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Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

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Impulse Invariance Method

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

H(z) (at z =e ST ) = ∑h(n)e - STn

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

If the real part is same, imaginary part is differ by integral multiple of  Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) this is the biggest disadvantage of Impulse Invariance method.

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

hA(t) =e-at  Cosbt    for t ≥ 0          s1 = -a-jb

= 0        otherwise

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

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.

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Desing a Chebyshev LPF using Impulse-Invariance Method.

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

[The freq response for analog filter we plotted over freq range 0 to 10000 π. To set the discrete-time freq range  Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) , therefore Ts =10-4

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Methods to convert analog filters into Digital filters:

1. By approximation of derivatives

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Or

Using forward-difference mapping based on first order approximation Z = e sTs≌ 1+STs

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

  Using backward- difference mapping is based on first order approximation

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Therefore H(z) = Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) using backward difference

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

lz - 0.5| = 0.5 is mapped into a circle of radius 0.5, centered at Z=0.5

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Using Forward-difference

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

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.

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Bilinear Transformation

  • Provides a non linear one to one mapping of the frequency points on the jw axis in s -plane to those on the unit circle in the z-plane.
  • This procedure also allows us to implement digital HP filters from their analog 
    counter parts.

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

{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  Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

The entire j Ω axis in the s-plane - ∞ <j Ω<∞ maps exactly once onto the unit circle - π< ≤ 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.

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

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  Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

For smaller value of frequency  Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

(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.

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

We now derive the rule by which the poles are mapped from the s-plane to the z-plane.

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

A pole at S=Sp in the s-plane gets mapped into a zero at z= -1 and a pole at Z = Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

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 

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) = 2*104 tan(0.0702π ) = 4484 rad/sec

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) = 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

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

C3(16) = 16301

N = 3 is sufficient

Using Impulse Invariance method a value of N=4 was required.

ρ=4.17

Major  R  = Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Since there are three poles, the angles are  Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

S1 = r cosθ + j Rsinθ = -2216

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Pole Mapping

At S=S1

In the Z-plane there is zero at Z = -1 and pole at Z = Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

S2,3 = there are two zeros at Z=-1

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Pole Mapping Rules:

Hz(z) = 1-CZ-1 zero at Z=C and pole at Z = 0

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) 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 

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

The low pass zero at z=c is transformed into a zero at z=C1 where C1 = Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

And pole at z=0 is Z=α

Similarly,

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Zero at z=0 => z =α

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

The document Impulse Invariance Method Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) is a part of the Electronics and Communication Engineering (ECE) Course Signals and Systems.
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