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Decomposing design problems . . . . . . . . . . . . . . . . . . . . .
4.4.2 Fast-sampling/fast-hold approximation . . . . . . . . . . . . . . . .
4.5 Design examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 The case of no compression (M = 1) . . . . . . . . . . . . . . . . .

41
41
41
43
45
45
47
49
49

v

vi

Contents

4.6

4.5.2 Compression effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5 Minimization of Quantization Errors
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Sampled-data control systems with quantization . . . . . . . . . . . . . . .
5.2.1 Additive noise model for quantizer . . . . . . . . . . . . . . . . . .
5.2.2 Stability of sampled-data systems with quantization . . . . . . . . .
5.2.3 Performance analysis of sampled-data systems with quantization . .
5.3 Differential pulse code modulation . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Differential pulse code modulation . . . . . . . . . . . . . . . . . . .
5.3.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Design example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61
61
62
62
62
66
69
69
69
71
73

6 Optimal FIR Approximation
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 FIR approximation problem . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Comparison of H ∞ design via LMI and the Nehari shuffle . . . . . .
6.3.2 Trade-off between pass-band and stop-band characteristics . . . . .
6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75
75
77
79
79
82
84

7 Conclusion

87

Notationvii

Notation
Z+ : non-negative integers.
N: natural numbers.
R: real numbers.
R+ : non-negative real numbers.
Rn : n-dimensional vector space over R.
C: complex numbers.
l2 : real-valued square summable sequences.
l∞ : real-valued bounded sequences.
L2 [0, ∞) and L2 [0, h): Lebesgue spaces consisting of square integrable real functions on[0, ∞) and [0, h), respectively. L2 [0, ∞) may be abbreviated to L2 .
lL2 2 [0,h) : square summable sequences whose values are in L2 [0, h).




A B
A B
: transfer function whose realization is {A, B, C, D}, that is,(λ) :=C DC DC(λI − A)−1 B + D, where λ := s in continuous-time and λ := z in discrete-time.

P11 P12Fl (P, K): linear fractional transformation of P and K, that is, if P =then
P21 P22
Fl (P, K) := P11 + P12 K(I − P22 K)−1 P21 .
Sh : ideal sampler with sampling period h.
Hh : zero order hold with sampling period h.
Lh : lifting for continuous-time signals with sampling period h.
LN : lifting for discrete-time signals by factor N (discrete-time lifting).
LN (P ): fast-discretizing and discrete-time lifting of continuous system P , that is, LN (P ) :=LN Sh/N P Hh/N L−1
N .
↑ M : upsampler with upsampling factor M.
↓ M : downsampler with downsampling factor M.
r(A): maximum absolute value of eigenvalues of matrix A.
AT : transpose of a matrix (or a vector) A.

Chapter 1
Introduction
This thesis presents a new method in multirate digital signal processing and digital communication systems.
When we execute these design procedures, we must first discretize the original analog
signal (e.g., speech, audio or visual image), then the discretized signal is processed in
the discrete-time domain (e.g., filtering, compressing, transmitting etc.), and finally wereconstruct an analog signal from the discrete signal.
Conventionally, the design is performed mostly in the discrete-time domain by assuming that the original analog signal is fully band-limited up to the Nyquist frequency.
Under this assumption, the sampling theorem gives a method for reconstructing an analog
signal from a sampled signal [16, 45].
Theorem 1.1 (Shannon et al. ). Let f (t) be a continuous-time signal ideally bandlimited to the range (−π/h, π/h), that is, its Fourier transform fˆ(ω) is zero outside
this interval. Then f (t) can uniquely be recovered from its sampled values f (nh), n =0, ±1, ±2, . . . via the formula∞X
sin π(t/h − n)f (t) =f (nh)=f (nh)sinc(t/h − n).
π(t/h − n)
n=−∞
n=−∞
∞
X

(1.1)

Although most of the conventional studies of digital signal processing are based on
Theorem 1.1 [9, 32, 33, 35, 45, 46], we encounter two questions in the implementation:
the question of D/A and A/D conversions.
We consider the first question. The process (1.1) is a kind of D/A conversion; weconvert sampled values {f (nh)} to modulated impulses {f (nh)δ(nh)}, then filter themby the ideal low-pass filter1) [46]. In practice, this conversion is physically impossible, andin reality a zero-order hold followed by a sharp low-pass filter is often used. We should
notice that the effect of such a real situation of D/A conversion can never be taken into
account only in the discrete-time domain.
The question that we must consider next is the assumption of full band-limitation.
The assumption that Theorem 1.1 requires is in reality impossible because no real analog
1)

The frequency response of this filter is equal to 1 up to the Nyquist frequency ωN and zero beyond

ωN .

1

2

CHAPTER 1. INTRODUCTION

signal is fully band-limited. In order to achieve the assumption, a sharp low-pass filter is
attached before sampling. However, such a sharp low-pass characteristic will deteriorate
the quality of the analog signal, and moreover even such a filter does not satisfy theassumption. Therefore we have to consider the effect of sampling (i.e., aliasing), which
can never be captured in the discrete-time domain.
To answer these questions, we must take account of not only discrete-time signals but
also continuous-time signals. Therefore, we propose to consider an analog performance
to design digital systems by using the modern sampled-data control theory. Sampled-data
control theory deals with control systems that consist of continuous-time plants to becontrolled, discrete-time controllers controlling them, and ideal A/D and D/A converters.
The modern theory can take the intersample behavior of sampled-data control systems
into account. The key idea is lifting [36, 2, 37]. Although sampled-data systems are nottime-invariant, the lifting method leads to time-invariant discrete-time models. Thesemodels are infinite-dimensional, which can be reduced to equivalent discrete-time finitedimensional ones [1, 4, 18].
There is another method for sampled-data systems: fast-sampling/fast-hold (FSFH)
method [19, 43]. The idea is to approximate the continuous-time inputs by step functionsof sufficiently small step size and also approximate the continuous-time outputs by taking
their samples by sufficiently fast ideal sampler. The approximated system will be a periodically time-varying discrete-time system (finite-dimensional), which can be transformed
a time-invariant discrete-time system by using the discrete-time lifting [25, 4]
Based on these studies, the H ∞ optimal sampled-data control has been studied [34,1, 18, 13]. The H ∞ optimality criterion is suitable for the frequency characteristic, whichis intuitive for one who designs the system.
In the last few years, several articles have studied digital signal processing via sampleddata control theory. Chen and Francis [5] studied a multirate filter bank design problem
with an H ∞ criterion. Although this design was done in the discrete-time domain, theybrought the modern H ∞ control theory to the digital signal processing, and thus they
threw new light on the subject. The first study of digital signal processing in the sampleddata setting was made by Khargonekar and Yamamoto [21]. They formulated a singlerate signal reconstruction problem by using sampled-data theory. After their study, thismethod was developed to multirate signal processing [40, 14, 15, 44, 27], digital communications [29, 30] and quantizer design [26].
The purpose of this thesis is to answer the questions mentioned above by sampled-data∞
H optimal control theory, and to show that this method is effective in designing digitalsystems.
The organization of this thesis is as follows.
• Chapter 2 surveys sampled-data control theory. We describe the fundamental difficulty in sampled-data systems, and introduce the lifting method, which resolved
this difficulty: by using it we can define the frequency response and the H ∞ optimal control. The fast-sampling/fast-hold method for multirate sampled-data control
systems is discussed in detail, because we mainly use this method in this thesis.
• Chapter 3 deals with multirate signal processing, in particular, interpolation, decimation and sampling rate conversion. We present a new method for designing these3

systems via the sampled-data H ∞ optimization. Design examples shows that our
method is superior to the conventional one.
• Chapter 4 presents a new design of digital communication systems. Under signal
compression and channel distortion, we design an optimal transmitting/receiving
filter by using the sampled-data H ∞ optimization. We show also design examples
to indicate that our method is effective in digital communication.
• Chapter 5 investigates issues of quantization. Since quantization is a nonlinear operation, we introduce a linearized model (i.e., additive noise model). By using thismodel, we discuss stability and performance of a quantized sampled-data control
system. Then we apply it to differential pulse code modulation (DPCM) systems.
Design examples are shown and our design is superior to the conventional ∆ modulation.
• Chapter 6 presents a new method to approximate an IIR filter by an FIR filter, which
directly yields an optimal approximation with respect to the H ∞ error norm. We
show that this design problem can be reduced to an LMI (Linear Matrix Inequality).
We will also make a comparison via a numerical example with an existing method,
known as the Nehari shuffle.
• Chapter 7 concludes this thesis with a summary of the results presented and future
perspectives.

Chapter 2
Sampled-Data Control Theory
2.1

Sampled-data control systems

A sampled-data control system is a system in which a continuous-time plant is to becontrolled by a discrete-time controller. Consider the unity-feedback sampled-data control
system shown in Figure 2.1. In this figure, P (s) is a continuous-time plant and K(z) is
r(t)
+

e(t)

y(t)
Sh

K(z)

Hh

P (s)

−

Figure 2.1: Sampled-data control systema discrete-time controller. In order to include K(z) in this control system, we need aninterface. Therefore we introduce the ideal sampler Sh and the zero-order hold Hh withsampling time h.
Definition 2.1. The ideal sampler Sh and the zero-order hold Hh are defined as follows:
Sh : L2 [0, ∞) 3 u 7−→ v ∈ l2 ,Hh : l2 3 v 7−→ u ∈ L2 [0, ∞),k = 0, 1, 2, . . . ,v[k] := u(kh),u(kh + θ) := H(θ)v[k],where H(·) is the hold function defined as follows:(1,
H(θ) :=0,θ ∈ [0, h),otherwise.

(2.1)

In practice, a quantization error occurs in the A/D conversion. We however omit thisquantization error here. The quantization effects are discussed in Chapter 5.
56

CHAPTER 2. SAMPLED-DATA CONTROL THEORY

The system contains both continuous-time and discrete-time signals and is theoretically regarded as a periodically time-varying system [4]. Since this system is not timeinvariant, it is difficult to analyze or design by using such conventional machinery astransfer functions or frequency responses.

2.2

Lifting

Conventionally there are two ways for designing sampled-data systems. One is the following: first design a continuous-time controller in the continuous-time domain, and thendiscretize the controller. A typical discretization method is the Tustin (or bilinear) transformation [4, 38, 10]. If the sampling period is sufficiently small, the designed system may
perform well. However, if the sampling period is not small enough, the performance not
only deteriorates, but also the closed-loop system may become unstable.
The other is to approximate the continuous-time signal by a discrete-time one byconsidering only the signals at sampling points. As a result the sampled-data system
becomes a discrete-time time-invariant system. This method preserves the closed-loop
stability. However, the output of the system may sometimes induce very large intersample
ripples despite a very small sampling period. The reason is that the method ignores theintersample behavior.
Recently, Yamamoto [36, 37] studied the problem of sampled-data control systems.
He developed what is now called the lifting method, which takes the intersample behavior
into account and gives an exact, not approximated, time-invariant discrete-time system
for a sampled-data control system.
We begin by defining the lifting operator.
Definition 2.2. Define the lifting operator Lh by
Lh : L2 [0, ∞) −→ lL2 2 [0,h) : {f (t)}t∈ + 7−→ {fe[k](θ)}∞
k=0 ,fe[k](θ) := f (kh + θ) ∈ L2 [0, h).

θ ∈ [0, h),By lifting, continuous-time signals in L2 [0, ∞) will become discrete-time signals whose
values are in L2 [0, h), hence the sampled-data system can be rewritten as a time-invariant
discrete-time system with infinite-dimensional signal spaces. As a result, we can introduce
the concept of transfer functions or the frequency response for sampled-data systems, andhence we can treat sampled-data systems as time-invariant systems without approximation.
Consider the standard sampled-data control system in Figure 2.2, where G is a continuoustime generalized plant, whose state-space equation is given as follows:



 w
ẋ = Ax + B1 B2
,
u
  




z
C1
D11 D12
w
=
x+
.
y
C2
0
0
u
The signal w is the exogenous input consisting of reference commands, disturbance or
sensor noise, while z is the signal to be controlled to have a desirable performance. Notethat both of these signals are continuous-time. The system Kd is a digital controller.

72.2. LIFTING



z

w



G

y

u



- Sh

yd

-

ud

- Hh

Kd

Figure 2.2: Sampled-data control system

The design problem of the sampled-data control system in Figure 2.2 is to obtain thecontroller Kd which stabilizes the closed-loop system and makes the performance from w
to z desirable.
This system is a sampled-data control system, and hence it is a periodically timevarying system. However, by lifting the continuous-time signals z and w, and taking
ze := Lh z and w
e := Lh w, the sampled-data control system in Figure 2.2 can be converted
to a time-invariant discrete-time system shown in Figure 2.3. Namely, the state-space
ze





G

yd



-

w
e

ud

Kd

Figure 2.3: Lifted sampled-data control systemequation of the lifted system G is obtained as follows:


 


x[k + 1]
Ad B1 Bd2x[k]
 ze[k]  =  C1 D11 D12   w[k],eyd [k]
Cd2 0
0
ud [k]
k = 0, 1, . . . ,8

CHAPTER 2. SAMPLED-DATA CONTROL THEORYZ heA(h−τ ) B2 dτ,
Cd2 := C2 ,Z h2n
B1 : L [0, h) −→ R : w 7→
eA(h−τ ) B1 w(τ )dτ,AhAd := e ,Bd2 :=

0

0

C1 : Rn −→ L2 [0, h) : x 7→ C1 eAθ x,
D11 : L2 [0, h) −→ L2 [0, h)Z θ
C1 eA(θ−τ ) B1 w(τ )dτ + D11 w(θ),: w 7→
0m

D12 : R −→ L2 [0, h)Z θ: ud 7→
C1 eA(θ−τ ) B2 H(τ )dτ ud + D12 H(θ)ud,0θ ∈ [0, h).

In this equation, n and m are the dimensions of x and ud , respectively, and H(·) is thehold function defined by (2.1). Note that B1 , C1 , D11 and D12 are operators in infinitedimensional spaces, while Ad , Bd2 and Cd2 are matrices. Therefore the lifted system
becomes a discrete-time time-invariant system with infinite-dimensional operators.

2.3

Frequency response and H ∞ optimization

In the previous section, we have shown that sampled-data systems can be represented
as time-invariant discrete-time systems. We can then define their transfer function or
frequency response. The concept of frequency response for sampled-data systems is introduced by Yamamoto and Khargonekar [41] and its computation is developed, for example,in [12].
Let T denote the system from w to z in Figure 2.2, and Te the lifted system of T .
Define the state-space equation for Te as follows:xs [k + 1] = Axs [k] + Bw[k],eze[k] = Cxs [k] + Dw[k],ek = 0, 1, 2, . . . .

(2.2)

Note that A is a matrix, while B, C and D are infinite-dimensional operators. We assume
that A is a power stable matrix, that is, An → 0 as n → ∞.
For the lifted signal {fe[k]}∞k=0 , define its z transform as
Z[fe](z) :=∞Xkfe[k]z −k .

It follows that the transfer function Te (z) of the sampled-data system can be defined asfollows:
Te (z) := D + C(zI − A)−1 B,z ∈ C.

