1. Chapter 4 LTI Discrete-Time Systems in the Transform Domain

2. 4.0 Introduction
Signal and system analysis may be implemented in the time-domain, frequency-domain and s- or z-domain.
In the time-domain, an LTI system can be uniquely specified by its unit impulse response h[n].
If the input x[n] applied to an LTI system with impulse response h[n], the corresponding output is

3. 4.0 Introduction
In the time-domain, an LTI system can also be described by a linear constant-coefficient difference equation:
By solving the equation, we can get the explicit response y[n] to an input signal.
By letting x[n]=δ[n], the impulse response can also be determined.

4. 4.0 Introduction
But some other important characteristics of LTI systems can not be obtained only in the time-domain.
For example, we can’t interpret filtering, we can’t know how shall we do, an unstable system could come to be stable.
All of these problems can not be solved in the time-domain, but can be solved in the transform domain easily.

5. 4.0 Introduction
In this chapter, we are going to introduce some concepts and methods for LTI system analysis in the transform domain including the frequency response and the system function.
And we will discuss the characteristics of several digital filters.

6. 4.1 Finite-Dimensional Discrete-Time Systems
Here, the finite-dimensional system means, I think, that the number of the order and the number of the inputs and outputs of the system are finite.
If a system has one input terminal and one output terminal, then the system is called the single-input single-output (SISO) system.

7. 4.1 Finite-Dimensional Discrete-Time Systems
For such an LTI discrete-time system, the input-output relationship is described by
By applying the Fourier transform to the equation we have the frequency-domain input-output relationship.

8. 4.1 Finite-Dimensional Discrete-Time Systems
Also by applying the z-transform to the difference equation we have the z-domain input-output relationship:

9. 4.2 The Frequency Response
4.2.1 Definition
In the time-domain representation, the convolution representation of an LTI discrete-time system with an impulse response h[n] is
Assuming:

10. 4.2 The Frequency Response
Here,
is called the frequency response of the LTI system. It is precisely the DTFT of the impulse response of the system.
Remember, if the input is an eigenfunction, the corresponding output signal is also the same eigenfunction but weighted by the frequency response?

11. 4.2 The Frequency Response
Thus, in the frequency-domain, we use the frequency response to represent an LTI system:
In general, H(ej) is complex valued and periodic with period 2p, and can be expressed in different forms.

12. 4.2 The Frequency Response
|H(ejω)| is called the magnitude response, also called the gain or attenuation. In some cases, it is specified in decibels as defined below:
called the phase response.

13. 4.2 The Frequency Response
For a discrete-time system characterized by a real impulse response h[n]
Even function
Odd function
Even function
Odd function

14. 4.2 The Frequency Response
4.2.2 Frequency Response Computation Using MATLAB
The M-file function freqz(h, w) can be used to determine the values of the frequency response of a prescribed impulse response vector h at a set of given frequency points w.

15. 4.2 The Frequency Response
Example 4.1 Consider the moving-average filter:
Its frequency response is given by
Program 4_1
moving_average

16. 4.2 The Frequency Response
4.2.3 Steady-State Response
What is the steady-state response?
The steady-state response is the response of the LTI system to a sinusoidal input when t→∞. In this case, the transient response approaches to 0.
The frequency response also determines the steady-state response.

17. 4.2 The Frequency Response
Example 4.2 Determine the steady-state response y[n] of a real coefficient LTI discrete-time system with a frequency response H(ej) for an input
From the Euler’s relation:

18. 4.2 The Frequency Response
The output response is determined from

19. 4.2 The Frequency Response
Again applying the Euler’s relation to the expression:
We have
This example tells us, if the input is a sinusoidal, the output is the same sinusoidal, but the amplitude is weighted by |H(ej)|, and the phase is shifted byθ().

20. 4.2 The Frequency Response
4.2.4 Response to a Causal Exponential Sequence
Causal system
From the convolution representation:

21. 4.2 The Frequency Response
Lets verify the sum
So that the system response is
Steady-state response
Transient response

22. 4.2 The Frequency Response
4.2.5 The Concept of Filtering
One application of an LTI discrete-time system is to pass certain frequency components in an input sequence without any distortion (if possible) and to block other frequency components.
Such systems are called digital filters and are one of the main subjects of discussion in this course.

