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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.
h_{A}(t) =e^{at }Cosbt for t ≥ 0 s_{1} = ajb
= 0 otherwise
The pole located at s=p is transformed into a pole in the Zplane at Z = e^{ PTS}, however, the finite zero located in the splane at s= a was not converted into a zero in the zplane at Z = e^{aTs }, although the zero at s=∞ was placed at z=0.
Desing a Chebyshev LPF using ImpulseInvariance Method.
[The freq response for analog filter we plotted over freq range 0 to 10000 π. To set the discretetime freq range , therefore Ts =10^{4}
Methods to convert analog filters into Digital filters:
1. By approximation of derivatives
Or
Using forwarddifference 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 Forwarddifference
if σ =0 u=1 and j Ω axis maps to Z=1
If σ >0, then u>1, the RHSplane maps to right of z=1.
If σ <0, then u<1, the LHSplane maps to left of z=1.
The stable analog filter may be unstable digital filter.
Bilinear Transformation
{Using trapezoidal rule y(n)=y(n1)+0.5T_{s}[x(n)+x(n1)]
H(Z)=2(Z1) / [T_{s}(Z+1)] }
To find H(z), each occurrence of S in HA(s) is replaced by
The entire j Ω axis in the splane  ∞ <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 splane maps on to the inside of the unit circle in the zplane and the right half of splane 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 wdomain.
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 splane to the zplane.
A pole at S=S_{p} in the splane 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*10^{4} tan(0.0702π ) = 4484 rad/sec
= 2*10^{4} 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
C_{3}(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
S_{1} = r cosθ + j Rsinθ = 2216
Pole Mapping
At S=S_{1}
In the Zplane there is zero at Z = 1 and pole at Z =
S2,3 = there are two zeros at Z=1
Pole Mapping Rules:
H_{z}(z) = 1CZ^{1 }zero at Z=C and pole at Z = 0
pole ar Z=d and zero at z=0
C and d can be complexvalued number.
Pole Mapping for LowPass to Low Pass Filters
Applying low pass to low pass transformation to H_{z}(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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