The z transform Te (z) is a linear operator on L2 [0, h) with a complex variable z. By
substituting ejωh for z in Te (z), we can define the frequency response operator.

2.3. FREQUENCY RESPONSE AND H ∞ OPTIMIZATION

9Definition 2.3. Define the frequency response operator of sampled-data system T by
Te (ejωh ) = D + C(ejωh I − A)−1 B : L2[0,h) −→ L2 [0, h),ω ∈ R.

The norm of the frequency response operator Te (ejωh )kTe (ejωh )k :=kTe(ejωh )vkL2 [0,h),kvkL2 [0,h)v∈L2 [0,h)
sup
v6=0

is called the gain at ω. The H ∞ -norm of the sampled-data system is then given bykTe k∞ :=supω∈[0,2π/h)kTe(ejωh )k.

The H ∞ -norm kTe k∞ is equivalent to the L2 induced norm of the sampled-data system(2.2), that is,kTe k∞ = kT k :=kT wkL2 [0,∞).
w∈L2 [0,∞) kwkL2 [0,∞)
sup
w6=0

Sampled-data H ∞ control problem is to find a discrete-time controller Kd which stabilizes the closed-loop system and makes the H ∞ -norm of the system small. The H ∞control has the following advantages:• Since the H ∞ -norm is the L2 induced norm of the sampled-data control system, we
can formulate the worst case design.
• Many robustness requirements for the design against the system uncertainty can be
described by H ∞ -norm constraint.
• We can shape the frequency characteristic with frequency weights, which is intuitive
to designers.
Although T is infinite-dimensional, the H ∞ optimization can be equivalently transformed to that for a finite-dimensional discrete-time system [34, 1, 18, 13, 4]. Note thatthe obtained finite-dimensional discrete-time optimization problem takes intersample behavior into account.
On the other hand, there is another method for obtaining a finite-dimensional discretetime system: fast-sampling/fast-hold approximation [19, 4]. This method provides not an
equivalent but an approximated model, however its computation is easier than that of themethod giving equivalent models.
Moreover, with regard to signal reconstruction problem (main theme of this thesis),the method giving equivalent models yields some conservative results. The reason is asfollows: we often allow signal reconstruction to have time delays since the system does
not have any feedback loop. In other words, a sampled value at a certain time can bereconstructed by using future sampled values, that is, the filter is allowed to be non-causal.
However, the method giving equivalent method does not readily apply to this situation,
while the fast-sampling/fast-hold method does.
For the reasons mentioned above, we adopt the approximation method in this thesis
to consider the problem. In the following section, we discuss this method in detail.

10

CHAPTER 2. SAMPLED-DATA CONTROL THEORY

2.4

Fast discretization of multirate sampled-data systems

In this section, we deal with multirate sampled-data control systems by using the fastsampling/fast-hold method. The fast-sampling/fast-hold technique is a method for approximating the performance of sampled-data systems. The procedure is as follows:• discretize the continuous-time input by a hold with sampling period h/N,
• discretize the continuous-time output by a sampler with sampling period h/N.
With large N ∈ N, the discretized signals may be good approximation of the continuous
signals.
Figure 2.4 illustrates the procedure. Assume that the controller K is (M1 , M2 )-periodic
zd

z

Sh/N

y

w
G(s)

Sh/M1
yd

Hh/N

wd

u

Hh/M2
K

ud

Figure 2.4: Fast-sampling/fast-hold discretization(M1 , M2 ∈ N) [25], that is, z M2 Kz −M1 = K where z is the unit advance and z −1 is theunit time delay.

2.4.1

Discrete-time lifting

By attaching a fast-sampler and a fast-hold as shown in Figure 2.4, the multirate sampleddata system will be converted to a finite-dimensional discrete-time system (we will show
this in the next section). However, this system has three sampling periods: h/M1 , h/M2
and h/N, and hence the system will be time-varying (to put it precisely, periodicallytime-varying). In order to equivalently convert a multirate system to a single-rate one,the discrete-time lifting [20, 25, 4] is useful. The definition is as follows:

2.4. FAST DISCRETIZATION OF MULTIRATE SAMPLED-DATA SYSTEMS

11

Definition 2.4. Define the discrete-time lifting LN and its inverse L−1
N by


 



v[0]
v[N]




 v[1]   v[N + 1] 


 

2
2
LN : l −→ l : {v[0], v[1], . . . } 7→ ,,... ,
..
..



 

.
.




 v[N − 1]v[2N − 1]2
2
L−1
N : l −→ l :


 



v0 [0]
v0 [1]

 v1 [0]   v1 [1] 


 

,
,
.
.
.
7→ {v0 [0], v1 [0], . . . , vN −1 [0], v0 [1], v1 [1], . . . } .

 

..
..



 

.
.




 v

vN −1 [1]N −1 [0]The discrete-time lifting LN converts a 1-dimensional signal into an N-dimensionalsignal and the sampling rate becomes N times slower. This operation makes it possible toequivalently convert multirate systems into single-rate systems, and hence its analysis anddesign become easier. Note that the discrete-time lifting LN is norm-preserving, namely,
kLN vk = kvk, v ∈ l2 and so is L−1
N .

2.4.2

Approximating multirate sampled-data systems

By using the discrete-time lifting, the multirate system shown in Figure 2.4 is converted
to a single-rate discrete-time system. Take
G11 (s) G12 (s)G(s) =,
G21 (s) G22 (s)where G11 , G12 and G22 are strictly proper and G21 is proper. Let their state-spacerealization beA Bj(s), i, j = 1, 2.
Gij (s) =Ci Dij
Let the system from wd to zd in Figure 2.4 be TdN . Then TdN can be rewritten as
follows:
TdN = Fl (GdN , K),




Sh/N 0G11 G12
Hh/N 0
TdN :=.
0I
G21 G220
I
Then we apply the discrete-time lifting to wd and zd as shown in Figure 2.5. Let the systemfrom w
ed to zed in Figure 2.5 be TedN . Note that since LN and L−1
N are norm-preservingeemappings, we have kTdN k = kTdN k. Then we rewrite TdN as follows:
bdN , K),
TedN = Fl (G




  −1
LN 0
Sh/N 0G11 G12
Hh/N 0LN 0b
GdN :=.
0 I
0I
G21 G22
0
I
0 I

12

CHAPTER 2. SAMPLED-DATA CONTROL THEORY

zed

LN

zd

z

Sh/N

w
G(s)

y

Sh/M1

Hh/N

wd

L−1
N

u

w
ed

Hh/M2

yd

K

ud

Figure 2.5: Lifted and fast-discretized sampled-data system
Let us turn to the controller K. By the assumption that K is (M1 , M2 )-periodic,e
Kd := LM2 KL−1
M1 is time-invariant [25]. By using this property, we rearrange GdN as
follows:
edN , Kd ),TedN = Fl (G




I
0
I
0
edN :=bdN
GG
0 LM1 Sh/M1
0 Hh/M2 L−1
M2




LN Sh/N0
G11 G12
Hh/N L−10N.
=
0
LM1 Sh/M 1
G21 G220
HM2 L−1
M2

edN to a simple time-invariant system by the following proposition.
Finally, we convert G

Proposition 2.1. Assume N = kM1 M2 , k ∈ N. Then we have the following identities:
LM1 Sh/M1 = SLN Sh/N ,
where




S := 
H := 
|pq−1
Hh/M2 L−1
M2 = Hh/N LN H,





..
 M1 ,
.

p 


..

.

{z

M2

q


,
}

p := [1, 0, . . . , 0],| {z }(2.3)

kM2 −1q := [1, . . . , 1]T .
| {z }
kM1

This proposition gives us the following equation:




LN Sh/N0
G11 G12
Hh/N L−1
0
N
e
GdN =
0
SLN Sh/N
G21 G220
Hh/N L−1
N H




I 0
LN (G11 ) LN (G12 )
I 0
=
,
0 S
LN (G21 ) LN (G22 )
0 H

(2.4)

2.4. FAST DISCRETIZATION OF MULTIRATE SAMPLED-DATA SYSTEMS

13

where LN (G) := LN Sh/N GHh/N L−1
N . Note that LN (Gij ) =: Gd,ij (i, j = 1, 2) is a timeinvariant discrete-time system [25, 4]. In fact, a state-space realization of Gd,ij is given as
follows:
−2
AN
AdN −1 Bd,jAN
Bd,j . . . Bd,j
d
d

Ci
Dij
0
...
0 



Ci Bd,jDij...
0 
Gd,ij =  Ci Ad
 , i, j = 1, 2,

..
..
..
.. 

.
.
.
. 
N −2
N −3
N −1
Ci Ad Bd,j Ci Ad Bd,j . . . Dij
Ci Ad
where

Ah/N

Ad := e,Bd,j :=Z h/N

eAt Bj dt.

0

We show the obtained time-invariant discrete-time system in Figure 2.6.
edN , Kd )When N becomes larger, the performance of the discrete-time system TdN := Fl (Gconverges to that of the original sampled-data system T := Fl (G, Hh/M2 KSh/M1 ) [43].
Therefore, if we take sufficiently large N, the error between T and TdN will small. The
error estimate for the fast-sampling factor N is discussed in [39].
We summarize the above discussion as a theorem:
Theorem 2.1. For the multirate sampled-data control system T := Fl (G, Hh/M2 KSh/M1 ),edN , Kd ) such that
there exists a time-invariant discrete-time system TdN := Fl (Glim kTdN k = kT k.

N →∞

zed





edN (z)Gwed- Kd (z)Figure 2.6: FSFH discretized system

Chapter 3
Multirate Signal Processing
3.1

Introduction

Multirate techniques are now very popular in digital signal processing. They are particularly effective in subband coding, and various techniques for economical information
saving have been developed [9, 35, 46].
One example is signal decoding in audio/speech processing. For example, in thecommercial CD format, the sampling frequency is 44.1 kHz, but one hardly employs thesame sampling period in decoding. A popular technique is interpolation. The process
is as follows: first upsample the encoded digital signal (i.e., inserting zeros between two
consecutive samples), remove the parasitic imaging spectra via a digital low-pass filter,
and then convert it back to an analog signal with a hold device and an analog low-passfilter. Imaging is a phenomenon due to zeros inserted by upsampling, and yields highfrequency noise.
The chief advantage here is that one can employ a fast hold device, and do not haveto use a very sharp analog filter (thereby avoiding much phase distortion induced by a
sharp analog filter).
Another example is signal compression. Due to the limitation of the bandwidth ofcommunication channels or of the size of storage devices, one often needs to compress thesignals. In signal compression, multirate processing plays a major role. The fundamental
operation for signal compression is decimation. Decimation is to reduce the sampling rateof a signal; the process is to first filter out the aliasing components by using a digital
low-pass filter, and then downsample the filtered signal. Downsampling is an operation
to keep every M-th sample (M is a natural number) and remove in-between samples.
The aliasing caused by upsampling is comparative to aliasing in A/D conversion, that is,removing samples causes frequency overlapping.
By combining interpolators and decimators, we can obtain a sampling rate converter.
In commercial applications, there are many different sampling rates employed: for example
48kHz for DAT and 44.1kHz for audio CD. The conversion from one sampling rate to theother becomes necessary. In such a process, it is clearly required that the information loss
be as little as possible.
The conventional way of doing this is as follows: Suppose we want to convert a signal
v with sampling frequency f1 Hz to another signal u with sampling frequency f2 Hz.
15

16

CHAPTER 3. MULTIRATE SIGNAL PROCESSING

Suppose also that there exist (coprime) integers L1 and L2 such that f1 L1 = f2 L2 . We
first upsample v by factor L1 , to make the sampling frequency f1 L1 . Suppose that theoriginal signal is perfectly band-limited in the range |ω| < f1 /2. We then introduce
a digital filter H(z) to filter out the undesirable imaging component. After this, theobtained signal is downsampled by factor L2 to become a signal with sampling frequency
f2 = f1 L1 /L2 Hz.
In the existing literature, it is a commonly accepted principle that one inserts a very
sharp digital low-pass filter after the upsampler or before the downsampler to eliminate
the effect of imaging or aliasing components [9, 35, 46]. This is based on the following
reasoning: Suppose that the original signal is fully band-limited. Then the imaging(aliasing) components induced by upsampling (downsampler) is not relevant to the original
analog signal and hence they must be removed by a low-pass filter. If the original signal
is band-limited, the closer is this filter to an ideal filter, the better.
In practice, however, no signals are fully band-limited in a practical range of a passband, and they obey only an approximate frequency characteristic. The argument aboveis thus valid only in an approximate sense. One may rephrase this as a problem of robustness: namely, when the original signals are not fully band-limited but obey only a certain
frequency characteristic, how close should the digital filter be to the ideal low-pass filter?
This type of question has been seldom addressed in the signal processing literature
until very recently. However, this can be properly placed in the framework of sampleddata control theory, and there are now several investigations that apply the sampled-data
control methodology to digital signal processing.
We formulate a multirate digital signal reconstruction problem under the assumptionthat the original analog signal is subject to a certain frequency characteristic, but not
fully band-limited. Under the assumption we will optimize the analog performance withan H ∞ optimality criterion.
This may also be regarded as an optimal D/A converter design. We will show that
performance improvement is possible over conventional low-pass filters. It is also seenthat the presented method can be used as a new design method for a low-pass filter.
In this chapter, we first introduce the fundamentals of multirate digital signal processing. The conventional idea of its design is also discussed and we point out that there aresome drawbacks in its idea. We then present an alternative method of designing multirate
systems, that is, sampled-data H ∞ design. The last section provides design examples andwe show the advantages of the present method.

3.2

Interpolators and decimators

In this section, we introduce mathematical definitions of interpolators and decimators.
An interpolator or a decimator is implemented with upsamplers ↑ M and downsamplers↓ M respectively. We begin by defining the upsampler and the downsampler.
Definition 3.1. For discrete-time signal {x[k]}∞k=0 , define the upsampler ↑ M and the17

3.2. INTERPOLATORS AND DECIMATORS

downsampler ↓ M by
↑ M : {x[k]}∞0, 0, . . . , 0, x[1], 0, . . . },k=0 7−→ {x[0], |
{z }

M −1
∞
↓ M : {x[k]}k=0 7−→ {x[0], x[M], x[2M], . . . }.

The upsampling operation is implemented by inserting M − 1 equidistant zero-valued
samples between two consecutive samples of x[k] before the sampling rate is multipliedby the factor M. Figure 3.1 indicates the upsampling operation.
x

t
y

t

Figure 3.1: Upsampling operation y = (↑ 2)x
On the other hand, the downsampling operation is implemented by keeping every Mth sample of x[k] and removing in-between samples to generate y[k], then the sampling
rate becomes multiplied by 1/M. This procedure is illustrated in Figure 3.2.
x

t
y

t

Figure 3.2: Downsampling operation y = (↓ 2)x
Note that the upsampler is left-invertible (and the downsampler is right-invertible),that is, (↓ M)(↑ M) = I, however (↑ M)(↓ M) is not the identity. In fact, by (↑ M)(↓ M)

18

CHAPTER 3. MULTIRATE SIGNAL PROCESSING

the discrete-time signal x = {x[0], x[1], . . . } is converted into(↑ M)(↓ M)x = {x[0], 0, . . . , 0, x[M], 0, . . . } 6= x.
| {z }M −1

To put it differently, downsampling is a lossy data compression, that is, the original signal
cannot be perfectly reconstructed from the downsampled signal. On the other hand, we
have the duality relation: (↑ M)∗ = (↓ M), (↓ M)∗ = (↑ M), that is, for any signal x ∈ l2
and y ∈ l2 , we have h(↑ M)x, yi = hx, (↓ M)yi, h(↓P M)x, yi = hx, (↑ M)yi, where h·, ·idenotes the inner product on l2 , that is, hx, yi := ∞k=0 y[k]x[k].
In addition, an upsampler and a decimator are represented with the discrete-time
lifting and its inverse (see Section 2.4):(↑ M) = L−1
M
and vice versa:
LM := (↓ M)

1 0 ... 0

1 z . . . z M −1

T

,

T

,

(↓ M) =

L−1
M :=





1 0 ...


