T. Saramäki and R. Bregovi
, ”Multirate Systems and Filter Banks,” Chapter 2 in Multirate Systems: Design
and Applications edited by G. Jovanovic-Dolecek. Hershey PA: Idea Group Publishing.
1
MULTIRATE SYSTEMS AND FILTER BANKS
Tapio Saramäki and Robert Bregovi
Signal Processing Laboratory, Tampere University of Technology
P.O. Box 553, FIN-33101 Tampere, Finland
1 Introduction
During the last two decades, multirate filter banks have found various applications in many
different areas, such as speech coding, scrambling, adaptive signal processing, image
compression, signal and image processing applications as well as transmission of several signals
through the same channel (Malvar, 1992a; Vaidyanathan, 1993; Vetterli amd Kova
evi
, 1995;
Fliege, 1994; Misiti, Misiti, Oppenheim, and Poggi, 1996). The main idea of using multirate
filter banks is the ability of the system to separate in the frequency domain the signal under
consideration into two or more signals or to compose two or more different signals into a single
signal.
When splitting the signal into two or more signals an analysis-synthesis system is used. The
analysis-synthesis systems under consideration in this chapter are critically sampled multi-
channel or M-channel uniform filter banks and octave filter banks as shown in Figures 1(a) and
1(b), respectively. In the analysis bank of the uniform bank, the signal is split with the aid of M
filters H
k
(z) for k
=
0,1,…,
M−1 into M bands of the same bandwidth and each sub-signal is
decimated by a factor of M. In the case of octave filter banks, the overall signal is first split into
two bands of the same bandwidth and both sub-signals are decimated by a factor of two. After
that, the decimated lowpass filtered signal is split into two bands and so on. Doing this three
times gives rise to a three-level octave filter bank corresponding to the structure shown in Figure
1(b). In this case, H
0
(z) is a highpass filter with bandwidth equal to half the baseband and the
decimation factor is 2, H
1
(z) and H
2
(z) are bandpass filters with bandwidths equal to one fourth
and one eighth of the baseband, respectively, and the corresponding decimation factors are 4 and
8, whereas H
3
(z) is a lowpass filter with the same bandwidth and decimation factor as H
2
(z).
In many applications, the processing unit corresponds to storing the signal into the memory or
transferring it through the channel. The main goal is to significantly reduce, with the aid of
proper coding schemes, the number of bits representing the original signal for storing or
transferring purposes. When splitting the signal into various frequency bands with the aid of the
analysis filter bank, the signal characteristics are different in each band and various numbers of
bits can be used for coding and decoding the sub-signals. In some applications, the processing
unit is used for treating the sub-signal in order to obtain the desired operation for the output
signal of the overall system. A typical example is the use of the overall system for making
adaptive signal processing more efficient. Another example is the de-noising of a signal
performed with the aid of a special octave filter bank, called a discrete-time wavelet bank
(Vetterli and Kova
evi
, 1995; Misiti et al., 1996).
T. Saramäki and R. Bregovi
, ”Multirate Systems and Filter Banks,” Chapter 2 in Multirate Systems: Design
and Applications edited by G. Jovanovic-Dolecek. Hershey PA: Idea Group Publishing.
2
+
y[n]
Processing Unit
H
0
(z)
H
1
(z)
v
0
[n]
v
1
[n]
M
M
H
2
(z)
H
M--1
(z)
v
2
[n]
v
M--1
[n]
M
M
x[n]
Analysis bank
Synthesis bank
f
s
/2
f
s
/(2M)
H
0
(z)
F
0
(z)/M
1
2f
s
/(2M)
H
1
(z)
F
1
(z)/M
3f
s
/(2M)
H
2
(z)
F
2
(z)/M
(M--2)f
s
/(2M) (M--1)f
s
/(2M)
H
M--2
(z)
F
M--2
(z)/M
H
M--1
(z)
F
M--1
(z)/M
w
0
[n]
w
1
[n]
w
2
[n]
w
M--1
[n]
M
M
M
M
F
0
(z)
F
1
(z)
F
2
(z)
F
M--1
(z)
(a)
+
y[n]
Processing Unit
H
0
(z)
H
1
(z)
v
0
[n]
v
1
[n]
2
4
H
2
(z)
v
2
[n]
8
8
x[n]
Analysis bank
Synthesis bank
f
s
/2
f
s
/16
H
3
(z)
F
3
(z)/8
1
f
s
/8
H
2
(z)
F
2
(z)/8
f
s
/4
H
1
(z)
F
1
(z)/4
w
0
[n]
w
1
[n]
w
2
[n]
2
4
8
8
F
0
(z)
F
1
(z)
F
2
(z)
H
0
(z)
F
0
(z)/2
H
3
(z)
F
3
(z)
v
3
[n]
w
3
[n]
(b)
Figure 1. Analysis-synthesis filter bank. (a) M-channel uniform filter bank. (b) Three-level
octave filter bank. Note that in the case of interpolation by a given factor, the corresponding filter
should approximate this factor in the passband in order to preserve the signal energy.
