U. Glaeser

FIR design methods are well known and well developed. The simplest design technique is called the
direct, or window method. The design process begins with a specification of the desired filter frequency
response H(e jϖ ). An M-harmonic (M >> 1) inverse Fourier transform (IFFT) of H(e jϖ ) is computed,
which defines an M-sample time-series h′[k] that is an approximation to the desired FIR impulse response.
Normally, the long M-sample symmetric time-series is symmetrically reduced to an N-sample impulse
response h[k], defined by the N central values of h′[k]. The major weakness of the direct design paradigm
is that the approximation errors in the frequency domain can be locally large about points of discontinuity
of H(e jϖ ), as shown in Fig. 24.1. A commonly used design criteria that overcomes this weakness is based
on a minimax error criterion. The minimax criterion requires that the maximum value of the approximation
error be minimized. A minimax FIR is characterized by the frequency domain errors having an
equirriple (equal ripple) envelope. Thus, this class of FIR is logically referred to as an equirriple filter
and has a typical magnitude frequency response shown in Fig. 24.1.
Windows are tools that are sometimes used to improve the shape of an FIR’s frequency domain envelope.
An N-sample data window is applied to an Nth-order FIR on a sample-by-sample basis according to the
rule h w[k] = h[k]w[k], where h[k] is an FIR’s impulse response, w[k] is a window function, and h w[k] is
the windowed FIR impulse response. In the frequency domain, the effect of a window is defined by the
convolution operation H w(n) = H(n)∗W(n), which results in a tendency to smooth the envelope of the parent
FIR’s frequency response. The attributes of a window are defined by the width of the center (main) lobe
and sideband suppression in the frequency domain (see Table 24.3). Common window functions are
rectangular, Hann, Hamming, Blackman, Kaiser, and Flat Top. The effect of a window on the direct FIR
frequency response shown in Fig. 24.1 is also displayed in Fig. 24.2.
Infinite Impulse Response (IIR) Filters
Filters containing feedback are called IIR filters. With feedback, an IIR’s impulse response can be infinitely
long. The presence of feedback allows an IIR to achieve very high frequency selectivity and near resonance
behavior. An Nth-order constant coefficient IIR filter can be modeled by the transfer function
where the filter’s zeros {z i} are the roots of N(z) = 0, and the filter’s poles {p i} are the roots of D(z) = 0.
© 2001 by CRC Press LLC
TABLE 24.3 Effects of Data Windows
Window Transition Width f s/N Highest Sidelobe in dB
Rectangular 0.9 −13
Hann 2.07 −31
Hamming 2.46 −41
Blackman 3.13 −58
Kaiser (β = 2.0) 1.21 −19
FIGURE 24.1 Comparison of direct and equirriple FIR designs.
M
∑
a / iz i
N
∑
i=0
H( z)
N( z)/D(
z)
biz i –
–
K z N−M
= = =
( ) z– zi i=0
M−1
∏(
N−1
) z– pi / ∏
i=0
i=0
( )