0 LM ,

1 z −1 · · · z −M +1(↑ M).

(3.1)

(3.2)

These relations are used to design interpolators or decimators in Section 3.3 and 3.4.
Having defined the upsampler and the downsampler, we can now explain the interpolator and the decimator.
An interpolator consists of two parts: an upsampler and a digital filter. Figure 3.3
shows the block diagram of an interpolator. First, the upsampler inserts zeros and inx

- ↑M

v

- K(z)y-Figure 3.3: Interpolator
creases the sampling rate. Then the filter K(z), called interpolation filter, operates on the
M − 1 zero-valued samples inserted by the upsampler ↑ M to yield nozero values between
the original samples, as illustrated in Figure 3.4.
Let us now consider the interpolation in the frequency domain. Assume the sampling
period of signal x in Figure 3.3 is 1, that is, the Nyquist frequency is π, and its Fouriertransform X(ω) has characteristic as shown above in Figure 3.5. By upsampling, theNyquist frequency of the upsampled signal v becomes 2πM, and there occur unwanted
frequency imaging components (the shaded portions below in Figure 3.5). In order toremove these imaging components, the frequency response K(ω) of the interpolation filter must be of low-pass characteristic with cut-off frequency π as shown in Figure 3.5.
Therefore, a very sharp low-pass filter close to the ideal one is often used.
On the other hand, a decimator is constructed by using a downsampler and a digital
filter, whose block diagram is shown in Figure 3.6. To see the role of the filter H(z), let us
look at the frequency domain. Assume the sampling period of signal v in Figure 3.6 is 1

19

3.2. INTERPOLATORS AND DECIMATORS

0
Figure 3.4: Signal interpolation by interpolator

X (ω)

ω
0

2π
Κ(ω)

V (ω)

images

ω

2π

0

2πΜ

Figure 3.5: Imaging components caused by upsamplerx- H(z)v- ↓M

yFigure 3.6: Decimator-

20

CHAPTER 3. MULTIRATE SIGNAL PROCESSING

V (ω)

ω
0

2π

Y (ω)

aliasing
0ω

2πFigure 3.7: Aliasing(the Nyquist Frequency is π) and its Fourier transform V (ω) has characteristic as indicatedabove in Figure 3.7. If V (ω) is not band-limited to |ω| ≤ π/M, the spectrum Y (ω)
obtained after downsampling will overlap as shown below in Figure 3.7. This overlapping(the shaded portions below in Figure 3.7) is called aliasing.
The filter H(z), called decimation filter or anti-aliasing filter, is connected before thedownsampler to avoid aliases in advance. To eliminate the aliasing completely, the filter
H(z) must be the ideal low-pass filter with cut-off frequency π/M. Therefore, similarly
to the case of interpolation, a very sharp low-pass filter close to the ideal one is often
employed.
As mentioned above, the interpolation filter K(z) or the decimation filter H(z) ideally
has the characteristic shown in Figure 3.8. In practice, we cannot realize a filter with
|K(ω)||H(ω)|

6

6

M

1
-ω

π/M

-ω

π/M

Figure 3.8: Ideal characteristic of interpolation filter K(z) and decimation filter H(z)such a frequency characteristic. Therefore the filter is conventionally designed such thatthe frequency response approximates that of the ideal filter.
As far as the discrete-time signals are concerned, that is, the spectra of the original
analog signals are completely band-limited by the Nyquist frequency, the design may be21

3.3. DESIGN OF INTERPOLATORS

correct. However, the real analog signals have the spectra beyond the Nyquist frequency,
and hence the ideal characteristic in Figure 3.8 will not be necessarily optimal. In thefollowing sections, we propose an alternative method that takes the analog performance,in particular, the frequency component over the Nyquist frequency, into account.

3.3

Design of interpolators

3.3.1

Problem formulation

We start by formulating a design problem for (sub)optimal interpolators. Consider theblock diagram shown in Figure 3.9. The incoming signal wc first goes through an antiwc-F (s)

yc

- Sh

yd

- ↑L

A/D

xd

-K(z)

ud

- Hh/Luc-P (s)zc-

D/A

Figure 3.9: Signal reconstruction by interpolator
aliasing filter F (s) and the filtered signal yc becomes nearly (but not entirely) bandlimited. The filter F (s) governs the frequency-domain characteristic of the analog signal
yc . This signal is then sampled by the sampler Sh to become a discrete-time signal yd
with sampling period h.
To restore yc we usually let it pass through a digital filter, a hold device and thenan analog filter. The present setup however places yet one more step: The discretetime signal yd is first upsampled by ↑ L , and becomes another discrete-time signal xd
with sampling period h/L. The discrete-time signal xd is then processed by a digital
filter K(z), becomes a continuous-time signal uc by going through the zero-order hold
Hh/L (that works in sampling period h/L), and then becomes the final signal by passing
through an analog filter P (s). An advantage here is that one can use a fast hold device
Hh/L thereby making more precise signal restoration possible. The objective here is todesign the digital filter K(z) for given F (s), L and P (s).
Figure 3.10 shows the block diagram of the error system for the design. The delay inthe upper portion of the diagram corresponds to the fact that we allow a certain amountof time delay for signal reconstruction. Let TI denote the input/output operator from wc
to ec := zc (t) − uc (t − mh). Our design problem is as follows:
Problem 3.1. Given a stable, strictly proper F (s), stable, proper P (s), upsampling factor
L ∈ N, delay step m ∈ N, sampling period h > 0 and an attenuation level γ > 0, find adigital filter K(z) such thatkTI k :=

kTI wc kL2 [0,∞)< γ.
wc ∈L2 [0,∞) kwc kL2 [0,∞)supwc 6=0(3.3)

22

CHAPTER 3. MULTIRATE SIGNAL PROCESSING- e−mhswc- F (s)

+

yc
- Sh

- ↑L

- K(z)- Hh/L- P (s)? ec
j6−

Figure 3.10: Signal reconstruction error system
The norm in (3.3) is an L2 induced norm, which is equal to the H ∞ -norm of the system
TI , so Problem 3.1 is the H ∞ optimization problem.

3.3.2

Reduction to a finite-dimensional problem

A difficulty in Problem 3.1 is that it involves a continuous-time delay, and hence it is an
infinite-dimensional problem. Another difficulty is that it contains the upsampler ↑ L sothat it makes the overall system time-varying (to be precise, periodically time-varying).
However we can reduce the problem to a finite-dimensional single-rate one. Let{AF , BF , CF , 0} be a realization of F (s), that is, the state equation of F (s):ẋF = AF xF + BF wc ,yc = CF xF .

Theorem 3.1. For the error system TI , there exist (finite-dimensional) discrete-time systems {TI,N : N = L, 2L, . . . } such that
lim kTI,N k = kTI k.

(3.4)

N →∞

Proof. We first reduce the problem to a single-rate one. Recall the property (3.2) of thediscrete-time lifting LL and its inverse L−1
L :LL := (↓ L)

1 z . . . z L−1

Then K(z)(↑ L) can be rewritten aseK(z)(↑ L) = L−1
L K(z),

T

,

L−1
L :=



1 z −1 · · · z −L+1eK(z):= LL K(z)L−1L

1 0 ... 0(↑ L).

T.

e
The lifted system K(z)is an LTI, single-input/L-output system that satisfies
K(z) =

1 z −1 · · · z −L+1

eh defined byUsing the generalized hold H
eh : l2 3 v 7→ u ∈ L2 ,Hu(kh + θ) = H(θ)v[k],e L ).
K(zθ ∈ [0, h),k = 0, 1, 2, . . . ,23

3.3. DESIGN OF INTERPOLATORS

-

e−mhs

? ec
j-

wc

-F (s)- Sh- K(z)e- Heh-P (s)

6−

Figure 3.11: Reduced single-rate problem
where H(·) is the hold function:ih
1 0 0 . . . 0 , θ ∈ [0, h/L),


h
i


 0 1 0 . . . 0 , θ ∈ [h/L, 2h/L),
H(θ) :=

···


h
i


 0 0 . . . 0 1 , θ ∈ [(L − 1)h/L, h),we obtain the identitye
Hh/L L−1
L = Hh .

This yields

eh K(z)Se
Hh/L K(z)(↑ L)Sh = H
h.

Hence Figure 3.10 is equivalent to Figure 3.11.
We modify the diagram in Figure 3.11 into the diagram in Figure 3.12, and we in-

ec



yd

wc



Sh





e−mhs F (s), −P (s)F (s),
0

-





eh
H



ud

e
K(z)

Figure 3.12: Sampled-data system TI
troduce the fast-sampling/fast-hold approximation [19, 43] in order to obtain a finitedimensional discrete-time system approximately. Figure 3.13 illustrates the procedure.

24

CHAPTER 3. MULTIRATE SIGNAL PROCESSING

ẽd



ec
Sh/N 

wc

TI



w̃d
Hh/N 

Figure 3.13: Fast-sampling/fast-hold discretization
By the fast-sampling/fast-hold approximation, we obtain the approximated discrete-timesystem TI,N (N := Ll, l ∈ N),

where

e
TI,N (z) = z −m GI11,N (z) + GI12,N (z)K(z)G
I21,N (z),GI11,N := z −m LN (F ),S := [1, 0, . . . , 0],| {z }
N −1

GI12,N := −LN (P )H, GI21,N := SLN (F ),


q


..
H := 
. . . , 1]T .
, q := [1,.
| {z }
q
l
|
{z
}
L

e
Figure 3.14 shows the obtained discrete-time system, where K(z)is an LTI, singleẽ

d

yd



z −m GI11,N GI12,N
GI21,N
0





w̃d



ud

- K(z)eFigure 3.14: Discrete-time system TI,N
input/L-output system that satisfies
e L ).
K(z) = 1 z −1 · · · z −L+1 K(zThe convergence of (3.4) is guaranteed in [43].

3.4

Design of decimators

3.4.1

Problem formulation

We now formulate a design problem for optimal decimators. While this can be considered
dually with interpolators, it is less studied in the literature. Downsampling occurs usuallyin the filter bank design, and its independent design has received less attention.

25

3.4. DESIGN OF DECIMATORS

wc- F (s)yc- Sh/M

yd

-H(z)

ud

- ↓M

xd

- Hh

A/D

xc

- P (s)

zc

-

D/A

Figure 3.15: Signal reconstruction with decimator
Consider the block diagram Figure 3.15. The incoming signal wc first goes throughan anti-aliasing filter F (s) and the filtered signal yc becomes nearly (but not entirely)
band-limited. This signal is then sampled by Sh/M to become a discrete-time signal yd
with sampling period h/M.
The discrete-time signal yd is first processed by a digital filter H(z). Then the filteredsignal xd is downsampled by ↓ M, and becomes another discrete-time signal ud withsampling period h. The discrete-time signal ud then becomes a continuous-time signal
uc by going through the zero-order hold Hh , and then becomes the final signal by going
through an analog filter P (s). The objective here is to design the digital filter H(z) for
given F (s), M and P (s).
Figure 3.16 shows the block diagram of the error system for the design. The delay inthe upper portion of the diagram corresponds to the fact that we allow a certain amountof time delay for signal reconstruction. Let TD denote the input/output operator from wc- e−mhswc- F (s)+yc- Sh/M- H(z)- ↓M- Hh- P (s)ec?
j6−

Figure 3.16: Signal reconstruction error system
to ec in Figure 3.16. Our design problem is then as follows:
Problem 3.2. Given a stable, strictly proper F (s), stable, proper P (s), downsampling
factor M ∈ N, delay step m ∈ N, sampling period h > 0 and an attenuation level γ > 0,
find a digital filter H(z) such thatkTD k :=

kTD wc kL2 [0,∞)< γ.
wc ∈L2 [0,∞) kwc kL2 [0,∞)supwc 6=0(3.5)

26

CHAPTER 3. MULTIRATE SIGNAL PROCESSING

-

e−mhs

? ec
j-

wc

-F (s)- Seh- H(z)e- Hh-P (s)

6−

Figure 3.17: Reduced single-rate problem

3.4.2

Reduction to a finite-dimensional problem

Theorem 3.2. For the error system TD , there exist (finite-dimensional) discrete-timesystems {TD,N : N = M, 2M, . . . } such that
lim kTD,N k = kTD k.

(3.6)

N →∞

Proof. Using the discrete-time lifting LM we rewrite (↓ M)H(z) asee(↓ M)H(z) = H(z)LH(z):= 1 0 · · · 0 LM H(z)L−1
M,
M ,

e
where the system H(z)is an LTI, M-input/single-output system that satisfiese M ) 1 z . . . z M −1 T .
H(z) = H(zUsing the generalized sampler Seh defined by
Seh : L2 3 u 7−→ v ∈ l2 ,we obtain the identity
v[k] := 



u kh

u kh + h/M..
.

u kh + (M −1)h/M







,


k = 0, 1, 2, . . . ,LM Sh/M = Seh .

Hence Figure 3.16 is equivalent to Figure 3.17. As has been mentioned above, this canbe reduced to a finite-dimensional discrete-time system.
We modify the block diagram in Figure 3.17 into the block diagram in Figure 3.18.
Then by using the fast-sampling/fast-hold method, the sampled-data system in Figure
3.18 is approximated to the following discrete-time system TD,N (N := Ml, l ∈ N),

where

e
TD,N (z) = z −m GD11,N (z) + GD12,N (z)K(z)G
D21,N (z),GD11,N := z −m LN (F ), GD12,N := −LN (P )H, GD21,N := SLN (F ),



p



.
.
S := 
. . , 0], H := [1, . . . , 1]T .
 M, p := [1, |0, .{z.
}| {z }

p 
l−1
N

27

3.4. DESIGN OF DECIMATORS

ec



yd

wc





Seh



e−mhs F (s), −P (s)F (s),
0

-







Hh

e
H(z)

Figure 3.18: Sampled-data system TD

ẽ

d

yd



z −m GD11,N GD12,N
GD21,N
0





w̃d



ud

- H(z)eFigure 3.19: Discrete-time system TD,N

ud

28

CHAPTER 3. MULTIRATE SIGNAL PROCESSING

Figure 3.19 shows the obtained finite-dimensional discrete-time system.
The convergence of (3.6) is guaranteed in [43].
Note that the decimation filtere M)H(z) = H(zTz M −1

1 z ...

may not be causal, thus we adopt the following filter:

3.5

e M)
H(z) = z −M H(z

1 z . . . z M −1

,

T

.

