FIR filter design - Continuous and Discrete Time Signals and Systems - page 708

variations (or ripples) within the pass and stop bands. In addition, a transition
band is included between the pass and stop bands so that the filter gain can drop
off smoothly.
Section 15.1 introduced the windowing approach used to design FIR filters
from the ideal frequency-selective filters. The windowing approach truncates
the impulse response h [ k ] of an ideal filter, with a linear-phase component of
exp( − j m Ω ), to a finite length N within the range 0 ≤ k ≤ ( N − 1). The value
of m in the phase component is selected to be ( N − 1) / 2 such that the filter
coefficients in the causal FIR filter are symmetrical with respect to m . Common
elementary windows used to design FIR filters are the rectangular, Bartlett,
Hamming, Hanning, and Blackman windows. The selection of type of window
depends upon the maximum value of the pass- and stop-band ripples. The length
N of the window is determined from the allowable width of the transition
band.
The minimum stop-band attenuation in the FIR filter obtained from the ele-
mentary windows is fixed. In most cases, the selected window surpasses the
given specification on the stop-band attenuation and the resulting FIR filter is
therefore of higher computational complexity than required. Section 15.2 intro-
duced the Kaiser window, which provides control over the stop-band attenuation
by including an additional design parameter, referred to as the shape control
parameter β . The order of the FIR filter designed by the Kaiser window is sig-
nificantly smaller than those of the FIR filters obtained using the elementary
window functions.
The FIR design techniques covered in Sections 15.1 and 15.2 are applicable to
all types of frequency-selective filters such as the lowpass, highpass, bandpass,
and bandstop filters. Common convention, however, is to express the transfer
functions of the highpass, bandpass, and bandstop filters in terms of the transfer
function of the lowpass filter. Using the resulting relationships, the design of any
type of filter can be reduced to the design of one or more lowpass filters. Section
15.3 covered design techniques for highpass FIR filters. We covered design
algorithms using the original highpass filter specifications as well as techniques
that transform the problem of designing a highpass FIR filter to designing
a lowpass FIR filter. Similarly, Section 15.4 presented design techniques for
bandpass FIR filters, while Section 15.5 designed bandstop FIR filters.
The windowing approaches produce a suboptimal design. Section 15.5 intro-
duced a computational procedure based on the Parks-McClellan algorithm that
exploits the inherent structure, expressed in Proposition 14.1 for the linear-
phase FIR filters. The Parks-McClellan algorithm computes the best FIR filter
of length N that minimizes the maximum absolute difference between the trans-
fer function H d ( Ω ) of the ideal filter and the transfer function H ( Ω ) of the cor-
responding FIR filter. Mathematically, the Parks-McClellan algorithm solves
the minimax optimization problem, which finds the set of filter coefficients
that minimizes the maximum error between the desired frequency response and
the actual frequency response. According to Proposition 14.1, the frequency