impulse

7
Finite response
(FIR) filter design

343
7.1 Introduction
349
7.2 FIR filter design
350
7.3 FIR filter specifications

7.4 FIR coefficient calculation methods 351

7.5 Window method 352

7.6 The optimal method 367

380
7.7 Frequency sampling method
7.8 Comparison of the window, optimum and frequency sampling methods 398

402
7.9 Special FIR filter design topics
7.10 Realization structures for FIR filters 407

7.11 Finite wordlength effects in FIR digital filters 411

7.12 FIR implementation techniques 420

7.13 422
Design example
7.14 Summary 425

7.15 Application examples of FIR filters 425

Problems 426

References 435

Bibliography 436

Appendices 437

This chapter is concerned with the design of FIR filters from specifications, through
coefficient calculation, to analysis of finite wordlength effects and implementations.
Several fully worked examples are given throughout the chapter to illustrate the
various design stages and to consolidate the important concepts. A complete filter
 7.1 Introduction 343

design is included to show how all the stages fit together and to assist
the readers
who wish to design their own filters. PC-based MATLAB programs
are available on the
web (see Preface for details). In addition, the MATLAB and C-language prog
rams are
available on the CD in the companion handbook and may be used to replicate the
results presented here or for designing user-specified filters.

7.1 Introduction

First, we will summarize important characteristics of FIR filters before devoting

attention to their design.

7.1.1 Summary of key characteristic features of FIR filters

(1) The basic FIR filter is characterized by the following two equations:
N-1

y(n)= Σ h(k)x(n-k) (7.1a)

k0

N-1
H(z) = h(k)z (7.1b)
20

where h(k), k = 0, 1,... N-1, are the impulse response coefficients of the
filter, H(2) is the transfer function of the filter and N is the filter length, that is
the number of filter coefficients. Equation 7.1a is the FIR difference equation.
It is a time domain equation and describes the FIR filter in its nonrecursive
form: the current output sample, y(n), is a function only of past and present
values of the input, x(n). When FIR filters are implemented in this form, that is
by direct evaluation of Equation 7.1a, they are always stable. Equation 7.1b is
the transfer function of the filter. It provides a means of analyzing the filter, for
example evaluating the frequency response.
(2) FIR filters can have an exactly linear phase response. The implications of this
will be discussed in the next section.

(3) FIR filters are very simple to implement. All DSP processors available have
architectures that suited to FIR filtering. Nonrecursive FIR filters suffer less
are

from the effects of finite wordlength than IIR filters. Recursive FIR filters also
exist and may offer significant computational advantages (see Section 7.7 for
details).

FIR filters should be used whenever we wish to exploit any of the advantages
above, in particular the advantage of linear phase. Issues to consider when choosing
between FIR and IIR filters are given in Section 6.3.
344 Chapter 7 Finite impulse response (FIR) filter design

7.1.2 Linear phase response and its implications

The ability exactly linear phase response is one of the most important
to have an
properties of FIR filters. For this reason we shall look more closely at this property.

When a signal passes through a filter, it is modified in amplitude and/or phase. The
nature and extent of the modification of the signal is dependent on the amplitude and
phase characteristics of the filter. The phase delay or group delay of the filter provides
a useful measure of how the filter modifies the phase characteristics of the signal. If

we consider a signal that consists of several frequency components (such as a speech

waveform or a modulated signal) the phase delay of the filter is the amount of time
delay each frequency component of the signal suffers in going through the filter. The
group delay on the other hand is the average time delay the composite signal suffers at
each frequency. Mathematically, the phase delay is the negative of the phase angle
divided by frequency whereas the group delay is the negative of the derivative of the
phase with respect to frequency:

T=-0()/ (7.2a)
T-do/da (7.2b)

A filter with a nonlinear phase characteristic will phase distortion in the signal
cause a

that passes through it. This is because the frequency components in the signal will
each be delayed by an amount not proportional to frequency thereby altering their
harmonic relationships. Such a distortion is undesirable in many applications, for
example music, data transmission, video, and biomedicine, and can be avoided by
using filters with linear phase characteristics over the frequency bands of interest.
A filter is said to have linear phase response if its phase response satisfies
a one of
the following relationships:

00)-co (7.3a)
θ(ω) = β- αω
(7.3b)
where a and B are constant. If a filter satisfies the condition
given in Equation 7.3a it
will have both constant group and constant phase delay responses. It
can be shown that
for condition 7.3a to be satisfied the impulse response of the filter must
have positive
symmetry. The phase response in this case is simply a function of the filter length:

h(n)=h(N-n-1), Jn 0, 1,.., (N-1)/2

= (N odd)
n =0, 1,... (N/2) -1 (N even)
a=(N-1)/2
When the condition given in Equation 7.3b only
is satisfied the filter will have a
constant group delay only. In this case, the impulse respon
se of the filter has negative
symmetry:

h(n)=-h(N-n-1)
a=(N-1)/2
B=n/2