1. 12015-10-161Zhongguo Liu_Biomedical Engineering_Shandong Univ. 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 ) 》 http://course.sdu.edu.cn/bdsp.htmlhttp://control.sdu.edu.cn/bdsp

2. 2 10/16/2015 2 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 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

3. 3 10/16/2015 3 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 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

4. 4 10/16/2015 4 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

5. 5 10/16/2015 5 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2.1 Discrete-Time Signals: Sequences  Discrete-Time signals are represented as  In sampling,  1/T (reciprocal of T) : sampling frequency Cumbersome, so just use

6. 6 10/16/2015 6 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Figure 2.1 Graphical representation of a discrete-time signal Abscissa: continuous line : is defined only at discrete instants

7. 7 Figure 2.2 EXAMPLE Sampling the analog waveform

8. 8 10/16/2015 8 Zhongguo Liu_Biomedical Engineering_Shandong Univ.  Sum of two sequences  Product of two sequences  Multiplication of a sequence by a number α  Delay (shift) of a sequence Basic Sequence Operations

9. 9 10/16/2015 9 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Basic sequences  Unit sample sequence (discrete-time impulse, impulse)

10. 10 10/16/2015 10 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Basic sequences  arbitrary sequence A sum of scaled, delayed impulses

11. 11 10/16/2015 11 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Basic sequences  Unit step sequence First backward difference

12. 12 10/16/2015 12 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

13. 13 10/16/2015 13 Zhongguo Liu_Biomedical Engineering_Shandong Univ. EX. 2.1 Combining Basic sequences  If we want an exponential sequences that is zero for n <0, then Cumbersome simpler

14. 14 10/16/2015 14 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Basic sequences  Sinusoidal sequence

15. 15 10/16/2015 15 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Exponential Sequences Complex Exponential Sequences Exponentially weighted sinusoids Exponentially growing envelope Exponentially decreasing envelope is refered to

16. 16 10/16/2015 16 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Frequency difference between continuous-time and discrete-time complex exponentials or sinusoids  : frequency of the complex sinusoid or complex exponential  : phase

17. 17 10/16/2015 17 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Periodic Sequences  A periodic sequence with integer period N

18. 18 10/16/2015 18 Zhongguo Liu_Biomedical Engineering_Shandong Univ. EX. 2.2 Examples of Periodic Sequences  Suppose it is periodic sequence with period N

19. 19 10/16/2015 19 Zhongguo Liu_Biomedical Engineering_Shandong Univ.  Suppose it is periodic sequence with period N EX. 2.2 Examples of Periodic Sequences

20. 20 10/16/2015 20 Zhongguo Liu_Biomedical Engineering_Shandong Univ. EX. 2.2 Non-Periodic Sequences  Suppose it is periodic sequence with period N

21. 21 10/16/2015 21 Zhongguo Liu_Biomedical Engineering_Shandong Univ. High and Low Frequencies in Discrete-time signal (b) w 0 =  /8 or 15  /8 (c) w 0 =  /4 or 7  /4 (d) w 0 =  Frequency: The rate at which a repeating event occurs. (a) w 0 = 0 or 2 

22. 22 10/16/2015 22 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 a unique y[n]  y[n]=T{x[n]}, x[n], y[n]: discrete-time signal T{ ‧ } x[n]y[n] Discrete-Time System

23. 23 10/16/2015 23 Zhongguo Liu_Biomedical Engineering_Shandong Univ. EX. 2.3 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.

24. 24 10/16/2015 24 Zhongguo Liu_Biomedical Engineering_Shandong Univ. EX. 2.4 Moving Average x[m] m n n-5 dummy index m y[n] for n=7, M 1 =0, M 2 =5

25. 25 10/16/2015 25 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 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

26. 26 10/16/2015 26 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Properties of Discrete-time systems 2.2.2 Linear Systems  If T{ ‧ } additivity property homogeneity or scaling 同 ( 齐 ) 次性 property  principle of superposition  and only If:

27. 27 10/16/2015 27 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Example of Linear System  Ex. 2.6 Accumulator system for arbitrary when

28. 28 10/16/2015 28 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Example 2.7 Nonlinear Systems  Method: find one counterexample  counterexample  For  counterexample  For

29. 29 10/16/2015 29 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Properties of Discrete-time systems 2.2.3 Time-Invariant Systems  Shift-Invariant Systems T{ ‧ }

30. 30 10/16/2015 30 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Example of Time-Invariant System  Ex. 2.8 Accumulator system

