U. Glaeser

frequency response involves some aliasing:
where H a(s) is the Laplace transform system function of the analog filter. The aliasing effect usually causes
only a slight perturbation of the digital filter with respect to the analog filter. The system function of the
analog filter can be expressed in partial fraction form
After sampling the digital filter has a frequency response that is also a rational form:
where b[n] and a[n]are the coefficients of the designed filter. Impulse invariance is equivalent to a linear
mapping of the analog frequency range [−π /T d, π /T d] into the digital frequency range [−π,π].
Bilinear Transform
On the other hand, the bilinear transformation performs a nonlinear mapping of the whole analog
frequency range [−∞, ∞]into the finite digital frequency range [−π, π]. The mapping of the s-plane to
the z-plane is done by the bilinear transform:
The resulting correspondence between the analog and digital frequency domains is a tangent function:
Despite the nonlinear nature of the mapping, it is relatively easy to turn the digital design specification
into an analog design specification. The resulting filter is IIR and the filter coefficients can be computed
with an algebraic form. The bilinear transform method is usually applied to four classical analog filter
frequency selective filters: Butterworth, Chebyshev-I, Chebyshev-II, and elliptic filters. All these are well
known for their frequency-selective behavior as lowpass, bandpass, or highpass filters. When using the
bilinear mapping, elliptic IIR filters turn out to have the best magnitude response for given filter order,
but elliptic filters have severe phase distortion, which can be a significant problem in advanced DSP
applications such as telecommunications.
Windowing
IIR filter designs have poor phase response, so interest in FIR filters has always been strong. If the
coefficients of an FIR filter are real and symmetric b[k] = b∗ [M − k] then the filter will have perfectly
linear phase. The first attempt to design FIR filters in the 1960s was to truncate the inverse DTFT of the
ideal frequency response (which is the impulse response h[n]of the ideal filter), so that the filter is
1
symmetric and linear-phase. This requires the ideal filter to have linear-phase with slope −-- M, where M
© 2001 by CRC Press LLC
H(ω) Ha j ω
∞
= ∑ Td k=−∞
Ha() s
=
---- j 2π ⎛ + -----k ⎞
⎝ ⎠
N
∑
k=1
A k
--------s–
sk T
DTFT: H(ω)
dAk 1 e Td s N
b[ n]e
∑-------------------------
k – jω
k=1 – e jωn –
k=0
1 a[ n]e
jωn –
= = --------------------------------------
N
+
s
ω a
1
2 1 z
----
Td –
–
1 z 1 –
= ⎛--------------- ⎞
⎝ + ⎠
=
2
---- tan
Td ω ⎛--- ⎞
⎝2⎠ T d
∑
N−1
∑
k=1
2