Design of sampling rate converters

By combining interpolators and decimators, we can construct a sampling rate converter.
Figure 3.20 shows a sampling rate converter, where an interpolation with factor M1 isfollowed by a decimation with factor M2 . By this converter, the sampling rate of the
input signal is changed by the factor M1 /M2 . In the application of digital audio, the- ↑ M1-L(z)- ↓ M2-

Figure 3.20: Sampling rate converter (M1 : M2 )conversion from CD signals at 44.1kHz to DAT signals at 48kHz is realized by a converterwith the factors M1 = 3 × 72 and M2 = 25 × 5. Conventionally, the digital filter L(z) is
designed to be a low-pass filter with the cut-off frequency ω = π/M, M := max(M1 , M2 )[9, 35]. In this section, we design the filter L(z) that combines the interpolation filter H(z)designed by the method discussed in Section 3.3 and the decimation filter H(z) in Section
3.4. The designed sampling rate converter will be in the form illustrated in Figure 3.21
The block diagrams of the error system for sampled-data H ∞ design of an interpolation- ↑ M1- K(z)- H(z)- ↓ M2-

Figure 3.21: Sampling rate converter with interpolator and decimatorfilter K(z) and a decimation filter H(z) are shown in Figure 3.22 and in Figure 3.23,
2respectively. In these diagrams, h2 = h · M
and the analog low-pass filters F1 (s) andM1F2 (s) have characteristics illustrated in Figure 3.24. The filters F1 and F2 take accountof the characteristic of the analog input signal, and hence we can design filters K(z)and H(z) that optimize the analog performance using the sampled-data system design
method.

29

3.5. DESIGN OF SAMPLING RATE CONVERTERS- e−m1 hs
wc- F1 (s)

+

yc
- Sh

- ↑ M1- K(z)- Hh/M1-P1 (s)? ec
j6−

Figure 3.22: Error system for designing interpolator- e−m2 h2 s
wc
- F2 (s)+yc- Sh2 /M2

- H(z)- ↓ M2- Hh2- P2 (s)ec?
j6−

Figure 3.23: Error system for designing decimator

F1

F2

ωN

M1 ωN

Figure 3.24: Characteristic of F1 and F2

ω30

CHAPTER 3. MULTIRATE SIGNAL PROCESSING

The advantage of the design mentioned above is that we can design a converter withlarge M1 or M2 . For example, to design a converter for CD/DAT (i.e., M1 = 3 × 72 andM2 = 25 × 5), we first design interpolators with M = 3, 7 and decimators with M = 2, 5
as shown in Figure 3.25, and then by combining the interpolation filters K3 (z), K7 (z) and-↑3- K3 (z)-- H2 (z)

-

↓2

-

-

↑7

- K7 (z)-- H5 (z)-↓5-interpolators

decimators

Figure 3.25: Interpolators and decimators for CD/DAT sampling rate conversion
the decimation filters H2 (z), H5 (z), we obtain the sampling rate converter as illustrated
in Figure 3.26.
- ↑3- K3 (z)- ↑7- K7 (z)- ↑7- K7 (z)- H2 (z)- ↓2-- H2 (z)

- ↓2

r r

- H5 (z)- ↓5-Figure 3.26: Sampling rate converter for CD/DAT
On the other hand, there is a design method for sampling rate converters by periodically time-varying systems [15]. However, the order of the design in the method will be
M1 × M2 , and hence the design of a converter with very large number of M1 or M2 suchas CD/DAT (M1 × M2 = 23520) has much difficulty in numerical computation.

3.6

Design examples

3.6.1

Design of interpolators

In this section, we compare the interpolator designed by the method discussed in Section
3.3 with that designed by Johnston’s method [17]. The parameters are as follows: inter-31

3.6. DESIGN EXAMPLES

polation ratio M = 2, sampling period h = 1 and delay step m = 2. The analog filters
F (s) and P (s) are
F (s) =

1,
(T s + 1)(0.1T s + 1)

T := 22.05/π ≈ 7.0187,

P (s) = 1.

Note that the low-pass filter F (s) has first order attenuation in the frequency range ω ∈[0.14248, 1.4248] [rad/sec], and second order attenuation in the range ω > 1.4248 [rad/sec].
The filter simulates the frequency energy distribution of a typical orchestral music. The
Johnston filter is taken to be of order 31.
The frequency responses of the obtained filters are shown in Figure 3.27. The sampleddata design filter is of order 7, which is lower than the Johnston filter. The Johnston filter
Frequency Response
10
0
−10

Gain [dB]

−20
−30
−40
−50
−60
−70
−80
0

0.5

1

1.5
2
Frequency [rad/sec]

2.5

3

3.5

Figure 3.27: Frequency responses of interpolation filters: sampled-data design (solid) andJohnston filter (dash)shows the sharper decay beyond the cut-off frequency ω = π/2, while the filter obtainedby the sampled-data design shows a rather slow decay.
On the other hand, the reconstruction error (see Figure 3.10) characteristic in Figure
3.28 exhibits quite an admirable performance in spite of the low-order of the sampled-data
design filter. It is almost comparable with 31st order Johnston filter.
While for those frequencies much below the cut-off frequency the gain characteristic of
the sampled-data design is not as good as the Johnston filter, the sampled-data designed
filter need not be inferior. To see this, let us see the time responses against rectangularwaves in Figure 3.29 (sampled-data designed) and in Figure 3.30 (Johnston filter).
The Johnston filter exhibits a very typical Gibbs phenomenon (i.e., we can see ringing
caused by the sharp characteristic of the filter), whereas the one by the sampled-data

32

CHAPTER 3. MULTIRATE SIGNAL PROCESSING

Frequency Response
−33
−34
−35

Gain [dB]

−36
−37
−38
−39
−40
−41
−42 −2
10

−1

10

0

10

1

10

Frequency [rad/sec]

Figure 3.28: Frequency responses of error system: sampled-data design (solid) and Johnston filter design (dash)

1.5

1

0.5

0

−0.5

−1

−1.5
0

5

10

Figure 3.29: Time response (sampled-data design)

15

33

3.6. DESIGN EXAMPLES

1.5

1

0.5

0

−0.5

−1

−1.5
0

5

10

15

Figure 3.30: Time response (Johnston filter)
design shows a much smaller peak near the edge. We also note that the sampled-data
designed filter has nearly linear phase as shown in Figure 3.31.

3.6.2

Design of decimators

We now present an example of the H ∞ design of decimators discussed in Section 3.4. For
comparison, we take the Johnston filter of order 31.
Let the decimation ratio M = 2 and the other parameters is the same as the interpolator design in the previous section.
Figure 3.32 shows the frequency response of the decimation filters. Note that thefilter designed by the sampled-data method is of order 6. The Johnston filter shows the
sharper decay beyond the cut-off frequency ω = π/2, while the filter by sampled-data
design shows a rather slow decay.
Figure 3.33 shows the frequency response of the error system (see Figure 3.16). We
can see from the frequency response of the error system that the decimator designed by
the sampled-data method exhibits a clear advantage over all frequency range, even though
the sampled-data designed filter is of lower order than the Johnston filter. In particular,
around the frequency ω ≈ 4 [rad/sec], the difference is about 20 dB.
Figure 3.34 and Figure 3.35 shows the time responses against rectangular waves.
The sampled-data designed decimator reconstructs the rectangular wave well, while thedecimator with the Johnston filter exhibits a large amount of ringing due to the Gibbs
phenomenon.

34

CHAPTER 3. MULTIRATE SIGNAL PROCESSING

Phase
0
-100
-200

Phase [degree]

-300
-400
-500
-600
-700
-800

0

0.5

1

1.5
2
Frequency [rad/sec]

2.5

3

3.5

Figure 3.31: Phase response (Sampled-data design)Frequency response
10
0
−10

Gain [dB]

−20
−30
−40
−50
−60
−70
−80
0

0.5

1

1.5
2
Frequency [rad/sec]

2.5

3

3.5

Figure 3.32: Frequency responses of decimation filters: sampled-data design (solid) andJohnston filter (dash)35

3.6. DESIGN EXAMPLES

Frequency response
−10
−15
−20

Gain [dB]

−25
−30
−35
−40
−45
−50 −2
10

−1

10

0

10

1

10

Frequency [rad/sec]

Figure 3.33: Frequency responses of error system: sampled-data design (solid) and Johnston filter design (dash)

1.5

1

0.5

0

−0.5

−1

−1.5
0

5

10

Figure 3.34: Time response (sampled-data design)

15

36

CHAPTER 3. MULTIRATE SIGNAL PROCESSING

1.5

1

0.5

0

−0.5

−1

−1.5
0

5

10

15

Figure 3.35: Time response (Johnston filter)

3.6.3

Design of sampling rate converters

In this section, we present a design example for the case of changing the sampling period
from h1 = 1 to h2 = 4/3. Then we have the sampling rate converter with M1 = 3 andM2 = 4 that are coprime (see Figure 3.20). Let the filter for the interpolator design (seeFigure 3.22) be
F1 (s) =1,(T s + 1)(0.1T s + 1)

P1 (s) = 1,

and that for the decimator design (see Figure 3.23) be
F2 (s) =1,(T2 s + 1)(0.1T2 s + 1)

P2 (s) = 1,

where T := 22.05/π, T2 := T /M1 . The filters simulate the frequency energy distributionof a typical orchestral music, which are observed by FFT analysis of analog records ofsome orchestral musics.
An approximate design is executed here for N = M1 × 4 = 12 (interpolator) andN = M2 × 4 = 16 (decimator). For comparison, we compare it with the equiripple filter
obtained by Parks-McClellan method [35, 46] of order 31. Parks-McClellan method is
widely used for designing FIR filters [46]. The delay stemps m1 and m2 are 2.
The obtained (sub) optimal interpolation filter K(z) is of order 11 and the decimation
filter H(z) of order 15. The sampling rate conversion filter L(z) = H(z)K(z) is of order221) .
1)

The order of L(z) is reduced by the minimal realization method.

37

3.6. DESIGN EXAMPLES

Figure 3.36 shows the gain characteristics of these filters. The equiripple filter shows
the sharper decay beyond the cut-off frequency (π/4 [rad/sec]) while the sampled-data
design shows a rather mild cut-off characteristic.
Frequency response
20
0
−20

Gain [dB]

−40
−60
−80
−100
−120
−140
0

0.5

1

1.5
2
Frequency [rad/sec]

2.5

3

3.5

Figure 3.36: Frequency responses of sampling rate conversion filters: sampled-data design
L(z) (solid) and quiripple filter Le (z) (dash)In spite of these superficial differences, the frequency response of the error system(illustrated in Figure 3.37, where m := m1 + m2 and P (s) = P1 (s) = P2 (s) = 1) ofsampling rate converter exhibits quite an admirable performance of the sampled-data
design as shown in Figure 3.38.
wc

Sh/M2

F1 (s)

↑ M1

L(z)

e−mhs
ec

+
−

P (s)

Hh/M1

↓ M2

Figure 3.37: Error system of sampling rate converter
It is interesting to observe that the slow decay need not yield an inferior design. To see
this, let us see the time responses against a rectangular wave in Figure 3.39 and Figure

38

CHAPTER 3. MULTIRATE SIGNAL PROCESSING

Frequecy response
−20

−30

Gain [dB]

−40

−50

−60

−70

−80 −2
10

−1

0

10

1

10
Frequency [rad/sec]

10

2

10

Figure 3.38: Frequency responses of error system: sampled-data design (solid) and equiripple design (dash)

1.5

1

0.5

0

−0.5

−1

−1.5
0

5

10

Figure 3.39: Time response (sampled-data design)

15

39

3.7. CONCLUSION

1.5

1

0.5

0

−0.5

−1

−1.5
0

5

10

15

Figure 3.40: Time response (equiripple design)
3.40. The response with the equiripple filter shows a large amount of ringing, whereasthat with the filter by the sampled-data design has much less peak around the edge. Notealso that L(z) is nearly linear phase up to a certain frequency as shown in Figure 3.41.

3.7

Conclusion

Conventional theories of digital signal processing assert that the ideal filter is the best forinterpolation or decimation. However, as we have shown above, a sharp filter characteristic
approximating the ideal filter does not necessarily behave well. In particular, such a filter
often exhibits a large amount of ringing as illustrated in the previous section. The ringing
is due to the Gibbs phenomenon, which is caused by the sharp characteristic of the filter.
On the other hand, our filter shows a slow decay. The reason is that due to the underlying analog characteristic (i.e., F (s)), there is an important information content beyond
the Nyquist frequency, and such a slow decay is necessary to recover such information.
Moreover, conventional design requires us to give a filter order in advance. The higher
the order is, the closer to the ideal characteristic the filter is, and hence filters of a very
high order are often used.
In contrast to the conventional design, our method is free from the choice of filter
order. Namely, the order of designed filter depends on the order of filter F (s) and P (s),delay step m and upsampling (or downsampling) ratio M. As indicated in the previous
section, in spite of the fact that the order of the obtained filter is not very high, theresponse is better than high-order conventional filters. This fact cannot be recognizedwithout considering the analog performance.

40

CHAPTER 3. MULTIRATE SIGNAL PROCESSING

Frequency response
0Phase [degrees]

−200

−400

−600

−800

−1000

−1200
0

0.5

1

1.5
2
Frequency [rad/sec]

2.5

Figure 3.41: Phase plot of L(z)33.5Chapter 4
Application to Communication
Systems
4.1

Introduction

The importance of digital communication is ever increasing owing to the rapid growth of
the Internet, cellular phones, and so on [32]. In digital communication, especially in pulse
amplitude modulation (PAM) or in pulse code modulation (PCM), the analog signal to betransmitted is sampled and becomes a discrete-time signal. In the conventional method,
the analog characteristics of the signals are not considered, and hence the total system isregarded as a discrete-time system. Namely, one usually assumes that the original analog
signal is band-limited up to the Nyquist frequency.
In [8], a discrete-time H ∞ design of receiving filters or equalizers is introduced. This
design is based on the assumption of full band-limitation, but in reality no signals are fully
band-limited. Moreover, it is difficult to attenuate both the signal reconstruction error
and the distortion caused by the channel only by equalizing after receiving. Therefore
an enhancer (or transmitting filter) that amplifies the signal before transmission is often
attached in order to increase signal-to-noise ratio [32].
In this chapter, we propose a new design of receiving/transmitting filters by using thesampled-data control theory. Moreover, we introduce the H ∞ method that takes accountof a tradeoff between the quality of signal reconstruction and the cost (i.e., the amount ofenergy of transmitting signals) with an appropriate weighting function. Design examples
are presented to illustrate the effectiveness of the proposed method.