31. 31 10/16/2015 31 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Ex. 2.9 The compressor system T{ ‧ } 0 0 0 0 0 0

32. 32 10/16/2015 32 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 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

33. 33 10/16/2015 33 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Ex. 2.10 Example for Causal System  Forward difference system is not Causal  Backward difference system is Causal

34. 34 10/16/2015 34 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 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

35. 35 10/16/2015 35 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Ex. 2.11 Testing for Stability or Instability if then is stable

36. 36 10/16/2015 36 Zhongguo Liu_Biomedical Engineering_Shandong Univ.  Accumulator system  Accumulator system is not stable Ex. 2.11 Testing for Stability or Instability

37. 37 10/16/2015 37 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2.3 Linear Time-Invariant (LTI) Systems  Impulse response T{ ‧ }

38. 38 10/16/2015 38 Zhongguo Liu_Biomedical Engineering_Shandong Univ. LTI Systems: Convolution  Representation of general sequence as a linear combination of delayed impulse  principle of superposition An Illustration Example （ interpretation 1 ）

39. 39 10/16/2015 39 Zhongguo Liu_Biomedical Engineering_Shandong Univ.

40. 40 10/16/2015 40 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Computation of the Convolution  reflecting h[k] about the origion to obtain h[-k]  Shifting the origin of the reflected sequence to k=n （ interpretation 2 ）

41. 41 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].

42. 42 10/16/2015 42 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Ex. 2.13 Analytical Evaluation of the Convolution For system with impulse response h(k) 0 Find the output at index n input

43. 43 10/16/2015 43 Zhongguo Liu_Biomedical Engineering_Shandong Univ. h(k) 0 0 h(n-k)x(k) h(-k) 0

44. 44 10/16/2015 44 Zhongguo Liu_Biomedical Engineering_Shandong Univ. h(-k) 0 h(k) 0

45. 45 10/16/2015 45 Zhongguo Liu_Biomedical Engineering_Shandong Univ. h(-k) 0 h(k) 0

46. 46 10/16/2015 46 Zhongguo Liu_Biomedical Engineering_Shandong Univ.

47. 47 10/16/2015 47 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 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

48. 48 10/16/2015 48 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Cascade connection of systems x [n]x [n] h1[n]h1[n] h2[n]h2[n] y [n]y [n] x [n]x [n] h2[n]h2[n] h1[n]h1[n] y [n]y [n] x [n]x [n] h 1 [n] ]  h 2 [n] y [n]y [n]

49. 49 10/16/2015 49 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Parallel connection of systems

50. 50 10/16/2015 50 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Stability of LTI Systems  LTI system is stable if the impulse response is absolutely summable. Causality of LTI systems HW: proof, Problem 2.62

51. 51 10/16/2015 51 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Impulse response of LTI systems  Impulse response of Ideal Delay systems  Impulse response of Accumulator

52. 52 10/16/2015 52 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Impulse response of Moving Average systems

53. 53 10/16/2015 53 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Impulse response of Moving Average systems

54. 54  Impulse response of Forward Difference  Impulse response of Backward Difference

55. 55 Finite-duration impulse response (FIR) systems  The impulse response of the system has only a finite number of nonzero samples.  The FIR systems always are stable. such as:

56. 56 Infinite-duration impulse response (IIR)  The impulse response of the system is infinite in duration Stable IIR System: Unstable system

57. 57 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. Any noncausal FIR system can be made causal by cascading it with a sufficiently long delay. Useful ideal delay system

58. 58 Equivalent systems

59. 59 Inverse system

60. 60 2.5 Linear Constant-Coefficient Difference Equations  An important subclass of linear time- invariant systems consist of those system for which the input x[n] and output y[n] satisfy an N th-order linear constant-coefficient difference equation.

61. 61 Ex. 2.14 Difference Equation Representation of the Accumulator

62. 62 Block diagram of a recursive ( 递推 ) difference equation representing an accumulator

63. 63 Ex. 2.15 Difference Equation Representation of the Moving- Average System with representation 1 another representation 1

64. 64

65. 65 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 input-output relation.

66. 66 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.

67. 67 Solving the difference equation  Output:  Particular solution: one output sequence for the given input  Homogenous solution: solution for the homogenous equation( ):  where is the roots of

68. 68 Solving the difference equation recursively  If the input and a set of auxiliary value are specified. y(n) can be written in a recurrence formula:

69. 69 Example 2.16 Recursive Computation of Difference Equation

70. 70 Example 2.16 Recursive Computation of Difference Equation

71. 71 10/16/2015 71 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Impulse response of Moving Average systems Review

72. 72 10/16/2015 72 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Impulse response of Moving Average systems Review

73. 73 Example 2.16 Recursive Computation of Difference Equation  The system is noncausal.  The system is not linear.  is not time invariant. When

74. 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. 75 Summary  If the auxiliary conditions are in the form of N sequential values of the output,  later value can be obtained by rearranging the difference equation as a recursive relation running forward in n,

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

77. 77 Difference Equation Representation of the System  If a system is characterized by a linear constant-coefficient difference equation and is further specified to be 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 ).  The auxiliary information is that if the input is zero for,then the output, is constrained to be zero for

78. 78 Summary  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, then the system will be linear, time invariant, and causal.

79. 79 Example 2.16 with initial-rest conditions  If the input is, again with initial- rest conditions, then the recursive solution is carried out using the initial condition

80. 80 Discussion  If the input is, with initial-rest conditions,  Note that for, initial rest implies that  It does mean that if.  Initial rest does not always means

81. 81 2.6 Frequency-Domain Representation of Discrete- Time Signals and systems  2.6.1 Eigenfunction and Eigenvalue for LTI  is called as the eigenfunction of the system, and the associated eigenvalue is  If

82. 82 Eigenfunction and Eigenvalue  Complex exponentials is the eigenfunction for LTI discrete-time systems: frequency responseeigenvalue eigenfunction

83. 83 Frequency response  is called as frequency response of the system.  Magnitude, phase  Real part, imaginary part

84. 84 Example 2.17 Frequency response of the ideal Delay From defination(2.109):

85. 85 Example 2.17 Frequency response of the ideal Delay

86. 86 Linear combination of complex exponential

87. 87 Example 2.18 Sinusoidal response of LTI systems

88. 88 Sinusoidal response of the ideal Delay

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

90. 90 Periodic Frequency Response  The “low frequencies” are frequencies close to zero  The “high frequencies” are frequencies close to  More generally, modify the frequency with, r is integer.  We need only specify over

91. 91 Example 2.19 Ideal Frequency-Selective Filters Frequency Response of Ideal Low-pass Filter

92. 92 Frequency Response of Ideal High-pass Filter

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

94. 94 Frequency Response of Ideal Band-pass Filter

95. 95 Example 2.20 Frequency Response of the Moving-Average System

96. 96

97. 97 Frequency Response of the Moving- Average System M 1 = 0 and M 2 = 4 相位也取决于符号，不仅与指数相关

98. 98 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] = e jwn  u[n]  i.e., x[n] suddenly applied at an arbitrary time, which for convenience we choose n=0.  For causal LTI system:

99. 99 2.6.2 Suddenly applied Complex Exponential Inputs For n≥0 For causal LTI system

100. 100 2.6.2 Suddenly applied Complex Exponential Inputs  Steady-state Response  Transient response

101. 101 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

102. 102  If h[n] = 0 except for 0  n  M (FIR), then the transient response y t [n] = 0 for n+1 > M.  For n  M, only the steady-state response exists 2.6.2 Suddenly Applied Complex Exponential Inputs (continue) Illustration of a real part of suddenly applied complex exponential Input with FIR

103. 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. 104 Fourier Transform rectangular form polar form

105. 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. 106 Impulse response and Frequency response  The frequency response of a LTI system is the Fourier transform of the impulse response.

107. 107 Example 2.21: Absolute Summability  The Fourier transform  Let

108. 108 2.6.2 Suddenly applied Complex Exponential Inputs For n≥0 For causal LTI system Illustration of a real part of suddenly applied complex exponential Input with IIR Review

109. 109 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 Review

110. 110 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.,

111. 111 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.

112. 112 Discussion of convergence  The error may not approach zero at each value of as, but total “energy” in the error does.  Mean-square convergence

113. 113 Example 2.22 : Square-summability for the ideal Lowpass Filter  Since is nonzero for, the ideal lowpass filter is noncausal.

114. 114 Example 2.22 Square-summability for the ideal Lowpass Filter  Define  is not absolutely summable. does not converge uniformly for all w.  approaches zero as, but only as.

115. 115 Example 2.22 Square-summability for the ideal Lowpass Filter  Define

116. 116 Gibbs Phenomenon M=1 M=3 M=7 M=19

117. 117 Example 2.22 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.

118. 118 Example 2.22 continued  However, is square summable, and converges in the mean- square sense to  does not converge uniformly to the discontinuous function.

119. 119 Example 2.23 Fourier Transform of a constant  The sequence is neither absolutely summable nor square summable.  The impulses are functions of a continuous variable and therefore are of “infinite height, zero width, and unit area.”  Define the Fourier transform of :

120. 120 Example 2.23 Fourier Transform of a constant: proof

121. 121 Example 2.24 Fourier Transform of Complex Exponential Sequences Proof

122. 122 Example: Fourier Transform of Complex Exponential Sequences

123. 123 Example: Fourier Transform of unit step sequence

124. 124 2.8 Symmetry Properties of the Fourier Transform  Conjugate-symmetric sequence  Conjugate-antisymmetric sequence

125. 125 Symmetry Properties of real sequence  even sequence: a real sequence that is Conjugate-symmetric  odd sequence: real, Conjugate-antisymmetric real sequence:

126. 126 Decomposition of a Fourier transform Conjugate-antisymmetric Conjugate-symmetric

127. 127 x[n] is complex

128. 128 x[n] is real Conjugate-symmetric

129. 129 Ex. 2.25 illustration of Symmetry Properties x[n], a is real

130. 130 Ex. 2.25 illustration of Symmetry Properties  a=0.75(solid curve) and a=0.5(dashed curve)  Real part  Imaginary part

131. 131  Its magnitude is an even function, and phase is odd. Ex. 2.25 illustration of Symmetry Properties  a=0.75(solid curve) and a=0.5(dashed curve)

132. 132 2.9 Fourier Transform Theorems  2.9.1 Linearity

133. 133 Fourier Transform Theorems  2.9.2 Time shifting and frequency shifting

134. 134 Fourier Transform Theorems  2.9.3 Time reversal  If is real,  If is real, even, is real, even.

135. 135 Fourier Transform Theorems  2.9.4 Differentiation in Frequency

136. 136 Fourier Transform Theorems  is called the energy density spectrum  2.9.5 Parseval’s Theorem

137. 137 Fourier Transform Theorems  2.9.6 Convolution Theorem if HW: proof

138. 138 Fourier Transform Theorems  2.9.7 Modulation or Windowing Theorem HW: proof

139. 139 Fourier transform pairs

140. 140 Fourier transform pairs

141. 141 Fourier transform pairs

142. 142 Ex. 2.26 Determine the Fourier Transform of sequence

143. 143 Ex. 2.27 Determine an inverse Fourier Transform of

144. 144 Ex. 2.28 Determine the impulse response from the frequency respone:

145. 145 Ex. 2.29 Determine the impulse response for a difference equation:  Impulse response

146. 146 Ex. 2.29 Determine the impulse response for a difference equation:

147. 147 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.

148. 148 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

149. 149 2.10 Discrete-Time Random Signals  is an outcome of some random variable ( not distinguished in notation).  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.

150. 150 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.

151. 151 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 ). If the input is stationary, then so is the output Consider a stable LTI system with real h[n].

152. 152 Stochastic signal as input  The mean of output process  m Xn = E{X n }, m Yn = E(Y n }, can be written as m x [n] = E{x[n]}, m y [n] =E(y[n]}.  In our discussion, no necessary to distinguish between the random variables X n andY n and their specific values x[n] and y[n]. wide-sense stationary

153. 153 Stochastic signal as input  The autocorrelation function of output  is called a deterministic autocorrelation sequence or autocorrelation sequence of

154. 154 Stochastic signal as input DTFT of the autocorrelation function of output power (density) spectrum real h[n]

155. 155 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

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

157. 157 For Ideal bandpass system  能量 非负  the power density function of a real signal is real, even, and nonnegative. the area under for can be taken to represent the mean square value of the input in that frequency band.

158. 158 Ex. 2.30 White Noise  The average power of a white noise is  A white-noise signal is a signal for which  Assume the signal has zero mean. The power spectrum of a white noise is

159. 159  A noise signal with power spectrum can be assumed to be the output of a LTI system with white-noise input.  A noise signal whose power spectrum is not constant with frequency. Color Noise

160. 160  Suppose Color Noise

161. 161 Cross-correlation between the input and output

162. 162 Cross-correlation between the input and output  If  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.

163. 163 Cross power spectrum between the input and output  The cross power spectrum is proportional to the frequency response of the system.

164. 164 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.

165. 165 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.

166. 166 2015-10-16 166 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Chapter 2 HW  2.1, 2.2, 2.4, 2.5, 2.7, 2.11, 2.12,2.15, 2.20, 2.62 上一页下一页 返 回