A framework for digital

filter design 6
6.1 Introduction to digital filters 318

6.2 Types of digital filters: FIR and IIR filters 319

6.3 Choosing between FIR and IIR filters 321

6.4 Filter design steps 324

6.5 Illustrative examples 334

6.6 Summary 339

Problems 339

Reference 341

Bibliography 341

The purpose of this chapter is to provide a common framework for digital filter design.
A simple step-by-step guide for designing digital filters, from specifications to
implementation, is described. The options open to the designer at each step of the
design process and factors that influence their choice are highlighted using several
illustrative examples. Most DSP texts devote substantial space to the theory of digital
filters, especially approximation methods, reflecting the considerable research effort
that has gone into finding useful methods of calculating filter coefficients and the
significant advances that have been made in filter design. However, such a coverage
often overwhelms the inexperienced filter designer and leaves him or her not know-
ing how actually to go about designing a filter or how it all fits together. Thus the
framework provided here, in our experience, is valuable to the designer who actually
wants to design digital filters, as opposed to just learning about them from a purely
theoretical point of view. This chapter sets the scene for Chapters 7 and 8 in which
actual digital filter design is fully covered.
318 Chapter 6 A framework for digital filter design

6.1 Introduction to digital filters

system or network that selectively changes the wave shape,

A filter is essentially a
amplitude-frequency and/or phase-frequency characteristics of a signal in a desired
manner. Common filtering objectives are to improve the quality of a signal (for
example, to remove or reduce noise), to extract information from signals or to separate
two or more signals previously combined to make, for example, efficient use of an
available communication channel.
A digital filter, as we shall see later, is a mathematical algorithm implemented in
hardware and/or software that operates on a digital input signal to produce a digital
output signal for the purpose of achieving a filtering objective. The term digital
filter refers to the specific hardware or software routine that performs the filtering
algorithm. Digital filters often operate on digitized analog signals or just numbers.
representing some variable, stored in acomputer memory.
A simplified block diagram of a real-time digital filter, with analog input and output
signals, is given in Figure 6.1. The bandlimited analog signal is sampled periodically
and converted into a series of digital samples, x(n), n=0, 1,.... The digital processor
implements the filtering operation, mapping the input sequence, x(n), into the output
sequence, y(n), in accordance with a computational algorithm for the filter. The DAC
converts the digitally filtered output into analog values which are then analog filtered
to smooth and remove unwanted high frequency components.
Digital filters play very important roles in DSP. Compared with analog filters they
are preferred inanumber of applications (for example data compression, biomedical

signal processing, speech processing, image processing, data transmission, digital audio,
telephone echo cancellation) because of one or more of the following advantages.

Digital filters can have characteristics which are not possible with analog filters.
such as a truly linear phase response.
Unlike analog filters, the performance of digital filters does not vary with
environmental changes, for example thermal variations, This eliminates the need
to calibrate periodically.
The frequency response of a be automatically adjusted if it is-
digital filter can

implemented using a programmable processor, which is why they are widely

used in adaptive filters.
Several input signals or channels can be filtered by one digital filter without the
need to replicate the hardware.

ADC withh x(n) y()

Input
Digital уи)
sample Output
Analog filter DAC
and hold processor filter Analog
input output

Figure 6.1 A simplified block diagram of a real-time digital filter with analog input and
output signals.
 6.2 Types of digital filters: FIR and IIR filters 319

Both filtered and unfiltered data can be saved for further use.
Advantage can be readily taken of the tremendous advancements in VLSI
technology fabricate digital filters and
to to make them small in size, to consume
low power, and to keep the cost down.

In practice, the precision achievable with analog filters is restricted; for example,
typically a maximum of only about 60 to 70 dB stopband attenuation is possible
with active filters designed with off-the-shelf components. With digital filters the
precision is limited only by the wordlength used.
The performance of digital filters is repeatable from unit to unit.
Digital filters be used at very low frequencies, found in many biomedical
can

applications for example, where the use of analog filters is impractical. Also,
digital filters can be made to work over a wide range of frequencies byamere
change to the sampling frequencyу.

The following are the main disadvantages of digital filters compared with analog
filters:

Speed limitation The maximum bandwidth of signals that digital filters can
handle, in real time, is much lower than for analog filters. In real-time situations.
the analog-digital-analog conversion speed constraint on
processes introduce a
the digital filter performance. The conversion time of the ADC and the settling
time of the DAC limit the highest frequency that can be processed. Further, the
speed of operation of a digital filter depends on the speed of the digital processor
used and on the number of arithmetic operations that must be performed for the
filtering algorithm, which increases as the filter response is made tighter.
Finite wordlength effects Digital filters are subject to ADC noise resulting
from quantizing a continuous signal, and to roundoff noise incurred during
computation. With higher order recursive filters, the accumulation of roundoff
noise could lead to instability.
Long design and development times The design and development times for
digital filters, especially hardware development, can be much longer than for
analog filters. However, once developed the hardware and/or software can be
used for other filtering or DSP tasks with little or no modifications (several
examples of this are given in subsequent chapters). Good computer aided design
(CAD) support can make the design of digital filters an enjoyable task, but some
expertise is required to make full and effective use of such design aids.

6.2 Types of digital filters: FIR and IIR filters

Digital filters are broadly divided into two classes, namely infinite impulse response
(IIR) and finite impulse response (FIR) filters. Either type of filter, in its basic form,
320 Chapter 6 A framework for digital filter design

h(). = 0.1.....
(impulse response) y(n)
(imput sequence) (output sequence)

Figure 6.2 A conceptual representation of a digital filter.

can be represented by its impulse response sequence, h(k) (k = 0, 1,...), as in

Figure 6.2. The input and output signals to the filter are related by the convolution
sum, which is given in Equations 6.1 for the IIR and 6.2 for the FIR filter.

y(n)= h(k)x(n-k) (6.1)

440

N-1

y(n)= h(k)x(n-k) (6.2)

It is evident from these equations that, for IIR filters, the impulse response is of
infinite duration whereas for FIR it is of finite duration, since h(k) for the FIR has only
N values. In practice, it is not feasible to compute the output of the IIR filter using
Equation 6.1 because the length of its impulse response is too long (infinite in theory).
Instead, the IIR filtering equation is expressed in a recursive form:

y(n) =Σ
{ h(&x(n
0
– k) =Σ b,xn-k)
6-0
- {axи-k) (6.3)

where the a and b, are the coefficients of the filter. Thus, Equations 6.2 and 6.3 are the
difference equations for the FIR and IIR filters respectively. These equations, and in
particular the values of h(k), for FIR, or a and b, for IIR, are often very important
objectives of most filter design problems. We note that, in Equation 6.3, the current
output sample, y(n), is a function of past outputs as well as present and past input
samples, that is the IIR is a feedback system of some sort. This should be compared
with the FIR equation in which the current output sample, y(n), is a funct
ion only of
past andpresent values of the input. Note, however, that when the b are set to zero.
Equation 6.3 reduces to the FIR Equation 6.2.
Alternative representations for the FIR and IIR filters are given in Equations 6.4a
and 6.4b respectively. These are the transfer functions for these filters and are very
useful in evaluating their frequency responses (see Chapters 4, 7 and 8 for details).
As will become clear in the next few sections, factors that influence the
choice
of options open to the digital filter designer at each stage of the
design process
are strongly linked to whether the filter in question is IIR or FIR. Thus, it is very

important to appreciate the differences between IIR and FIR, their peculiar char-
acteristics, and more importantly, how to choose between them.