23. 4.2 The Frequency Response
The key to the filtering process is
It expresses an arbitrary input as a linear weighted sum of an infinite number of exponential sequences, or equivalently, as a linear weighted sum of sinusoidal sequences.

24. 4.2 The Frequency Response
From the convolution property of DTFT:
Thus, by appropriately choosing the values of the magnitude function |H(ej)| at frequencies, we can get the desired output sequence from the input sequence.

25. 4.2 The Frequency Response
And, by appropriately choosing the values of the magnitude function of the LTI digital filter at frequencies corresponding to the frequencies of the sinusoidal components of the input, some of these components can be selectively heavily attenuated or filtered with respect to the others.

26. 4.2 The Frequency Response
To understand the mechanism behind the design of frequency-selective filters, consider a real-coefficient LTI discrete-time system characterized by a magnitude function
x[n]=Acos1n+Bcos2n, < 1< c< 2<

27. 4.2 The Frequency Response
From the result obtained early, we can write the expression of the output sequence y[n] As
So that the output sequence is obtained by

28. 4.2 The Frequency Response
Example 4.3 In this example, we consider the design of a very simple digital filter.
The input signal consists of a sum of two sinusoidal sequences of angular frequencies 0.1 rad/sample and 0.4 rad/sample.
We need to design a highpass filter that will pass the high-frequency component of the input but block the low-frequency part.

29. 4.2 The Frequency Response
For simplicity, assume the filter be an FIR filter of length 3 with an impulse response:
h[0] = h[2] = a, h[1] = b
i.e., or
Replacing with

30. 4.2 The Frequency Response
Design Objective: Choose suitable values of a and b so that the output is a sinusoidal sequence with a frequency 0.4 rad/sample.
The frequency response is computed below:

31. 4.2 The Frequency Response
i.e.,
The magnitude and phase functions are
In order to block the low-frequency component of =0.1 and to pass the high-frequency component at =0.4, |H(ej)| must satisfy the two equations:

32. 4.2 The Frequency Response
Solving the two equations we arrive at:
Thus the FIR filter can be expressed by the impulse response:

33. 4.2 The Frequency Response
or by the difference equation:
Substituting the input signal
x[n] = {cos(0.1n) + cos(0.4n)}
into the difference equation, we have
program 4_2

34. 4.2 The Frequency Response
Reducing the expression of the output sequence, we arrive at:
program 4_2.m

35. 4.2 The Frequency Response
4.2.6 Phase and Group Delays
As indicated early, when a sequence x[n] passes through an LTI discrete-time system, the system will effect on the signal by weighting |H(ej)| to the magnitudes of each frequency components in x[n] and shifting x[n] by a certain samples.

36. 4.2 The Frequency Response
Again consider the input sequence
From the above conclusion, the output sequence y[n] is
This means that a phase shift corresponds to a time delay.

37. 4.2 The Frequency Response
This expression indicates a time delay, at = 0 given by
If the input signal contains many sinusoidal components with different frequencies that are not harmonically related, each component will go through different phase delays when processed by the LTI discrete-time system.

38. 4.2 The Frequency Response
Then, the output signal, in general, will not look like the input signal.
In general, the group delay is defined by the function of :
Some times, it is also called the envelope delay.

39. 4.2 The Frequency Response
The physical meaning of the group delay
The group delay at a frequency c is the slope of the phase function at c.
It displays time delayed by the system to a frequency component with frequency c.

40. 4.2 The Frequency Response
Right figure shows the waveform of an amplitude-modulated input and the output generated by an LTI system

41. 4.2 The Frequency Response
In discrete-time LTI systems, the unit of the group delay is samples.
Again consider the moving-average filter discussed in Example 4.1.
Its frequency response is given by

42. 4.2 The Frequency Response
4.2.7 Frequency-Domain Characterization of the LTI Discrete-Time System
In this part, we discuss the frequency-domain representation of an LTI discrete-time system.

