**for**

**9.3.1 Introduction to
Filtering**

In the field of signal processing the design
of digital signal filters involves the process of suppressing certain
frequencies and boosting others. A simplified filter model
is

(9-22) ,

where the input signal is
is modified to obtain the output signal
using the recursion formula

(9-23) .

The implementation of (9-23) is
straightforward and only requires starting values, then
is obtained by simple iteration. Since the signals must
have a starting point, it is common to require
that and for . We
emphasize this concept by making the following
definition.

**Definition 9.3 (Causal
Sequence)** Given the the input
and output
sequences. If and for ,
the sequence is said to be causal.

Given the causal sequence ,
it is easy to calculate the solution
to (9-23). Use the fact that
these sequences are causal:

(9-24) .

Then compute

(9-25)

The general iterative step is

(9-26) .

**9.3.2 The Basic
Filters**

The following three simplified basic filters
serve as illustrations.

(i) Zeroing Out
Filter , (note
that ).

(ii) Boosting Up
Filter , (note
that ).

(iii) Combination
Filter .

The transfer
function
for these model filters has the
following general form

(9-27) ,

where the z-transforms of the input and output sequences are
and ,
respectively. In the previous section we mentioned
that the general solution to a homogeneous difference
equation is
stable only if the zeros of the characteristic equation lie inside
the unit circle. Similarly, if a filter is stable then the
poles of the transfer function must
all lie inside the unit circle.

Before developing the general theory, we
would like to investigate the amplitude
response
when the input signal is a linear combination
of
and . The
amplitude response for the frequency
uses the complex unit signal ,
and is defined to be

(9-28) .

The formula for will be rigorously explained after a few introductory examples.

**Example 9.21.** Given
the filter .

**9.21 (a).** Show that it
is a zeroing out filter for the signals
and
and calculate the amplitude response .

**9.21 (b).** Calculate
the amplitude responses
and investigate the the filtered signal for .

**9.21 (c).** Calculate
the amplitude responses
and investigate the the filtered signal for .

**Figure 9.4.** The
amplitude response
for .

**Figure 9.5.** The
input and
output .

**Figure 9.6.** The
input
and output .

**Example 9.22.** Given
the filter .

**9.22 (a).** Show that
it is a boosting up filter for the signals
and
and calculate the amplitude response .

**9.22 (b).** Calculate
the amplitude responses
and investigate the the filtered signal for .

**Figure 9.7.** The
amplitude response
for .

**Figure 9.8.** The
input and
output .

**9.3.3 The
General Filter Equation**

The general form of a
order filter difference equation
is

(9-29)

where
and
are constants. Note carefully that the terms involved are of
the form
and
where
and ,
which makes these terms time delayed. The
compact form of writing the difference equation is

(9-30) ,

where the input signal
is modified to obtain the output signal
using the recursion formula

(9-31) .

The portion
will "zero out" signals and
will "boost up" signals.

**Remark
9.14.** Formula
(9-31) is
called the recursion equation and the recursion coefficients are
and . It
explicitly shows that the present output is
a function of the past values , for
,
the present input , and
the previous inputs for . The
sequences can be regarded as signals and they are zero for negative
indices. With this information we can now define the
general formula for the transfer function . Using
the time delayed-shift property for causal sequences and taking the
z-transform of each term in (9-31), we
obtain

(9-32) .

We can factor out
of the summations and write this in an equivalent form

(9-33) .

From equation (9-33) we obtain

which leads to the following important definition.

**Definition 9.4
(Transfer Function)** The
transfer function corresponding to the
order difference equation (8) is given by

(9-34) .

Formula (9-34) is the transfer function
for an infinite impulse response filter (IIR
filter). In the special case when the denominator is
unity it becomes the transfer function for a finite impulse response
filter (FIR filter).

**Definition 9.5 (Unit-Sample
Response)** The sequence
corresponding to the transfer function
is called the unit-sample response.

**Theorem 9.6 (Output
Response)** The output
response of
a the filter (10) given an input signal is
given by the inverse z-transformation

(9-35) ,

and in convolution form it is given by

(9-36) .

Another important use of the transfer
function is to study how a filter affects various
frequencies. In practice, a continuous time signal is
sampled at a frequency that
is at least twice the highest input signal frequency to avoid
frequency fold-over, or aliasing. That is because the Fourier
transform of a sampled signal is periodic with
period , though
we will not prove this here. Aliasing prevents accurate
recovery of the original signal from its samples.

Now it can be shown that the argument of the
Fourier transform maps onto the z-plane unit circle via the
formula

(9-37) , where is
called the normalized frequency.

Therefore the z-transform evaluated on the unit circle is also
periodic, except with period .

**Definition 9.6 (****Amplitude
Response****)** The
amplitude response
is defined to be the magnitude of the transfer function evaluated at
the complex unit signal . The
formula is

(9-38)
over the interval .

