1. 山东省精品课程《生物医学信号处理(双语)》
2018年11月8日11时26分
Biomedical Signal processing Chapter 2 Discrete-Time Signals and Systems
Zhongguo Liu
Biomedical Engineering
School of Control Science and Engineering, Shandong University
山东省精品课程《生物医学信号处理(双语)》
2018/11/8 1
Zhongguo Liu_Biomedical Engineering_Shandong Univ.

2. Chapter 2 Discrete-Time Signals and Systems
Review of Chapter 2
2.1 Discrete-Time Signals: Sequences
High and Low Frequencies in Discrete-time signal
2.2 Discrete-Time Systems
Memoryless (memory); Linear; Time-Invariant; Causality; Stability(BIBO)
2.3 Linear Time-Invariant (LTI) Systems
LTI Systems:Convolution( 系统适用吗？)
2.4 Properties of LTI Systems
Stability and Causality of LTI systems;FIR and IIR systems;
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3. Chapter 2 Discrete-Time Signals and Systems
Review of Chapter 2
2.5 Linear Constant-Coefficient Difference Equations
The output for a given input is not uniquely specified. Auxiliary conditions are required; initial-rest conditions
2.6 Frequency-Domain Representation of Discrete-Time Signals and systems
Eigenfunction and Eigenvalue for LTI systems
2.7 Representation of Sequences by Fourier Transforms
2.8 Symmetry Properties of the Fourier Transform
2.9 Fourier Transform Theorems
2.10 Discrete-Time Random Signals
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4. Chapter 2 Discrete-Time Signals and Systems
2.0 Introduction
2.1 Discrete-Time Signals: Sequences
2.2 Discrete-Time Systems
2.3 Linear Time-Invariant (LTI) Systems
2.4 Properties of LTI Systems
2.5 Linear Constant-Coefficient
Difference Equations
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5. Chapter 2 Discrete-Time Signals and Systems
2.6 Frequency-Domain Representation of Discrete-Time Signals and systems
2.7 Representation of Sequences by Fourier Transforms
2.8 Symmetry Properties of the Fourier Transform
2.9 Fourier Transform Theorems
2.10 Discrete-Time Random Signals
2.11 Summary
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6. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
2.0 Introduction
Signal: something conveys information, represented mathematically as functions of one or more independent variables. Classified as:
Continuous-time (analog) signals, discrete-time signals, digital signals
Signal-processing systems are classified along the same lines as signals: Continuous-time (analog) systems, discrete-time systems, digital systems
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7. 2.1 Discrete-Time Signals: Sequences
Discrete-Time signals are represented as
In sampling of an analog signal xa(t):
1/T (reciprocal of T) : sampling frequency
Cumbersome, so just use
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8. Figure 2.1 Graphical representation of a discrete-time signal
Abscissa: continuous line
: is defined only at discrete instants
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9. Sampling the analog waveform
EXAMPLE
Figure 2.2

