优化FIR数字滤波器在降采样与插值中的实现

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"该论文主要探讨了用于降采样和升采样的最优FIR数字滤波器实现方法，提出了多级降采样器和插值器的一般理论，并给出了一套优化设计多级降采样器的曲线和关系。论文指出，降采样和插值过程是互逆的，因此相同的优化设计曲线适用于这两个问题。此外，通过级联降采样和插值过程，可以得到高效实现的窄带FIR滤波器，其效率与递归椭圆滤波器设计相当。" 在信号处理领域，特别是在数字技术的应用日益增多的情况下，不同采样频率之间的转换需求变得越来越重要。例如，在通信、音频处理和图像处理中，降采样和升采样是常见的操作，用于调整信号的采样率以适应不同的系统需求或进行数据压缩。 FIR（有限脉冲响应）滤波器是一种线性相位的数字滤波器，由于其灵活性和可设计性，常用于信号滤波、降噪和信号整形等任务。在本论文中，作者关注的是如何优化设计FIR滤波器以实现高效的降采样和升采样。多级降采样器和插值器的设计理论为实现这一目标提供了基础，它们可以通过级联多个简单的滤波器来实现更复杂的滤波功能，从而提高计算效率。论文中给出的优化设计曲线和关系为工程师提供了指导，帮助他们在设计过程中找到最佳的滤波器参数，以达到特定的采样率转换效果。同时，通过将降采样和插值结合，可以实现对窄带信号的高效处理。窄带FIR滤波器在保持较高效率的同时，能够精确地通过或阻塞特定频带内的信号，这对于频率选择性的滤波任务至关重要。比较了这种通过级联降采样和插值获得的FIR滤波器实现与递归椭圆滤波器设计的效率，表明这两种方法在性能上是相当的。递归椭圆滤波器以其高选择性和优良的频率响应特性而闻名，但通常需要更多的计算资源。因此，本文提出的方法提供了一种替代方案，可以在保持高性能的同时，降低计算复杂度，对于资源受限的系统尤其具有价值。这篇论文为信号处理领域的研究人员和工程师提供了一种新的工具，用于设计和实现高效且适应性强的FIR滤波器，以应对各种采样率转换的需求。通过优化的多级降采样和插值策略，不仅可以实现窄带滤波，而且在效率上与传统方法相比毫不逊色，对于实际应用具有重要的意义。

446

IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, OCTOBER

1975

put

w(n)

by the relation

u(n)=

h(m)w(n-

N-

m=o

where

h(m),

1) are the filter coefficients

and

is the duration of the unit sample response of the filter.

From (1) it is seen that the computation of an output point

depends only upon past and present values of

w(n)

and not

upon past values of any internal filter variables. Thus the filter

computations need only be made once for every Mth output

point. Furthermore,

is known that

w(n)

is nonzero

only

for

every Lth input point. Thus only one multiplication and addi-

tion must be performed for every Lth input point. Therefore

the effective number of multiplications and additions is

N/(LM)

per output sample: instead of

as predicted by a simple ap-

plication of (1). For

this

and other reasons to be discussed

later, we will assume for the remainder of this paper that the

low-pass filter is

linear phase FIR design used in a direct

form implementation.

111.

OPTIMUM

DESIGN

MULTISTAGE

DECIMATORS

FOR SAMPLING

RATE

REDUCTION

In the previous section a general one-stage technique for

changing the sampling rate of a signal by the factor LIM was

discussed. For large changes in sampling rate, however, it is

generally more efficient to reduce the sampling rate

with

series of decimation stages rather than making the entire rate

reduction with one stage. In

this

way the simpling rate

reduced gradually resulting

much less severe filtering require-

ments on the low-pass filters at each stage. Bellanger

al.

[SI

and Nelson

al.

[8]

also have implemented sampling rate re-

ductions using several decimation stages; however they re-

stricted their results by only using factors of

at each stage.

Shively

[9]

considered a more general approach with integer

decimation for a two-stage design similar to that of Fig. l(b).

He also suggested a procedure for optimizing the two-stage de-

sign by properly choosing the decimation ratios of each

the

stages. In

this

section we extend and generalize the work of

Shively using the more general decimation stage shown in

Fig. 3(b). Design curves and formulas are presented for imple-

menting optimum decimators for a wide range of parameters

and conditions.

The basic multistage process for sampling rate reduction

with

stages is illustrated in Fig. 4(a) and a frequency domain

interpretation of this process

given in Fig. 4(b). The initial

sampling rate is

fro

and the fmal sampling rate is

fui

with in-

termediate sampling frequencies designated as

frl

fr2,

fr(K-1).

The sampling rate reduction achieved by each stage

of the decimation process is denoted as

Di,

1,2,

K and

therefore the intermediate sampling frequencies are given by

2This

does not imply that greater efficiencies can be obtained by

si-

multaneously increasing

and

both

Nand the

sampling

rate in

(1)

are proportional

LPF-

STAGE

FINPUT

SAMPLING

RATE

OUTPUT SAMPLING RATE

frK fr(K-1)

f,,

f--

(b)

Fig.

(a) Illustration of a K-stage decimator and (b) frequency re-

sponse interpretation of this process.

The overall sampling rate reduction achieved

this process is

denoted as

and is given as

fro

/frK

Di.

(3)

i=1

In Fig. 4(b)

X[exp

j(27rflfu,,)J

represents the magnitude

the spectrum of

x(n)

and

Y[exp

j(hflfrK)]

corresponds to

the magnitude

the spectrum

y(n).

From the sampling

theorem, we recognize that the highest frequency

(n)

which

f,<-.

frK

Because of practical considerations in the design of the low-

pass filters used in the decimation process, this bandwidth can

never be fully realized and the usable portion of the base-

band will always be somewhat less than

f,.

We denote this

bandwidth as being froh zero to

corresponding to the band

over which the magnitude response of the composite

the

low-pass filters remains flat within specified tolerance limits of

6,.

Also

because of practical considerations a inakimum

toierance must be allowed for the magnitude response of the

low-pass filters in the stopband and this will be denoted as

6,.

We can now focus on the design requirements for each stage.

We will permit the most general

form

for each stage as dis-

cussed in the previous section [see Fig. 3(b)]

This form for

stage

is illustrated in Fig. 5(a). Li corresponds to the sam-

pling rate increase and

corresponds to the sampling rate de-

crease for stage

there

a net sampling rate reduction

this

implies,Mi

Li.

Then

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