Multirate数字滤波器：基础与应用探索

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"这篇文章是关于多速率数字滤波器、滤波器银行和多相网络在通信、语音处理、图像压缩、天线系统、模拟语音隐私系统以及数字音频产业中的应用的一个教程。它概述了多速率数字信号处理（DSP）的基础概念和构建模块，并讨论了近年来的研究进展，包括降采样和升采样滤波器的设计、分析/合成滤波器银行（又称正交镜像滤波器，或QMF）以及新的采样定理的发展。此外，还介绍了一些具体的应用场景，如波形的子带编码、语音隐私系统、整数和分数采样率转换（如在数字音频中）、数字交叉网络以及窄带滤波器系数的多速率编码。M带QMF滤波器银行也得到了特别的讨论。" 在多速率数字信号处理中，滤波器和滤波器银行是关键组件。滤波器用于修改信号的频率特性，而滤波器银行则可以将信号分解为多个子带，每个子带通过独立的滤波器处理，这样可以实现高效的数据压缩、信号分析和编码。多相网络是一种用于实现多速率滤波器的结构，它可以将单个滤波器分解为多个更简单的阶段，这些阶段在不同的时钟速率下运行，从而提高处理效率并简化设计。正交镜像滤波器（QMF）是滤波器银行的一种特殊形式，它在保持信号的低频成分的同时，可以有效地对高频成分进行处理。QMF在音频编码、图像压缩等领域有着广泛的应用，因为它们能提供良好的频谱局部化和较低的计算复杂度。多速率系统的采样率转换是另一个重要主题。无论是降采样还是升采样，都涉及到重新调整信号的采样频率，以适应不同的系统要求。在数字音频领域，这通常用于改变音频的播放速度而不改变音调，或者适应不同的存储和传输带宽。此外，文章还探讨了子带编码技术，这是一种将信号分解为多个频带然后分别编码的方法，常用于语音压缩和图像压缩，以减少数据量同时保持可接受的信号质量。最后，多速率编码在窄带滤波器系数的应用中也很重要，这种技术允许在不同速率下处理滤波器的参数，对于实时通信和自适应滤波等应用场景非常有用。这篇教程涵盖了多速率数字信号处理的多个核心概念和技术，对于理解和应用这些技术在实际工程问题中具有很高的价值。

be of good quality retaining naturalness and voice char-

acteristics. These features have indeed been achieved by

Cox et

al.

[29].

At the receiver end, y(n)

again split into subbands, and

the time-segments of the subbands unshuffled to get xk(n),

which can then be interpolated and recombined through

the synthesis filters.

Iv.

THE

POLYPHASE

DECOMPOSITION

4ND

ITS

APPLICATIONS

The polyphase decomposition, which originated from the

work by Bellanger et

a/.

[Ill,

very fundamental to many

applications in multirate DSP. These include efficient real-

time implementation of decimation and interpolation fil-

ters, fractional sampling-rate changing devices, uniform

DFT filter banks, and perfect-reconstruction analysiskyn-

thesis systems. More recently, the polyphase idea has been

applied for the derivation of new sampling theorems [30],

and for recovering bandlimited signalsfrom non-uniformly

sampled versions. In this section the basic concept

intro-

duced and some of the aforementioned applications are

elaborated. Section

deals with the remaining applica-

tions. To avoid repetitions, results presented in [3], [4] are

reviewed only briefly here.

A. The Polyphase Decomposition

senting a digital filter. We can rewrite H(z) in the form

Let H(z)

E:=-,

h(n)z-" be a transfer function repre-

H(z)

h(-4)z4

h(-2)1

h(0)

z-'[

* *

h(-3)z4

h(-l)z2

h(2)zT2

h(4)~-~

h(1)

h(3)z-Z

. .

.].

(5)

we use the abbreviations

m m

Eo(z)

h(2n)z-", El(z)

h(2n

l)z-"

(6)

n=-m

we can re-express (5) in the form

H(z)

Eo(z2)

z-'E1(z2).

