640 C H A P T E R 11: Introduction to the Design of Discrete Filters
FIGURE 11.1
Eigenfunction property of LTI systems.H(ejω)x[n]=ejω^0 n y[n]=ejω^0 n H(ejω^0 )LTIapproximation—these filters are called recursive or infinite-impulse response (IIR) filters. The other
is the nonrecursive or finite-impulse response (FIR) filters that result from a polynomial approxima-
tion. In the continuous-time domain, depending on their implementation, filters are either passive
or active. Passive filters are implemented using resistors, capacitors, and inductors, while active fil-
ters are implemented with resistors, capacitors, and operational amplifiers. The implementation of
discrete or digital filters is done by means of software or dedicated hardware.As we will see, the discrete filter specifications can be in the frequency or in the time domain. For
recursive or IIR filters, the specifications are typically given in the form of magnitude and phase
specifications, while the specifications for nonrecursive or FIR filters can be in the time domain as a
desired impulse response. The discrete filter design problem then consists in: Given the specifications
of a filter we look for a polynomial or rational (ratio of polynomials) approximation to the specifica-
tions. The resulting filter should be realizable, which besides causality and stability requires that the
filter coefficients be real valued.The typical approach in filter design is to consider low-pass prototypes with normalized frequency
and magnitude responses that may be transformed into other filters with the desired frequency
response. Thus, a great deal of effort is put into the design of low-pass filters and into developing fre-
quency transformations to map low-pass filters into other types of filters. Using cascade and parallel
connections of filters also provides a way to obtain different types of filters.There are different ways to obtain the rational approximation for discrete IIR filters—by transforma-
tion of analog filters, or by optimization methods that include stability as a constraint. We will see
that the classical analog design methods (Butterworth, Chebyshev, Elliptic, etc.) can be used to design
discrete filters by means of the bilinear transformation that maps the analogs-plane into thez-plane.
Given that the FIR filters are unique to the discrete domain, the approximation procedures for FIR
filters are unique to that domain.
The difference between discrete and digital filters is in the quantization and coding. For a discrete filter
we assume that the input and the coefficients of the filter are represented with infinite precision—that
is, using an infinite number of quantization levels—and thus no coding is performed. The coefficients
of a digital filter are binary and the input and output are quantized and coded. Thus, quantization
affects the performance of a digital filter, while it has no effect in discrete filters.Considering continuous-to-discrete converters (CDCs) and discrete-to-continuous converters
(DCCs) as simply samplers and reconstruction filters, it is possible to implement the filtering of
band-limited analog signals using discrete filters (Figure 11.2). In such an application, an additional
specification for the filter design is the sampling period. In this process it is crucial that the sampling
period in the CDCs and DCCs be synchronized. In practice, filtering of analog signals is done using
analog-to-digital (ADC) and digital-to-analog (DAC) together with digital filters.