Multirate Digital Filters, Filter Banks,
Polyphase Networks,
and
Applications:
A
Tutorial
Multirate digital filters and filter banks find application in com-
munications, speech processing, image compression, antenna
sys-
tems, analog voice privacy systems, and in the digital audio indus-
try. During the last several years there has been substantial progress
in multirate system research. This includes design of decimation
and interpolation filters, analysis/synthesis filter banks (also called
quadrature mirror filters, or QMFJ, and the development
of
new
sampling theorems. First, the basic concepts and building blocks
in multirate digital signal processing (DSPJ, including the digital
polyphase representation, are reviewed. Next, recent progress as
reported by several authors in this area is discussed. Several appli-
cations are described, including the following: subband coding of
waveforms, voice privacy systems, integral and fractional sampling
rate conversion (such as in digital audio), digital crossover net-
works, and multirate coding of narrow-band filter coefficients. The
M-band QMF bank is discussed in considerable detail, including
an analysis of various errors and imperfections. Recent techniques
for perfect signal reconstruction in such systems are reviewed. The
connection between
QMF
banks and other related topics, such
as
block digital filtering and periodically time-varying systems, based
on
a
pseudo-circulant matrix framework, is covered. Unconven-
tional applications of the polyphase concept are discussed.
I.
INTRODUCTION
In recent years there has been tremendous progress in
the multirate processing of digital signals. Unlike the sin-
gle-rate system, the sample spacing in
a
multirate system
can vary from point to point
[I], [2].
This often results in more
efficient processing of signals because the sampling rates
at various internal points can be kept as small as possible.
Unfortunately, thisalso results in the introduction of
a
new
type of error, i.e., aliasing, which should somehow be can-
celed eventually.
The basic building blocks in a multirate digital signal pro-
cessing
(DSP)
system are decimators and interpolators. In
1981,
an excellent tutorial article on decimation and inter-
polation appeared in
[3].
Subsequent to this
a
text on the
subject
of
multirate systems has also been published by the
same authors
[4].
Since then,
a
number of new develop-
ments have taken place in the area, particularly in multirate
Manuscript received October12,1988; revised Junel3,1989.This
work was supported
in
part by the National Science Foundation
under grants DCI 8552579 and MIP 8604456.
The author
is
with the Department
of
Electrical Engineering, Cal-
ifornia Institute
of
Technology, Pasadena,
CA
91125,
USA.
IEEE
Log
Number 8933329.
digital filter bankdesigns.Ashort summaryof someofthese
developments was reported recently by this author at an
IEEE
international conference
[5].
The purpose of this article
is
to provide a self-contained and more complete exposure
to many recent contributions on multirate systems, includ-
ing filter bank design.
As
mentioned in
[3],
multirate systems find application
in communications, speech processing, spectrum analysis
[6],
radar systems, and antenna systems. In this tutorial,
two
sections are devoted to a review of applications. In Section
111,
we point out applications in digital audio systems, in
subband coding techniques (used in speech and image
compression), and in analog voice privacy systems (for stan-
dard telephone communications). In Section
V-E,
appli-
cations of special transfer functions (such
as
complemen-
tary functions) in digital audio is reviewed. In Section
IX,
several unconventional applications of multirate systems
and polyphase theory are indicated. These include a) deri-
vation of new sampling theorems for efficient compression
of signals, b) derivation of new techniques for efficient cod-
ing of impulse response sequences of narrow band filters,
c) design of
FIR
filters with adjustable multilevel responses,
and d) adaptive filtering in subbands.
A.
Paper
Outline
In section
II,
basic tools, such as decimators, interpola-
tors, decimation and interpolation filters, and digital filter
banks, are reviewed, along with the interconnection prop-
erties of the building blocks. In section
Ill,
some applica-
tions of multirate
DSP
are indicated, in digital audio sys-
tems, in subband coding, and in voice privacy systems.
Section
IV
reviews the digital polyphase decomposition due
to Bellanger, along with applications such as the uniform
DFT filter bank. The concept of multilevel polyphase
decomposition
is
also introduced here
as
a
tool for efficient
implementation of fractional decimation filters. Several
special types of filter banks, such as Nyquist filters, power-
complementary systems and Euclidean filter-banks, are
studied in section
V.