10. Basic Sequence Operations
Sum of two sequences
Product of two sequences
Multiplication of a sequence by a number α
Delay (shift) of a sequence
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11. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Basic sequences
Unit sample sequence (discrete-time impulse, impulse, Unit impulse)
离散时间单位脉冲(样本)序列, 区别连续时间单位冲激函数(continuous-time unit impulse function δ(t) )。
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12. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Basic sequences
A sum of scaled, delayed impulses
arbitrary sequence
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13. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Basic sequences
Unit step sequence
First backward difference
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14. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Basic Sequences
Exponential sequences
A and α are real: x[n] is real
A is positive and 0<α<1, x[n] is positive and decrease with increasing n
-1<α<0, x[n] alternate in sign, but decrease in magnitude with increasing n
: x[n] grows in magnitude as n increases
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15. EX. 2.1(第二版) Combining Basic sequences
If we want an exponential sequences that is zero for n <0, then
Cumbersome
simpler
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16. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Basic sequences
Sinusoidal sequence
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17. Exponential Sequences
Exponentially weighted sinusoids
Exponentially growing envelope
Exponentially decreasing envelope
is refered to
Complex Exponential Sequences
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18. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
difference between continuous-time and discrete-time complex exponentials or sinusoids
: frequency of the complex sinusoid or complex exponential
: phase
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19. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Periodic Sequences
A periodic sequence with integer period N
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20. EX. 2.1 Examples of Periodic Sequences
Suppose it is periodic sequence with period N
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21. EX. 2.1 Examples of Periodic Sequences
Suppose it is periodic sequence with period N
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22. EX. 2.1 Non-Periodic Sequences
Suppose it is periodic sequence with period N
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23. High and Low Frequencies in Discrete-time signal
(a) w0 = 0 or 2
Frequency: The rate at which a repeating event occurs.
(b) w0 = /8 or 15/8
(c) w0 = /4 or 7/4
(d) w0 = 
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24. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
2.2 Discrete-Time System
Discrete-Time System is a trasformation or operator that maps input sequence x[n] into an output sequence y[n].
y[n]=T{x[n]};
x[n], y[n]: discrete-time signal
T{‧}
x[n]
y[n]
Discrete-Time System
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25. EX. 2.2 The Ideal Delay System
If is a positive integer: the delay of the system, Shift the input sequence to the right by samples to form the output .
If is a negative integer: the system will shift the input to the left by samples, corresponding to a time advance.
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26. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
EX Moving Average
x[m] m n
n-5
dummy index m
y[7]
for n=7, M1=0, M2=5
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27. Properties of Discrete-time systems 2.2.1 Memoryless (memory) system
Memoryless systems:
the output y[n] at every value of n depends only on the input x[n] at the same value of n
Example 2.4 A Memoryless System
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28. Properties of Discrete-time systems 2.2.2 Linear SystemsIf
T{‧}
and only If:
additivity property
T{‧}
homogeneity or scaling
同(齐)次性 property
T{‧}
principle of superposition
T{‧}
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29. Example of Linear System
Ex. 2.5 Accumulator system
for arbitrary
when
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30. Example 2.6 Nonlinear Systems
Method: find one counterexample
For
counterexample
For
counterexample
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31. Properties of Discrete-time systems 2.2.3 Time-Invariant Systems
Shift-Invariant Systems
T{‧}
T{‧}
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32. Example of Time-Invariant System
Ex. 2.7 The Accumulator as a Time-Invariant System
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33. Ex. 2.8 The compressor system
T{‧}
T{‧}
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34. Properties of Discrete-time systems 2.2.4 Causality
A system is causal if, for every choice of , the output sequence value at the index depends only on the input sequence value for
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35. Ex. 2.9 The Forward and Backward Difference Systems
Forward difference system is not Causal
Backward difference system is Causal
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36. Properties of Discrete-time systems 2.2.5 Stability
Bounded-Input Bounded-Output (BIBO) Stability: every bounded input sequence produces a bounded output sequence. if
then
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37. Ex. 2.10 Testing for Stability or Instability
is stable if
then
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38. Ex. 2.10 Testing for Stability or Instability
Accumulator system
Accumulator system is not stable
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39. 2.3 Linear Time-Invariant (LTI) Systems
2018年11月8日11时26分
2.3 Linear Time-Invariant (LTI) Systems
Impulse response
T{‧}
T{‧}
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40. LTI Systems: Convolution
Representation of general sequence as a linear combination of delayed impulse
principle of superposition
An Illustration Example(interpretation 1)
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41. Zhongguo Liu_Biomedical Engineering_Shandong Univ.41
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42. Computation of the Convolution
（interpretation 2）
reflecting h[k] about the origion to obtain h[-k]
Shifting the origin of the reflected sequence to k=n
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43. Reflecting(reversing) h[k] about the origin to obtain h[-k].
Convolution can be realized by
Reflecting(reversing) h[k] about the origin to obtain h[-k].
Shifting the origin of the reflected sequences to k=n.
Computing the weighted moving average of x[k] by using the weights given by h[n-k].