The role of the filters in the synthesis part is to approximately reconstruct the original signal.
This is performed in two steps. First, for the uniform filter bank, the M sub-signals at the output
of the processing unit are interpolated by a factor of M and filtered by M synthesis filters F
k
(z)
for k
=
0,1,…,
M−1, whereas for the octave filter bank, the interpolation factors for the sub-
signals are the same as for the analysis part. Second, the outputs of these filters are added. In the
transferring and storing applications, the ultimate goal is to design the overall system such that,
despite of a significantly reduced number of bits used in the processing unit, the reconstructed
signal is either a delayed version of the original signal or suffers from a negligible lost of
information carried by the sub-signals.
There are two types of coding techniques, namely, lossy and lossless codings. For the lossless
coding, it is desired to design the overall system such that the output signal is si mply a delayed
version of the input signal or suffers from some phase distortion being tolerable in some
applications. For the lossy coding, it is beneficial to design the analysis-synthesis filter bank such
that some distortions, including amplitude distortion and aliasing errors, being less than those
caused by coding distortions are allowed. This increases the overall filter bank performance or,
alternatively, the same performance can be achieved by shorter filter orders or a shorter delay
caused for the input signal by the overall filter bank. These facts are very crucial for speech,
T. Saramäki and R. Bregovi
, ”Multirate Systems and Filter Banks,” Chapter 2 in Multirate Systems: Design
and Applications edited by G. Jovanovic-Dolecek. Hershey PA: Idea Group Publishing.
3
audio, and communication applications. The coding techniques are not considered at all in this
chapter and we concentrate on the case where the processing unit does not cause any errors to the
sub-signals.
In the case of audio or speech signals, the goal is to design the overall system together with
coding such that our ears are not able to notice the errors caused by reducing the number of bits
used for storing or transferring purposes. In the case of images our eyes serve as “referees”, that
is, the purpose is to reduce the number of bits to represent the image to the limit that is still
satisfactory to our eyes.
Depending on how many channels are used for the signal separation, there are two groups of
uniform filter banks, namely, multi-channel or M-channel filter banks (M
>
2) and two-channel
filter banks (M
=
2). In the first group, the signal is separated into M different channels and in the
second group into two channels. Using a tree-structure, two-channel filter banks can be used for
building M-channel filter banks in the case where M is a power of two. A more effective way of
building M-channel filter banks is to first design a prototype filter in a proper manner. The filters
in the analysis and synthesis banks are then generated with the aid of this prototype filter by
using a cosine-modulation or a modified discrete Fourier transform (MDFT) technique (Malvar,
1992b; Vaidyanathan, 1993; Fliege, 1993; Heller, Karp, and Nguyen 1999; Karp, Mertins, and
Schuller 2001).
Two-channel filter banks are very useful in generating octave filter banks. In this case, the
overall signal is first split with the aid of a two-channel filter bank into two bands. After that, the
decimated lowpass filtered signal is split into two bands using the same two-channel filter bank
and so on. There are two basic types of octave filter banks, namely, frequency-selective filter
banks mostly used for audio and telecommunications applications and discrete-time wavelet
banks used in applications where the signal waveform is desired to be preserved, like in the case
of images. For discrete-time wavelet banks, the frequency selectivity of the filters in the octave
analysis-synthesis filter banks is not so important due to their different applications. There are
other properties that are more important, as will be discussed in Subsection 4.2 and in more
details in Chapter 3. In these cases, the main goal is to preserve the waveform of the input signal
after treating it in an appropriate manner in the processing unit.