43. 4.2 The Frequency Response
By making the DTFT to the convolution representation:
The frequency response can be expressed as:

44. 4.3 The Transfer Function
A generalization of the frequency response function leads to the concept of transfer function.
4.3.1 Definition
Given an LTI discrete-time system with impulse response h[n], and if the input signal is x[n], then the output signal is

45. 4.3 The Transfer Function
By applying the z-transform to the convolution representation we have:
The transfer function
4.3.2 Derivation of the Transfer Function
Like the frequency response is the DTFT of the impulse response h[n], the transfer function is the z-transform of h[n].

46. 4.3 The Transfer Function
The definition of the transfer function also follows from the convolution property of the z-transform:
For an FIR system, its transfer function is defined by:

47. 4.3 The Transfer Function
For an IIR system, its transfer function is a rational ratio of two polynomials of z-1.
It can also be expressed in the form:
Used to determine the zeros and poles.

48. 4.3 The Transfer Function
On the other hands, the transfer function can be expressed in other forms, zero-pole form and partial-fraction expansion form.
Example 4.4 Consider the moving-arerage filter of Example 4.1 with an impulse response h[n] given by
Determine its transfer function.

49. 4.3 The Transfer Function
Its transfer function is then given by

50. 4.3 The Transfer Function Lets determine the zeros and poles.
From the expression of H(z), we see
(M-1)th-order pole:
A single pole:
There are M zeros on the unit circle at z=ej2pk/M. But the pole z=1 exactly cancels the zero at the same place.

51. 4.3 The Transfer Function
Example 4.5 A causal LTI IIR digital filter is characterized by a difference equation given by:
The transfer function is directly written as:

52. 4.3 The Transfer Function
Multiplying the numerator and denominator of the right-hand side by z3 we obtain
By the use of the M-file: zplane(num,den) and [r,p,k]=tf2zp(num,den), the zeros and poles can be determined.

53. 4.3 The Transfer Function
The zero-pole diagram of the IIR transfer function:
Gain
Zeros at j0.8000
j0.8000
Poles at j0.7000
j0.8000
0.3000

54. 4.3 The Transfer Function
4.3.3 Frequency Response from Transfer Function
As the z-transform is a generalization of DTFT, to be exact, the DTFT is the z-transform on the unit circle.
So, if the ROC of the transfer function includes the unit circle, then the frequency response exists, and can be determined from the transfer function by replacing z of H(z) with ej.

55. 4.3 The Transfer Function
4.3.5 Stability Condition in Terms of Pole Locations
The stable system: for an LTI discrete-time system, if its impulse response h[n] satisfies the absolutely summable condition, then the system is stable.

56. 4.3 The Transfer Function
In fact, the above condition is the condition for which the DTFT exists.
And associated with the relationship between the DTFT and the z-transform, we can conclude that if the ROC includes the unit circle, then the system is stable.

57. 4.3 The Transfer Function
Example 4.6 Consider the causal second-order IIR transfer function given by

58. 4.3 The Transfer Function
By quantizing the coefficients of H(z):

59. 4.3 The Transfer Function Conclusion:
If the ROC of the system’s transfer function includes the unit circle, then the LTI discrete-time system is stable.

60. 4.4 Types of Transfer Function
The time-domain classification of an LTI digital transfer function is based on the length of its impulse response:
FIR transfer function (FIR filter);
IIR transfer function (IIR filter).
A very common classification is based on the shape of the magnitude function of H(ei) or the form of the phase function q(w).

61. 4.4 Types of Transfer Function
4.41 Ideal Filters
Based on shape of |H(ei)|, four types of ideal filters are usually defined.
Ideal highpass
Ideal lowpass
Ideal bandpass
Ideal bandstop

62. 4.4 Types of Transfer Function
Earlier in the course we derived the impulse response h[n] of the ideal lowpass filter:
Here, h[n] is not absolutely summable, hence, the corresponding transfer function is not BIBO stable. And because h[n] is noncausal and infinite, thus the filter can not be realized.

