Transform Analysis of LTI systems

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Transform Analysis ofLTI systems

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Content

• The Frequency Response of LTI systems
• Systems Characterized by Constant-Coefficient
  Difference Equations
• Frequency Response for Rational System Functions
• Relationship btw Magnitude and Phase
• Allpass Systems
• Minimum-Phase Systems
• Generalized Linear-Phase Systems

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Transform Analysis ofLTI systems

• Frequency Response of LTI systems

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Time-Invariant System
y(n)x(n)h(n)
x(n)
H(z)
X(z)
Y(z)X(z)H(z)
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Frequency Response
Magnitude
Phase
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Ideal Frequency-Selective Filters
Ideal Lowpass Filter
Computationally Unrealizable
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Ideal Frequency-Selective Filters
Ideal Highpass Filter
Computationally Unrealizable
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Ideal Frequency-Selective Filters

• Such filters are
• Noncausal
• Zero phase
• Not Computationally realizable
• Causal approximation of ideal frequency-selective
  filters must have nonzero phase response.

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Phase Distortion and Delay ---Ideal Delay
Delay Distortion Linear Phase
Delay Distortion would be considered a rather
mild form of phase distortion.
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Phase Distortion and Delay ---A Linear Phase
Ideal Filter
Still a noncausal one. Not computationally
realizable.
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Phase Distortion and Delay ---Group Delay

• A convenient measure of the linearity of phase.
• Definition

• Linear Phase ? ?(?)constant
• The deviation of ?(?) away from a constant
  indicates the degree of nonlinearity of the phase.

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Transform Analysis ofLTI systems

• Systems Characterized by
• Constant-Coefficient Difference Equations

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Nth-Order Difference Equation
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Representation in Factored Form
Contributes poles at 0 and zeros at cr
Contributes zeros at 0 and poles at dr
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Example
Two zeros at z ?1
poles at z 1/2 and z ? 3/4
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Stability and Causality

• For a given ration of polynomials, different
  choice of ROC will lead to different impulse
  response.
• We want to find the proper one to build a causal
  and stable system.
• How?

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Stability and Causality

• For Causality
• ROC of H(z) must be outside the outermost pole
• For Stability
• ROC includes the unit circle
• For both
• All poles are inside the unit circle

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Stability and Causality

• Example

Discuss its stability and causality
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Inverse Systems
X(z)
Y(z)
X(z)
G(z) H(z)Hi(z)1
g(n) h(n) hi(n) ?(n)
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Inverse Systems
Does every system have an inverse system?
Give an example.
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Inverse Systems
Zeros
Zeros
Poles
Poles
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Minimum-Phase Systems

• A LTI system is stable and causal and also has a
  stable and causal inverse iff both poles and
  zeros of H(z) are inside the unit circle.
• Such systems are referred to as minimum-phase
  systems.

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Impulse Response for Rational System Functions

• By partial fraction expansion

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FIR and IIR
Zero poles
nonzero poles
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FIR and IIR
FIR The system contains only zero poles.
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FIR and IIR
IIR The system contains nonzero poles (not
canceled by zeros).
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FIR
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ExampleFIR
One pole is canceled by zero here.
Does this system have nonzero pole?
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ExampleFIR
Write its system function.
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ExampleIIR
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Transform Analysis ofLTI systems

• Frequency Response of For Rational System
  Functions

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Rational Systems
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Log Magnitude of H(ej?) ---Decibels (dBs)
Gain in dB 20log10H(ej?)
Contributed by zeros
Contributed by poles
Scaling
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Advantages of Representing the magnitude in dB
The Magnitude Of Impulse Response
The magnitude of Output FT
The magnitude of Input FT
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Phase for Rational Systems
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Systems with a Single Zero or Pole
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Frequency Response of a Single Zero or Pole
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Frequency Response of a Single Zero
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Frequency Response of a Single Zero
H(ej?)2 Its maximum is at ????. max
H(ej?)2 (1r)2 Its minimum is at ???0. min
H(ej?)2 (1?r)2
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Frequency Response of a Single Zero
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Frequency Response of a Single Zero
r 0.9 ? 0
r 0.9 ? ?/2
r 0.9 ? ?
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Frequency Response of a Single Zero
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Frequency Response of a Single Zero
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Frequency Response of a Single Zero
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Frequency Response of a Single Zero
Zero outside the unit circle
Note that the group delay is always positive when
rgt1
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Frequency Response of a Single Zero
Some zeros inside the unit circle And some outside
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Frequency Response of a Single Pole

• The converse of the single-zero case.
• Why?
• A stable system r lt 1
• Excise Use matlab to plot the frequency
  responses for various cases.