In section
VI,
the two-band
QMF
bank
is
studied in sufficient detail along with procedures for
eliminating aliasing in such systems. Procedures for elim-
ination of amplitude and/or phase distortion are discussed.
00189219/90/0100-0056$01.00
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1990
Perfect-reconstruction two-channel QMF banks are intro-
duced by blending the polyphase concept with the classical
network-theoretic concept of losslessness.
The relation between M-band QMF banks and two other
related topics (block filtering and periodically time-varying
systems)
is
reviewed in section
VII,
based on an algebraic
structure called the pseudo-circulant matrix. In section
VIII,
M-band QMF banks are discussed in greater detail, and
techniques for elimination of aliasing, amplitude, and phase
distortions are reviewed. Section
IX
discusses unconven-
tional applications, and Section
X
discusses some exten-
sions of multirate ideas to cases of multidimensional sig-
nals. The paper concludes with a discussion of open
problems
in
multirate DSP.
B.
Notations
and
Terminology
and
w
are used as frequency variables for
continuous-time and discrete-time cases, respectively. In
the discrete-time case the term normalized frequency
is
used to denote
f
=
d2s.
The frequency response of a trans-
fer function
H(z)
is
expressed as H(e'")
=
IH(e/")l
e
/$("),
where
IH(e/")l
is
the magnitude response and
+(U)
the phase
response. The quantity
~(w)
=
-d+(w)/dw
is
the group delay
of
H(z).
If (H(e/")l
is
constant for all
a,
H(z)
is
all-pass. If
+(a)
has the form
ko
-
klo,
then
H(z)
is
said to have linear phase
and the group delay is a constant
k,;
physically, if the input
to such a filter
H(z)
has energy only in the passband of
H(z),
then the output
is
a delayed version of the input, by
kl
sam-
ples. Unless mentioned otherwise,
a
low-pass filter has real
coefficients,
so
that IH(e/")l
is
symmetric and
+(U)
is
anti-
symmetric with respect too
=
0.
Usually (H(e/")(
is
plotted
for
0
I
f
5
0.5
(i.e., for
0
5
w
5
s).
If
up
and
us
denote the
passband and stopband edges of
a
low-pass filter, the quan-
tity
wc
=
(ap
+
wJ2
is
said to be the cutoff frequency.
Bold-faced quantities denote matrices and vectors, as in
A,
H(z)
etc. The symbol
/k
denotes the
k
X
k
identity matrix
(with subscript often omitted). The quantitiesAr,At and A*
denote, respectively, the transpose, transpose conjugate,
and conjugate of
A.
For functions
H(z),
the notation
H,(z)
denotes conjugation of the coefficients without conjugat-
ing
z.
For example if
H(z)
=
a
+
bz-',
then
HJz)
=
a*
+
b*z-l.
Thus, H*(z)
=
H,(z*). The notation
fi(z)
stands for
H;(z-').
In other words, conjugate the coefficients, take
transpose (if matrix), and replace
z
with
z-'.
When
z
=
e/"
(i.e., on the unit circle), we have
A(z)
=
Ht(z).
Linear time-
invariant systems
[7l
are abbreviated as LTI and linear
periodically time-varying systems as
LPW.
A
p
x
r
matrix
A is said to be unitary (orthogonal if
it
is
real) if AtA
=
c/~,
c
#
0.
Note that A
is
not restricted to be square. For exam-
ple,
[::$)]
is
unitary for any real
0.
The symbol
WM
stands
for The subscript
M
is
usually deleted because
its
value
is
often clear from the context. This quantity appears
in the definition of the discrete Fourier transform (DFT)
[I,
[81.
Thus an M-point sequence
[xo,
xl,
. .
,
xM-,l
has the
M-point DFT sequence
The variables
M-1
xk
=
XnWkn,
0
5
k
I
M
-
1.
(1
a)
ll=O
The inverse DFT (IDFT)
is
given by
(1
b)
The most crucial property of
W
that finds repeated use in
multirate
DSP
is the following:
M,
ll=O
0,
otherwise.
k
=
integer multiple of
M
c
Wk"=
(IC)
M-l
I
For any pair of integers
k,
n,
we have
WK
=
W",
if and only
if k
-
n
is an integer multiple of
M.
