1. Chapter 5. Transform Analysis of LTI Systems Section 5.1-5.7
5.1 Frequency Response of LTI Systems
5.2 Systems Functions from LCCDE
5.3 Rational System Functions
5.4 All-pass, Minimum and Linear Phase Systems

2. 5.1.1 Frequency Response of LTI Systems
Impulse response can fully characterize a LTI system.
Further is often called the system function,
and if , then
Frequency response at is valid if ROC
includes and
Consider and , then
magnitude
phase
We will model and analyze systems based on magnitude
and phase response.

3. 5.1.2 Ideal Frequency-Selective Filters
Consider ideal frequency selective filters
Four basic types can be easily defined which are symmetric.
Arbitrary shapes can also be specified. -  - 
Low Pass Filter
High Pass Filter -  - 
Band Rejection Filter
Band Pass Filter

4. 5.1.3 LPF and HPF Low Pass Filter (LPF) High Pass Filter (HPF)
Not stable
Not realizable
Not causal
High Pass Filter (HPF)

5. 5.1.4 Phase Response of LTI System
Phase delay
Observation: if phase is specified to be zero (as in ideal filters) the impulse response will be centered at n=0, response will be non-causal and cannot be implemented in real-time.
Consider the “delay system”
This system exhibits linear phase. We will often be satisfied with linear phase distortion since it is just a time delay.

6. 5.1.5 Phase Response Example: Ideal LPF
Let
now linear phase – not zero phase
all n
The center of h[n] is shifted to and the phase is linear.
Filter will always be non-causal, however.

7. 5.1.6 Important Phase Information
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8. 5.1.7 Phase Group Delay
If () is constant, then phase is linear or zero.
The deviation from constant indicates nonlinearity.
Unusually undesirable and very hard to compensate.
phase spectrum
group delay

9. 5.1.8 Group Delay Example

10. 5.2.1 System Functions from LCCDE
General form of LCCDE
Compute Z-transform
Pole/zero factorization
Stability requirements
Choice of ROC determines causality
Location of zeros and poles determines the frequency response and phase

11. 5.2.2 Inverse Systems is an inverse system for , if A common problem
Useful for canceling the effects of another system
See the discussion in Sec regarding ROC
A common problem
The ROC of must overlap H (z). To give a valid result – pole/zero cancellation can make this happen
An LTI system and its inverse are both stable and causal iff all the poles and zeros of H (z) are inside the unit circle called a Minimum Phase System.
Unknown (want) H
Observed
Some Knowledge
H-1
Estimate of x

12. 5.2.3 Inverse System Example-1

13. 5.2.3 Inverse System Example-2

14. 5.2.4 Impulse Response of Rational System Functions
Consider the partial fraction expansion (PFE) representation:
where there are only first order poles
Each pole (second-term) contributes an exponential to h[n], such that
assuming H(z) is causal so all poles are inside unite circle.
If only terms like the first are present: Finite Impulse Response (FIR)
If any of second type are present: Infinite Impulse Response (IIR)

15. 5.2.5 FIR Example

16. 5.3.1 Frequency Response for Rational Systems
If a stable LTI system has a rational system function, then its frequency response has the form

17. 5.3.2 Interpretations of Frequency Response
is the product of the magnitude of the zero terms of H(z) evaluated on the unit circle divided by the product of the magnitude of all the poles terms of H(z) evaluated on the unit circle.
Expressing in decibels (dB)
Also note
Zero factors – add phase
Pole factors – subtract phase

18. 5.3.3 Group Delay for A Rational System Function
Principal value – the value usually returned by your calculator (extra factors of are removed ) “ complex Analysis” ?

19. 5.3.4 Group Delay Example can be computed from
except at discontinuities.

20. 5.3.5 Frequency Response of a Single Zero or Pole
We just looked at equations for frequency response which were functions of poles and zeros.
Consider a single pole or zero factor of form
Consider maximum and minimum values of
Maximum
Minimum

21. 5.3.5 Frequency Response of a Single Zero or Pole
Let’s rewrite the factor
In terms of z-plane
The contribution of a single zero factor to the magnitude response at frequency  is just the length of the zero vector from the zero to the point on unit circle.
has minimum length when =, corresponds to the dip in frequency at =.
, the pole vector from the pole at z=0, always has unit length and does no affect the response.

22. 5.3.5 Frequency Response of a Single Zero or Pole
In terms of vectors:
r<1

24. 5.3.6 Discussions and Observations
The factor
Represent a zero of H(z): curves with positive algebraic sign
Represent a pole of H(z): curves with negative sign
Observations:
Zero factors near unit circle suppress the magnitude response
A zero on unit circle: |H|0
Pole factors near unit circle the magnitude response
A pole very close to unit circle |H|
Multiple poles and zeros can be placed so as to form somewhat arbitrary shape magnitude response (poles inside unit circle).
If zeros are also inside unit circle, then the inverse system will also be stable and causal – often desired property.

25. 5.4.1 All Pass Systems A system of the form (or cascade of these)
In general, all pass systems have form
All-pass systems have non-positive phase response for 0<<.
All-pass systems have always positive group delays.
All-pass systems can be designed to perform equalization and others.
Causal/stable:
real poles
complex poles

26. 5.4.2 All Pass System Examples
Unit
circle
z-plane
0.8
0.5 2

27. 5.4.3 Minimum Phase System
A system with all its poles and zeros inside the unit circle (stable and causal) is called minimum phase. Their causal and stable inverse also exists.
Given
For H(z) to be minimum phase, it will consist of the poles and zeros of C(z) that lie inside the unit circle.
We know that a Z-transform is not uniquely determined without specifying ROC. But if a system is known to be minimum phase, it is uniquely determined due to the requirements of its poles and zeros.

28. 5.4.4 All-pass Decomposition and Minimum Phase System Properties
Any rational system function H(z) can be written as
contains the poles and zeros of H(z) that lie inside the unit circle, plus zeros that are the conjugate reciprocal of the zeros of H(z) that lie outside the unit circle.
is comprised of all the zeros of H(z) that lie outside the unit circle, and poles to cancel the reflected conjugate reciprocal zeros in
Three important properties:
The minimum phase-lag property
The minimum group-delay property
The minimum energy-delay property

30. 5.4.5 Frequency-Response Compensation
Distorting
system
Compensating
system
After compensation:
G(z) corresponds to an all-pass system.
The frequency-response magnitude is exactly compensated.
The phase response is modified to

31. 5.4.6 Linear Phase Filtering
In general, a system of the form below has linear phase
If  or 2  is integer, then it has a linear phase.
This condition is sufficient but not necessary (have other possibilities).
Examples of ideal low-pass filter with linear phases

32. 5.4.7 Four Types of FIR Linear Phase Systems
Type I Linear Phase System:
Type II Linear Phase System:
Type III Linear Phase System:
Type IV Linear Phase System:
with M an even integer, then h[n] is symmetric and M/2 is an integer delay.
with M an odd integer, then h[n] is symmetric and M/2 is the time delay.
with M an even integer, then h[n] is anti-symmetric and M/2 is an integer delay.
with M an odd integer, then h[n] is anti-symmetric and M/2 is the delay.
Textbook p298-p300.