44. Ex. 2.11 Analytical Evaluation of the Convolution
For system with impulse response
input
h(k)
Find the output at index n
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45. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
h(-k)
h(k)
h(n-k)
x(k)
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46. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
h(-k)
h(k)
x(k)
h(n-k)
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47. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
h(-k)
h(k)
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48. Zhongguo Liu_Biomedical Engineering_Shandong Univ.48
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49. 2.4 Properties of LTI Systems
Convolution is commutative(可交换的)
h[n]
x[n]
y[n]
x[n]
h[n]
y[n]
Convolution is distributed over addition
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50. Parallel connection of systems
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51. Cascade connection of systems
associative property 结合性
commutative
x [n]
h1[n]
h2[n]
y [n]
x [n]
h2[n]
h1[n]
y [n]
x [n]
h1[n] ]h2[n]
y [n]
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52. Stability of LTI Systems
LTI system is stable if the impulse response is absolutely summable .
Causality of LTI systems
HW: proof, Problem 2.62 52
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53. Impulse response of LTI systems
Impulse response of Ideal Delay systems
Impulse response of Accumulator
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54. Impulse response of Moving Average systems
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55. Impulse response of Moving Average systems
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56. Impulse response of Forward Difference
Impulse response of Backward Difference

57. Finite-duration impulse response (FIR) systems
The impulse response of the system has only a finite number of nonzero samples.
such as:
The FIR systems always are stable.

58. Infinite-duration impulse response (IIR)
The impulse response of the system is infinite in duration
Unstable system
Stable IIR System:

59. Output of the ideal delay system
Useful ideal delay system
Output of the ideal delay system
The convolution of a shifted impulse sequence with any signal x[n] is easily evaluated by simply shifting x[n] by the displacement of the impulse shifted.
Any noncausal FIR system can be made causal by cascading it with a sufficiently long delay.

60. Equivalent systems

61. Inverse system
x[n]
h[n]
hi[n]
y[n]= x[n]*]= x[n]
δ[n]

62. 2.5 Linear Constant-Coefficient Difference Equations
An important subclass of LTI systems consist of those system for which the input x[n] and output y[n] satisfy an Nth-order linear constant-coefficient difference equation.

63. Ex. 2.12 Difference Equation Representation of the Accumulator

64. Block diagram of a recursive (递推) difference equation representing an accumulator

65. Ex. 2.13 Difference Equation Representation of the Moving-Average System with
another representation 1

67. Difference Equation Representation of the System
In Chapter 6, we will see that many (unlimited number of ) distinct difference equations can be used to represent a given linear time-invariant (LTI) input-output relation.

68. Solving the difference equation
Without additional constraints or information, a linear constant-coefficient difference equation for discrete-time systems
does not provide a unique specification of the output for a given input.

69. Solving the difference equation
Output form :
: Particular solution, one output sequence for the given input
齐次解
Homogenous solution: solution for the homogenous equation( ):
where is the distinct roots of
can be chosen for a set of auxiliary value of

70. Solving the difference equation recursively
2018年11月8日11时26分
If the input and a set of auxiliary value
are specified for
y(n) can be written in a recurrence formula:

71. Example 2.16(第二版) Recursive Computation of Difference Equation

72. Example 2.16 (第二版) Recursive Computation of Difference Equation

73. Example 2.16 (第二版) Recursive Computation of Difference Equation
The system is noncausal.
The system is not linear.
is not time invariant.
When

74. Summary
The system for which the input and output satisfy a linear constant-coefficient difference equation:
The output for a given input is not uniquely specified. Auxiliary conditions are required.

75. Summary
If auxiliary conditions are given as N sequential values of the output, for
later value can be obtained by rearranging the difference equation as a recursive relation running forward in n,

76. Summary
Given
for
prior values can be obtained by rearranging the difference equation as a recursive relation running backward in n.

77. initial-rest conditions (初始松弛条件)
If a system is characterized by a linear constant-coefficient difference equation and is further specified to be LTI (Linear, Time Invariant), and causal, the solution is unique.
In this case, the auxiliary conditions are stated as initial-rest conditions(初始松弛条件):
i.e.: The auxiliary information is that if the input is zero for , then the output
is constrained to be zero for
( )

78. Summary
LTI (Linearity, Time Invariance), and causality of the system will depend on the auxiliary conditions.
If an additional condition is that the system is initially at rest (初始松弛), then the system will be linear, time invariant, and causal.

