Discrete-Time Signals and Systems 主講人：虞台文

Content Introduction Discrete-Time Signals---Sequences Linear Shift-Invariant Systems Stability and Causality Linear Constant-Coefficient Difference Equations Frequency-Domain Representation of Discrete-Time Signals and Systems Representation of Sequences by Fourier Transform Symmetry Properties of Fourier Transform Fourier Transform Theorems The Existence of Fourier Transform Important Transform Pairs

Discrete-Time Signals and Systems Introduction

The Taxonomy of Signals Signal: A function that conveys information Time Amplitude analog signals continuous-time signals discrete-time digital signals Continuous Discrete

Signal Process Systems Facilitate the extraction of desired information e.g., Filters Parameter estimation Signal Processing System signal output

Signal Process Systems analog system signal output continuous-time signal discrete- time system signal output discrete-time signal digital system signal output digital signal

Signal Process Systems A important class of systems Linear Shift-Invariant Systems. In particular, we’ll discuss Linear Shift-Invariant Discrete-Time Systems.

Discrete-Time Signals and Systems Discrete-Time Signals---Sequences

Representation by a Sequence Discrete-time system theory Concerned with processing signals that are represented by sequences. 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 n x(n)

Important Sequences Unit-sample sequence (n) Sometime call (n) a discrete-time impulse; or an impulse 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 n (n)

Important Sequences Unit-step sequence u(n) Fact: u(n) n 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 n u(n)

Important Sequences Real exponential sequence . . . x(n) n 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 n x(n) . . .

Important Sequences Sinusoidal sequence n x(n)

Important Sequences Complex exponential sequence

Important Sequences A sequence x(n) is defined to be periodic with period N if Example: consider must be a rational number

Energy of a Sequence Energy of a sequence is defined by

Operations on Sequences Sum Product Multiplication Shift

Sequence Representation Using delay unit 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 n x(n) a1 a2 a7 a-3

Discrete-Time Signals and Systems Linear Shift-Invariant Systems

Mathematically modeled as a unique transformation or operator. Systems T [ ] y(n)=T[x(n)] x(n) Mathematically modeled as a unique transformation or operator.

Linear Systems T [ ] x(n) y(n)=T[x(n)]

Examples: y(n)=T[x(n)] x(n) T [ ] Ideal Delay System Moving Average Accumulator

Examples: Are these system linear? y(n)=T[x(n)] x(n) T [ ] Ideal Delay System Accumulator Moving Average T [ ] x(n) y(n)=T[x(n)] Are these system linear?

Examples: y(n)=T[x(n)] x(n) Is this system linear? T [ ] A Memoryless System Is this system linear?

Linear Systems T [ ] x(n) y(n)=T[x(n)] 時間k之impulse 於時間n時之輸出值

Shift-Invariant Systems x(n) y(n)=T[x(n)] T [ ] x(nk) y(nk) y(n) x(n) y(n-1) x(n-1) x(n-2) y(n-2)

Shift-Invariant Systems x(n) y(n)=T[x(n)] T [ ] x(n-k) y(n-k) y(n) x(n-1) y(n-1) x(n-2) y(n-2) 輸入/輸出關係僅與時間差有關

Linear Shift-Invariant Systems x(n) y(n)=T[x(n)] 時間k之impulse 於時間n時之輸出值 僅與時間差有關

Impulse Response h(n)=T[(n)] x(n)=(n) T [ ]

Convolution Sum h(n) (n) x(n) y(n) T [ ] convolution A linear shift-invariant system is completely characterized by its impulse response.

Characterize a System h(n) x(n) x(n)*h(n)

Properties of Convolution Math

Properties of Convolution Math h1(n) x(n) h2(n) y(n) h2(n) x(n) h1(n) y(n) h1(n)*h2(n) x(n) y(n) These systems are identical.

Properties of Convolution Math h1(n) x(n) h2(n) y(n) + h1(n)+h2(n) x(n) y(n) These two systems are identical.

Example y(n)=? 1 2 3 4 5 6 1 2 3 4 5 6

Example 1 2 3 4 5 6 k x(k) 1 2 3 4 5 6 k h(k) 1 2 3 4 5 6 k h(0k)

Example compute y(0) compute y(1) How to computer y(n)? x(k) k h(0k) 1 2 3 4 5 6 k x(k) compute y(0) 1 2 3 4 5 6 k h(0k) compute y(1) 1 2 3 4 5 6 k h(1k) How to computer y(n)?

Example Two conditions have to be considered. n<N and nN. 1 2 3 4 5 6 k x(k) h(0k) h(1k) compute y(0) compute y(1) How to computer y(n)? n<N and nN.

Example n < N n  N

Example n < N n  N

Impulse Response of the Ideal Delay System By letting x(n)=(n) and y(n)=h(n), (n nd) 1 2 3 4 5 6 nd

Impulse Response of the Ideal Delay System 你必須知道 (n nd)扮演如下功能： Shift; or Copy (n nd) 1 2 3 4 5 6 nd

Impulse Response of the Moving Average M1  0  M2 . . . 你能以(n k)解釋嗎?