4.2

Digital communication systems

Figure 4.1 illustrates a typical digital communication system. In this figure, the source
is assumed to be an analog signal (e.g., audio, speech or image). The analog signal will
be discretized with an A/D converter, which contains sampler, quantizer and encoder (orcoder).
Sampling is a discretization in time, while quantization is that in amplitude.
Encoder converts the sampled and quantized signal to a binary valued signal. It
41

42

CHAPTER 4. APPLICATION TO COMMUNICATION SYSTEMS

Quantizer

Sampler

SourceEncoder
Channel

OutputHold

FilterDecoder

Figure 4.1: Digital communication system
often contains compressing and filtering for efficiency of signal transmission. For example,
subband coding is often used for efficient communication. Figure 4.2 illustrates a simple
subband coding system. The input signal y is divided into two subband signals1) . By
H1 (z)

↓2

y1

y

v1
Q

H2 (z)↓2y2

v2

Figure 4.2: Subband coding
taking the filters H1 (z) and H2 (z) appropriately, we can divide the frequency into two
subbands, for example, y1 is a signal with low frequency and y2 with high frequency. If
we need not reconstruct the information in the high frequency range precisely, we cancompress the signal by applying fewer bits to the high-frequency signal (i.e. the signal y2
in Figure 4.2). For simplicity, we apply infinitely many bits to y1 and no bit to y2 , thatis, we assume Q in Figure 4.2 as


1 0
.
(4.1)
Q=
0 0
This means that we transmit only the signal v1 .
The signal then goes through the communication channel. Since modeling for the chan-

v

C(z)

n +
+
ν

Figure 4.3: Communication channel model1)

In real applications, the signal will be subdivided into 16 or 32 signals.

434.3. DESIGN PROBLEM FORMULATION

nel is an important and difficult issue, we assume a simple model: a linear time-invariant
system and additive noise as shown in Figure 4.3. Note that communication channels are
generally time-varying and nonlinear, in particular, in wireless communication, channel
modeling is very complicated [32, 33].
Then, the signal distorted by the channel enters the receiver. The receiver decodes
the binary signal, and then filters the decoded signal. The filter reduces distortion of thereceived signal and expands the compressed signal. The process is shown in Figure 4.4.
Since we take the encoding (4.1), we similarly assume Q0 to be the same as Q, that is,


1 0
0
Q =
.
0 0
ν1

↑2

F1 (z)

u1
+

0

Q
ν2

↑2

F2 (z)u2u+Figure 4.4: Decoding
Then the decoded and filtered signal is converted to an analog signal by a hold device,and finally we obtain a received signal.
The conventional design is executed in the discrete-time domain by assuming that theoriginal analog signal contains no frequency beyond the Nyquist frequency. To satisfy thisassumption, we often use a low-pass filter, which deteriorates the quality of communication.
In contrast to this, we introduce the sampled-data H ∞ control theory to take the analog performance into account. We will show that our design is superior to the conventional
design in the following sections.

4.3

Design problem formulation

We consider a digital communication system as shown in Figure 4.5. The incoming signal
wc ∈ L2 [0, ∞) goes through an analog low-pass filter Fc and becomes yc that is nearly(but not entirely) band limited. The filter Fc governs the frequency-domain characteristicof the analog signal yc 2) . The signal yc is then sampled by the sampler Sh/M to become adiscrete-time signal (i.e., PAM signal) yd with sampling period h/M. Then the signal iscompressed by the downsampler with the factor M, and becomes a discrete-time signal
with sampling period h. The downsampled signal is then shaped or enhanced by the2)

In the conventional design, Fc is considered to be an ideal filter that has the cut-off frequency up to
the Nyquist frequency.

44

CHAPTER 4. APPLICATION TO COMMUNICATION SYSTEMS

wc

Sh/M

F (s)

yd

↓M

KT (z)

vd
C(z)
+

zc

P (s)

uc

Hh/M

ud

n

+
↑M

KR (z)

Figure 4.5: Digital communication systemtransmitting digital filter KT to the signal vd to be transmitted to a communication
channel.
The transmitted signal vd is corrupted by the communication channel C and the
additive noise nd . In PCM communication, nd is also considered as the noise generated
by quantization and coding errors. The received signal goes through the upsampler andreceiving digital filter KR that attempts to attenuate the imaging components caused by
the upsampler and the distortion by the channel. Then the signal becomes an analog
signal uc by the hold device Hh/M with sampling period h/M and this analog signal issmoothed by an analog low-pass filter Pc . Finally we have the output signal zc .
Our objective is to reconstruct the original analog signal yc by the transmitting filter
KT and the receiving filter KR against data compression effect caused by the downsampler
and the distortion caused by the channel.
We thus consider the block diagram shown in Figure 4.6 that is the signal reconstruction error system for the design. In the diagram the following points are taken into
account:• The time delay e−Ls is introduced because we allow a certain amount of time delay
for signal reconstruction.
• The transmitted signal vd is estimated with a weighting function Wz because theenergy or the amplitude of the transmitted signal vd is usually limited.
• The noise obeys a frequency characteristic Wn .
Our design problem is defined as follows:
Problem 4.1. Given a stable, strictly proper, continuous-time F (s), stable, proper, continuoustime P (s), stable proper discrete-time weighting functions Wn (z) and Wz (z), stable proper
channel model C(z), downsampling factor M, delay time L and a sampling period h, finddigital filters KT (z) and KR (z) that minimizes
J(KR , KT ) :=supwc ∈L2 ,nd ∈l2
kwc kL2 +knd kl2 6=0

kec k2L2 + kzd k2l2.
kwc k2L2 + knd k2l2

(4.2)

45

4.4. DESIGN ALGORITHM

wc

Sh/M

F (s)

yd

KT (z)

↓M

vd

Wz

zd

C(z)
−Ls

e
ec

+
+zc
−

P (s)

uc

Hh/M

ud

+
KR (z)

↑M

Wn

nd

ν

Figure 4.6: Signal reconstruction error system

4.4

Design algorithm

4.4.1

Decomposing design problems

Problem 4.1 is a simultaneous design problem of a transmitting filter and a receiving filter,and it is difficult to solve this problem directly. We thus decompose the design problem
into two steps, that is, the design of the receiving filter and that of the transmitting filter.
Obviously the transmitting filter KT cannot attenuate the additive noise nd , hence thereceiving filter KR must play this role. Moreover KR must reconstruct the original signal
from the corrupted signal (if KR did not have to reconstruct, the optimal filter will beclearly KR = 0). Therefore we first design the receiving filter KR in order to reconstruct
the original signal and to attenuate the noise by the block diagram shown in Figure 4.6
with Wz = 0 and with KT = 1. We then design the transmitting filter by the block
diagram shown in Figure 4.6 with Wn = 0 and with KR that is obtained in the previous
design, that is, we consider the channel as KR (↑ M)C.
Denote TR the system from [wc , nd ]T to ec (shown in Figure 4.7), and TT the systemfrom wc to [ec , zd ]T (shown in Figure 4.8). The design procedure is then as follows:
Step 1 (Design of a receiving filter) Find a receiving filter KR that minimizes
J1 (KR ) := kTR k2 :=supwc ∈L2 ,nd ∈l2
kwc kL2 +knd kl2 6=0

kec k2L2,
kwc k2L2 + knd k2l2

(4.3)

with fixed KT .
Step 2 (Design of a transmitting filter) Find a transmitting filter KT that minimizes
kec k2L2 + kzd k2l2,
kwc k2L2
wc ∈L2J2 (KT ) := kTT k2 := sup(4.4)

wc 6=0with KR obtained in the previous step. We iterate Step 1 and Step 2 alternately withinitial condition KT = 1.
The filter KT designed in Step 2 cannot have any influence on the performance of thesystem from nd to ec . Therefore, the objective function J(KR , KT ) in (4.2) monotonicallydecreases by each step. In fact, we have the following proposition:

46

CHAPTER 4. APPLICATION TO COMMUNICATION SYSTEMS

wc

Sh/M

F (s)

yd

vd

↓M

KT (z)

C(z)
−mhs

e

+

+zc

ec

−

P (s)

uc

Hh/M

ud

Wn (z)+↑M

KR (z)ndνdFigure 4.7: Error system TR for receiving filter design
wc

Sh/M

F (s)

yd

KT (z)vd↓M

Wz (z)

zd

C(z)
−mhs

e

+zc

ec

−

P (s)

uc

Hh/M

ud

KR (z)↑M

Figure 4.8: Error system TT for transmitting filter design
Proposition 4.1. For any integer n ≥ 1, the following inequality holds:
(n−1)

J(KR
(n)

(n)

where KR and KT

(n−1)

, KT

(n)

(n)

) ≥ J(KR , KT ) ≥ Jopt ,are filters obtained at the n-th design step, and define
Jopt := min J(KR , KT ).

(4.5)

KR ,KT

Proof. Let T1 (KR , KT ) and T2 (KR ) be the system from wc to [ec , zd ]T and the systemfrom nd to [ec , zd ]T , respectively, in Figure 4.6. At the n-th step of KR design (Step 1),
we have(n)(n−1)

KR = arg min J1 (KR ) = arg min J(KR , KT
KR

KR

),

and hence
(n−1)

J(KR

(n−1)

, KT

(n)

(n−1)

) ≥ J(KR , KT),

holds. Then at the n-th step of KT design (Step 2), we have
(n)

(n)

KT = arg min J2 (KT ) = arg min kT1 (KR , KT )k.
KT

KT

(4.6)

47

4.4. DESIGN ALGORITHM

By using this equation, we have
(n)

(n−1)

kT1 (KR , KT

(n)

(n)

)k ≥ kT1 (KR , KT )k.

It follows that
h
i
h
i
(n)
(n)
(n)
(n)
(n−1)
(n)
,
≥
T1 (KR , KT ), T2 (KR )T1 (KR , KT), T2 (KR )holds. This implies

(n)

(n−1)

J(KR , KT

(n)

(n)

(n)

(n)

) ≥ J(KR , KT ).

(4.7)

By (4.6) and (4.7), we have
(n−1)

J(KR

(n−1)

, KT

) ≥ J(KR , KT ).

Then, by the definition (4.5),
(n)

(n)

J(KR , KT ) ≥ Jopt ,is obvious.

4.4.2

Fast-sampling/fast-hold approximation

The design problems (4.3) and (4.4) involve a continuous-time delay component e−Ls ,and hence they are infinite-dimensional sampled-data problems. To avoid this difficulty,
we employ the fast-sampling/fast-hold approximation method [19, 43]. By the method,
our design problems (4.3) and (4.4) are approximated by finite-dimensional discrete-time
problems assuming that the delay time L is mh (m ∈ N).
Theorem 4.1. Assume that L = mh, m ∈ N. Then,
1. for the error system TR in Step 1, there exist finite-dimensional discrete-time systems{TR,N : N = M, 2M, . . . } such that
lim kTR,N k = kTR k,

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

N →∞Proof. By the fast-sampling/fast-hold method, we approximate continuous-time inputs
and outputs to discrete-time ones via the ideal sampler and the zero-order hold that
operate in the period h/N (Figure 4.9). Assume N = Ml, l ∈ N. Then apply thediscrete-time lifting LN (see Section 2.4) to the discretized input/output signal edN and
wdN , we can obtain the lifted signals
edN := LN (edN ) = LN Sh/N ec ,ew
edN := LN (wdN ) = LN Sh/N wc .

48

CHAPTER 4. APPLICATION TO COMMUNICATION SYSTEMS

edN



ec
Sh/N 

wc
wdN
T  Hh/N 

Figure 4.9: Fast-sampling/fast-hold discretization
We next denote TR,N the system from [wedN , nd ]T to e
edN and the system TT,N from w
edN
T
to [e
edN , zd ] , and define their norm
kTR,N k2 :=supwdN ,nd ∈l2
kwdN kl2 +knd kl2 6=0ke
edN k2l2,kw
edN k2l2 + Nh knd k2l2

ke
edN k2l2 + Nh kzd k2l2
kTT,N k := sup.
kw
edN k2l2wdN ∈l22

wdN 6=0

Then we can show kTR,N k → kTR k, kTT,N k → kTT k, as N → ∞ by using the method asshown in [43] under the assumption L = mh.
Once the problems have been reduced to discrete-time problems, they can be solved bystandard softwares such as MATLAB. The resulting discrete-time approximant is givenby the following:
Theorem 4.2. The approximated discrete-time systems TR,N and TT,N are given as follows:e R , TT,N := Fl GT,N , K
eT ,TR,N := Fl GR,N , K"  −m

#
z
F
,
0
,
−P
H
dN
dN
h
i
GR,N :=,
e T SFdN , Wn ,
0
CK
 
 
  −me RC
z FdN−PdN H K,
,0GT,N := Wz
SFdN ,
0



p



.
..
S := 
. . , 0],
 M, p := [1, |0, .{z
}


p
k−1


q


..
H := 
. . . , 1]T ,, q := [1,.
| {z }
q
k
{z
}
|
M

FdN := LN (F ),PdN (z) := LN (P ).

Proof. First, consider the receiving filter design in Figure 4.7. The relation between the

49

4.5. DESIGN EXAMPLES

input and the output is as follows:
ec = e−mhs F wc − P Hh/M ud ,
νd = C(↓ M)KT Sh/M F wc + Wn nd ,ud = KR (↑ M)νd .

(4.8)
(4.9)
(4.10)

Then applying the fast-sampling and the discrete-time lifting LN to the input wc and theoutput wc , we have from (4.8)
LN Sh/N ec = LN Sh/N e−mhs F Hh/N L−1
eN d − LN Sh/N P Hh/M ud
N w−m= z LN (F )weN d − LN (P )HLM ud ,and from (4.9)νd = C(↓ M)KT Sh/M F Hh/N L−1
eN d + Wn nd
N w−1= C(↓ M)KT LM SLN (F )w
e + Wn nd .

In these equations we have used Proposition 2.1. Then by using the relation (3.1), weconvert the multirate systems to single rate ones:Te
LM ud = LM KR (↑ M)νd = LM KR L−1
M [1, 0, . . . , 0] νd = KR νd ,eT .
C(↓ M)KT L−1 = C[1, 0, . . . , 0]LM KT L−1 = C K
MConsequently, we have

M




w
eN d
edN = z FdN 0e− PdN H u
ed,
nd

h
i
w
e
N
d
e T SFdN W n
νd = C K,nde R νd ,u
ed = K−me R )wwhere ued := LM ud . It implies that e
edN = Fl (GR,N , K
edN .
The proof for GT,N is almost similar to that for GR,N and hence we omit the proof.
Then our design problems (4.3) and (4.4) are reduced to finite-dimensional discrete-time
H ∞ design problems, which are shown in Figure 4.10 and Figure 4.11.

4.5

Design examples

4.5.1

The case of no compression (M = 1)

Design for Wz = 0
In this section, we present a design example for1
F (s) :=, P (s) := 1, Wn (z) := 1,
10s + 1C(z) := 1 + 0.65z −1 − 0.52z −2 − 0.2975z −3 ,

50

CHAPTER 4. APPLICATION TO COMMUNICATION SYSTEMS

edN
e

GR,N

nd

w
edN

KR
Figure 4.10: Discrete-time system for designing receiving filter KR
eedN
zd

GT,N

w
edN

KT
Figure 4.11: Discrete-time system for designing transmitting filter KT
with sampling period h = 1 and time delay L = mh = 2. We here consider the caseof no compression (i.e., M = 1) and no limitation on transmission (i.e., Wz = 0). An
approximate design is executed here for N = 8. For comparison, the discrete-time H ∞design [8] is also executed.
Figure 4.12 shows the gain responses of the filters, and Figure 4.13 shows the frequency
responses of Tew (the system from the input wc to the error ec ), and Figure 4.14 shows
those of Tzn (the system from the additive noise nd to the output zc ). Compared with thediscrete-time design, the sampled-data one shows better frequency response both in Tew
and in Tzn . Moreover, we can say that an equalizer alone cannot attenuate the corruption
caused by the channel and the additive noise, that is, we need an appropriate transmitterfor transmission.
To explain this fact, we show a simulation of these communication systems. The input
signal yc is a rectangular wave with amplitude 1, and the noise disturbance nd is a discretetime sinusoid: nd [k] = sin(2k). Figure 4.15 shows the output zc with the receiving filter
and the transmitting filter designed via sampled-data method, and Figure 4.16 shows thatwith the receiving filter designed in discrete-time (and without any transmitting filter).
We see that the former shows much better reconstruction against the noise than the latter.