79. Finite-duration impulse response (FIR) systems
Review
Finite-duration impulse response (FIR) systems
The impulse response of the system has only a finite number of nonzero samples.
such as:
Moving Average systems
The FIR systems always are stable.

80. Infinite-duration impulse response (IIR)
Review
Infinite-duration impulse response (IIR)
The impulse response of the system is infinite in duration
Accumulator system:
Unstable system
Stable IIR System:

81. Ex. 2.16 (第二版) with initial-rest conditions
by recursing
backward
causal，
linear
If the input is , again with initial-rest conditions, then recursive solution is carried out using the initial condition
time invariant

82. Discussion Note that for , initial rest implies that
If input is , with initial-rest conditions:
Note that for , initial rest implies that
Initial rest does not always means
It does mean that if

83. 2.6 Frequency-Domain Representation of Discrete-Time Signals and systems
2.6.1 Eigenfunction and Eigenvalue for LTI If
is called as the eigenfunction of the system , and the associated eigenvalue is

84. Eigenfunction and Eigenvalue
2018年11月8日11时26分
Eigenfunction and Eigenvalue
Complex exponentials is the eigenfunction for LTI discrete-time systems :
eigenfunction
frequency response
eigenvalue

85. Frequency response is called as frequency response of the system.
i.e. DTFT, (Discrete-Time) Fourier Transform
Real part, imaginary part
Magnitude, phase

86. Example 2.14 Find Frequency response of the ideal Delay
solution 1:
solution 2:
From defination of frequency response:

87. Example 2.14 Frequency response of the ideal Delay
2018年11月8日11时26分
Example Frequency response of the ideal Delay

88. x[n]: Linear combination of complex exponentialIf
, output of LTI system?
From the principle of superposition, and
the corresponding output of an LTI system is

89. Periodic Frequency Response
The frequency response of discrete-time LTI systems is always a periodic function of the frequency variable with period

90. Periodic Frequency Response
We need only specify over
The “low frequencies” are frequencies close to zero
or an even multiple of π.
The “high frequencies” are frequencies close to ±π
or an odd multiple of π.
More generally, modify the frequency with
, r is integer.

91. Ideal Frequency-Selective Filters
periodicity
Frequency Response of Ideal Low-pass Filter
repeats periodically with period 2πr outside the plotted interval
one period

92. Frequency Response of Ideal High-pass Filter
Ideal Band-pass Filter

93. Frequency Response of Ideal Band-stop Filter
Ideal Low-pass Filter
periodicity

94. Example 2.16 Frequency Response of the Moving-Average System
solution:

96. Frequency Response of the Moving-Average System
相位也取决于符号，不仅与指数相关
M1 = 0 and M2 = 4

97. 2.6.2 Suddenly applied Complex Exponential Inputs
In practice, we may not apply the complex exponential inputs e jwn to a system, but the more practical-appearing inputs of the form
x[n] = ejwn  u[n]
i.e., x[n] suddenly applied at an arbitrary time, which for convenience we choose n=0.
For causal LTI system:

98. 2.6.2 Suddenly applied Complex Exponential Inputs
For causal LTI system
For n≥0

99. 2.6.2 Suddenly applied Complex Exponential Inputs
Steady-state Response
Transient response

100. 2.6.2 Suddenly Applied Complex Exponential Inputs (continue)
For infinite-duration impulse response (IIR)
For stable system, transient response must become increasingly smaller as n  ,
Illustration of a real part of suddenly applied complex exponential Input with IIR

101. 2.6.2 Suddenly Applied Complex Exponential Inputs (continue)
If h[n] = 0 except for 0 n  M (FIR), then the transient response yt[n] = 0 for n+1 > M.
For n  M, only the steady-state response exists
Illustration of a real part of suddenly applied complex exponential Input with FIR

102. 2.6.2 Suddenly Applied Complex Exponential Inputs (continue)
The condition for stability is also a sufficient condition for the existence of the frequency response function, as

103. 2.7 Representation of Sequences by Fourier Transforms
(Discrete-Time) Fourier Transform, DTFT，
analyzing
If is absolutely summable, i.e then exists. (Stability)
Inverse Fourier Transform, synthesis

104. Fourier Transform
rectangular form
polar form

105. Principal Value（主值）
is not unique because any may be added to without affecting the result of the complex exponentiation.
Principle value: is restricted to the range of values between It is denoted as
: phase function is referred as a continuous function of for

106. Impulse response and Frequency response
The frequency response of a LTI system is the Fourier transform of the impulse response of the LTI .

