584 Practical MATLAB® Applications for Engineers
b. Transform the analog fi lter into its digital version by obtaining the discrete fi lter
transfer function H(z) or its impulse response h(n).R.6.88 One fundamental difference exists between H(s) and its discrete mapping H(z).
Since the phase function of the digital prototype is given by
∠H(z) = tan−^1 [Imag(H(z)/Real(H(z)] + / 2 ( 1 + sign(Real(H(z)))
where z = jW.
Then the gain presents a periodic behavior (with period 2π), then the principal
value is limited over the digital frequencies over the range −π ≤ W ≤ +π.R.6.89 The steps involved in the design of IIR fi lters using MATLAB are summarized as
follows:
- Design an analog LP normalized prototype fi lter
- Use frequency transformation to transform the LP analog prototype into the
desired fi lter type: LP, HP, BP, or BS fi lter depending on the requirements of the
design (using the MATLAB functions: lp2lp, lp2hp, etc.) - Transform the analog fi lter into its digital version by
a. Bilinear transformation
b. Impulse invariant method
c. Yule–Walker algorithm
These techniques are presented and discussed in the following.
R.6.90 The bilinear transformation is a one-to-one mapping of the complex analog fre-
quency s(s = σ + jw) of the transfer function H(s) (s-plane) into the complex digital
frequency z(z = esT) (z-plane).
R.6.91 Mathematically, the mapping s (a nalog) → z (digital) or z → s is accomplished by
the following substitution:
s→A
z
z1
(^2)
or
z→
As
As
where A = 2 /T, in which T is the sampling period.
The objective of the mapping is to move the poles and zeros of the desired analog
transfer function H(s) from the s-plane into the z-plane.
R.6.92 The effect of the bilinear transformation results in moving the poles and zeros from
the left half of the s-plane into the unit circle in the z-plane, where the imaginary
axis in the s-plane is mapped into the outside of the unit circle of the z-plane and
the right half of the s-plane is mapped into the outside of the unit circle of the
z-plane guaranteeing stability in this way.
Therefore, a stable analog fi lter H(s) results in a stable digital fi lter H(z).
Recall that to analyze the stability of a digital fi lter it is suffi cient to compute the
magnitude of the poles, and if they are smaller in magnitude than 1 (inside the unit
circle), then the system is stable otherwise the system is unstable.
R.6.93 Since the analog frequency w (radians per second) is related to the digital frequency
W (radians) by the relation W = wT, the infi nite range of w(−∞ to +∞) in the s-plane
is compressed into the range −π ≤ W ≤ π (referred to as frequency wrapping).