When two or more different signals are composed into a single signal, then a uniform
synthesis-analysis system is used, as shown in Figure 2. This system is also called a
transmultiplexer. In this system, all the M signals are interpolated by a factor of M and filtered by
M synthesis filters F
k
(z) for k
=
0,1,…,
M−1. Then, the outputs are added to give a single signal
with sampling rate being M times that of the input signals. The next step is to transfer the signal
through a channel. Finally, in the analysis bank the original signals are reconstructed with the aid
of M analysis filters H
k
(z) for k
=
0,1,…,
M−1. These signals have the original sampling rates due
to the decimation by a factor of M. If the output signal in the analysis-synthesis system is just a
delayed version of the input signal, then for the corresponding transmultiplexer the output signals
in the case of the ideal channel are delayed versions of the inputs. Therefore, the design of a
transmultiplexer can be converted to the design of an analysis-synthesis filter bank. Based on this
fact, this chapter does not consider the design of transmultiplexers. An interested reader is
referred to the textbook written by Fliege (1993).
T. Saramäki and R. Bregovi
, ”Multirate Systems and Filter Banks,” Chapter 2 in Multirate Systems: Design
and Applications edited by G. Jovanovic-Dolecek. Hershey PA: Idea Group Publishing.
4
+
Synthesis bank
x
0
[n]
x
1
[n]
x
2
[n]
x
M--1
[n]
M
M
M
M
F
0
(z)
F
1
(z)
F
2
(z)
F
M--1
(z)
Channel
H
0
(z)
H
1
(z)
y
0
[n]
y
1
[n}
M
M
H
2
(z)
H
M--1
(z)
y
2
[n]
y
M--1
[n]
M
M
Analysis bank
Figure 2. Synthesis-analysis filter bank: Transmultiplexer.
The outline of this chapter is as follows. Section 2 reviews various types of existing finite
impulse response (FIR) and infinite impulse response (IIR) two-channel filter banks. The basic
operations of these filter banks are considered and the requirements are stated for alias-free,
perfect-reconstruction (PR), and nearly perfect-reconstruction (NPR) filter banks. Also some
efficient synthesis techniques are referred to. Furthermore, examples are included to compare
various two-channel filter banks with each other. Section 3 concentrates on the design of multi-
channel (M-channel) uniform filter banks. The main emphasis is laid on designing these banks
using tree-structured filter banks with the aid of two-channel filter banks and on generating the
overall bank with the aid of a single prototype filter and a proper cosine-modulation or MDFT
technique. In Section 4, it is shown how octave filter banks can be generated using a single two-
channel filter bank as the basic building block. Also, the relations between the frequency-
selective octave filter banks and discrete-time wavelet banks are briefly discussed. Finally,
concluding remarks are given in Section 5.
2 Two-Channel Filter Banks
This section considers the synthesis of two-channel filter banks based on the use of FIR and
IIR filters. First, basic operation principles are discussed and the necessary requirements for
alias-free, perfect-reconstruction (PR), and nearly PR (NPR) filter banks are stated. Second, an
overview of the most important filter bank types and references to some existing synthesis
schemes are given.
2.1 Basic Operation of a Two-Channel Filter Bank
The block diagram of a two-channel filter bank is shown in Figure 3. This system consists of
an analysis and a synthesis bank as well as a processing unit between these two banks.
H
1
(z)
F
0
(z)
F
1
(z)
↑2
↑2↓2
H
0
(z)
↓2
y[n]
x[n]
v
0
[n]
v
1
[n]
x
0
[n]
x
1
[n]
w
0
[n]
w
1
[n]
u
0
[n]
u
1
[n]
y
0
[n]
y
1
[n]
Processing unit
Analysis bank Synthesis bank
Figure 3. Two-channel filter bank.
2.1.1 Operation of the analysis bank
The role of the analysis bank is to split the input signal x[n] into lowpass and highpass filtered
channel signals, denoted by x
0
[n] and x
1
[n] in Figure 3, using a lowpass−highpass filter pair with
transfer functions H
0
(z) and H
1
(z). Hence, the z-transforms of these signals are expressible as
(
)
(
)
(
)
1,0for
=
=
kzXzHzX
k
k
. (1)
T. Saramäki and R. Bregovi
, ”Multirate Systems and Filter Banks,” Chapter 2 in Multirate Systems: Design
and Applications edited by G. Jovanovic-Dolecek. Hershey PA: Idea Group Publishing.