107. Example 2.17: Absolute Summability
Let , find condition of existence of the fourier transform or Absolute Summability.
Soltution:
Absolute Summability

108. Discussion of convergence
Absolute summability is a sufficient condition for the existence of a Fourier transform representation, and it also guarantees uniform convergence.
Some sequences are not absolutely summable, but are square summable, i.e.,

109. Discussion of convergence
Sequences which are square summable, can be represented by a Fourier transform, if we are willing to relax the condition of uniform convergence of the infinite sum defining
Then we have Mean-square Convergence.

110. Discussion of convergence
Mean-square convergence
The error may not approach zero at each value of as , but total “energy” in the error does.

111. Example 2.18 : Square-summability for the ideal Lowpass Filter
Soltution:
Since is nonzero for , the ideal lowpass filter is noncausal.

112. Example 2.18 Square-summability for the ideal Lowpass Filter
approaches zero as ,
but only as
is not absolutely summable.
does not converge uniformly for all w.
Define

113. Gibbs Phenomenon
M=3
M=1
M=19
M=7

114. Example continued
As M increases, oscillatory behavior at
is more rapid, but the size of the ripple does not decrease. (Gibbs Phenomenon)
As , the maximum amplitude of the oscillation does not approach zero, but the oscillations converge in location toward the point

115. However, is square summable, and converges in the mean-square sense to
Example continued
does not converge uniformly to the discontinuous function
However, is square summable, and converges in the mean-square sense to

116. Example 2.19 Fourier Transform of a constant
The sequence is neither absolutely summable nor square summable.
Define the Fourier transform of :
the periodic impulse train
The impulses are functions of a continuous variable and therefore are of “infinite height, zero width, and unit area.”

117. Example 2.19 Fourier Transform of a constant: proof

118. Example 2.20 Fourier Transform of Complex Exponential Sequences
Proof

119. Example: Fourier Transform of Complex Exponential Sequences

120. Example: Fourier Transform of unit step sequence

121. 2.8 Symmetry Properties of the Fourier Transform
Conjugate-symmetric sequence
Conjugate-antisymmetric sequence

122. Symmetry Properties of real sequence
even sequence: a real sequence that is Conjugate-symmetric
odd sequence: real, Conjugate-antisymmetric
real sequence:

123. Decomposition of a Fourier transform
Conjugate-symmetric
Conjugate-antisymmetric

124. x[n] is complex

125. x[n] is real, Conjugate-symmetric
the real part is even, the imaginary part is odd.

126. Ex. 2.21 illustration of Symmetry Properties
x[n] , a is real

127. Ex. 2.21 illustration of Symmetry Properties
Real part
Imaginary part
a=0.75(solid curve) and a=0.5(dashed curve)

128. Ex. 2.21 illustration of Symmetry Properties
Its magnitude is an even function, and phase is odd.
a=0.75(solid curve) and a=0.5(dashed curve)

129. 2.9 Fourier Transform Theorems
Linearity

130. Fourier Transform Theorems
2.9.2 Time shifting and frequency shifting

131. Fourier Transform Theorems
2.9.3 Time reversal
If is real,
If is real, even, is real, even.

132. Fourier Transform Theorems
2.9.4 Differentiation in Frequency

133. Fourier Transform Theorems
Parseval’s Theorem
is called the energy density spectrum

134. Fourier Transform Theorems
2.9.6 Convolution Theorem if
HW: proof

135. Fourier Transform Theorems
2.9.7 Modulation or Windowing Theorem
HW: proof

136. Fourier transform pairs

137. Fourier transform pairs

138. Fourier transform pairs

139. Ex. 2.22 Determine the Fourier Transform of sequence
Solution:

140. Ex. 2.23 Determine an inverse Fourier Transform of
Solution:

141. Ex. 2.24 Determine the impulse response from the frequency respone:
Solution:

142. h[n] can be computed recursively, or by FT:
Ex Determine the impulse response for a causal LTI system with difference equation:
Solution:
Impulse response
h[n] can be computed recursively, or by FT:

