Biomedical Signal processing Chapter 2 Discrete-Time Signals and Systems Zhongguo Liu Biomedical Engineering School of Control Science and Engineering, Shandong University 2017/4/22 1 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 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2 4/22/2017

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 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 3 4/22/2017

Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2.0 Introduction Signal: something conveys information Signals are represented mathematically as functions of one or more independent variables. 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 Discrete-time signal Sampling a continuous-time signal Generated directly by some discrete-time process Zhongguo Liu_Biomedical Engineering_Shandong Univ. 4 4/22/2017

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 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 5 4/22/2017

Figure 2.1 Graphical representation of a discrete-time signal Abscissa: continuous line : is defined only at discrete instants Zhongguo Liu_Biomedical Engineering_Shandong Univ. 6 4/22/2017

Sampling the analog waveform EXAMPLE Figure 2.2

Basic Sequence Operations Sum of two sequences Product of two sequences Multiplication of a sequence by a numberα Delay (shift) of a sequence Zhongguo Liu_Biomedical Engineering_Shandong Univ. 8 4/22/2017

Zhongguo Liu_Biomedical Engineering_Shandong Univ. Basic sequences Unit sample sequence (discrete-time impulse, impulse) Zhongguo Liu_Biomedical Engineering_Shandong Univ. 9 4/22/2017

Zhongguo Liu_Biomedical Engineering_Shandong Univ. Basic sequences A sum of scaled, delayed impulses arbitrary sequence Zhongguo Liu_Biomedical Engineering_Shandong Univ. 10 4/22/2017

Zhongguo Liu_Biomedical Engineering_Shandong Univ. Basic sequences Unit step sequence First backward difference Zhongguo Liu_Biomedical Engineering_Shandong Univ. 11 4/22/2017

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 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 12 4/22/2017

EX. 2.1 Combining Basic sequences If we want an exponential sequences that is zero for n <0, then Cumbersome simpler Zhongguo Liu_Biomedical Engineering_Shandong Univ. 13 4/22/2017

Zhongguo Liu_Biomedical Engineering_Shandong Univ. Basic sequences Sinusoidal sequence Zhongguo Liu_Biomedical Engineering_Shandong Univ. 14 4/22/2017

Exponential Sequences Exponentially weighted sinusoids Exponentially growing envelope Exponentially decreasing envelope is refered to Complex Exponential Sequences Zhongguo Liu_Biomedical Engineering_Shandong Univ. 15 4/22/2017

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 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 16 4/22/2017

Zhongguo Liu_Biomedical Engineering_Shandong Univ. Periodic Sequences A periodic sequence with integer period N Zhongguo Liu_Biomedical Engineering_Shandong Univ. 17 4/22/2017

EX. 2.2 Examples of Periodic Sequences Suppose it is periodic sequence with period N Zhongguo Liu_Biomedical Engineering_Shandong Univ. 18 4/22/2017

EX. 2.2 Examples of Periodic Sequences Suppose it is periodic sequence with period N Zhongguo Liu_Biomedical Engineering_Shandong Univ. 19 4/22/2017

EX. 2.2 Non-Periodic Sequences Suppose it is periodic sequence with period N Zhongguo Liu_Biomedical Engineering_Shandong Univ. 20 4/22/2017

High and Low Frequencies in Discrete-time signal (a) w0 = 0 or 2 (b) w0 = /8 or 15/8 (c) w0 = /4 or 7/4 (d) w0 =  Zhongguo Liu_Biomedical Engineering_Shandong Univ. 21 4/22/2017

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 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 22 4/22/2017

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. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 23 4/22/2017

Zhongguo Liu_Biomedical Engineering_Shandong Univ. EX. 2.4 Moving Average x[m] m n n-5 dummy index m for n=7, M1=0, M2=5 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 24 4/22/2017

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 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 25 4/22/2017