Design for Wz (z) 6= 0
We now consider the design with the estimation of the transmitting signal vd , that is
Wz (z) 6= 0.
We observe from Figure 4.12 that the transmitting filter shows high gains around the

51

4.5. DESIGN EXAMPLES

Frequency Response30

20

Gain [dB]

10

0

−10

−20

−30

−40
0

0.5

1

1.5
2
Frequency [rad/sec]

2.5

3

3.5

Figure 4.12: Gain responses of filters: sampled-data H ∞ design (transmitting filter: solid,
receiving filter: dots) and discrete-time H ∞ design (dash)
Nyquist frequency (i.e. ω ≈ π), and hence we takeWz (z) = r ·z−1,
z + 0.5

as the weighting function of the transmitting signal, where the parameter r = 0.21. The
other design parameters are the same as the example above.
Figure 4.17 shows the H ∞ -norm of Tew and Tvw (the system from wc to vd in Figure 4.6)
that varies with r ∈ [0, 5]. We can take account of a trade-off between the error attenuationlevel and the amount of the transmitting signal with Figure 4.17. For example, we chooser = 0.21 in order to attenuate the error less than −26dB.
Figure 4.18 shows the gain responses of transmitting filters designed for r = 0 andr = 0.21. We can see that the new filter shows better attenuation than the filter designed
for r = 0 at high frequency.
Figure 4.19 shows the frequency responses of the error system Tew . We see that theattenuation level of Tew designed for r = 0.21 is less than −26dB. Figure 4.20 shows thefrequency responses of Tvw . We can see that the amount of the transmitting signal isattenuated at high frequency.

4.5.2

Compression effects

In this section, we consider compression effects. First, we take compression with the downsampling factor M = 1, 2, 4, 8. It means that the size of data transmitted is compressed

52

CHAPTER 4. APPLICATION TO COMMUNICATION SYSTEMSFrequency Response
0−5Gain [dB]

−10

−15

−20

−25

−30

−35 −3
10

−2

10

−1

10
Frequency [rad/sec]

0

10

1

10

Figure 4.13: Frequency responses of Tew : sampled-data H ∞ design (solid) and discretetime H ∞ design (dash)to 1, 1/2, 1/4, 1/8, respectively. For simplicity, we put Wz = 0. The other parameters
are the same as the previous example. Denote Tew the system from wc to ec and Tzn thesystem from nd to ec .
We show the frequency responses of Tew in Figure 4.21 and those of Tzn in Figure 4.22
for M = 1, 2, 4, 8. From Figure 4.21, we can see that when we compress the signal to
1/M, the gain of kTew k (i.e., the maximum gain) increases about 6 × M dB. In particular,
the compression of M = 2 is almost comparative with the case of no compression in the
low frequency range; the difference is about 0.5dB. Figure 4.22 indicates that the larger
the compression ratio M, the smaller kTzn k.
Next we simulate a transmission with compression ratio M = 4. For simplicity, we
put Cd = 1. The source is a speech signal with the sampling frequency 22.05kHz. The
frequency of the downsampled signal transmitted is 5.5125kHz.
Figure 4.23 shows the FFT of the source speech signal. Then we show the FFT ofthe reconstructed signal ud (see Figure 4.5) that is designed by the sampled-data H ∞optimization. For comparison, we take the equiripple filter [9, 35, 46] with the cut-off
frequency ωc = 22.05/8 ≈ 2.76kHz. This filter is often used in multirate signal processing.
The FFT of the reconstructed signal is shown in Figure 4.25. We can see from Figure
4.25 that the FFT of the signal processed by the equiripple filter is sharply cut at thefrequency about 2.76kHz, while that of sampled-data design shows slow decay. The FFTplot of the source signal shown in Figure 4.23 indicates that the frequency of the original
signal does not very decay up to 8kHz, and we can say that the response by the sampled-53

4.6. CONCLUSION

Frequency Response−5−10Gain [dB]

−15

−20

−25

−30

−35 −3
10

−2

10

−1

10
Frequency [rad/sec]

0

10

1

10

Figure 4.14: Frequency responses of Tzn : sampled-data H ∞ design (solid) and discretetime H ∞ design (dash)data design is better than that by the conventional design. In fact, the reconstructedvoice by equiripple filter sounds blur because of lack of high frequency, while that by the
sampled-data design sounds clearer. However, the sound by the sampled-data design has
high frequency noise. To reduce the noise, we have to introduce a low-pass filter, whose
design needs further discussion.

4.6

Conclusion

In this chapter, we have treated communication systems which contains signal compression. Under distortion by a channel, we have presented a design method of transmitting/receiving filters by using sampled-data H ∞ optimization. By iterating a transmitting filter design and a receiving filter design, we can obtain sub optimal filters. We haveshown that the objective function monotonically decreases by the iteration.
In this design, the channel is assumed to be time-invariant. However, the real channel
often contains time-varying systems, in particular, in the case of wireless communication.
Moreover, the real channel is very complicated and we should notice that the model ofthe channel always contains a modeling error. To overcome this, we have to choose anadaptive filter. Design for adaptive filters by using sampled-data theory is an important
subject for the future.

54

CHAPTER 4. APPLICATION TO COMMUNICATION SYSTEMSTime Response
1.5

1

0.5

0

−0.5

−1

−1.5
0

20

40

60

80

100

Figure 4.15: Time response with sampled-data design

Time Response
1.5

1

0.5

0

−0.5

−1

−1.5
0

20

40

60

80

Figure 4.16: Time response with discrete-time design

100

55

4.6. CONCLUSION

105

H norm [dB]

0
−5

∞

−10
−15
−20
−25
−30
0

1

2

3

4

5

r

Figure 4.17: Relation between r and kTew k (solid), kTvw k (dash)

Fequency Response30

25

Gain [dB]

20

15

10

5

0
0

0.5

1

1.5
2
Frequency [rad/sec]

2.5

3

3.5

Figure 4.18: Gain responses of transmitting filters designed for r = 0.21 (solid) and r = 0(dash)

56

CHAPTER 4. APPLICATION TO COMMUNICATION SYSTEMSFrequency Response
−25.5
−26
−26.5

Gain [dB]

−27
−27.5
−28
−28.5
−29
−29.5
−30
−30.5 −3
10

−2

10

−1

10
Frequency [rad/sec]

0

10

1

10

Figure 4.19: Frequency responses of Tew designed for r = 0.21 (solid) and r = 0 (dash)

Frequency Response
20

10Gain [dB]

0

−10

−20

−30

−40 −3
10

−2

10

−1

10
Frequency [rad/sec]

0

10

1

10

Figure 4.20: Frequency responses of Tvw designed for r = 0.21 (solid) and r = 0 (dash)57

4.6. CONCLUSION

Frequency Response
-12
-14
-16
-18

Gain [dB]

-20
-22
-24
-26
-28
-30
-32
10

-2

-1

10
Frequency [rad/sec]

10

0

10

1

Figure 4.21: Frequency responses of Tew : compression ratio M = 1 (solid), M = 2 (dash),
M = 4 (dot) and M = 8 (dash-dot)Frequency Response-5-10Gain [dB]

-15

-20

-25

-30

-35

-40
10

-2

-1

10
Frequency [rad/sec]

10

0

10

1

Figure 4.22: Frequency responses of Tzn : compression ratio M = 1 (solid), M = 2 (dash),
M = 4 (dot) and M = 8 (dash-dot)

58

CHAPTER 4. APPLICATION TO COMMUNICATION SYSTEMS

Original
60

40

Magnitude [dB]

20

0

-20

-40

-60

0

2

4
6
Frequency [kHz]8

10

Figure 4.23: FFT of the source
Sampled-data design
60

40

Magnitude [dB]

20

0

-20

-40

-60

0

2

4
6
Frequency [kHz]8

10

Figure 4.24: FFT of the reconstructed signal (sampled-data design)59

4.6. CONCLUSION

Discrete-time design
60

40

Magnitude [dB]

20

0

-20

-40

-60

0

2

4
6
Frequency [kHz]8

10

Figure 4.25: FFT of the reconstructed signal (equiripple design)Chapter 5
Minimization of Quantization Errors
5.1

Introduction

In digital signal processing, digital communications and digital control, analog signals haveto be discretized by an A/D converters to become digital signals. In discretization, we
have two operations; sampling and quantization. Sampling is discretization in time, andthe model is described as a linear one, which is relatively easy to analyze mathematically(e.g., via lifting discussed in Chapter 2). On the other hand, quantization, discretizationin amplitude, is a nonlinear operation, and its analysis is much more difficult.
For analyzing such a nonlinearity, an additive noise model has been widely used [10,6]. The model is then linear and hence we can easily analyze or design a system withquantization, but there has not been much established knowledge on the characteristic of
the nonlinear system designed by the linearization method.
In this chapter, we first discuss the stability of quantized sampled-data control systems.
Then we investigate how much quantization influences the performance of a sampled-datasystem. We will show that
• if the linear model is stable, the states of the quantized system are bounded,• if the linear model has a small L2 gain (i.e., the H ∞ -norm of the system), thequantized system has a small power gain.
It follows that the linearization method is valid for analyzing and designing systems withquantizations.
Next we apply the above results to the design of a quantizer that is called differential
pulse code modulation (DPCM) [32]. Although a large number of studies have been madeon DPCM systems, little is known on the stability and the performance, which we will
discuss by means of linearization.
In this chapter, we measure the performance by the power norm defined as follows:
Z T
12
pow(z) := lim|z(t)|2 dt (continuous-time),
T →∞ 2T −T
N
1 X
2|zd [k]|2 (discrete-time),
pow[zd ] := lim
N →∞ 2N
k=−N
61

62

CHAPTER 5. MINIMIZATION OF QUANTIZATION ERRORS

and the supremum norm
kzd k∞ := sup |zd [k]|.
k∈

+

5.2

Sampled-data control systems with quantization

5.2.1

Additive noise model for quantizer

We will begin with a sampled-data system with quantization shown in Figure 5.1. In the
z

w
G

Sh
y

Hh
Q

K

u

Figure 5.1: Sampled-data control system with quantization
figure, Q is a uniform quantizer with a quantization level ∆, and K(z) is a discrete-time
controller. Let {AK , BK , CK , DK } be a realization of the controller K(z), and setA B1 B2
G11 (s) G12 (s)G(s) :==:  C1 0 D12  (s).
G21 (s) G22 (s)C2 00

We use the additive noise model for the uniform quantizer; the quantizer Q is modeledby
Qy = y + d,

kdk∞ ≤ ∆/2.

Since we have kdk∞ = ky−Qyk∞ ≤ ∆/2, the additive noise model covers the input/output
relation of the uniform quantization with a belt-like region as shown in Figure 5.2.

5.2.2

Stability of sampled-data systems with quantization

Let us consider the block diagram shown in Figure 5.3 to check the stability of the closed
e B,
e C,
e D}
e
loop system. In the figure, Sh G22 Hh K is a discrete-time system, of which let {A,eeeebe a realization. Note that A, B, C, D are given by






AK
0
BK
e
e
e := 0 C2 , D
e := 0,A :=, B :=, C
B2d CK Ad
B2d DK63

5.2. SAMPLED-DATA CONTROL SYSTEMS WITH QUANTIZATIONQyν =y+d∆yy

kdk∞ ≤ ∆2
y

Q

y

Qy

+ ν =y+d+

Figure 5.2: Uniform quantization (left) and its additive noise model (right)

G22
Sh

Hh
K
d

Figure 5.3: Closed loop system

64

CHAPTER 5. MINIMIZATION OF QUANTIZATION ERRORS

where
Ah

Ad := e ,B2d :=Z heAt B2 dt.

0Then, we have the state-space representation of the system with the additive noisemodel shown in Figure 5.3 as follows:
e+Be C)x[n]eex[n + 1] = (A
+ Bd[n],kdk∞ ≤ ∆/2.

(5.1)

We should notice that without quantization (i.e., d = 0) the stability of the feedback
e+B
e C.
e We have the following theorem ofsystem is equal to the stability of the matrix A
the stability of the system in Figure 5.1.

e+B
eC
e =: F ∈ Rn×n is stable, and let γ > 0 be a real
Theorem 5.1. Assume that A
number such that r(F ) < γ < 1, where r(F ) denotes the maximum absolute value of theeigenvalues of F . Then there exists c > 0 such that|x[k] − F k x0 | ≤ cfor all x0 := x[0] ∈ Rn and k ≥ 0, andlim |x[k]| ≤ ck→∞for all x0 ∈ Rn .

1 − γk e ∆kBk =: rk ,
1−γ
2

(5.2)

1
e ∆ =: r∞ ,
kBk
1−γ
2

(5.3)

Proof. First of all, let J be the Jordan canonical form of F . That is, for a nonsingularmatrix P , we have
J = P −1F P = Λ + U,
where




λ1 0 . . . 0
.
.. ..

.
. .. 
 0
Λ :=  . .
,.
.. .. 0  ..
0 . . . 0 λn




0 ∗
0 ... 0 .. . . . . . . .. 
.
.
. . 
 .
 .

.
.
,
.
.
.
U := 
.
.
.
0


 .

..
 ..
. ∗ 
0 ... ... ... 0∗ = 0 or 1,

and λi (i = 1, . . . , n) denotes the i-th eigenvalue of F . Take δ > 0 such that r(F ) + δ =γ < 1, and define
1 0 ...
0
.. 
.

. 
 0 δ −1 . .
D :=  . .
.
.. ...
 ..
0 
0 . . . 0 δ −1+n

5.2. SAMPLED-DATA CONTROL SYSTEMS WITH QUANTIZATION

65

Then we haveD −1 JD = Λ + δU.
Put T := DP . Then for any x ∈ Rn ,|T −1 F T x| = |(Λ + δU)x| ≤ (r(F ) + δ)|x| = γ|x|.
It follows that|x[k] − F k x0 | ≤
=
≤

k
X

l=1
k
X

l=1
k
X

e
|F k−l Bd[l]|e|T (T −1 F T )k−l T −1 Bd[l]|e ∆kT k · γ k−l · kT −1 kkBk
2
l=1

= kT kkT −1k

1 − γk e ∆
kBk .
1−γ
2

Then put c := kT kkT −1k, we have (5.2). Since F is stable, for any x0 ∈ Rn , we havelim F k x0 = 0.

k→∞Hence by taking n → ∞ for the inequality (5.2), we obtain the second inequality (5.3).
From this theorem, we conclude that if we stabilize the linearized model, the state of
the sampled-data system with quantization shown in Figure 5.1 are bounded, that is, thesystem is bounded-input bounded-output (BIBO) stable.
Moreover, the set
D := {x ∈ Rn : |x| ≤ r∞ }is a positive invariant set of the system (5.1). In fact, if x[0] ∈ D ∈ Rn , that is, |x[0]| ≤ r∞ ,we have for any k ≥ 1|x[k]| ≤ cγ k |x[0]| + rk
≤ cγ k r∞ + rk
1 − γk
1
k
e ∆
+
=c γ
kBk
1−γ
1−γ
2
1
e ∆
=c
kBk
1−γ
2
= r∞ ,
that is, x[k] ∈ D.