143. Ex. 2.25 Determine the impulse response for a difference equation:

144. 2.10 Discrete-Time Random Signals
Deterministic: each value of a sequence is uniquely determined by a mathematically expression, a table of data, or a rule of some type.
Stochastic signal: a member of an ensemble of discrete-time signals that is characterized by a set of probability density function.

145. 2.10 Discrete-Time Random Signals
For a particular signal at a particular time, the amplitude of the signal sample at that time is assumed to have been determined by an underlying scheme of probability.
That is, is an outcome of some random variable Xn .

146. 2.10 Discrete-Time Random Signals
x[n] is an outcome of some random variable
Xn (no necessary to distinguish x[n] and Xn ).
The collection of random variables is called a random process(随机过程).
The stochastic signals do not directly have Fourier transform, but the Fourier transform of the autocorrelation and autocovariance sequece often exist.

147. Fourier transform in stochastic signals
The Fourier transform of autocorrelation sequence has a useful interpretation in terms of the frequency distribution of the power in the signal.
The effect of processing stochastic signals with a discrete-time LTI system can be described in terms of the effect of the system on the autocorrelation sequence.

148. Stochastic signal as input
Let be a real-valued sequence that is a sample sequence of a wide-sense stationary discrete-time random process (随机过程).
Consider a stable LTI system with real h[n].
If the input is stationary, then so is the output

149. Stochastic signal as input
In our discussion, no necessary to distinguish between the random variables Xn andYn and their specific values x[n] and y[n].
mXn = E{Xn }, mYn= E(Yn}, can be written as
mx[n] = E{x[n]}, my[n] =E(y[n]}.
x[n] stationary, mx[n] independent of n, mx[n]→mx
The mean of output process
wide-sense stationary

150. Stochastic signal as input
The autocorrelation function of output
is called a deterministic autocorrelation sequence or autocorrelation sequence of

151. Stochastic signal as input
real h[n]
power (density) spectrum
DTFT of the autocorrelation function of output

152. Total average power in output
provides the motivation for the term power density spectrum.
能量无限
Parseval’s Theorem
能量有限

153. For Ideal bandpass system
Suppose that H(e jw ) is an ideal bandpass filter, as shown in Figure .
Since is a real, even, its FT is also real and even, i.e.,
so is

154. For Ideal bandpass system
the area under for can be taken to represent the mean square
value of the input in that frequency band.
能量非负
the power density function of a real signal is real, even, and nonnegative.

155. Ex White Noise
A white-noise signal is a signal for which
Assume the signal has zero mean. The power spectrum of a white noise is
The average power of a white noise is

156. Color Noise
A noise signal whose power spectrum is not constant with frequency.
A noise signal with power spectrum can be assumed to be the output of a LTI system with white-noise input.

157. Color Noise
Suppose

158. Cross-correlation between the input and output

159. Cross-correlation between the input and outputIf
That is, for a zero mean white-noise input, the cross-correlation between input and output of a LTI system is proportional to the impulse response of the system.

160. Cross power spectrum between the input and output
The cross power spectrum is proportional to the frequency response of the system.

161. 2.11 Summary
Define a set of basic sequence.
Define and represent the LTI systems in terms of the convolution, stability and causality.
Introduce the linear constant-coefficient difference equation with initial rest conditions for LTI , causal system.
Recursive solution of linear constant-coefficient difference equations.

162. 2.11 Summary
Define FIR and IIR systems
Define frequency response of the LTI system.
Define Fourier transform.
Introduce the properties and theorems of Fourier transform. (Symmetry)
Introduce the discrete-time random signals.

163. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Chapter 2 HW
2.4, 2.5, 2.11, 2.20
2.2, 2.5, 2.7, 2.12, 2.62
Zhongguo Liu_Biomedical Engineering_Shandong Univ.
171
2018/11/8
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