Properties of Discrete-time systems 2.2.2 Linear Systems If T{‧} and only If: additivity property T{‧} homogeneity or scaling 同(齐)次性 property T{‧} principle of superposition T{‧} Zhongguo Liu_Biomedical Engineering_Shandong Univ. 26 4/22/2017

Example of Linear System Ex. 2.6 Accumulator system for arbitrary when Zhongguo Liu_Biomedical Engineering_Shandong Univ. 27 4/22/2017

Example 2.7 Nonlinear Systems Method: find one counterexample For counterexample counterexample For Zhongguo Liu_Biomedical Engineering_Shandong Univ. 28 4/22/2017

Properties of Discrete-time systems 2.2.3 Time-Invariant Systems Shift-Invariant Systems T{‧} T{‧} Zhongguo Liu_Biomedical Engineering_Shandong Univ. 29 4/22/2017

Example of Time-Invariant System Ex. 2.8 Accumulator system Zhongguo Liu_Biomedical Engineering_Shandong Univ. 30 4/22/2017

Example of Time-varying System Ex. 2.9 The compressor system T{‧} T{‧} T{‧} Zhongguo Liu_Biomedical Engineering_Shandong Univ. 31 4/22/2017

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 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 32 4/22/2017

Ex. 2.10 Example for Causal System Forward difference system is not Causal Backward difference system is Causal Zhongguo Liu_Biomedical Engineering_Shandong Univ. 33 4/22/2017

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 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 34 4/22/2017

Ex. 2.11 Test for Stability or Instability is stable if then Zhongguo Liu_Biomedical Engineering_Shandong Univ. 35 4/22/2017

Ex. 2.11 Test for Stability or Instability Accumulator system Accumulator system is not stable Zhongguo Liu_Biomedical Engineering_Shandong Univ. 36 4/22/2017

2.3 Linear Time-Invariant (LTI) Systems Impulse response T{‧} T{‧} Zhongguo Liu_Biomedical Engineering_Shandong Univ. 37 4/22/2017

LTI Systems: Convolution Representation of general sequence as a linear combination of delayed impulse principle of superposition An Illustration Example（interpretation 1） Zhongguo Liu_Biomedical Engineering_Shandong Univ. 38 4/22/2017

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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 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 40 4/22/2017

Zhongguo Liu_Biomedical Engineering_Shandong Univ. Ex. 2.12 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 41 4/22/2017

Reflecting h[k] about the origin to obtain h[-k]. 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].

Ex. 2.13 Analytical Evaluation of the Convolution For system with impulse response h(k) input Find the output at index n Zhongguo Liu_Biomedical Engineering_Shandong Univ. 43 4/22/2017

Zhongguo Liu_Biomedical Engineering_Shandong Univ. h(-k) h(k) h(n-k) x(k) Zhongguo Liu_Biomedical Engineering_Shandong Univ. 44 4/22/2017

Zhongguo Liu_Biomedical Engineering_Shandong Univ. h(-k) h(k) Zhongguo Liu_Biomedical Engineering_Shandong Univ. 45 4/22/2017

Zhongguo Liu_Biomedical Engineering_Shandong Univ. h(-k) h(k) Zhongguo Liu_Biomedical Engineering_Shandong Univ. 46 4/22/2017

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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 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 48 4/22/2017

Cascade connection of systems x [n] h1[n] h2[n] y [n] x [n] h2[n] h1[n] y [n] x [n] h1[n] ]h2[n] y [n] Zhongguo Liu_Biomedical Engineering_Shandong Univ. 49 4/22/2017

Parallel connection of systems Zhongguo Liu_Biomedical Engineering_Shandong Univ. 50 4/22/2017

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 4/22/2017 Zhongguo Liu_Biomedical Engineering_Shandong Univ.

Impulse response of LTI systems Impulse response of Ideal Delay systems Impulse response of Accumulator Zhongguo Liu_Biomedical Engineering_Shandong Univ. 52 4/22/2017

Impulse response of Moving Average systems Zhongguo Liu_Biomedical Engineering_Shandong Univ. 53 4/22/2017

Impulse response of Forward Difference Impulse response of Backward Difference

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.