In
particular, therefore,
Wk
#
W"for0
5
k
<
n
I
M
-
1.
State Space Descriptions: Consider a discrete time trans-
fer matrix
H(z)
with input vector
u(n)
and output vector
y(n).
Suppose we have implemented this transfer matrix using
a structure, and let
N
denote the number of delay elements
used. Label the outputs of the delay elements as the state
variables
xk(n),
0
I
k
5
N
-
1,
and define the state vector
x(n)
=
[x,(n) xl(n)
.
xN-,(n)lr.
With
u(n)
and
y(n)
denoting
the input and output (vector) sequences to the structure,
one can always find equations of the form
[9]
x(n
+
1)
=
Ax(n)
+
Bu(n),
y(n)
=
Cx(n)
+
Dub),
to describe the structure. This
is
called the state-space
description of the structure. The matrix A, called the state
transition matrix, has size
N
x
NI
where
N
is the number
of delays in the structure. The transfer function
H(z)
of the
structure
is
given by
H(z)
=
D
+
C(z/
-
A)-'B. The smallest
number
of
delay elements (i.e.,
z-l
elements) required to
implement
H(z)
is
called the McMillan degree (or simply,
the degree) of
H(z).
If the number of delays
N
in the struc-
ture
is
equal to the degree, then the structure
is
said to be
a minimal realization of H(z). This
is
equivalent to saying
that A
is
as small as possible.
A
summary of acronyms and common notations used in
this paper
is
found in the Nomenclature, which follows sec-
tion
XI.
II.
BASIC
BUILDING BLOCKS
AND
TOOLS
In
this
section we introduce the basic multirate building
blocks, along with their frequency-domain characteriza-
tions, and interconnection behaviors.
A.
Decimators
and
Interpolators
decimator
is
characterized by the input-output relation
Fig.
1
shows block diagrams of these building blocks. The
ydn)
=
x(Mn) (2a)
(a)
(b)
Fig.
1.
Building blocks.
(a)
M-fold decimator.
(b)
L-fold
interpolator.
which says that the output at time
n
is
equal to the input
at time
Mn.
As
a consequence, only the input samples with
sample numbers equal to a multiple of
M
are retained. This
sampling-rate reduction by a factor of
M
is
demonstrated
in Fig.
2
for the case of
M
=
2.
The L-fold interpolator
is
char-
VAIDYANATHAN:
MULTIRATE
DIGITAL FILTERS
57
I
,
I
,
~ ...
_,,
0
1234
Fig.
2.
Demonstration of decimation for
M
=
2.
acterized by the input-output relation
That
is,
the output yI(n) is obtained by inserting
L
-
1
zero-
valued samples between adjacent samples of x(n),
as
dem-
onstrated in Fig. 3for
L
=
2. The decimator and interpolator
Sincedecimation corresponds tocompression in the time
domain, one might expect a stretching effect in the fre-
quencydomain. To be more precise, theztransform of y,(n)
is
given by
(3b)
which means MY,(e/")
=
E::,,'
X(e/'"-2"k"M
).
The term with
k
=
0
is
indeed the M-fold stretched version of X(e/"). The
M
-
1
terms with
k
>
0
are uniformly shifted versions of
this stretched version. These
M
terms together make up
a
function with period
27r
in
w,
which
is
the basic property
of the Fourier transform of any sequence [8]. Fig. 4(c) dem-
onstrates this effect for
M
=
2.
The terms with
k
>
0
are
called the aliasing terms. As long as x(n)
is
bandlimited to
IwI
<
dM,
there is no overlap of these terms with the
k
=
0
term.
The fundamental difference between aliasing and imag-
ing is important to notice. Aliasing can cause
loss
of infor-
mation because of the possible overlap of the shifted ver-
sionsof the stretched version of X(e1"). Imaging, on the other
hand, does not lead to any
loss
of
information (which iscon-
sistent with the fact that no time-domain samples are lost).
1
M-'
yD(z)
=
-
X(Z"MWk)
M
k=O
B.
Interconnections
Fig.
5
shows acascade connection which is often encoun-
tered in filter-bank systems. The signal v(n) here
is
equal to
Fig. 3. Demonstration of interpolation for
L
=
2.