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Frequency Response of Multiple Zero and Poles

• Using additive method to compute
• Magnitude
• Phase
• Group Delay

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Example Multiple Zero and Poles
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Example Multiple Zero and Poles
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Transform Analysis ofLTI systems

• Relationship btw
• Magnitude and Phase

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Magnitude and Phase
In general, knowledge about the magnitude
provides no information about the phase, and vice
versa. Except when
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Magnitude
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Magnitude
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Magnitude
Conjugate reciprocal pairs
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Magnitude
Given C(z), H(z)?
How many choices if the numbers of zeros and
poles are fixed?
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Allpass Factors
Pole at a
Zero at 1/a
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Allpass Factors
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Allpass Factors
There are infinite many systems to have the same
frequency-response magnitude?
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Transform Analysis ofLTI systems

• Allpass Systems

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General Form
Complex Poles
Real Poles
Hap(ej?)1
?Hap(ej?)?
grdHap(ej?)?
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AllPass Factor
Consider arej?
Always positive for a stable and causal system.
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Example AllPass FactorReal poles
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Example AllPass FactorReal Poles
Phase is nonpositive for 0lt?lt?.
Group delay is positive
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Example AllPass FactorComplex Poles
Continuous phase is nonpositive for 0lt?lt?.
Group delay is positive
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Example AllPass FactorComplex Poles
Continuous phase is nonpositive for 0lt?lt?.
Group delay is positive
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Transform Analysis ofLTI systems

• Minimum-Phase Systems

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Properties of Minimum-Phase Systems

• To have a stable and causal inverse systems
• Minimum phase delay
• Minimum group delay
• Minimum energy delay

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Rational Systems vs. Minimum-Phase Systems
How?
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Rational Systems vs. Minimum-Phase Systems
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Rational Systems vs. Minimum-Phase Systems
Pole/zero Canceled
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Frequency-Response Compensation
s(n)
sd(n)
s(n)
The inverse system of Hd(z) iff it is a
minimum-phase system.
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Frequency-Response Compensation
s(n)
sd(n)
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Frequency-Response Compensation
Hd(z)
Hc(z)
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ExampleFrequency-Response Compensation
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ExampleFrequency-Response Compensation
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ExampleFrequency-Response Compensation
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ExampleFrequency-Response Compensation
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ExampleFrequency-Response Compensation
Minimum Phase
Nonminimum Phase
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Minimum Phase-Lag
Nonpositive For 0????
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Minimum Group-Delay
Nonnegative For 0????
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Minimum-Energy Delay
Apply initial value theorem
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Transform Analysis ofLTI systems

• Generalized
• Linear-Phase Systems

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Linear Phase

• Linear phase with integer (negative slope) ---
  simple delay
• Generalization constant group delay

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Example Ideal Delay
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Example Ideal Delay
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Example Ideal Delay
Impulse response is symmetric about n nd ,
i.e., h(2nd ?n)h(n).
If ?nd (e.g., ?5) is an integer, h(n)?(n?nd).
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Example Ideal Delay
h(2??n)h(n).
The case for 2? (e.g., ?4.5) is an integer.
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Example Ideal Delay
Asymmetry
? as an arbitrary number (e.g., ?4.3).
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More General Frequency Response with Linear Phase
Zero-phase filter
Ideal delay
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More General Frequency Response with Linear Phase
Zero-phase filter
Ideal delay
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Example Ideal Lowpass Filter
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Example Ideal Lowpass Filter
Show that
If 2? is an interger, h(2? ?n)h(n).
That is, it has the same symmetric property as an
ideal delay.
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Generalized Linear Phase Systems

• and ?
• are constants

Real function. Possibly bipolar.
constant group delay
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h(n) vs. ? and ?
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h(n) vs. ? and ?
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h(n) vs. ? and ?
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h(n) vs. ? and ?
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Necessary Condition for Generalized Linear Phase
Systems
Lets consider special cases.
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Necessary Condition for Generalized Linear Phase
Systems
?0 or ?
2? M an integer
?0 or ?
Such a condition must hold for all ? and ?
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Necessary Condition for Generalized Linear Phase
Systems
?0 or ?
2? M an integer
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Necessary Condition for Generalized Linear Phase
Systems
??/2 or 3?/2
2? M an integer
??/2 or 3?/2
Such a condition must hold for all ? and ?
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Necessary Condition for Generalized Linear Phase
Systems
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CausalGeneralized Linear Phase Systems
Generalized Linear Phase System
Causal Generalized Linear Phase System
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CausalGeneralized Linear Phase Systems
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CausalGeneralized Linear Phase Systems
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Type I FIR Linear Phase Systems
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ExampleType I FIR Linear Phase Systems
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ExampleType I FIR Linear Phase Systems
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ExampleType II FIR Linear Phase Systems
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ExampleType III FIR Linear Phase Systems
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ExampleType IV FIR Linear Phase Systems
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Zeros Locations for FIR Linear Phase Systems
(Type I and II)
Let z0 be a zero of H(z)
1/z0 is a zero
If h(n) is real
z0 and 1/ z0 are zeros
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Zeros Locations for FIR Linear Phase Systems
(Type I and II)
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Zeros Locations for FIR Linear Phase Systems
(Type I and II)
Consider z ?1
if M is odd, z ?1 must be a zero.
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Zeros Locations for FIR Linear Phase Systems
(Type III and IV)
Let z0 be a zero of H(z)
1/z0 is a zero
If h(n) is real
z0 and 1/ z0 are zeros
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Zeros Locations for FIR Linear Phase Systems
(Type III and IV)
z 1 must be a zero.
Consider z 1
Consider z ?1
if M is even, z ?1 must be a zero.