Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Digital Signal Processing

Electrical Engineering (EE) : Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

The document Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev is a part of the Electrical Engineering (EE) Course Digital Signal Processing.
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Impulse Invariance Method

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

H(z) (at z =e ST ) = ∑h(n)e - STn

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRevImpulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRevImpulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRevImpulse Invariance Method Electrical Engineering (EE) Notes | EduRev

If the real part is same, imaginary part is differ by integral multiple of  Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev this is the biggest disadvantage of Impulse Invariance method.

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRevImpulse Invariance Method Electrical Engineering (EE) Notes | EduRev

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

= 0        otherwise

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRevImpulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRevImpulse Invariance Method Electrical Engineering (EE) Notes | EduRev

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 Electrical Engineering (EE) Notes | EduRev

Desing a Chebyshev LPF using Impulse-Invariance Method.

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

[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 Electrical Engineering (EE) Notes | EduRev , therefore Ts =10-4

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Methods to convert analog filters into Digital filters:

1. By approximation of derivatives

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRevImpulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Or

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

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

  Using backward- difference mapping is based on first order approximation

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Therefore H(z) = Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev using backward difference

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

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

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRevImpulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Using Forward-difference

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

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 Electrical Engineering (EE) Notes | EduRev

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 Electrical Engineering (EE) Notes | EduRev

{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 Electrical Engineering (EE) Notes | EduRev

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRevImpulse Invariance Method Electrical Engineering (EE) Notes | EduRev

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 Electrical Engineering (EE) Notes | EduRev

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 Electrical Engineering (EE) Notes | EduRev

For smaller value of frequency  Impulse Invariance Method Electrical Engineering (EE) Notes | EduRevImpulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

(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 Electrical Engineering (EE) Notes | EduRev

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

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRevImpulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRevImpulse Invariance Method Electrical Engineering (EE) Notes | EduRev

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 Electrical Engineering (EE) Notes | EduRev

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 Electrical Engineering (EE) Notes | EduRev = 2*104 tan(0.0702π ) = 4484 rad/sec

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev = 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 Electrical Engineering (EE) Notes | EduRev

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

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 Electrical Engineering (EE) Notes | EduRevImpulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Since there are three poles, the angles are  Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

S1 = r cosθ + j Rsinθ = -2216

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRevImpulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Pole Mapping

At S=S1

In the Z-plane there is zero at Z = -1 and pole at Z = Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

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

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRevImpulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRevImpulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Pole Mapping Rules:

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

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev 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 Electrical Engineering (EE) Notes | EduRevImpulse Invariance Method Electrical Engineering (EE) Notes | EduRev

The low pass zero at z=c is transformed into a zero at z=C1 where C1 = Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

And pole at z=0 is Z=α

Similarly,

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Zero at z=0 => z =α

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

Impulse Invariance Method Electrical Engineering (EE) Notes | EduRev

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