The fundamental theorem of algebra implies that the numerator has roots (called zeros) and the denominator has roots (called poles). The zeros may be chosen in conjugate pairs on the unit circle and for . For stability, all the poles must inside the unit circle and for . Furthermore, the poles are chosen to be real numbers and/or in conjugate pairs. This will guarantee that the recursion coefficients are all real numbers. IIR filters may be all pole or zero-pole and stability is a concern; FIR filters and all zero-filters are always stable.

**9.3.4 Design
of Filters**

In practice recursion formula (10) is used to
calculate the output signal. However, digital filter
design is based on the above theory. One starts by
selecting the location of zeros and poles corresponding to filter
design requirements and constructing the transfer
function . Since
the coefficients in
are real, all zeros and poles having an imaginary component must
occur in conjugate pairs. Then the recursion
coefficients are identified in (13) and used in (10) to write
the recursive filter. Both the numerator and denominator
of
can be factored into quadratic factors with real coefficients and
possibly one or two linear factors with real
coefficients. The following principles are used to
construct .

(i) Zeroing Out
Factors

To filter out the signals
and ,
use factors of the form

if , and

if ,

in the numerator of . They
will contribute to the term

(9-42) .

(ii) Boosting Up
Factors

To amplify the signals
and ,
use factors of the form

if and , and

if and , and

if and ,

in the denominator of . They
will contribute to the term

(9-43) .

(iii) Attenuating
Factors

To attenuate the signals
and , use
factors of the form

if and , and

if and .

The factor

if and ,

is a special case that attenuates low frequency
signals. These factors will contribute to the term
(9-42).

(iv) Combination of
Factors

The transfer function
could have a zero or pole at the origin, but this has no net effect
on the output signal. The other zeros and poles determine
the nature of the filter. A conjugate pair of zeros
of
on the unit circle will "zero-out" the signals
and . If
the conjugate pair of zeros
of
will attenuate the signals
and ,
and the conjugate pair of poles
of
will amplify the signals
and . It
is useful to plot the location of the zeros and poles and notice
their magnitude and argument. As a general rule, zeros are
used to attenuate signals and poles are used to amplify
signals. The primary goal of filter design is to
construct
so that the amplitude response
has a desired shape. The following examples have been
chosen to illustrate these concepts. Books on digital
signal filter design will explain the process in detail.

**Example 9.23
(a).** The filter is
designed to zero out .

**9.23 (b).** The
moving average filter is
designed to zero out .

**Figure
9.9.** Amplitude response ,
and zero-pole plot for .

**Example 9.24.** The
moving average filter

is designed to zero out .

**Figure
9.10.** Amplitude response ,
and zero-pole plot for .

**Example 9.25
(a).** Design a filter with
poles for
boosting up signals near
and .

**9.25
(b).** Include the additional pole
at
to the filter design in (a) so that it also boosts up low frequency
signals.

**Figure
9.11.** Amplitude response ,
and zero-pole plot for .

**Figure
9.12.** Amplitude response ,
and zero-pole plot for .

**Example
9.26.** Design a combination filter using the
zeros
for zeroing out
and poles
boosting up some of the low frequencies.

**Figure
9.13.** Amplitude response ,
and zero-pole plot for .

**Low Frequency Filter**

The previous examples emphasize high
frequency filters. To boost up high frequencies place the
poles near the higher frequencies. The following example is similar
to Example 9.25. The poles have been changed
from to
.

**Extra Example 1
(a).** Design a filter with
poles for
boosting up signals near
and .

**1
(b).** Include the additional pole
at
to the filter design in (a) so that it also boosts up high frequency
signals.

**Band Pass Filter**

Poles can be placed near frequencies to boost
them up and zeros for attenuating the other frequencies.

**Extra Example
2.** Design a combination filter boosting up some
of the low frequencies where
and attenuating frequencies and .

A signal processing engineer uses complex analysis to construct filters with the desired amplitude and phase response characteristics. Finite impulse response (FIR) filters have only zeros, whereas infinite impulse response (IIR) filters have poles and may have zeros as well. The area of filter design involves many types, such as: low pass, high pass, all pass, band pass and band stop. Special forms of such filters include, but are not limited to Bessel, Butterworth, Chebyshev, Gaussian, moving average, single pole, Remez, etc. More information about filter design can be found in books on digital signal processing.

**Algorithm (Analysis of
Filters)** The following subroutine will analyze
the transfer function for . The
z-transform is

The functions and must
be defined.

**Mathematica
Subroutine (Analysis of Filters)**

**Exercises
for Section 9.3. Digital Signal Filters**

**The Next Module
is**

**Return to the Complex
Analysis Modules **

__Return
to the Complex Analysis Project__

This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

(c) 2012 John H. Mathews, Russell W. Howell