5
After filtering, the signals in both channels are down-sampled by a factor of two by picking up
every second sample, resulting in two subband signal components, denoted by v
0
[n] and v
1
[n] in
Figure 3. If the input sampling rate is F
s
, then the sampling rates of v
0
[n] and v
1
[n] are F
s
/2. The
z-transforms of these components are given by
( )
(
)
(
)
(
)
(
)
[
]
.1,0for
2
1
2/12/12/12/1
=−−+= kzXzHzXzHzV
kkk
(2)
Typically, H
0
(z) and H
1
(z) have the same transition band region with the band edges being
located around f = F
s
/4 at f = (1−
ρ
1
)F
s
/4 and f = (1+
ρ
2
)F
s
/4 with
ρ
1
> 0 and
ρ
2
> 0, as shown in
Figure 4(b). In order to give a pictorial viewpoint of what is happening in the frequency domain,
Figure 4(a) shows the Fourier transforms of an input signal x[n], whereas Figure 5 shows those
of signals x
0
[n], x
1
[n], v
0
[n], and v
1
[n]. These transforms are obtained from the corresponding z-
transform by simply using the substitution z
=
e
j2
π
f
/
Fs
. It is seen that after decimation V
k
(z) for
k = 0, 1 contain two overlapping components X
k
(z
1/2
) and X
k
(−z
1/2
). This overlapping can,
however, be eliminated in the overall system of Figure 3 by properly designing the transfer
functions H
0
(z), H
1
(z), F
0
(z), and F
1
(z), as will be seen later on.
|
X
(
e
j2
π
f/F
s
)|
(a)
|
H
0
(
e
j2
π
f/F
s
)|
(b)
|
H
1
(
e
j2
π
f/F
s
)|
|
F
0
(
e
j2
π
f/F
s
)|
(c)
|
F
1
(
e
j2
π
f/F
s
)|
F
s
/4
2
3F
s
/4F
s
/2 F
s
1
(1
−
ρ
1
)
F
s
/4 (1+
ρ
2
)
F
s
/4
F
s
/2
F
s
F
s
/4 3
F
s
/4
F
s
/2
F
s
1
(1
−
ρ
2
)
F
s
/4 (1+
ρ
1
)
F
s
/4
Figure 4. (a) Magnitude of the Fourier transform of an input signal x[n]. (b) Amplitude responses
for H
0
(z) and H
1
(z). (c) Amplitude responses for F
0
(z) and F
1
(z).
2.1.2 Operation of the processing unit
In the processing unit, the signals v
0
[n] and v
1
[n] are compressed and coded suitably for either
transmission or storage purposes. Before using the synthesis part, signals in both channels are
decoded. The resulting signals denoted by w
0
[n] and w
1
[n] in Figure 3 may differ from the
original signals v
0
[n] and v
1
[n] due to possible distortions caused by coding and quantization
errors as well as channel impairments. In the sequel, it is supposed, for simplicity, that there are
no coding, quantization, or channel degradations, that is, w
0
[n]
≡
v
0
[n] and w
1
[n]
≡
v
1
[n].
T. Saramäki and R. Bregovi
, ”Multirate Systems and Filter Banks,” Chapter 2 in Multirate Systems: Design
and Applications edited by G. Jovanovic-Dolecek. Hershey PA: Idea Group Publishing.