66

CHAPTER 5. MINIMIZATION OF QUANTIZATION ERRORS

5.2.3

Performance analysis of sampled-data systems with quantization

Generally, quantization deteriorates the performance of control systems, and it is important to know how much quantization affects on the systems. The aim of this section
is to analyze the quantization effect by using the additive noise model discussed above.
Consider the block diagram shown in Figure 5.4. Although the quantization noise d is
z

w
G

Sh
y

Hh
+
d

u

K

+

Figure 5.4: Sampled-data control system with additive noise
usually taken as the white noise, we assume the noise d in l2 . The reason is as follows:• generally the quantization noise is not white; the noise will depend on the input
signal of the quantizer,• we can use the H ∞ -norm that has a connection with the worst case analysis.
We denote by Tzw and Tzd the system shown in Figure 5.4 from w to z and from d to z
respectively. Assume Tzw and Tzd are stable andkTzw wkL2= γ1 ,
kwkL2
w∈L2

kTzd dkL2= γ2 .
kdkl2
d∈l2

kTzw k := supkTzd k = sup

w6=0

d6=0

We will now discuss the performance of the system in Figure 5.1.
zw
Tzw , TzddFigure 5.5: Additive noise model for sampled-data system with quantization

Lemma 5.1. Let T be a stable sampled-data system with discrete-time inputs and continuoustime outputs. We have the following inequality for any discrete-time, power signal u:pow(T u) ≤ kT k pow[u].

5.2. SAMPLED-DATA CONTROL SYSTEMS WITH QUANTIZATION

67

Proof. First of all, define the power norm of lifted signal ye = Lh y as follows:N
1 X
key [k]k2L2 [0,h) ,pow{ey } := lim
N →∞ 2Nk=−N
2Note that for y and ye = Lh y, we haveye[k] ∈ L2 [0, h).

pow(y) = pow{e
y }.

(5.4)

For a lifted power signal ye (i.e., pow{ey } < ∞), define the autocorrelation functionN
1 X
hey [n], ye[n + k]i,
N →∞ 2Nn=−N

Ry [k] := lim

where h·, ·i is the inner product on L2[0,h) , that is, hx, yi :=the (discrete) Fourier transform of Ry :
Sy (e

jωh

) :=

∞
X

Rh
0

y(t)T x(t)dt. Let Sy denote

Ry [k]e−jωkh .

k=−∞Similarly, for a discrete-time u, define Ru and Su as
N
1 X
u[n + k]T u[n],N →∞ 2Nn=−NRu [k] := lim
Su (ejωh ) :=∞X

Ru [k]e−jωkh .

k=−∞Then we havehpow{ey } = Ry [0] =2π2hpow[u] = Ru [0] =
2π
2

Z 2π
h

Sy (ejωh )dω,

0

Z 2π

(5.5)

h

Su (e

jωh

)dω.

0

For the sampled-data system T , we denote by Te (ejωh ) the frequency response [41] of thesystem T . Let u and y be the input and output of T respectively, that is, y = T u. Then
we obtain
Sy (ejωh ) = Te (ejωh )Tf∗ (ejωh )Su (ejωh ) = kTe (ejωh )k2 Su (ejωh ),(5.6)

where T ∗ is the dual system [41] of T . The equation (5.6) can be proven by the same

68

CHAPTER 5. MINIMIZATION OF QUANTIZATION ERRORS

method as the continuous-time version [7]. By using (5.4), (5.5) and (5.6), we havehpow(y) = pow{e
y} =
2π
2

2

=

h
2π

Z 2π
h

Sy (ejωh )dω

0

Z 2π
h

0

kTe(ejωh )k2 Su (ejωh )dωZ 2π
hjωh 2 he
≤ sup kT (e )kSu (ejωh )dω
2π 0
ω∈(0,2π/h)= kTe k2 pow[u]2 = kT k2 pow[u]2 .
∞Using Lemma 5.1, we have the following theorem.
Theorem 5.2. For any input w ∈ L2 and d ∈ l∞ of the sampled-data system [Tzw , Tzd ]defined above, the output z satisfies
pow(z) ≤γ2 ∆.
2

Proof. Since Tzw w ∈ L2 , we have pow(Tzw w) = 0. Generally, if kdk∞ ≤ 1 then pow[d] ≤ 1,
and hence
sup pow(Tzd d) ≤kdk∞ ≤1

sup pow(Tzd d).
pow[d]≤1

From Lemma 5.1 we havesup pow(Tzd d) ≤ kTzd k.
pow[d]≤1

Thereforepow(z) = pow(Tzw w + Tzd d)≤ pow(Tzw w) + pow(Tzd d)= pow(Tzd d) ≤ kTzd kkdk∞ ≤

γ2 ∆
.
2

The theorem leads us to the conclusion that if we take the H ∞ design to attenuate
kTzd k∞ , the quantization has a small effect on the output z with respect to the power,
and hence the H ∞ design will be valid.

69

5.3. DIFFERENTIAL PULSE CODE MODULATION

5.3

Differential pulse code modulation

5.3.1

Differential pulse code modulation

Differential pulse code modulation (DPCM) is a quantization system, which is used, forexample, in the telephone communication systems. Figure 5.6 shows a DPCM system.
The encoder quantizes the error e = r − u, where u is a prediction of the input signal r.
If the filter K1 (z) well predicts r from the quantizer output ê := Qe, the error e will besmaller than the input r, and hence fewer bits are required to quantize the signal. Assume
that the quantizer Q is a uniform quantizer with a quantization level ∆.
r

e
+

Q

ê

r̂
+

−
u

K1 (z)

K1 (z)

Encoder

Decoder+

Figure 5.6: DPCM system
The signal ê is transmitted into a communication channel or stored in digital media,
and then the decoder reconstructs the original signal r and makes the output r̂. Notethat the filter in the decoder is the same as that in the encoder, and we have the following
error estimatekr̂ − rk∞ = kê − ek∞ ≤∆.
2That is to say, the reconstruction error is less than ∆/2. Therefore, DPCM can transmit
data with fewer bits.
However, the model does not take account of the channel noise that adds the transmitted signal ê, and hence the estimate is not valid. For example, in the ∆ modulation,
the filter K1 is the adder K1 (z) = 1/(z − 1). Since this filter is unstable, the channel noise
will be amplified in the decoder.
Therefore, we use a model taking the channel noise into account for designing theencoder and the decoder.

5.3.2

Problem formulation

The block diagram of our DPCM system to be designed is illustrated in Figure 5.7. In
the figure, n is the channel noise. Our purpose is to attenuate the prediction error e tobe quantized and the reconstruction error. For this purpose, we design the filter K1 (z)and K2 (z) in Figure 5.7 by using the sampled-data H ∞ optimization method.
The block diagram of the error system for designing these filters is shown is Figure 5.8.
We first take the quantization error caused by Q for the additive noise d. Then, assume
that the analog signal to be quantized has frequency characteristic W (s), and introduce a

70

CHAPTER 5. MINIMIZATION OF QUANTIZATION ERRORS

r

e
+

ê

Q

K2 (z)

+

−
u

n
+

r̂

K1 (z)
Encoder

Decoder

Figure 5.7: DPCM system with channel noise
Wd
w

Sh

W

+ e
−

+
K1

e−Ls
er

d

+

−
+

Hh

Wn

n

K2

Figure 5.8: Error system for designing filters

time delay e−Ls . This delay time L is the time that K1 (z) and K2 (z) will take to process
signals. In the error system, let T1 be the system from [w, d]T to e, and T2 be the systemfrom [w, d, n]T to er . Then the design problem is formulated as follows:
Problem 5.1. Given an analog filter W (s), time delay L and a sampling period h, findfilters K1 (z) and K2 (z) that minimize
kT1 k2 :=supw∈L2 ,d∈l2
kwkL2 +kdkl2 6=0

kT2 k2 :=kek2l2,kwk2L2 + kdk2l2

supw∈L2 ,d,n∈l2
kwkL2 +kdkl2 +knkl2 6=0

ker k2l2,
kwk2L2 + kdk2l2 + knk2l2

respectively.
This is a sampled-data H ∞ optimization problem and assuming L = mh ( m ∈ N ),the solution can be obtained by using the fast-sampling/fast-hold method discussed in
Chapter 2.
Theorem 5.3. Assume that L = mh, m ∈ N. Then, for the sampled-data systems T1
and T2 , there exist finite-dimensional discrete-time systems {T1,N : N = 1, 2, . . . } and71

5.4. DESIGN EXAMPLE{T2,N : N = 1, 2, . . . } such that
lim kT1,N k = kT1 k,

N →∞lim kT2,N k = kT2 k.

N →∞The proof of this theorem is almost the same as the proof of Theorem 5 in Chapter 4.
The approximated discrete-time systems T1,N and T2,N are as follows: hifN , 0W−1i,T1,N = Fl (G1,N , K1 ), G1,N :=  hfN , Wd −1
W
"
#
−m
[z
,
0,
0]
−1
h
i
T2,N = Fl (G2,N , K2 ), G2,N :=,fN , Sd Wd , Wn
Sd W0
fN := [1, 0, . . . , 0]LN (W ),W| {z }

Sd := (1 + K1 )−1 .

N −1

5.4

Design example

In this section, we present a design example of DPCM. The design parameters are as
follows: the sampling period h = 1, the reconstruction delay L = 2, the weighting
functions are
Wd = 0.5,

Wn (z) = 0.1 ×

0.01753z 2 − 0.03506z + 0.01753,
z 2 + 0.572z + 0.3147

where Wn (z) is a Chebyshev type I high-pass filter [9, 35, 46], and the analog filter W (s)is
W (s) =

1.
(10s + 1)2

For comparison, we take the ∆ modulation, that is, K1 (z) = 1/(z − 1) and K2 (z) =1 + K1 (z) = z/(z − 1).
Figure 5.9 shows the frequency responses of the system T1 . We can see that thesystem T1 designed by H ∞ optimization has lower gain than that of the ∆ modulation
over the whole frequency, in particular, in the high frequency range, our system shows
better attenuation.
Figure 5.10 shows the frequency responses of the system T2 . Since the ∆ modulation is
not stable (i.e., the system has a pole at z = 1), the frequency response of T2 is indefinite.
Therefore in Figure 5.10, we take a decoder with the decoding filter K2 = 1+K1 where K1
is the H ∞ (sub) optimal filter of the encoder, and compare it with the optimal decoding
filter designed by sampled-data H ∞ optimization. We can see that the proposed system
attenuates the gain over the whole frequency. It follows that the channel noise on thereceived signal will be attenuated more than the conventional system.
Then we show a simulation for the obtained DPCM systems. The parameters areπt), the channel noise
as follows: the quantization level ∆ = 0.125, the input r = sin( 1072

CHAPTER 5. MINIMIZATION OF QUANTIZATION ERRORSFrequency response of T

1

-5

-10

Gain [dB]

-15

-20

-25

-30

-35

-40 -3
10

10

-2

-1

10
Frequency [rad/sec]

10

0

Figure 5.9: Frequency responses of T1 : proposed (solid) and conventional (dash)

Frequency response of T

2

-10
-15
-20
-25

Gain [dB]

-30
-35
-40
-45
-50
-55
-60 -3
10

10

-2

-1

10
Frequency [rad/sec]

10

0

Figure 5.10: Frequency responses of T2 : proposed (solid) and conventional (dash)73

5.5. CONCLUSIONn = 0.1 sin(2t) and the sampling period h = 1. Figure 5.11 shows the time response of thesampled-data designed DPCM system, and Figure 5.12 shows that of the conventional ∆modulation system. We can see that the proposed system attenuates the channel noise
considerably better than the conventional system.
Time response
1.5

1

Amplitude

0.5

0

-0.5

-1

-1.5

0

10

20

30
40
Time [sec]

50

60

70

Figure 5.11: Time response of sampled-data designed system

5.5

Conclusion

In this chapter, we have discussed the stability and the performance of quantized sampleddata control systems. We have adopted the additive noise model, and by using it wehave shown the BIBO stability and the performance of quantized systems. Moreover, wehave proposed a new method for designing DPCM systems. Since the conventional ∆modulation system is not stable, a channel noise can be amplified at the encoder, while
our system will attenuate the channel noise.
However, there remains an issue. We have not dealt with saturation in a quantizer.
The additive noise model can be applied effectively in the case that the noise is small, butwith saturation, the noise can be enormous. In this case, we have to treat the quantizer
as a nonlinear system. We consider that hybrid system theory, in particular, switchingsystem theory may be applicable to that case.

74

CHAPTER 5. MINIMIZATION OF QUANTIZATION ERRORSTime response
1.5

1

Amplitude

0.5

0

-0.5

-1

-1.5

0

10

20

30
40
Time [sec]

50

60

70

Figure 5.12: Time response of conventional ∆ modulation system

Chapter 6
Optimal FIR Approximation
6.1

Introduction

According to the method we have discussed in the previous chapters, the filter we obtainis an IIR (Infinite Impulse Response) filter:PM−k
k=0 ak z.
(6.1)
F (z) =
P
−k1+ N
k=1 bk z

The structure of an IIR filter is shown in Figure 6.1.
In practice, FIR (Finite Impulse Response) filters are often preferred to IIR ones.
They have finitely many nonzero Markov parameters:
F (z) =MX

ak z −k .