X(n)*v(n)
are linear systems even though they are time-varying [4],
[SI,
[IO].
The z transform of the interpolator output yl(n)
is
given
Y/(Z)
=
X(ZL).
(3-3)
This means YI(e/")
=
X(eluL) i.e., YI(e/") is an L-fold com-
pressed version of X(e"?, as demonstrated in Fig. 4(b). The
appearance
of
multiple copies of the basic spectrum in Fig.
by
[41:
-n/M
n/M
Fig.
5.
Effect of decimation followed
by
interpolation.
x(n) whenever
n
is
a
multiple of
M,
and zero otherwise. The
transform-domain relation is
(4)
4 is called the imaging effect and the extra copies are the
1
M-'
V(Z)
=
-
c
X(zWk)
images created by the interpolator.
M
k=O
-2n
-x:
0
n
2n
(a)
t
,w
(C)
Fig.
4.
Transform-domain effects of decimation and inter-
polation. (a) The
z
transform.
(b)
L-fold compressed version.
(c) Demonstration of effect when
M
=
2.
which means that MV(e1") is
a
sum of X(e/") with the M
-
1
uniformly shifted versions X(el(w-2"k'M'
1.
From the figure
we see that x(n) can be recovered from
v(n)
by eliminating
the images by filtering, provided none of the images has an
overlap with X(e? If such an overlap occurs, it implies
aliasing and
x(n)
cannot be recovered. Notice that in order
for x(n) to be recoverable it
is
not necessary for X(e/") to be
restricted to
IwI
<
dM.
It is sufficient for the total band-
widthofX(e1")to
belessthan2dM.Thusageneral
bandpass
signal with energy in the region
a
5
w
5
a
+
2a/M can be
decimated by
M
without creating overlap of the alias com-
ponents, and the decimated signal in general is
a
full-band
signal.
A different type of cascade is shown in Fig. 6(a). We shall
have occasion to use this in section
IV-B,
which concen-
trates on multilevel-polyphase decompositions. It should
becautionedthatthetwo building blocks in Fig.6(a)are not,
in general, interchangeable, i.e., the systems in Fig. 6(a) and
6(b) are not equivalent. For example, with
M
=
L,
the system
of Fig. 6(a) is an identity system, whereas the system of Fig.
6(b) causes
a
loss
of
M
-
1
out of M samples. It can be shown
sa
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x(n)*yt(n)
(a)
x(n) y2(n)
(b)
Fig.
6.
Two ways to cascade decimator and interpolator.
These
are
equivalent
if
and only if
M
and
L
are relatively
prime.
(a)
Example of identity system. (b) Example of
loss
of
M
-
1
out of
M
samples.
(Appendix A) that the systems of
Fig.
6(a) and 6(b) are iden-
tical if and only if
L
and
M
are relatively prime.
Decimation Filters and Interpolation Filters: In most
applications a decimator
is
preceded
by
a bandlimiting fil-
ter
H(z)
whose purpose
is
to avoid aliasing. For example, a
low-pass filter with stopband edge
U,
=
TIM
can serve as
such a filter. The cascade shown in
Fig.
7(a)
is
commonly
called a decimation filter. An interpolation filter, on the
other hand,
is
a device which follows an interpolator (Fig.
7(b)), the purpose being to eliminate the images. The low-
pass filter of
Fig.
7(c) again serves as an example (with
L
=
M).
-y(n)
The decimation filter
(a)
-y(n)
(b)
-
-+ilH(eJw;
The interpOlatiOn filter
w
M
m
(C)
Fig.
7.
(a)
Decimation filter.
(b)
Interpolation filter. (c) Low-
pass filter.
Fractional Sampling Rate Alterations:
Fig.
8(a) shows a
scheme for reducing the sampling rate
by
a nonintegral
(rational) number
MIL.
Fractional reduction of sampling rate
often results in data compression without
loss
of infor-
mation. As an example, if X(eJw)
is
as in
Fig.
8(b),
then a frac-
tional reduction by
312
is
possible. This can be accom-
w
-&
0
x
3
(C)
Fig.
8.
Decimation by rational fraction of
MIL. (a)
General
structure. (b) Exampleof bandlimited signal.
(c)
Effect of frac-
tional decimation of this signal
(L
=
2,
M
=
3).
plished
by
setting
M
=
3,
L
=
2
in Fig. 8(a). The filter
H(z)
is
then taken to be
low
pass, with passband edge at *I3 and
stopband edge at
2~13.
Notice that
in
this application, the
transition bandwidth of
H(z)
need not be unduly narrow.
The various signals in Fig. 8(a) have transforms as in Fig. 8(c),
so
that Y(e/")
is
a fractionally stretched version of X(e'").
Two Noble Identities: In
Fig.
9(a) we have a decimator fol-
lowed
by
a transfer function
G(z).
It can be proved, based
(C)
(d)
Fig.
9.
Noble identities for multirate systems. (a) Decimator
followed by transfer function
G(z).
(b) Equivalent cascade.
(c)
Example of transfer function preceding. (d) Equivalent
cascade.
on (3b), that
this
cascade
is
equivalent to the one in Fig. 9(b)
provided
G(z)
is
a rational transfer function (i.e., a ratio of
polynomials
in
z-').
In
a similar manner, the two cascades
in
Figs.
9(c) and 9(d) are equivalent (provided
G(z)
is
rational),
as can be proved from (3a). These identities are very val-
uable in practically all applications for efficient implemen-
tation of filters and filter banks. We shall call these the
"noble identities."
C.
Analysis and Synthesis Banks
These are the two basic types of filter banks. An analysis
bank
is
a set of analysis filters
Hk(Z)
which splits a signal into
M
subband signals xk(n) as shown in
Fig.
10(a). What we do
(C)
Fig.
10.
Analysis and synthesis filter banks.
(a)
Analysis
bank. (b) Synthesis bank.
(c)
Typical response of uniform
DFT filter bank; here
M
=
4.
with the subband signals depends on the application, aswe
shall see in sections
Ill,
VI,
and
IX.
Next, a synthesis bank
(Fig. 10(b)) consists of
M
synthesis filters
Fk(Z),
which com-
bine
M
signals yk(n) (possibly from an analysis bank) into
a reconstructed signal i(n). There are several types of filter
banks, i.e., the complementary type, the Nyquist type, etc.,
to be described in Section
V
along with applications.
Uniform DFT Filter Banks: An analysis bank with
M
filters
(M
>
1)
is
said to be a uniform DFTfilter bank if all the filters
are derived from
H&)
according to
Hk(z)
=
H,(zWk),
0
5
k
5
M
-
1.
Here
H&)
is
called the prototype filter. Note
VAI DYANATHAN: MULTI RATE DIGITAL FILTERS
59
~
that Hk(e/")
=
Ho(el(w-2*k'M)
),
which means that the fre-
quency responses of
HJz)
are uniformly shifted versions
of Ho(elw). Fig.
IOW
shows a typical set of responses, where
Ho(z)
is
taken to be low pass. More details can be found in
section
IV-C
and in [4] and [Ill.
111.
SOME
APPLICATIONS
OF
MULTIRATE
SYSTEMS
We shall now review a number of important applications
of multirate filters and filter banks, with pointers to the lit-
erature for details, examples, and demonstrations. In sec-
tion
IX,
several unconventional applications are also out-
lined.
Applications in the design of transmultiplexers (which
are devices for conversion between frequency division
multiplexing (FDM) and time-division multiplexing (TDM))
are not discussed here in detail, primarily because of the
excellent treatment already available in [13]. Also see [I41
for the correspondence between transmultiplexers and
analysiskynthesis filter banks. The input to a TDM-to-FDM
converter
is
a signal y(n), which
is
the time-multiplexed ver-
sion of
M
signals y&),
0
5
k
5
M
-
1.
Given y(n), the com-
ponents yk(n) can easily be separated out by use of a com-
mutator switch [4], [13]. TheseM signals are then modulated
using distinct carrier frequencies. The carrier frequencies
uk,
0
5
k
5
M
-
1
are chosen
so
that there
is
sufficient spec-
tral gap between the messages. A sum of these
M
signals
(which
is
the FDM signal)
is
then transmitted through the
channel. The total channel bandwidth
is
therefore required
to exceed the sum of signal bandwidths because of the safe-
guard gap between adjacent spectra. The gap enables one
to obtain perfect recovery of the multiplexed signals yk(n)
at a future point.
A novel approach to transmultiplexing was suggested in
[36] and cited in [14], based on synthesis and analysis filter
banks. This approach permits overlap between the spectra
of successive messages in the frequency domain. The total
required channel bandwidth
is
therefore less than that in
conventional FDM channels. Conditions are derived under
which cross-talk can be avoided and the set of
M
original
signals can
still
be recovered from
this
version. Details can
be found in reference [36] cited in [14].
A. Digital Audio Systems
In the digital audio industry,
it
is
a common requirement
to change the sampling rates of band-limited sequences.
This arises for example when an analog music waveform
x,(t) is to be digitized. Assuming that the significant infor-
mation
is
in the band
0
5
lQ(I27r
5
22
kHz [15], a minimum
sampling rate of 44 kHz
is
suggested (Fig. Il(a)). It
is,
how-
ever, necessary to perform analog filtering before sampling
to eliminate aliasing of out-of-band noise. Now the require-
ments on the analog filter
/-/@)
(Fig. Il(b)) are strigent:
it
should have a fairly flat passband
(so
that
XJjQ)
is
not dis-
torted) and a narrow transition band
(so
that only a small
amount of unwanted energy
is
let in). Optimal filters for this
purpose (such as elliptic filters [9], which are optimal in the
minimax sense) have a very nonlinear phase response [16,
page82laround the bandedge (i.e., around
22
kHz). In high-
quality music this
is
considered to be objectionable [15]. A
common strategy to solve this problem
is
to oversample
x,(t) by a factor of two (and often four). The filter
H,(jQ)
now
has a much wider transition band, as in Fig. Il(c),
so
that
minimum over-sampling
sampling rate
*:
=
4
:/:kHz
-44
-22 22
44 88
t
*
kHz
-22
0
22
(b)
A
1
4
kHz
-44
-22
0
22
44
(C)
linear-phase
LPF
FIR
filter
(d)
Fig.
11. (a)
Spectrumofx,(t). (b)Antialiasingfilter response
for sampling
at
44
kHz.
(c)
Antialiasing filter response for
sampling
at
88
kHz.
(d) Improved scheme
for
A/D stage
of
digital audio system.
the phase-response nonlinearity
is
acceptably low. A simple
analogBessel filter (which has linear phase in the passband
[9]) can be used in practice. The sequencex,(n)
so
generated
is
then lowpass filtered (Fig. Il(d)) by a digital filter
H(z)
and
then decimated by the same factor of two to obtain the final
digital signal x(n). The crucial point
is
that since
H(z)
is
dig-
ital,
it
can be designed to have linear phase [71, [161, [171,
while at the same time providing the desired degree of
sharpness.
A
similar problem arises after the DIA conversion stage,
where the digital music signal y(n) should be converted to
an analog signal by lowpass filtering. To eliminate the
images of Y(e'") in the region outside 22 kHz, a sharp cutoff
(hence nonlinear phase) analog low-pass filter
is
required.
This problem is avoided by using an interpolation filter, as
in Fig. 7(b),which increasesthesampling ratedigitally.After
this, DIA conversion
is
performed followed by analog fil-
tering. The interpolation filter
H(z)
is
once again a linear-
phase
FIR
low-pass filter and introduces no phase distor-
tion.
The obvious price paid in these systems
is
the increased
internal rate of computation. However, by using the poly-
phase framework (section
IV)
the efficiency of these mul-
tirate systems can be dramatically improved.
In digital audio,
it
is
relativelyeconomic (compared to the
analog case) to produce high-quality copies of material from
one medium to another [15]. Perhaps to discourage such
practice, the sampling rates used for various mediaareoften
made different from each other. It
is
therefore necessary
in studios to design efficient nonintegral sampling rate con-
verters (such as the one in Fig. 8(a)). See section
IV-B
for
further detai
Is.
Further applications of mu Iti rate fi Iter banks
in digital audio can be found in section
V-E.
B.
Subband Coding
of
Speech
and
Image Signals
In practice, one often encounters signals with energy
dominantly concentrated in a particular region
of
fre-
60
PROCEEDINGS
OF
THE
IEEE,
VOL.
78,
NO.
1,
JANUARY
1990