6
|
X
1
(
e
j2
π
f/F
s
)|
(f)
F
s
/4 3F
s
/4F
s
/2 F
s
1
|
V
1
(
e
j2
π
f/(2F
s)
)|
(g)
F
s
/4 3
F
s
/4
F
s
/2
F
s
1/2
|
X
1
(
e
j2
π
f/F
s
)|
|
X
1
(
e
j2
π(
f+F
s
/2)/F
s
)|
|
U
1
(
e
j2
π
f/F
s
)|
(h)
F
s
/4 3F
s
/4F
s
/2 F
s
1/2
|
X
1
(
e
j2
π
f/F
s
)|
|
X
1
(
e
j2
π(
f+F
s
/2)/F
s
)|
(i)
F
s
/4 3
F
s
/4
F
s
/2
F
s
1
|
F
1
(
e
j2
π
f/F
s
)
X
1
(
e
j2
π
f/F
s
)|
1
|
F
1
(
e
j2
π
f/F
s
)
X
1
(
e
j2
π(
f+F
s
/2)/F
s
)|
(j)
F
s
/4 3F
s
/4F
s
/2 F
s
|
X
0
(
e
j2
π
f/F
s
)|
(a)
F
s
/4 3F
s
/4F
s
/2 F
s
1
|
V
0
(
e
j2
π
f/(2F
s)
)|
(b)
F
s
/4 3
F
s
/4
F
s
/2
F
s
1/2
|
X
0
(
e
j2
π
f/F
s
)| |
X
0
(
e
j2
π(
f+F
s
/2)/F
s
)|
|
U
0
(
e
j2
π
f/F
s
)|
(c)
F
s
/4 3F
s
/4F
s
/2 F
s
1/2
|
X
0
(
e
j2
π
f/F
s
)| |
X
0
(
e
j2
π(
f+F
s
/2)/F
s
)|
(d)
F
s
/4 3
F
s
/4
F
s
/2
F
s
1
|
F
0
(
e
j2
π
f/F
s
)
X
0
(
e
j2
π
f/F
s
)|
F
s
/4 3F
s
/4F
s
/2 F
s
1
|
F
0
(
e
j2
π
f/F
s
)
X
0
(
e
j2
π(
f+F
s
/2)/F
s
)|
(e)
Low-pass channel High-pass channel
Figure 5. Magnitudes of the Fourier transforms of the signals in the two-channel filter bank of
Figure 3. (a), (f) Transforms of x
0
[n] and x
1
[n]. (b), (g) Transforms of v
0
[n] and v
1
[n]. (c), (h)
Transforms of u
0
[n] and u
1
[n]. (d), (i) Transforms of unaliased components of y
0
[n] and y
1
[n].
(e), (j) Transforms of aliased components of y
0
[n] and y
1
[n].
2.1.3 Operation of the synthesis bank
The role of the synthesis bank is to approximately reconstruct in three steps the delayed
version of the original signal from the signal components w
0
[n] and w
1
[n]. In the first step, these
signals are up-sampled by a factor of two by inserting zero-valued samples between the existing
samples yielding two components, denoted by u
0
[n] and u
1
[n] in Figure 3. In the w
0
[n] ≡ v
0
[n]
and w
1
[n] ≡ v
1
[n] case, the z-transforms of these signals are expressible as
( )
(
)
( ) ( )
[ ]
1,0for
2
1
2
=−+== kzXzXzVzU
kkkk
. (3)
Simultaneously, the sampling rate is increased from F
s
/2 to F
s
and the baseband from [0, F
s
/4] to
[0, F
s
/2]. Therefore, u
0
[n] and u
1
[n] contain in their basebands, in addition to the frequency
components of v
0
[n] and v
1
[n] in their baseband [0, F
s
/4], the components in [F
s
/4, F
s
/2], as
illustrated in Figure 5. In Equation (3), X
k
(z) is the z-transform of the desired unaliased signal
component, whereas X
k
(−z) is the z-transform of the unwanted aliased signal component that
should be eliminated.
T. Saramäki and R. Bregovi
, ”Multirate Systems and Filter Banks,” Chapter 2 in Multirate Systems: Design
and Applications edited by G. Jovanovic-Dolecek. Hershey PA: Idea Group Publishing.
7
The second step involves processing u
0
[n] and u
1
[n] by a lowpass−highpass filter pair with
transfer functions F
0
(z) and F
1
(z), whereas the third step is to add the filtered signals, denoted by
y
0
[n] and y
1
[n] in Figure 3, to yield the overall output y[n]. The z-transform of y[n] is given by
(
)
(
)
(
)
zYzYzY
2
1
+
=
, (4)
where
( ) ( )
(
)
( ) ( ) ( ) ( )
[ ]
1,0for
2
1
2
=−+== kzXzFzXzFzVzFzY
kkkkkkk
. (5)
The role of the synthesis filters with transfer functions F
0
(z) and F
1
(z) is twofold. First, it is
desired that Y(z) does not contain the terms X
0
(
−
z) and X
1
(
−
z ). This is achieved by requiring that
F
0
(z)X
0
(
−
z) = −F
1
(z)X
1
(
−
z). Second, if it is desired that y[n] is approximately a delayed version of
x[n], that is, y[n] ≈ x[n − K], then F
0
(z)X
0
(z) + F
1
(z)X
1
(z) ≈ z
−
K
X(z) should be satisfied.
In order to satisfy these requirements, F
0
(z) and F
1
(z) should generate a lowpass−highpass
filter pair in a manner similar to H
0
(z) and H
1
(z). There exist two main differences. First, due to
the alias-free conditions to be considered in the next subsection, the lower and upper edges of
this filter pair are located at f = (1−
ρ
2
)F
s
/4 and f = (1+
ρ
1
)F
s
/4, as shown in Figure 4(c). Second,
because of interpolation, the amplitude responses should approximate two in the passbands. The
last subfigures on the left and right sides of Figure 5 show how the above conditions can be
satisfied in the frequency domain. The exact simultaneous conditions for H
0
(z), H
1
(z), F
0
(z), and
F
1
(z) to satisfy the above-mentioned two conditions will be given in the following subsections.
2.2 Alias-Free Filter Banks
Combining Equations (1), (4), and (5) the relation between the input and the output signals of
Figure 3 is expressible as
(
)
(
)
(
)
(
)
(
)
zXzAzXzTzY
−
+
=
, (6)
where
( )
[ ]
)()()()(
2
1
1100
zFzHzFzHzT += (7)
is the overall distortion transfer function and
( )
[ ]
)()()()(
2
1
1100
zFzHzFzHzA −+−= (8)
is the aliasing transfer function. In order to generate an alias-free filter bank, this term has to be
canceled, that is, A(z)
≡
0. The most straightforward way of achieving this is to select F
0
(z) and
F
1
(z) as follows:
(
)
(
)
zHzF
−
=
1
0
2 (9)
(
)
(
)
zHzF
−
−
=
0
1
2 . (10)
In the sequel, these conditions are used except for Subsection 2.5.2 where the filter bank is
constructed with the aid of causal and anti-causal IIR filters. After fixing F
0
(z) and F
1
(z) in the
above manner, the input-output relation of Equation (6) takes the following simplified form:
(
)
(
)
(
)
zXzTzY
=
, (11)
where
(
)
)()()()(
1
0
1
0
zHzHzHzHzT
−
−
−
=
. (12)
T. Saramäki and R. Bregovi
, ”Multirate Systems and Filter Banks,” Chapter 2 in Multirate Systems: Design
and Applications edited by G. Jovanovic-Dolecek. Hershey PA: Idea Group Publishing.
8
There are two important reasons for concentrating on the synthesis of two-channel filter banks
in such a manner that they are alias-free. First, relating F
0
(z) and F
1
(z) to H
0
(z) and H
1
(z)
according to Equations (9) and (10) makes both the design and implementation of the overall
system more efficient compared to the nearly alias-free case. Second, it has been observed that
when synthesizing the bank without the exact alias-free condition leads to a system that is
practically alias-free. This fact can be seen, for instance, from the results given by Nayebi,
Barnwell III, and Smith in (1992).
2.3 Perfect-Reconstruction (PR) and Nearly Perfect-Reconstruction (NPR) Filter Banks
The following subsection states the necessary and sufficient conditions for an FIR two-
channel filter bank to satisfy the PR conditions and describes their connections to the linear-
phase half-band FIR filters. Furthermore, these conditions are extended to their IIR counterparts.