(6.2)

k=0

The structure of an FIR filter is shown in Figure 6.2.
The reasons why FIR filters are often preferred to IIR filters are as follows [46]:• FIR filters are intrinsically stable; the stability issue is a non-issue.
• They can easily realize various features that are not possible or are difficult toachieve with IIR filters, e.g., linear phase property.
• They can be free from certain problems in implementation, for example, limit cycles,
attributed to quantization and the existence of a feedback loop in IIR filters.
On the other hand, a design process may have to start with an IIR filter for varietyof reasons. For example, we have a large number of continuous-time filters available, anda digital filter may be obtained by discretizing one of them. It is then desired that suchan IIR filter be approximated by an FIR filter. An easy way of doing this is to justtruncate the Markov parameters of the IIR filter at a desired number of steps. This may
however lead to either a very high-dimensional filter with good approximation, or a lower
dimensional filter with an unsatisfactory approximation, depending on the truncation
point.
The following problem is thus very natural and of importance:75

76

CHAPTER 6. OPTIMAL FIR APPROXIMATION

[Q
+
+

E

+

E

E1

] −
]

−

] −

D

\Q
+

D

+

D

+

D1

Figure 6.1: IIR filter

[Q
] −
]

−

] −

D

\Q
+

D

+

D

+

D1

Figure 6.2: FIR filter

776.2. FIR APPROXIMATION PROBLEM

Problem 6.1. Given an IIR filter K(z) and a positive integer N, find an optimal FIR
approximant Kf (z) that has order N and approximates K(z) with respect to a certain
performance measure.
There is a very elegant method called the Nehari shuffle, proposed by Kootsookos et al.
[22, 23]. Its basic idea may be described as follows: For a given IIR filter G and a desireddegree r−1 that an approximating FIR filter should assume, one first truncate the impulse
response of G to the first r steps. This is a mere truncation, and it may induce a large
error. One then takes its residual G1 , and suitably shifting and taking the mirror image,
one can reduce this to the situation of the Nehari extension (approximation) problem.
This will induce a truncation in the second step. By taking the residual further, thisprocess can be continued, and the approximation can be improved in each step. (Details
may be found in [31]. An advantage here is that this procedure gives rise to certain apriori and a posteriori error bounds. On the other hand, it does not necessarily give an
optimal approximation with respect to the H ∞ -norm.
In contrast to the Nehari shuffle, we here propose a method that directly deals with(sub)optimal approximants with respect to the H ∞ error norm. It is shown that• the design problem is reducible to a Linear Matrix Inequality(LMI) [3]; and• the obtained filter can be made close to be optimal by an iterative procedure.
A comparison with the Nehari shuffle is made for the Chebyshev filter of order 8, which
has been studied in detail in [23].

6.2

FIR approximation problem

Consider the block diagram Figure 6.3. K(z) is a given (rational and stable) IIR filter,
K(z)
w

−

W (z)
Kf (z)e+Figure 6.3: Error system
W (z) is a proper and rational weighting function, and Kf (z) is an FIR filter of order N.
Denote by Tew (z) the transfer function from w to e in Figure 6.3. The objective here is
to find Kf (z) that makes the H ∞ error norm less than a prespecified bound γ > 0, thatis,kTew wk2
< γ.
kwk2
w∈l2

kTew k∞ := sup
w6=0

78

CHAPTER 6. OPTIMAL FIR APPROXIMATION

Introduce state space realizations
W (z) := CW (zI − AW )−1 BW + DW ,K(z) := CK (zI − AK )−1 BK + DK ,and
Kf (z) =NX

ak z −k = Cf (α)(zI − Af )−1 Bf + Df (α),

k=0Cf (α) = aN aN −1 . . . a1 , Df (α) = a0 ,
α = aN aN −1 . . . a0 ,where α = aN aN −1 . . . a0 denote the Markov parameters of the filter Kf (z) to
be designed. The matrices Af and Bf are defined as follows:
0 1
0 ··· 0 
 .. . .
0..
. . .. 
.
.
. . 
 .
 .. 

 .
 . 
..
..
Af = 
,.
. 0 
 , Bf = 
 ..
 0 

 .
.
.. 1 
 ..
10 ··· ··· ··· 0
and they contain just zeros and ones.
A realization of Tew is given as follows:

Tew (z) =: C(α)(zI − A)−1 B + D(α),




AW
0
0
BW
A =  BK CW AK 0  , B =  BK DW  ,
Bf CW
0 Af
Bf DW


C(α) = (Df (α) − DK )CW −CK Cf (α) ,D(α) = (DK + Df (α))DW .

The important asset here is that the design parameter α appears only in the C and D
matrices linearly, and the underlying structure is of the one-block type. Hence the overall
transfer operator is linear in α, and the design problem of choosing α to minimize the
H∞ -norm can be expected to become a linear matrix inequality. In fact, the bounded reallemma [3] readily yields the following:
Theorem 6.1. kTew k∞ < γ if and only if there exists P > 0 such that
 T
A PA − P
AT P BC(α)T BT P A−γI + B T P B D(α)T  < 0.
C(α)
D(α)
−γI

(6.3)

Proof. By the bounded real lemma [3], kTew k∞ < γ is equivalent to the condition that
there exists a matrix P̃ > 0 such that




P̃ 0
P̃ 0
T
,
(6.4)
Q<
Q
0 γ 2I
0 I

79

6.3. NUMERICAL EXAMPLE

where
Q :=



A
B
C(α) D(α)

.

Although the inequality (6.4) is not affine in α, it can be converted to an affine one by
the Schur complement [3]:Φ11 Φ12< 0,
ΦT12 Φ22T
is equivalent to Φ22 < 0 and Φ11 < Φ12 Φ−1
22 Φ12 . By dividing the inequality (6.4) by γ > 0,
we get T −1
  T 
A γ P̃ A − γ −1 P̃
AT γ −1 P̃ B
C
−1
C
D
(−γ
I)
.
<
DT
B T γ −1 P̃ A
B T γ −1 P̃ B − γI

Then by using the Sure complement forT T
A PA − P
AT P B,Φ11 :=BT P AB T P B − γI
Φ22 := −γI,TC D
Φ12 :=,
where P := γ −1 P̃ > 0, we get the inequality (6.3).

The obtained condition is an LMI in α, and can be effectively solved by standard
MATLAB routines [11].

6.3

Numerical example

6.3.1

Comparison of H ∞ design via LMI and the Nehari shuffle

Take the following Chebyshev filter of order 8K(z) = 10−3
×0.04705z 8 + 0.3764z 7 + 1.317z 6 + 2.635z 5 + 3.294z 4 + 2.635z 3 + 1.317z 2 + 0.3764z + 0.04705
z 8 − 4.953z 7 + 11.71z 6 − 16.95z 5 + 16.29z 4 − 10.58z 3 + 4.552z 2 − 1.161z + 0.1369

as a target filter to be approximated. This has been studied thoroughly by Kootsookos
and Bitmead [23] for the Nehari shuffle, and is suitable for comparison with the present
method. For simplicity, we confine ourselves to approximations by FIR filters with 32 tap
coefficients (of order 31).
The design depends crucially on the choice of the weight W (z). One natural choice([31]) would be to take W (z) to be equal to K −1 (z) (or some variant of it having thesame gain on the imaginary axis, since K is not minimum phase). This is relative error
approximation, where (approximately) dB and phase errors are weighted uniformly with

80

CHAPTER 6. OPTIMAL FIR APPROXIMATION

Magnitude Response of Weighting
1.4

1.2

Magnitude

1

0.8

0.6

0.4

0.2

0

0

0.1

0.2
0.3
Frequency [rad/sec]

0.4

0.5

Figure 6.4: Inverse of the weighting function
frequency. Since the optimal overall error in Figure 6.3 will become all-pass, this will havethe effect of attenuating the stop-band error with the weight of K −1 (z) (which is verylarge) while maintaining reasonable pass-band characteristic. Unfortunately, however,
due to the very small gain of K(z) in the stop-band, this will make the solution of theapproximation problem Figure 6.3 numerically hard. Neither the Nehari shuffle nor the
LMI method gave a satisfactory result in this case. Hence one should sacrifice the stopband attenuation to obtain a reasonable W (z). There is also a trade-off, empiricallyobserved, between the stop-band attenuation and the pass-band ripples.
Kootsookos and Bitmead [23] thus employed the weight as depicted in Figure 6.4.
To be precise, the frequency response shown here is the inverse of the para-Hermitian
conjugate of the weight function. The reason for taking the para-Hermitian conjugate isthat the Nehari shuffle makes use of causal approximation for anti-causal transfer function,
so that we must reciprocate the poles and zeros. Then by taking the inverse, the weightattenuates the stop-band by the inverse of its gain and approximately shapes the passband as it is in the pass band. On the other hand, for the FIR approximation as in
Figure 6.3, we simply take the inverse of this weight, since we do not need to make theweight anti-stable.
The gain responses of obtained FIR filters based on the Nehari shuffle and Theorem
6.1 are given in Figure 6.5. Figure 6.6 shows their phase plots.
We see that the gain of the H ∞ approximant shows smaller pass-band ripples andbetter stop-band attenuation than those by the Nehari shuffle. The phase characteristicsof these are about the same up to the edge of the transition band.

81

6.3. NUMERICAL EXAMPLE

Magnitude Response
1.4
Nehari Shuffle
1.2

1

Magnitude

Original
0.8

0.6

0.4

0.2
LMI
0

0

0.1

0.2
0.3
Frequency [rad/sec]

0.4

0.5

Nehari Shuffle

Figure 6.5: Gain responses of FIR approximants with weight function in Figure 6.4: H ∞via LMI (solid), Nehari shuffle (dots) and original IIR Chebyshev filter (dash)Phase Response
5000
Original-500

Phase [degree]-1000
Nehari Shuffle

-1500

LMI

-2000
-2500
-3000
-3500
-4000
-4500

0

0.1

0.2

0.3

0.4

0.5

Frequency [rad/sec]

Figure 6.6: Phase plots of FIR approximants with weight function in Figure 6.4: H ∞ via
LMI (solid), Nehari shuffle (dots) and Chebyshev (dash)

82

CHAPTER 6. OPTIMAL FIR APPROXIMATION

Figure 6.7 shows the error magnitude responses. The design by the LMI method has
the advantage of 5–7 dB over the one by the Nehari shuffle.
Error Magnitude Response-5

Nehari Shuffle-10

Error Magnitude [dB]

-15

-20
LMI
-25

-30

-35

0

0.1

0.2
0.3
Frequency [rad/sec]

0.4

0.5

Figure 6.7: Gain of the error K − Kf : H ∞ design via LMI (solid); Nehari shuffle (dots)

6.3.2

Trade-off between pass-band and stop-band characteristics

The design in the previous subsection depends crucially on the weighting function. It
is desirable to obtain smaller pass-band ripples while maintaining reasonable stop-band
attenuation. In this section we attempt to see how the choice of a weighting function
affects the overall approximation.
We consider the following three weighting functions:
W1 =0.7661z2 − 1.305z + 0.675,
z 2 − 1.735z + 0.9289

W2 =0.2831z4 − 0.5515z3 + 0.5416z2 − 0.2708z + 0.05882,
z 4 − 2.865z3 + 3.6z2 − 2.268z + 0.6056

W3 = 10−3 ×

14.44z7 − 7.838z6 + 19.02z5 − 4.448 + 6.697z3 − 0.1857z2 + 0.5287z + 0.01134.
z 7 − 4.229z6 + 8.561z5 − 10.43z4 + 8.172z3 − 4.089z2 + 1.206z − 0.1613

83

6.3. NUMERICAL EXAMPLE

Table 6.1: H ∞ and H 2 error norm

H ∞ error norm
H 2 error norm

W1
W2
0.0954 0.1838
0.0713 0.0840

W3
0.26270.1333

These functions W1 , W2 , W3 are, respectively, obtained as the 2nd, 4th, 7th-order Hankel
norm approximations [31] of the IIR Chebyshev filter to be approximated. The weight
W2 is the same as that used in the previous section. Their magnitude frequency responses
are shown in Figure 6.8.
1.4

1.2

W1

1

0.8

0.6

W2

0.4

0.2
W3
0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Figure 6.8: Gain of inverse of weighting functions
Figure 6.9 and Figure 6.10 show the resulting gain and phase responses of the FIR
filters designed with respective weighting functions. Figure 6.9 in particular shows that
there is a clear trade-off between the magnitude of the pass-band ripples and the stop-band
attenuation. That is, if we attempt to decrease the stop-band error, we must sacrifice thepass-band characteristic (i.e., larger ripples), and vice versa.
Figure 6.11 shows the error magnitude responses. Table 1 also shows the H ∞ and H 2error norms. Interestingly, the design via W1 exhibits the best overall approximation inboth performance measures, although its stop-band attenuation is not as good as thoseby W2 and W3 . Note also that the one by W3 approximates the phase characteristic of
the original filter up to the edge of the stop-band as Figure 6.10 shows.

84

CHAPTER 6. OPTIMAL FIR APPROXIMATION

1.4

W3

1.2

1
original

0.8

0.6

0.4
W1

W2

0.2

0

0

0.5

1

1.5

2

2.5

3

Figure 6.9: Gain responses of FIR filters via LMI

6.4

Conclusion

We have given an LMI solution to the optimal H ∞ approximation of IIR filters via FIR
filters. A comparison with the Nehari shuffle is made with a numerical example, and it isobserved that the LMI solution generally performs better. Another numerical study also
indicates that there is a trade-off between the pass-band and stop-band approximation
characteristics.

85

6.4. CONCLUSION

0

−500

−1000
orignal

W1
−1500

W3
−2000

W2

−2500

−3000

−3500

−4000

−4500

0

0.5

1

1.5

2

2.5

3

Figure 6.10: Phase responses of FIR filters via LMI

0.3

W3

0.2

W1
0.1

W2

0

0

0.1

0.2

0.3

0.4

Figure 6.11: Gain responses of the error Kf − K

0.5

Chapter 7
Conclusion
In this thesis, we proposed a new design method for multirate digital signal processing anddigital communication systems. Conventionally, they are designed under the assumptionthat the original analog signal is fully band-limited, while our method takes the analog
characteristic into account that goes often beyond the Nyquist frequency, and optimizes
the analog performance via the sampled-data H ∞ optimization.
In Chapter 3, we have presented a sampled-data design for multirate signal processors,in particular, interpolation, decimation and sampling rate conversion. Conventionally, afilter used in these systems is designed to have a sharp characteristic which approximatesthe ideal filter. However, as shown by design examples, such a sharp filter is not necessarily
optimal for reconstruction. This fact will not be recognized without taking the analog
signal into account.
In Chapter 4, we have treated communication systems which contains signal compression. Under distortions by a channel, we have presented a design method of a transmitting
filter and a receiving filter by using sampled-data H ∞ optimization. By iterating a transmitting filter design and a receiving filter design, we can obtain sub-optimal filters. We
have shown that the objective function monotonically decreases by the iteration.
In Chapter 5, we have investigated the stability and the performance of quantizedsampled-data control systems. By using an additive noise model (linearized model) for
the quantization, we have shown that if the linearized model is stable, the states of thequantized system are bounded, and that if the linearized model has small L2 gain, thequantized system has small power gain. Then we have applied the results to DPCM
design. We have pointed out that the conventional ∆ modulation is not stable and achannel noise will be amplified at the decoder. Therefore we have proposed a designof the decoder and the encoder; the decoder reduces the quantization noise, while theencoder reduces the channel noise.
In Chapter 6, We have given an LMI solution to the optimal H ∞ approximation of
IIR filters via FIR filters. A comparison with the Nehari shuffle is made with a numericalexample, and it is observed that the LMI solution generally performs better. Anothernumerical study also indicates that there is a trade-off between the pass-band and stopband approximation characteristics.
We conclude by giving future directions of research as follows:• We have treated one-dimensional signals (e.g., audio/speech) and we believe that the87

88

CHAPTER 7. CONCLUSION

present method can be effectively extended to image processing, which is a future
direction of our research. In particular, the popular format JPEG or MPEG is amultirate filter bank system, which can be designed by using the method discussed
in Chapter 3.
• We have discussed a design of communication systems, where the channel is timeinvariant. However, the real channel often contains time-varying systems, in particular, in the case of wireless communication. Moreover, the real channel is very
complicated and we should notice that the model of the channel always contains amodeling error. To overcome this, we have to choose an adaptive filter. Design foradaptive filters by using sampled-data theory is an important subject for the future.

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