63. 4.4 Types of Transfer Function
For a stable and realizable transfer function, the frequency response has a transition band between the passband and the stopband.
Typical magnitude response specifications of a lowpass filter are shown as:

64. 4.4 Types of Transfer Function
4.4.2 Zero-Phase and Linear-Phase Transfer Functions
Based on the form of the phase function q(w), two types of digital filters can be defined:
Linear-phase filters (including zero-phase filters)
Nonlinear phase filters

65. 4.4 Types of Transfer Function
Remember the conditions for which signal is transmitted with no any distortion?
The gain is a constant in the passband;
The phase is a linear function of w.
In this section, we only pay respect to the phase function of digital filters.
But it is impossible to design a causal digital filter with a zero phase. Why?

66. 4.4 Types of Transfer Function
The reason is that processing a sequence takes time even the sequence only includes one sample.
For non-real-time processing of real-valued input signals of finite length, zero-phase filtering can be very simply implemented by relaxing the causality requirement.

67. 4.4 Types of Transfer Function
One zero-phase filtering scheme is sketched below
x[n]
v[n]
u[n]
w[n]
H(z)
u[n]=v[-n], y[n]=w[-n]
V(ej)= H(ej)X(ej) , W(ej)=H(ej)U(ej)
U(ej)= V*(ej), Y(ej)= W*(ej)
Y(ej) = H*(ej)H(ej)X(ej)
= |H(ej)|2X(ej)

68. 4.4 Types of Transfer Function
In the case of a causal transfer function with a nonzero phase response, the phase distortion can be avoided by ensuring that the transfer function has a unity magnitude and a linear-phase characteristic in the frequency band of interest.

69. 4.4 Types of Transfer Function
The most general type of a filter with a linear phase has a frequency response given by
H(ej)= ejD
which has a linear phase from w = 0 to w = 2p.
Note also |H(ej)|=1
()=D

70. 4.4 Types of Transfer Function
Figure right shows the frequency response if a lowpass filter with a linear-phase characteristic in the passband.
Since the signal components in the
stopband are blocked, the phase response in the stopband can be of any shape.

71. 4.4 Types of Transfer Function
Example 4.8 Determine the impulse response of an ideal lowpass filter with a linear phase response:
We have known that

72. 4.4 Types of Transfer Function
Applying the frequency-shifting property of the DTFT to the impulse response we arrive at
This transfer function can not be realized because of its noncausality.

73. 4.4 Types of Transfer Function
Truncate the impulse response h[n] to a finite number of terms by a window of length N+1, we can get a causal realizable FIR filter with impulse response being the truncated version of h[n].

74. 4.4 Types of Transfer Function
The truncated approximation may or may not exhibit linear phase, depending on the value of n0 chosen.
If we choose n0= N/2 with N a positive integer, the truncated and shifted approximation, the impulse response of the causal realizable FIR filter is

75. 4.4 Types of Transfer Function
Because of the symmetry of the impulse response coefficients, the frequency response of the truncated approximation can be expressed as:
zero-phase response

76. 4.4 Types of Transfer Function
4.4.3 Types of Linear-Phase FIR Transfer Functions
It is nearly impossible to design a linear-phase IIR transfer function.
It is always possible to design an FIR transfer function with an exact linear-phase response.
But an FIR digital filter may be or not with a linear-phase.

77. 4.4 Types of Transfer Function
Consider a causal FIR transfer function H(z) of length N+1, i.e., of order N:
If its impulse response h[n] satisfies
symmetric condition or
antisymmetric condition
Then it must have a linear phase.

78. 4.4 Types of Transfer Function
Since the length of the impulse response can be either even or odd, we can define four types of linear-phase FIR transfer functions.
The case of the length N+1 being odd:
Symmetry
N = 8
Antisymmetry
N = 8

79. 4.4 Types of Transfer Function
The case of the length N+1 being even:
Symmetry
N = 7
Antisymmetry
N = 7

80. 4.4 Types of Transfer Function
Type 1: Symmetric Impulse Response with Odd Length (Even order Symmetry)
Assuming that the degree N=8, in this case:
Symmetry
N = 8
The transfer function is

81. 4.4 Types of Transfer Function
Reducing H(z):

82. 4.4 Types of Transfer Function
The corresponding frequency response is then determined by replacing z with ejω