(7)

Basically, we have merely grouped the impulse-response

coefficients h(n) into even numbered samples eo(n)

h(2n)

and odd-numbered samples e,(n)

h(2n

1). Denoting the

z-transforms of eo(n) and el(n) by Eo(z) and E,(z), respectively,

we obtain the relation (7). This representation will be called

the two-component polyphase decomposition of H(z). The

quantities

Eo(z)

and E,(z) are called the polyphase compo-

nents of H(z).

an example, let H(z)

2z-'

5z-'

4~-~. Then

H(z)

5z-'

z-'[2

4z-*],

that Eo(z)

5z-' and E,(z)

4z-l.

Note that the representation of (7) holds whether H(z)

FIR or

IIR.

For example, let

-l

By using the identity

x')/(l

x),

we can rewrite

(8)

that Eo(z)

1/(1

a'z-') and

El(z)

a41

a'z-').

In an exactly analogous manner,

possible to repre-

sent H(z) in the M-component polyphase form

M-I

H(z)

z-kEk(zM).

k=O

Once again Ek(z) are called polyphase components and the

coefficients ek(n) and

(11)

In other words, the impulse response sequence

{h(n)}

has

now been grouped into

subsequences ek(n).The kth sub-

sequence ek(n)

mererly the M-fold decimated version of

h(n

k). Note that for a given H(z), the quantity Ek(z)

depends on

For example Eo(z) in (7)

totally different

from that in (IO). A second subscript (such as Ek,M(~)), which

would make this distinction clear,

avoided in this paper

in the interest of simplicity. In most applications

fixed,

and there

no room for confusion.

An important property of the representation

(IO)

given

by the following relation:

are given by

ek(n)

h(nM

k),

M-I

H(ZWk)

ME,(zM) (12)

k=O

which follows from the property (IC)

the quantity

The

z transform Eo(z) of eo(n)

thus seen to be equal to

H(z"~W~)IM,

and since eo(n)

nothing but the M-fold deci-

mated version of h(n), this gives a proof of the formula (3b)!

Before getting into applications, we shall mention a sec-

ond type of polyphase decomposition which

notationally

more convenient in some situations. This

given by

M-I

H(z)

z-(~-'-~)R k(ZM). (13)

evident that RJz)

related to Ek(z) by Rk(z)

EM-'

-&),

which

just a renumbering of the components. The rep-

resentations (IO) and (13) will be, respectively, called Type

and Type 2 polyphase decompositions. Correspondingly,

Ek(z)

and Rk(z),

are called Type

and Type

2 polyphase components. It

important to notice that the

polyphase decomposition can be applied

for

any sequence

x(n)

and

not restricted to impulse response coefficients.

Application

Sampling-Rate Alteration

In Section

Il-B we mentioned three standard kinds of

sampling-rate alterations, namely decimation, interpola-

tion, and fractional decimation. Each of these involves adig-

ita1 filter H(z), which unfortunatelyoperates at apointwhere

the sampling rate

highest (see Figs. 7(a), 7(b) and 8(a)).

These filtering operations can be made more efficient as

described next.

Decimation Filters: Consider again the decimation filter

of Fig. 7(a) with

2. Supposewe implement H(z) (assumed

FlRof 1engthN)directlyasin Fig.14.Thesystem hastocom-

puteonlytheeven numbered output samples y(2n). Foreach

computed sample we require

multiplications and

additions.Assumethatx(n) isarriving in real-timeatthe rate

Fig.

14.

Direct implementation

FIR

decimation filter.

PROCEEDINGS

THE

IEEE,

VOL.

78,

NO.

JANUARY

1990

of one sample per unit time. The scheme of Fig. 14 would

then require that the

multiplications and

additions

be performed in one unit of time (because,

the next input

sample arrives, the internal delay elementsz-' are updated

to the new set of values).

So,

whenever y(2n) is being com-

puted, the system is working at the rate of

MPU and

APU.' And at odd intervals, the system

sitting idle

(because y(2n

need not be computed). This is clearly

an inefficient use of computing resources.

On the other hand, if we implement the system in poly-

phase form, we can redraw Fig. 7(a)

in Fig. 15(a). Next,

(a) (b)

Fig.

15.

Polyphase implementation of decimation filter. (a)

Redrawing based on Fig. 7(a). (b) Redrawing by use of first

noble identity (Fig.

9).

with the helpof the first noble identity(Fig.9)wecan redraw

Fig.

