Impulse Invariance Method | Signals and Systems - Electronics and Communication Engineering (ECE) PDF Download

Impulse Invariance Method

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

H(z) (at z =e ST ) = ∑h(n)e - STn

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

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

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

Impulse Invariance Method | Signals and Systems - Electronics and Communication Engineering (ECE)Impulse Invariance Method | 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 | Signals and Systems - Electronics and Communication Engineering (ECE) this is the biggest disadvantage of Impulse Invariance method.

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

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

= 0        otherwise

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

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

Desing a Chebyshev LPF using Impulse-Invariance Method.

Impulse Invariance Method | 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 | Signals and Systems - Electronics and Communication Engineering (ECE) , therefore Ts =10-4

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

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

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

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

Methods to convert analog filters into Digital filters:

1. By approximation of derivatives

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

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

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

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

  Using backward- difference mapping is based on first order approximation

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

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

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

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

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

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

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

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

Using Forward-difference

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

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

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

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

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

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

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

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

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

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

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

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

S1 = r cosθ + j Rsinθ = -2216

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

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

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

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

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

And pole at z=0 is Z=α

Similarly,

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

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

Zero at z=0 => z =α

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

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

The document Impulse Invariance Method | 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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FAQs on Impulse Invariance Method - Signals and Systems - Electronics and Communication Engineering (ECE)

1. What is the impulse invariance method?
Ans. The impulse invariance method is a technique used in digital signal processing to convert an analog filter into a digital filter. It involves sampling the impulse response of the analog filter and then using these samples to design the digital filter.
2. How does the impulse invariance method work?
Ans. The impulse invariance method works by first obtaining the impulse response of the analog filter. This impulse response is then sampled at a certain rate to generate a set of discrete samples. These samples are then used to design the digital filter by applying a suitable discrete-time filter design technique.
3. What are the advantages of using the impulse invariance method?
Ans. The impulse invariance method has several advantages. Firstly, it provides a simple and straightforward way to convert an analog filter into a digital filter. Secondly, it preserves the frequency response characteristics of the analog filter, making it suitable for applications where maintaining the same frequency response is important. Lastly, it can be easily implemented using digital signal processing techniques.
4. What are the limitations of the impulse invariance method?
Ans. The impulse invariance method has some limitations. One limitation is that it can only be applied to linear time-invariant systems. Additionally, the method can introduce aliasing effects if the analog filter has a high cutoff frequency or if the sampling rate is not adequately chosen. Lastly, the method does not guarantee an exact match between the analog and digital filters, and some error in the frequency response may occur.
5. How can the impulse invariance method be used in practical applications?
Ans. The impulse invariance method is commonly used in various practical applications. It is often used in audio processing, where analog filters need to be implemented in digital audio systems. It is also used in communication systems, where analog filters need to be converted to digital filters for signal processing. Additionally, the impulse invariance method finds applications in control systems, image processing, and many other areas of digital signal processing.
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