Transform Analysis of LTI System

1
CHAPTER 5

• Transform Analysis of LTI System

2
Definition of LTI Systems

• As described in Chapter 2, LTI is the systems
  which is linear and time-invariant.
• This chapter will describe in details the
  analysis of LTI systems using Fourier
  z-Transform.
• The LTI systems are characterized in time-domain
  as the convolution of impulse response, hn and
  input, xn as shown below
  
  8
• yn xnhn S xkhn-k
• k -8

3
Definition of LTI Systems

• The z-Transform of the output of LTI systems
  described above will result,
• 
  Y(z) H(z)X(z)
• The Transfer Function of the LTI system is
• H(z) Y(z)
• X(z)

4
Definition of LTI Systems

• A LTI system can be described by the difference
  equation of the form
• Example 1
• a. yn 2xn xn 1 5yn 1
• b. yn xn xn 1
• c. yn 0.9yn 1 0.8yn 2 xn

5
Frequency Response of LTI Systems

• The LTI systems that are operating in Frequency
  Domain, will have a Fourier Transform of the
  system input and output as
• Y(ej?) H(ej?) X(ej?)
  H(?)X(?)
• H(ej?) or H(?) is called a Frequency Response
  of
• the LTI systems.

6
Frequency Response of LTI Systems

• For the Transfer Function, H(z) in the LTI
  system, z is referred as complex frequency which
  can be expressed in polar form as shown below
• z rej?, where r is a
  magnitude of z and ? is the phase angle.
• In the unit circle of z-domain, r 1 (unit
  circle). Thus,
• z ej?
• H(z) H(ej?) is called the Frequency Response of
  LTI system and defined as the response of the
  system to a sinusoidal of varying frequency.
• The Transfer Function of the system that
  evaluated at the unit circle gives the Frequency
  Response of the system.

7
Frequency Response of LTI Systems

• The Magnitude Phase Response of the Frequency
  Response, H(ej?) or H(j?)
• 1. Magnitude Response
• 2. Phase Response
• H(ej?) tan-1 ImH(ej?)
• ReH(ej?)
• In Polar form gt H(ej?)ej?

8
Frequency Response of LTI Systems

• The Frequency Response also can be defined as a
  relation between Fourier Transform z-Transform
  
• H(ej?) H(z)
• z
  ej?
• Example 2
• If the impulse response of the LTI systems
  are defined as
• hn anµn
• Determine the Frequency Response of the
  systems.

9
Frequency Response of LTI Systems

• Example 3
• The Transfer Function of the LTI system is
  defined as
• H(z) (1 z-1)2
• (1 ½z-1) (1
  ¾z-1)
• Determine the difference equation of the
  system.

10
Frequency Response of LTI Systems

• Example 4
• Find the amplitude and phase response for the
  system characterized by the difference equation
• yn 1/6 xn 1/3 xn - 1 1/6 xn - 2

11
Frequency Response of LTI Systems

• The Properties of Systems Frequency Response
• 1. H(ej?) takes on value for all ? on a
• continuous basis.
• 2. H(ej?) is periodic with period of 2p.
• 3. The Magnitude response H(ej?) is an even
• function of ? and is symmetrical about p.
• 4. The Phase response ? H(ej?) is an odd
• function of ? and is anti-symmetrical
  about p.

12
Relationship Between Frequency Response Function
and System Poles Zeros

• The transfer function, H(z), is normally
  expressed as a ratio of 2 polynomials in z.
• where z ?l are zeroes and z ?l are poles
• For a causal system, the degree of the Numerator
  polynomial is less than the degree of the
  Denominator.
• Otherwise, the system will be non-causal.

13
Relationship Between Frequency Response Function
and System Poles Zeros

• Example 5
• If the transfer function of the system is
  defined as
• H(z) 1 2z-1
• 1 2z-1 z-2
• Determine the systems poles and zeros, sketch
  poles-zeros plot and calculate its Frequency
  Response.

14
Relationship Between Frequency Response Function
and System Poles Zeros

• Example 6
• The LTI system is described by the differential
  equation as below
• yn yn - 1 0.5yn - 2 3xn - 2xn - 1
  
• Determine the transfer function of the system,
• systems poles zeros sketch pole-zero plot.

15
Relationship Between Frequency Response Function
and System Poles Zeros

• Example 7
• The LTI system is described by the differential
  equation as below
• yn 0.8yn 1 0.2xn
• Determine the transfer function of the system,
  systems poles zeros, sketch pole-zero plot and
  Frequency Response.

16
System Stability Causality

• A LTI system is said to be BIBO STABLE
• 1. If all the poles of its Transfer Function
  lie within the
• unit circle in the z-plane.
• 2. The degree of the Numerator of the Transfer
• Function MUST NOT be larger than of the
• Denominator (M lt N).
• 3. The input signal is bounded of its
  z-transform
• contains poles, ?l where ?l lt 1 for all
  l.
• 4. Its ROC of the system function includes the
• unit circle.
• 5. Bounded Input will produce bounded
• Output. Otherwise, the system will be
  unstable.

17
System Stability Causality

• A LTI system is said Casual
• 1. If its impulse response, hn satisfy
• this condition
• hn 0, n lt 0
• 2. ROC of the z-transform is the exterior
• of a circle.
• A Casual LTI system is BIBO stable if and only if
  all the poles are inside the unit circle.

18
System Stability Causality

• A stable casual LTI systems will satisfy this
  equation
• ?l lt 1 for every pole ?l of the Transfer
  Function of system, H(z).
• Example 8
• The System Transfer Function is defined as
• H(z) 3 2z-1 ROC, z gt a
• 1 az-1
• Determine the condition that affect the system
  stability.

19
System Stability Causality

• Example 9
• The Transfer Function of the LTI system is
  defined as
• H(z) 1 2z-2 2z-2 z-3
• (1 z-1)(1 0.5z-1)(1
  0.2z-1)
• ROC 0.5 lt z lt 1
• a) Sketch the pole-zero pattern. Is the
  system stable?
• b) Determine the Impulse Response of the
  system.

20
Impulse Response

• The impulse response of a LTI system is one
  method of modeling linear system whether discrete
  time or continuous time.
• The Impulse Response sequence or the discrete
  impulse response also called the sampled impulse
  response of the system when the input to a system
  is a unit impulse. It is denoted by hn.
• The Impulse Response of the system, hn can be
  obtained directly by solving the difference
  equation describing the system.

21
Impulse Response

• There are 2 type of Impulse Response of LTI
  system. The response are
• 1. FIR Finite Impulse Response
• Criteria
• a. It has NO Pole. Only has Zero.
• b. It has NO Feedback.
• c. System always stable.
• It is also called as Non-Recursive.

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Impulse Response

• 2. IIR Infinite Impulse Response
• Criteria
• a. It has Pole.
• b. It has Feedback.
• c. The system stability depend on the
  poles, pk.
• It is also called as Recursive.

23
Impulse Response

• Example 10
• FIR Impulse Response
• a. y(n) x(n) x(n - 2) - x(n 1)
• b. y(n) x(n)
• gt All the term of dk in the difference
  equation of
• y(n) are zero
• IIR Impulse Response
• a. y(n) x(n) 0.9y(n - 1) - 2y(n - 2)
• b. y(n) x(n) y(n 1)
• gt All the term of dk in the difference
  equation of
• y(n) are non-zero

24
Impulse Response

• Example 11
• If the difference equation of LTI system is
  defined as
• y(n) x(n) 0.5x(n - 2)
• Determine the type Response of the system.

25
Impulse Response

• Example 12
• If the difference equation of LTI system is
  defined as
• y(n) x(n) 0.5x(n - 1) 0.9y(n 1)
• Determine the type of Response of the system.

26
Convolution

• The convolution is the multiplication of 2
  discrete signals either in time domain or
  frequency domain.
• If the conversion of the discrete signals from
  time domain to frequency domain and vice versa is
  stated below
• x1n X1(?)
• x2n X2(?)
• then, the Convolution will be
• xn x1nx2n X(?)
  X1(?)X2(?)

27
Convolution

• The symbol is used to denote Convolution.
• The convolution of discrete input signal with
  unit impulse will produce discrete input signal
• x(n) d(n) x(n)
• The convolution of discrete input signal with
  impulse response will produce discrete output
  signal
• x(n) h(n) y(n)

28
Convolution

• The Properties of Convolution are stated below
• 1. Commutative
• x(n) h(n) h(n) x(n)
• 2. Associative
• x(n) h(n) g(n) x(n) h(n)
  g(n)
• 3. Distributive
• x(n) h(n) g(n) x(n) h(n) x(n)
  g(n)

29
Convolution

• If the Impulse Response, h(n) is arranged in
  Cascade as shown below, the final Impulse
  Response is the convolution of each Impulse
  Response.
• x(n)
  y(n)
• x(n)
  y(n)

h1(n)
h2(n)
h1(n) h2(n)
30
Convolution

• If the Impulse Response, h(n) is arranged in
  Parallel as shown below, the final Impulse
  Response is the Summation of each Impulse
  Response.

h2(n)
x(n)
y(n)

h1(n)
h1(n) h2(n)
x(n)
y(n)