15(a)

as in Fig. 15(b). Now the building blocks

€&z)

and

E,(z)

are receiving the samples at half the rate (i.e., one sam-

ple in two units of time). We therefore need to perform only

NI2 MPUsand

1)/2APUson theaverage.Thepolyphase

representation

thus a neat tool which enables us to rear-

range the computations of the filtering operation,

as to

minimize the computational load per unit time. Looking at

it in another way, each multiplier in Fig. 14 is required to

complete

its

operation in one unit of time, whereas in Fig.

I5(b), two units of time are available for the same job.

For general

the decimation filter can be implemented

using the M-component decomposition of

(IO).

The com-

putational saving

M-fold.

lnterpolation Filters: Recall the interpolation filter of Fig.

7(b).AssumefirstthatH(z)

isdirectlyimplemented

in Fig.

14. The input

x,(n)

to the filter has all the odd numbered

samples equal to zero. As a result, only about

percent

of the multipliers in Fig. 14 have nonzero inputs. As such,

the implementation does not exploit this fact to minimize

the required computation per unittime. For example, when

even,

h(0)

has to work at two times the input rate

(because its input will change after this duration), while at

the same time

h(1)

is resting. Once again

polyphase

approach will give a more efficient design. For this we use

(13)

with

2, and invoke the second noble identity (Fig.

9) to obtain the implementation of Fig.

16.

Here each mul-

Fig.

16.

Polyphase implementation

interpolation filter.

tiplier in each building block is constantly occupied, and

gets

two

units of time to finish its job.

Commutator Models: A very convenient way to represent

the polyphase decomposition

offered bythe commutator

'The abbreviations MPU and APU represent multiplications per

unit time and additions per unit time, respectively, where a unit

of time is the separation between the samples of the input signal

x(n).

switch model. The basic principles of this are discussed

clearly in

[3]

and [4]. The applications of the commutator

model in multirate analysislsynthesis systems is also indi-

cated in [31].

can be verified that the Type-I polyphase

decomposition can be represented using the counterclock-

wise commutator model [4] and theType-2 polyphase using

theclockwise model. In this paperweshall not makeexplicit

reference to this type of a model, and to minimize dupli-

cation of the results described in [3] and [4], we refrain from

further elaboration.

Fractional Decimators: Consider Fig. 8(a), which is a

scheme to obtain

sampling rate reduction by the factor

MIL.

If we implement

H(z)

directly, as in Fig. 14, then it

inefficient due to two reasons: first the input to H(z) con-

tains

zero-valued samples out of every

samples

that only one out of

multipliers

working at

time. Sec-

ond, only one out of

computed samples in

xdn)

going

to be used. These two factors are related to the fact that the

computations in Fig. 8(a) are done at the point where the

sampling rate

the highest possible.

To obtain a more efficient implementation, we begin by

considering the

2 example again. By using the

Type-I polyphase representation and rearranging as we did

in Fig.

15,

we obtain the implementation of Fig. 17(a). On

(b)

Fig.

17.

Two ways

improve the efficiency of fractional-

decimation filter. (a) Using Type-I decomposition, based on

Fig.

15.

(b) Using Type-2 decomposition, based on Fig. 16.

the other hand, if we use the Type-2 decomposition and

rearrange as in Fig. 16, we get Fig. 17(b). Clearly Fig. 17(a)

and Fig. 17(b) are more efficient than Fig. 8(a), by factors of

and

respectively. Can we do better than these? Notice

that Fig. 17(a) exploits the presence of the decimator

whereas Fig. 17(b) exploits the presence of the interpolator.

possible to exploit both?

Theanswer, indeed,

in the affirmative. Such an efficient

scheme is obtained in

[3,

page 3111 by the use of filter struc-

tures with time-varying coefficients. The derivation in [3]

based on time-domain reasonings.

We shall now describe an implementation with the high-

est possible efficiency, based on z-domain reasonings [32]

using the polyphase decomposition. First notice that we are

stuck in Fig. 17 mainly because we cannot move the inter-

polator any more to the right (and decimator any more to

the left). This

because the noble identities simply cannot

be applied anymore! However, here

the nice trick which

comes to our rescue: we can write

z-'

z-33

that Fig.

17(b) can be redrawn

in Fig. 18(a). With the help of the

noble identities, this becomes Fig. 18(b). Next we can inter-

change the decimator with the interpolator (which is valid

because 2 and

are relatively prime; see section

11-B)

VAIDYANATHAN: MULTIRATE DIGITAL FILTERS

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