2.3.1 Theorem for the PR Property
The PR property, that is, y[n]
=
x[n
−
K], is the ability of a system to produce an output signal
that is a delayed replica of the input signal. The necessary conditions for the PR property are
given for a two-channel FIR filter bank by the following theorem:
Theorem for the PR property: Consider the alias-free two-channel filter bank shown in Figure
3 with w
0
[n] ≡ v
0
[n] and w
1
[n] ≡ v
1
[n] and let H
0
(z) and H
1
(z) be the transfer functions of FIR
filters given by
[
]
∑
=
−
=
0
0
00
)(
N
n
n
znhzH and
[
]
∑
=
−
=
1
0
11
)(
N
n
n
znhzH . Then, y[n] = x[n
−
K] with K
being an odd integer, that is, T(z)
=
z
−K
, is met provided that the impulse-response coefficients of
( )
[ ]
∑
+
=
−
=−=
10
0
10
)()(
NN
n
n
znezHzHzE (13)
satisfy
[ ]
≠
=
=
. and odd is for 0
for 2/1
Knn
Kn
ne
(14)
In order to prove this theorem, Equation (12) is rewritten as
( )
[ ] [ ] [ ] [ ]
( )
∑∑
+
=
−
+
=
−
+=−−+==
1010
00
ˆ
)()(
NN
n
n
NN
n
n
znenezEzEzntzT , (15)
where
[ ]
[
]
[ ]
−
=
. odd for
even for
ˆ
nne
nne
ne
(16)
The impulse-response coefficients of this T(z) satisfy
[ ] [ ] [ ]
≠
=
=+=
,for 0
for 1
ˆ
Kn
Kn
nenent
(17)
yielding
(
)
K
zzEzEzT
−
=−−= )()(
. (18)
This implies that the output signal is the replica of the input signal delayed by K samples, as is
desired.
T. Saramäki and R. Bregovi
, ”Multirate Systems and Filter Banks,” Chapter 2 in Multirate Systems: Design
and Applications edited by G. Jovanovic-Dolecek. Hershey PA: Idea Group Publishing.
9
It is well known that the PR property can be satisfied only when K is an odd integer and
N
0
+N
1
is two times an odd integer (see, e.g., (Vaidyanathan, 1993)). There are two basic
alternatives to achieve the PR property, namely, K
=
(N
0
+N
1
)/2 and K
<
(N
0
+N
1
)/2, as illustrated
in Figures 6(a) and 6(b), respectively. In the first case, E(z) is an FIR filter transfer function with
a symmetric impulse response and the impulse-response value occurring at the odd central point
n
=
K being equal to 1/2, whereas the other values occurring at odd values of n are zero. Hence,
E(z) is the transfer function of a linear-phase FIR half-band filter (see, e.g., (Saramäki, 1993)). In
the second case, the impulse-response values at odd values of n are also zero except for one odd
value n
=
K, where the impulse response takes on the value of 1/2. The K
=
(N
0
+N
1
)/2 case is
attractive when the overall delay of K samples is tolerable, whereas the K
<
(N
0
+N
1
)/2 case is
used for reducing the delay caused by the filter bank to the overall signal.
0 5 10 15 20
−0.5
0
0.5
Impulse response for E(z): K=11
n in samples
impulse response
0 5 10 15 20
−0.5
0
0.5
Impulse response for −E(−z)
n in samples
impulse response
0 5 10 15 20
0
0.5
1
Impulse response for T(z)=E(z)−E(−z)
n in samples
impulse response
0 5 10 15 20
−0.5
0
0.5
Impulse response for E(z): K=7
n in samples
impulse response
0 5 10 15 20
−0.5
0
0.5
Impulse response for −E(−z)
n in samples
impulse response
0 5 10 15 20
0
0.5
1
Impulse response for T(z)=E(z)−E(−z)
n in samples
impulse response
(a) (b)
Figure 6. Impulse responses for E(z), E(−z), and T(z) for PR filter banks with N
0
+N
1
=
22. (a)
K
=
11
=
(N
0
+N
1
)/2. (b) K
=
7
<
(N
0
+N
1
)/2.
In the PR case, there is no amplitude or phase distortion. This is due to the fact that t[n] is
nonzero only at n
=
K achieving the value of unity. In the NPR case, the impulse response values
t[n] differ slightly from zero for n
≠
K and slightly from 1 for n
=
K so that there exists some
amplitude and/or phase distortions. These distortions are tolerable in many practical applications
(lossy channel coding and quantization) provided that they are smaller than the errors introduced
in the processing unit. Moreover, by slightly releasing the PR condition, filter banks with better
selectivities can be synthesized, as will be seen later on.
The above criteria for the PR and NPR property are also valid for IIR filter banks to be
considered in Subsection 2.5 (except for the banks to be described in Subsection 2.5.1). The
main difference is that the impulse responses of IIR analysis filters are of infinite length.
Therefore, the impulse response e[n] of E(z) and the response t[n] of T(z) are also of infinite
length. For a PR system e[n] and t[n] must satisfy the conditions of Equations (14) and (17) for
0
≤
n
<
∞ , whereas in the NPR case these conditions should be approximately satisfied.
2.3.2 PR Filter Bank Design and the Theorem for the PR Reconstruction
Consider
T. Saramäki and R. Bregovi
, ”Multirate Systems and Filter Banks,” Chapter 2 in Multirate Systems: Design
and Applications edited by G. Jovanovic-Dolecek. Hershey PA: Idea Group Publishing.
10
( )
[ ]
( )
∏∑
+
=
−−
+
=
−==
1010
1
1
0
1
NN
k
k
n
NN
n
zzSznezE (19)
with the impulse-response coefficients satisfying the conditions of Equations (14). The
factorization of E(z) as E(z)
=
H
0
(z) H
1
(−z) can be performed as follows. First, the zeros z
k
of E(z)
for k
=
1,2,…, N
0
+
N
1
are divided into two groups
α
k
for k
=
1,2,…, N
0
and
β
k
for k
=
1,2,…, N
1
in
such a manner that the zeros in both groups are either real or occur in complex-conjugate pairs.
Second, the constant S of E(z) is factorized as S = S
0
⋅S
1
. Then, according to the theorem of the
previous subsection, the transfer functions given by
( )
[ ]
( )
∏∑
=
−
=
−
−==
00
1
1
0
0
00
1
N
k
k
N
n
n
zSznhzH
α
(20)
( ) ( )
[ ]
( )
∏
∑
=
−
=
−
−==−≡
1
1
1
1
1
0
111
1
ˆ
ˆ
N
k
k
N
n
n
zSznhzHzH
β
(21)
can be used for generating a PR two-channel filter bank. In the above, H
0
(z) is directly the
transfer function of the lowpass analysis filter. The corresponding highpass filter transfer
function H
1
(z) is obtained from
(
)
zH
1
ˆ
by selecting the impulse-response coefficients to be
[
]
(
)
[
]
nhnh
n
11
ˆ
1−= for n
=
0,1,…, N
1
. The amplitude responses of these two filters are related
through
(
)
( )
(
)
ωπω
−
=
jj
eHeH
11
ˆ
so that
(
)
zH
1
ˆ
and H
1
(z) form a lowpass−highpass filter pair
with the amplitude responses obtained from each other using a lowpass-to-highpass
transformation
ω
→
π
−
ω
.
According to the above discussion, the synthesis of a PR two-channel filter bank can be stated
as follows: Given an odd integer K and integers N
0
and N
1
such that their sum is two times an
odd integer, find E(z) of order N
0
+
N
1
such that its impulse-response coefficients satisfy the
conditions of Equation (14) and H
0
(z) and H
1
(z) generated using the above factorization scheme
form a lowpass−highpass filter pair with the desired properties.
In general, this problem statement cannot be exploited in a straightforward manner for
designing two-channel FIR filter banks. However, there are two exceptional cases. In the first
case, K
=
(N
0
+N
1
)/2 and this problem statement is widely used for designing start-up two-channel
filter banks for generating discrete-time wavelet banks (Vetterli and Herley, 1992; Vetterli and
Kova
evi
1995; Misiti et al., 1996). In this case, half-band linear-phase FIR filter transfer
functions E(z) with the maximum numbers of zeros at z
=
−1 are of great importance. As a
curiosity, these filters are special cases of the maximally flat FIR filters introduced by Herrmann
(1971).
In the second exceptional case, the impulse-response coefficients of H
0
(z) and
(
)
zH
1
ˆ
are
time-reversed version of each other, that is,
[
]
[
]
nNhnh −=
001
ˆ
for n
=
0,1,…, N
0
(see Figure 7).
Furthermore N
0
=
N
1
=
K. These conditions imply the following (see, e.g., (Herrmann and
Schussler, 1970; Saramäki, 1993)). If H
0
(z) has a real zero or a complex-conjugate zero pair
inside (outside) the unit circle, then
(
)
zH
1
ˆ
has a reciprocal real zero or a complex-conjugate zero
pair outside (inside) the unit circle. Most importantly, if H
0
(z) has zeros on the unit circle, then
(
)
zH
1
ˆ
has the same zeros on the unit circle meaning that
(
)
zHzHzE
10
ˆ
)()( = is a linear-phase