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

Equivalent systems

Inverse system

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 Nth-order linear constant-coefficient difference equation.

Ex. 2.14 Difference Equation Representation of the Accumulator

Block diagram of a recursive difference equation representing an accumulator

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

Difference Equation Representation of the System An unlimited number of distinct difference equations can be used to represent a given linear time-invariant input-output relation.

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.

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

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:

Example 2.16 Recursive Computation of Difference Equation

Example 2.16 Recursive Computation of Difference Equation

Example for Recursive Computation of Difference Equation The system is noncausal. The system is not linear. The system is not time invariant.

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

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.

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,

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

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.

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

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

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

Eigenfunction and Eigenvalue Complex exponentials is the eigenfunction for discrete-time systems. For LTI systems: eigenfunction frequency response eigenvalue

Frequency response is called as frequency response of the system. Real part, imagine part Magnitude, phase

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

Example 2.17 Frequency response of the ideal Delay

Linear combination of complex exponential

Example 2.18 Sinusoidal response of LTI systems

Sinusoidal response of the ideal Delay

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

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

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

Frequency Response of Ideal High-pass Filter

Frequency Response of Ideal Band-stop Filter

Frequency Response of Ideal Band-pass Filter

Example 2.20 Frequency Response of the Moving-Average System

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

2.6.2 Suddenly applied Complex Exponential Inputs 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] i.e., x[n] suddenly applied at an arbitrary time, which for convenience we choose n=0. For causal LTI system:

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

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

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

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

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

Fourier Transform rectangular form polar form

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

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

Example 2.21: Absolute Summability Let The Fourier transform

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.,

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 . Is called Mean-square Cconvergence

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

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

Example 2.22 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

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

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 .

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

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

Example 2.23 Fourier Transform of a constant: proof

Example 2.24 Fourier Transform of Complex Exponential Sequences

Example: Fourier Transform of Complex Exponential Sequences

Example: Fourier Transform of unit step sequence

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

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

Decomposition of a Fourier transform Conjugate-symmetric Conjugate-antisymmetric

x[n] is complex

x[n] is real

Ex. 2.25 illustration of Symmetry Properties

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

Ex. 2.25 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)

2.9 Fourier Transform Theorems 2.9.1 Linearity

Fourier Transform Theorems 2.9.2 Time shifting and frequency shifting

Fourier Transform Theorems 2.9.3 Time reversal If is real,

Fourier Transform Theorems 2.9.4 Differentiation in Frequency

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

Fourier Transform Theorems 2.9.6 Convolution Theorem if HW: proof

Fourier Transform Theorems 2.9.7 Modulation or Windowing Theorem HW: proof

Fourier transform pairs

Fourier transform pairs

Fourier transform pairs

Ex. 2.26 Determine the Fourier Transform of sequence

Ex. 2.27 Determine an inverse Fourier Transform of

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

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

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

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.

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

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. The stochastic signals do not directly have Fourier transform, but the Fourier transform of the autocorrelation and autocovariance sequece often exist.

Fourier transform in stochastic signals The Fourier transform of autocovariance 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 autocovariance sequence.

Stochastic signal as input Let be a real-valued sequence that is a sample sequence of a wide-sense stationary discrete-time random process.

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]}. The mean of output process

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

Stochastic signal as input the power spectrum DTFT of the autocorrelation function of output

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

For Ideal bandpass system Since is a real, even, its FT is also real and even, i.e., so is 能量非负 the power density function of a real signal is real, even, and nonnegative.

Ex. 2.30 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

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.

Color Noise Suppose ,

Cross-correlation between the input and output

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.

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

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.

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.

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, Zhongguo Liu_Biomedical Engineering_Shandong Univ. 159 2017/4/22 返 回 上一页 下一页