83. 4.4 Types of Transfer Function
So that
Linear phase

84. 4.4 Types of Transfer Function
The group delay is given by
indicating a constant group delay of 4 samples.
In general case for type 1 FIR filters,

85. 4.4 Types of Transfer Function
Example 4.9 Consider
The above transfer function has a symmetric impulse response and therefore a linear phase response.
The phase function is obviously as
The group delay:
The group delay:

86. 4.4 Types of Transfer Function
And the magnitude function is given by:

87. 4.4 Types of Transfer Function
Type 2: Symmetric Impulse Response with Even Length (Odd order Symmetry)
Assuming that the degree N=7, in this case:
Symmetry
N = 7
The transfer function is

88. 4.4 Types of Transfer Function

89. 4.4 Types of Transfer Function
The frequency response is:
Linear phase

90. 4.4 Types of Transfer Function
In general case for type 2 FIR filters,

91. 4.4 Types of Transfer Function
Similarly, for Type 3 FIR filters (Antisymmetric Impulse Response with Odd Length):
Antisymmetry
N = 8

92. 4.4 Types of Transfer Function
For Type 4 FIR filters (Antisymmetric Impulse Response with Even Length):
Antisymmetry
N = 7

93. 4.4 Types of Transfer Function
Conclusion:
For an FIR filter, if its impulse response h[n] satisfies the symmetry or the antisymmetry condition, then it has a linear phase characteristic.

94. 4.5 Simple Digital Filters
In this section, we describe several low-order FIR and IIR digital filters.
4.5.1 Simple FIR Digital Filters
Note: Here, FIR filters have integer-valued impulse response coefficients. These filters are employed in a number of practical applications, primarily, because of their simplicity.

95. 4.5 Simple Digital Filters
Lowpass FIR Filters
The simplest FIR filter is the moving-average filter with M=2 which has a transfer function
It is the first-order FIR filter with a linear phase.
Zero:
Pole:

96. 4.5 Simple Digital Filters
Plots of the moving-average FIR filter.
moving_average.m

97. 4.5 Simple Digital Filters
The concept of 3-dB cutoff frequency
Some times, our interest is the 3-dB cutoff frequency.
Definition: given a frequency wc for which the following relationship is guaranteed:
The maximum value of the magnitude function

98. 4.5 Simple Digital Filters
For this moving-average FIR filter, we can readily determine its 3-dB cutoff frequency wc.

99. 4.5 Simple Digital Filters
Highpass FIR Digital Filters
The simplest highpass FIR filter is obtained by replacing z with -z to the moving-average filter.
This is a type 4 FIR filter.

100. 4.5 Simple Digital Filters
and
simple_highpass.m

101. 4.5 Simple Digital Filters
4.5.2 Simple IIR Digital Filters
Lowpass IIR Digital Filters
A first-order lowpass IIR digital filter has a transfer function given by
By long division:

102. 4.5 Simple Digital Filters
The impulse response can be determined from:

103. 4.5 Simple Digital Filters
Highpass IIR Digital Filters
A first-order highpass IIR filter is described by the transfer function:

104. 4.5 Simple Digital Filters
The plots of the magnitude, phase function are given by:

105. 4.6 Allpass Transfer Function
We now turn our attention to a very special type of IIR transfer function that is characterized by unity magnitude for all frequencies.
Such a transfer function, called an allpass function, has many useful applications in digital signal processing.

106. 4.6 Allpass Transfer Function
Definition: An IIR transfer function A(z) with unity magnitude response for all frequencies, i.e.,
is called an allpass transfer function
An M-th order causal real-coefficient allpass transfer function is of the form

107. 4.6 Allpass Transfer Function
If we denote the denominator polynomials of AM(z) as DM(z) : If
is a pole
must be a zero
then

108. 4.6 Allpass Transfer Function
This means that the numerator of an allpass transfer function is the mirror-image polynomial of the denominator, and vice versa.
For example, given an allpass transfer function as

109. 4.6 Allpass Transfer Function
Using MATLAB, we can plot the zero-pole diagram, the magnitude and the phase function given by:

110. 4.6 Allpass Transfer Function
It’s easy to show that the magnitude of A(ejw) is indeed equal to one for all w.
Hence: