Biomedical Signal processing Chapter 2 DiscreteTime Signals and
Biomedical Signal processing Chapter 2 Discrete-Time Signals and Systems Zhongguo Liu Biomedical Engineering School of Control Science and Engineering, Shandong University 2020/10/28 1 1 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Chapter 2 Discrete-Time Signals and Systems u 2. 0 Introduction u 2. 1 Discrete-Time Signals: Sequences u 2. 2 Discrete-Time Systems u 2. 3 Linear Time-Invariant (LTI) Systems u 2. 4 Properties of LTI Systems u 2. 5 Linear Constant-Coefficient Difference Equations 2 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Chapter 2 Discrete-Time Signals and Systems u 2. 6 Frequency-Domain Representation of Discrete-Time Signals and systems u 2. 7 Representation of Sequences by Fourier Transforms u 2. 8 Symmetry Properties of the Fourier Transform u 2. 9 Fourier Transform Theorems u 2. 10 Discrete-Time Random Signals u 2. 11 Summary 3 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
2. 0 Introduction u. Signal: something conveys information u. Signals are represented mathematically as functions of one or more independent variables. u. Continuous-time (analog) signals, discretetime signals, digital signals u. Signal-processing systems are classified along the same lines as signals: Continuous-time (analog) systems, discrete-time systems, digital systems u. Discrete-time signal u. Sampling a continuous-time signal u. Generated directly by some discrete-time process 4 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
2. 1 Discrete-Time Signals: Sequences u. Discrete-Time signals are represented as Cumbersome, so just use u. In sampling, u 1/T (reciprocal of T) : sampling frequency 5 10/28/2020 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 6 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
EXAMPLE 7 Sampling the analog waveform Figure 2. 2
Basic Sequence Operations u. Sum of two sequences u. Product of two sequences u. Multiplication of a sequence by a numberα u. Delay (shift) of a sequence 8 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Basic sequences u. Unit sample sequence (discrete-time impulse, impulse) 9 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Basic sequences A sum of scaled, delayed impulses uarbitrary sequence 10 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Basic sequences u. Unit step sequence First backward difference 11 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Basic Sequences u. Exponential sequences u. A and α are real: x[n] is real u. A is positive and 0<α<1, x[n] is positive and decrease with increasing n u-1<α<0, x[n] alternate in sign, but decrease in magnitude with increasing n u : x[n] grows in magnitude as n increases 12 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
EX. 2. 1 Combining Basic sequences u. If we want an exponential sequences that is zero for n <0, then Cumbersome simpler 13 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Basic sequences u. Sinusoidal sequence 14 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Exponential Sequences Exponentially weighted sinusoids Exponentially growing envelope Exponentially decreasing envelope is refered to Complex Exponential Sequences 15 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Frequency difference between continuous-time and discrete-time complex exponentials or sinusoids u : frequency of the complex sinusoid or complex exponential u : phase 16 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Periodic Sequences u. A periodic sequence with integer period N 17 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
EX. 2. 2 Examples of Periodic Sequences u. Suppose it is periodic sequence with period N 18 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
EX. 2. 2 Examples of Periodic Sequences u. Suppose it is periodic sequence with period N 19 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
EX. 2. 2 Non-Periodic Sequences u. Suppose it is periodic sequence with period N 20 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
High and Low Frequencies in Discrete-time signal (a) w 0 = 0 or 2 (b) w 0 = /8 or 15 /8 (c) w 0 = /4 or 7 /4 (d) w 0 = 21 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
2. 2 Discrete-Time System u. Discrete-Time System is a trasformation or operator that maps input sequence x[n] into a unique y[n] uy[n]=T{x[n]}, x[n], y[n]: discrete-time signal x[n] T{‧} y[n] Discrete-Time System 22 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
EX. 2. 3 The Ideal Delay System u. If is a positive integer: the delay of the system. Shift the input sequence to the right by samples to form the output. u. If is a negative integer: the system will shift the input to the left by samples, corresponding to a time advance. 23 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
EX. 2. 4 Moving Average for n=7, M 1=0, M 2=5 dummy index m x[m] n-5 n 24 10/28/2020 m Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Properties of Discrete-time systems 2. 2. 1 Memoryless (memory) system u. Memoryless systems: the output y[n] at every value of n depends only on the input x[n] at the same value of n 25 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Properties of Discrete-time systems 2. 2. 2 Linear Systems u. If T{‧} uand only If: additivity property T{‧} homogeneity or scaling 同(齐)次性 property T{‧} uprinciple of superposition T{‧} 26 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Example of Linear System u. Ex. 2. 6 Accumulator system for arbitrary when 27 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Example 2. 7 Nonlinear Systems u. Method: find one counterexample u For u counterexample 28 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Properties of Discrete-time systems 2. 2. 3 Time-Invariant Systems u. Shift-Invariant Systems T{‧} 29 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Example of Time-Invariant System u. Ex. 2. 8 Accumulator system 30 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Example of Time-varying System u. Ex. 2. 9 The compressor system T{‧} 0 0 T{‧} 0 31 10/28/2020 0 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Properties of Discrete-time systems 2. 2. 4 Causality u. A system is causal if, for every choice of , the output sequence value at the index depends only on the input sequence value for 32 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Ex. 2. 10 Example for Causal System u. Forward difference system is not Causal u. Backward difference system is Causal 33 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Properties of Discrete-time systems 2. 2. 5 Stability u. Bounded-Input Bounded-Output (BIBO) Stability: every bounded input sequence produces a bounded output sequence. if then 34 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Ex. 2. 11 Test for Stability or Instability is stable if then 35 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Ex. 2. 11 Test for Stability or Instability u. Accumulator system is not stable 36 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
2. 3 Linear Time-Invariant (LTI) Systems u. Impulse response T{‧} 37 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
LTI Systems: Convolution u. Representation of general sequence as a linear combination of delayed impulse uprinciple of superposition An Illustration Example(interpretation 1) 38 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
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Computation of the Convolution (interpretation 2) ureflecting h[k] about the origion to obtain h[-k] u. Shifting the origin of the reflected sequence to k=n 40 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Ex. 2. 12 41 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Convolution can be realized by –Reflecting 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
Ex. 2. 13 Analytical Evaluation of the Convolution For system with impulse response h(k) input 0 Find the output at index n 43 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
h(k) h(-k) 0 0 h(n-k) x(k) 0 44 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
h(-k) 0 45 10/28/2020 h(k) 0 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
h(-k) 0 46 10/28/2020 h(k) 0 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
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2. 4 Properties of LTI Systems u. Convolution is commutative(可交换的) x[n] h[n] y[n] h[n] x[n] y[n] u. Convolution is distributed over addition 48 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Cascade connection of systems x [ n] h 1[ n] h 2[ n] y [ n] x [ n] h 2[ n] h 1[ n] y [ n] x [ n] 49 10/28/2020 h 1[n] ] h 2[n] y [ n] Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Parallel connection of systems 50 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Stability of LTI Systems u. LTI system is stable if the impulse response is absolutely summable. Causality of LTI systems HW: proof, Problem 2. 62 51 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Impulse response of LTI systems u. Impulse response of Ideal Delay systems u. Impulse response of Accumulator 52 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Impulse response of Moving Average systems 53 10/28/2020 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
u. Impulse response of Forward Difference u. Impulse response of Backward Difference 54
Finite-duration impulse response (FIR) systems u. The impulse response of the system has only a finite number of nonzero samples. such as: u. The FIR systems always are stable. 55
Infinite-duration impulse response (IIR) u. The impulse response of the system is infinite in duration. Stable IIR System: 56
Equivalent systems 57
Inverse system 58
2. 5 Linear Constant-Coefficient Difference Equations u. An important subclass of linear timeinvariant systems consist of those system for which the input x[n] and output y[n] satisfy an Nth-order linear constant-coefficient difference equation. 59
Ex. 2. 14 Difference Equation Representation of the Accumulator 60
Block diagram of a recursive difference equation representing an accumulator 61
Ex. 2. 15 Difference Equation Representation of the Moving. Average System with representation 1 another representation 1 62
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Difference Equation Representation of the System u. An unlimited number of distinct difference equations can be used to represent a given linear time-invariant input-output relation. 64
Solving the difference equation u. Without additional constraints or information, a linear constantcoefficient difference equation for discrete-time systems does not provide a unique specification of the output for a given input. 65
Solving the difference equation u. Output: u Particular solution: one output sequence for the given input u Homogenous solution: solution for the homogenous equation( ): u where 66 is the roots of
Solving the difference equation recursively u. If the input and a set of auxiliary value are specified. y(n) can be written in a recurrence formula: 67
Example 2. 16 Recursive Computation of Difference Equation 68
Example 2. 16 Recursive Computation of Difference Equation 69
Example for Recursive Computation of Difference Equation u. The system is noncausal. u. The system is not linear. u. The system is not time invariant. 70
Difference Equation Representation of the System u. 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. u. In this case, the auxiliary conditions are stated as initial-rest conditions(初始松弛条件). u. The auxiliary information is that if the input is zero for , then the output, is constrained to be zero for 71
Summary u. The system for which the input and output satisfy a linear constantcoefficient difference equation: The output for a given input is not uniquely specified. Auxiliary conditions are required. 72
Summary u. If the auxiliary conditions are in the form of N sequential values of the output, ulater value can be obtained by rearranging the difference equation as a recursive relation running forward in n, 73
Summary uand prior values can be obtained by rearranging the difference equation as a recursive relation running backward in n. 74
Summary u. Linearity, time invariance, and causality of the system will depend on the auxiliary conditions. u. If an additional condition is that the system is initially at rest, then the system will be linear, time invariant, and causal. 75
Example 2. 16 with initial-rest conditions u. If the input is , again with initialrest conditions, then the recursive solution is carried out using the initial condition 76
Discussion u. If the input is conditions, , with initial-rest u. Note that for , initial rest implies that u. Initial rest does not always means u. It does mean that if. 77
2. 6 Frequency-Domain Representation of Discrete. Time Signals and systems u 2. 6. 1 Eigenfunction and Eigenvalue for LTI u. If u is called as the eigenfunction of the system , and the associated eigenvalue is 78
Eigenfunction and Eigenvalue u. Complex exponentials is the eigenfunction for discrete-time systems. For LTI systems: eigenfunction eigenvalue 79 frequency response
Frequency response u is called as frequency response of the system. u. Real part, imagine part u. Magnitude, phase 80
Example 2. 17 Frequency response of the ideal Delay From defination(2. 109): 81
Example 2. 17 Frequency response of the ideal Delay 82
Linear combination of complex exponential 83
Example 2. 18 Sinusoidal response of LTI systems 84
Sinusoidal response of the ideal Delay 85
Periodic Frequency Response u. The frequency response of discrete-time LTI systems is always a periodic function of the frequency variable with period 86
Periodic Frequency Response u. We need only specify over u. The “low frequencies” are frequencies close to zero u. The “high frequencies” are frequencies close to u. More generally, modify the frequency with , r is integer. 87
Example 2. 19 Ideal Frequency-Selective Filters Frequency Response of Ideal Low-pass Filter 88
Frequency Response of Ideal High-pass Filter 89
Frequency Response of Ideal Band-stop Filter 90
Frequency Response of Ideal Band-pass Filter 91
Example 2. 20 Frequency Response of the Moving-Average System 92
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Frequency Response of the Moving. Average System M 1 = 0 and M 2 = 4 94 相位也取决于符号,不仅与指数相关
2. 6. 2 Suddenly applied Complex Exponential Inputs u. In practice, we may not apply the complex exponential inputs ejwn to a system, but the more practical-appearing inputs of the form x[n] = ejwn u[n] ui. e. , x[n] suddenly applied at an arbitrary time, which for convenience we choose n=0. u. For causal LTI system: 95
2. 6. 2 Suddenly applied Complex Exponential Inputs For causal LTI system u. For n≥ 0 96
2. 6. 2 Suddenly applied Complex Exponential Inputs u. Steady-state Response u. Transient response 97
2. 6. 2 Suddenly Applied Complex Exponential Inputs (continue) u. For infinite-duration impulse response (IIR) u. For stable system, transient response must become increasingly smaller as n , Illustration of a real part of suddenly applied complex exponential Input with IIR 98
2. 6. 2 Suddenly Applied Complex Exponential Inputs (continue) u. If h[n] = 0 except for 0 n M (FIR), then the transient response yt[n] = 0 for n+1 > M. Ø For n M, only the steady-state response exists Illustration of a real part of suddenly applied complex exponential Input with FIR 99
2. 7 Representation of Sequences by Fourier Transforms u(Discrete-Time) Fourier Transform, DTFT, analyzing u. If then is absolutely summable, i. e. exists. (Stability) u. Inverse Fourier Transform, synthesis 100
Fourier Transform rectangular form 101 polar form
Principal Value(主值) u is not unique because any may be added to without affecting the result of the complex exponentiation. u. Principle value: is restricted to the range of values between. It is denoted as u : phase function is referred as a continuous function of for 102
Impulse response and Frequency response u. The frequency response of a LTI system is the Fourier transform of the impulse response. 103
Example 2. 21: Absolute Summability u. Let u. The Fourier transform 104
Discussion of convergence u. Absolute summability is a sufficient condition for the existence of a Fourier transform representation, and it also guarantees uniform convergence. u. Some sequences are not absolutely summable, but are square summable, i. e. , 105
Discussion of convergence u. 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. u. Is called Mean-square Cconvergence 106
Discussion of convergence u. Mean-square convergence u. The error may not approach zero at each value of as , but total “energy” in the error does. 107
Example 2. 22 : Square-summability for the ideal Lowpass Filter u. Since is nonzero for lowpass filter is noncausal. 108 , the ideal
Example 2. 22 Square-summability for the ideal Lowpass Filter approaches zero as u but only as u . is not absolutely summable. does not converge uniformly for all w. u. Define 109 ,
Gibbs Phenomenon 110 M=1 M=3 M=7 M=19
Example 2. 22 continued u. As M increases, oscillatory behavior at is more rapid, but the size of the ripple does not decrease. (Gibbs Phenomenon) u. As , the maximum amplitude of the oscillation does not approach zero, but the oscillations converge in location toward the point. 111
Example 2. 22 continued u does not converge uniformly to the discontinuous function . u. However, is square summable, and converges in the meansquare sense to 112
Example 2. 23 Fourier Transform of a constant u. The sequence is neither absolutely summable nor square summable. u. The Fourier transform of is u. The impulses are functions of a continuous variable and therefore are of “infinite height, zero width, and unit area. ” 113
Example 2. 23 Fourier Transform of a constant: proof 114
Example 2. 24 Fourier Transform of Complex Exponential Sequences 115
Example: Fourier Transform of Complex Exponential Sequences 116
Example: Fourier Transform of unit step sequence 117
2. 8 Symmetry Properties of the Fourier Transform u. Conjugate-symmetric sequence u. Conjugate-antisymmetric sequence 118
Symmetry Properties of real sequence ueven sequence: a real sequence that is Conjugate-symmetric uodd sequence: real, Conjugate-antisymmetric real sequence: 119
Decomposition of a Fourier transform Conjugate-symmetric 120 Conjugate-antisymmetric
x[n] is complex 121
x[n] is real 122
Ex. 2. 25 illustration of Symmetry Properties 123
Ex. 2. 25 illustration of Symmetry Properties u. Real part u. Imaginary part ua=0. 75(solid curve) and a=0. 5(dashed curve) 124
Ex. 2. 25 illustration of Symmetry Properties u. Its magnitude is an even function, and phase is odd. 125 ua=0. 75(solid curve) and a=0. 5(dashed curve)
2. 9 Fourier Transform Theorems u 2. 9. 1 Linearity 126
Fourier Transform Theorems u 2. 9. 2 Time shifting and frequency shifting 127
Fourier Transform Theorems u 2. 9. 3 Time reversal u. If 128 is real,
Fourier Transform Theorems u 2. 9. 4 Differentiation in Frequency 129
Fourier Transform Theorems u 2. 9. 5 Parseval’s Theorem u 130 is called the energy density spectrum
Fourier Transform Theorems u 2. 9. 6 Convolution Theorem if 131 HW: proof
Fourier Transform Theorems u 2. 9. 7 Modulation or Windowing Theorem HW: proof 132
Fourier transform pairs 133
Fourier transform pairs 134
Fourier transform pairs 135
Ex. 2. 26 Determine the Fourier Transform of sequence 136
Ex. 2. 27 Determine an inverse Fourier Transform of 137
Ex. 2. 28 Determine the impulse response from the frequency respone: 138
Ex. 2. 29 Determine the impulse response for a difference equation: u. Impulse response 139
Ex. 2. 29 Determine the impulse response for a difference equation: 140
2. 10 Discrete-Time Random Signals u. Deterministic: each value of a sequence is uniquely determined by a mathematically expression, a table of data, or a rule of some type. u. Stochastic signal: a member of an ensemble of discrete-time signals that is characterized by a set of probability density function. 141
2. 10 Discrete-Time Random Signals u. 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. u. That is, is an outcome of some random variable 142
2. 10 Discrete-Time Random Signals u is an outcome of some random variable ( not distinguished in notation). u. The collection of random variables is called a random process. u The stochastic signals do not directly have Fourier transform, but the Fourier transform of the autocorrelation and autocovariance sequece often exist. 143
Fourier transform in stochastic signals u. The Fourier transform of autocovariance sequence has a useful interpretation in terms of the frequency distribution of the power in the signal. u. 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 autocovariance sequence. 144
Stochastic signal as input u. Let be a real-valued sequence that is a sample sequence of a wide-sense stationary discrete-time random process. 145
Stochastic signal as input u. In our discussion, no necessary to distinguish between the random variables Xn and. Yn and their specific values x[n] and y[n]. um. Xn = E{xn }, m. Yn= E(Yn}, can be written as mx[n] = E{x[n]}, my[n] =E(y[n]}. u. The mean of output process 146
Stochastic signal as input u. The autocorrelation function of output u is called a deterministic autocorrelation sequence or autocorrelation sequence of 147
Stochastic signal as input the power spectrum DTFT of the autocorrelation function of output 148
Total average power in output Ø provides the motivation for the term power density spectrum. u能量 无限 u能量有限 Parseval’s Theorem 149
For Ideal bandpass system Since is a real, even, its FT also real and even, i. e. , is so is u能量 非负 u the power density function of a real signal is 150 real, even, and nonnegative.
Ex. 2. 30 White Noise u. A white-noise signal is a signal for which u. Assume the signal has zero mean. The power spectrum of a white noise is u. The average power of a white noise is 151
Color Noise u. A noise signal whose power spectrum is not constant with frequency. u. A noise signal with power spectrum can be assumed to be the output of a LTI system with white-noise input. 152
Color Noise u. Suppose 153 ,
Cross-correlation between the input and output 154
Cross-correlation between the input and output u. If , u. 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. 155
Cross power spectrum between the input and output u. The cross power spectrum is proportional to the frequency response of the system. 156
2. 11 Summary u. Define a set of basic sequence. u. Define and represent the LTI systems in terms of the convolution, stability and causality. u. Introduce the linear constant-coefficient difference equation with initial rest conditions for LTI , causal system. u. Recursive solution of linear constantcoefficient difference equations. 157
2. 11 Summary u. Define FIR and IIR systems u. Define frequency response of the LTI system. u. Define Fourier transform. u. Introduce the properties and theorems of Fourier transform. (Symmetry) u. Introduce the discrete-time random signals. 158
Chapter 2 HW u 2. 1, 2. 2, 2. 4, 2. 5, 2. 7, 2. 11, 2. 12, 2. 15, 2. 20, 2. 62, 159 返 回 2020/10/28 上一页 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 下一页
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