Impulse Invariance Method
H(z) (at z =e ST ) = ∑h(n)e - STn
If the real part is same, imaginary part is differ by integral multiple of this is the biggest disadvantage of Impulse Invariance method.
hA(t) =e-at Cosbt for t ≥ 0 s1 = -a-jb
= 0 otherwise
The pole located at s=p is transformed into a pole in the Z-plane at Z = e PTS, however, the finite zero located in the s-plane at s= -a was not converted into a zero in the z-plane at Z = e-aTs , although the zero at s=∞ was placed at z=0.
Desing a Chebyshev LPF using Impulse-Invariance Method.
[The freq response for analog filter we plotted over freq range 0 to 10000 π. To set the discrete-time freq range , therefore Ts =10-4
Methods to convert analog filters into Digital filters:
1. By approximation of derivatives
Or
Using forward-difference mapping based on first order approximation Z = e sTs≌ 1+STs
Using backward- difference mapping is based on first order approximation
=
Therefore H(z) = using backward difference
lz - 0.5| = 0.5 is mapped into a circle of radius 0.5, centered at Z=0.5
Using Forward-difference
if σ =0 u=1 and j Ω axis maps to Z=1
If σ >0, then u>1, the RHS-plane maps to right of z=1.
If σ <0, then u<1, the LHS-plane maps to left of z=1.
The stable analog filter may be unstable digital filter.
Bilinear Transformation
{Using trapezoidal rule y(n)=y(n-1)+0.5Ts[x(n)+x(n-1)]
H(Z)=2(Z-1) / [Ts(Z+1)] }
To find H(z), each occurrence of S in HA(s) is replaced by
The entire j Ω axis in the s-plane - ∞ <j Ω<∞ maps exactly once onto the unit circle - π< w ≤ π such that there is a one to one correspondence between the continuous -time and discrete time frequency points. It is this one to one mapping that allows analog HPF to be implemented in digital filter form.
As in the impulse invariance method, the left half of s-plane maps on to the inside of the unit circle in the z-plane and the right half of s-plane maps onto the outside.
In Inverse relationship is
For smaller value of frequency
(B.W of higher freq pass band will tend to reduce disproportionately)
The mapping is ≌ linear for small Ω and w. For larger freq values, the non linear compression that occurs in the mapping of Ω to w is more apparent. This compression causes the transfer function at the high Ω freq to be highly distorted when it is translated to
the w-domain.
Prewarping Procedure:
When the desired magnitude response is piece wise constant over frequency, this compression can be compensated by introducing a suitable prescaling or prewarping to the Ω freq scale. Ω scale is converted into Ω * scale.
We now derive the rule by which the poles are mapped from the s-plane to the z-plane.
A pole at S=Sp in the s-plane gets mapped into a zero at z= -1 and a pole at Z =
Ex:
Chebyshev LPF design using the Bilinear Transformation
Pass band:
-1<|H ( jΩ)|dB≤0 for 0 ≤ Ω ≤ 1404π=4411 rad
Stop band:
|H ( jΩ)| dB < -60 for Ω ≥ 8268 π rad/sec =25975 rad/s
Let the Ts = 10-4 sec
Prewarping values are
= 2*104 tan(0.0702π ) = 4484 rad/sec
= 2*104 tan(0.4134π ) = 71690 rad/sec
The modified specifications are
Pass band:
-1<lH ( jΩ*)|dB≤ 0 for 0 ≤ Ω * ≤ 4484 rad/s
Stop band:
|H ( jΩ*)| dB < - 60 for Ω *≥ 71690rad/sec
Value of μ : is determined from the pass band ripple 10log (1 + m -2 ) -1 > -1dB
μ= 0.508
Value of N: is determined from
C3(16) = 16301
N = 3 is sufficient
Using Impulse Invariance method a value of N=4 was required.
ρ=4.17
Major R =
Since there are three poles, the angles are
S1 = r cosθ + j Rsinθ = -2216
Pole Mapping
At S=S1
In the Z-plane there is zero at Z = -1 and pole at Z =
S2,3 = there are two zeros at Z=-1
Pole Mapping Rules:
Hz(z) = 1-CZ-1 zero at Z=C and pole at Z = 0
pole ar Z=d and zero at z=0
C and d can be complex-valued number.
Pole Mapping for Low-Pass to Low Pass Filters
Applying low pass to low pass transformation to Hz(z) α we get
The low pass zero at z=c is transformed into a zero at z=C1 where C1 =
And pole at z=0 is Z=α
Similarly,
Zero at z=0 => z =α
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