Impulse Response of the Accumulator . . . 你能解釋嗎?

Discrete-Time Signals and Systems Stability and Causality

Stability Stable systems --- every bounded input produce a bounded output (BIBO) Necessary and sufficient condition for a BIBO

Prove Necessary Condition for Stability Show that if x is bounded and S < , then y is bounded. where M = max x(n)

Prove Sufficient Condition for Stablility Show that if S = , then one can find a bounded sequence x such that y is unbounded. Define

Example: Show that the linear shift-invariant system with impulse response h(n)=anu(n) where |a|<1 is stable.

Causality Causal systems --- output for y(n0) depends only on x(n) with n n0. A causal system whose impulse response h(n) satisfies

Discrete-Time Signals and Systems Linear Constant-Coefficient Difference Equations

N-th Order Difference Equations Examples: Ideal Delay System Moving Average Accumulator

Compute y(n)

The Ideal Delay System x(n) y(n) y(n) x(n) . . . Delay nd sample delays x(n) y(n)

The Moving Average

The Moving Average Attenuator + M+1 sample delay Accumulator system _

Discrete-Time Signals and Systems Frequency-Domain Representation of Discrete-Time Signals and Systems

Sinusoidal and Complex Exponential Sequences Play an important role in DSP LTI h(n)

Frequency Response eigenvalue eigenfunction

Frequency Response phase magnitude

Example: The Ideal Delay System magnitude phase

Example: The Ideal Delay System

Periodic Nature of Frequency Response

Periodic Nature of Frequency Response   2 3 4 2 3 4

Periodic Nature of Frequency Response Generally, we choose  To represent one period in frequency domain.   2 3 4 2 3 4

Periodic Nature of Frequency Response   High Frequency Low Frequency

Ideal Frequency-Selective Filters   c c 1 a a b b Lowpass Filter Bandstop Filter Highpass Filter

Moving Average h(n) M

Moving Average

M=4 Lowpass Try larger M Moving Average

Discrete-Time Signals and Systems Representation of Sequences by Fourier Transform

Fourier Transform Pair Synthesis Inverse Fourier Transform (IFT) Analysis Fourier Transform (FT)

Prove n = m

Prove n  m

Prove = x(n)

Inverse Fourier Transform Notations Synthesis Inverse Fourier Transform (IFT) Analysis Fourier Transform (FT)

Real and Imaginary Parts Fourier Transform (FT) is a complex-valued function

Magnitude and Phase magnitude phase

Discrete-Time Signals and Systems Symmetry Properties of Fourier Transform

Conjugate-Symmetric and Conjugate-Antisymmetric Sequences Conjugate-Symmetric Sequence Conjugate-Antisymmetric Sequence an even sequence if it is real. an odd sequence if it is real.

Sequence Decomposition Any sequence can be expressed as the sum of a conjugate-symmetric one and a conjugate-antisymmetric one, i.e., Conjugate Symmetric Conjugate Antisymmetric

Function Decomposition Any function can be expressed as the sum of a conjugate-symmetric one and a conjugate-antisymmetric one, i.e., Conjugate Symmetric Conjugate Antiymmetric

Conjugate-Symmetric and Conjugate-Antiymmetric Functions Conjugate-Symmetric Function Conjugate-Antisymmetric Function an even function if it is real. an odd function if it is real.

Symmetric Properties   magnitude phase   magnitude phase

Symmetric Properties   magnitude phase   magnitude phase

Symmetric Properties   magnitude phase   magnitude phase

Symmetric Properties

Symmetric Properties

Symmetric Properties for Real Sequence x(n)  Facts: 1. real part is even 2. Img. part is odd 3. Magnitude is even 4. Phase is odd   magnitude phase

Discrete-Time Signals and Systems Fourier Transform Theorems

Linearity

Time Shifting  Phase Change

Frequency Shifting Signal Modulation

Time Reversal

Differentiation in Frequency

The Convolution Theorem

The Modulation or Window Theorem

Parseval’s Theorem Facts: Letting =0, then proven.

Parseval’s Theorem Energy Preserving

Example: Ideal Lowpass Filter

Example: Ideal Lowpass Filter The ideal lowpass fileter Is noncausal.

Example: Ideal Lowpass Filter The ideal lowpass fileter Is noncausal. To approximate the ideal lowpass filter using a window.

Example: Ideal Lowpass Filter -4 -3 -2 -1 1 2 3 4 M =3 =5 =19

Discrete-Time Signals and Systems The Existence of Fourier Transform

Key Issue Synthesis Analysis Does X(ej) exist for all ? We need that |X(ej)| <  for all  Analysis

Sufficient Condition for Convergence

More On Convergence Define Uniform Convergence Mean-Square Convergence

Discrete-Time Signals and Systems Important Transform Pairs

Fourier Transform Pairs Sequence Fourier Transform

Fourier Transform Pairs Sequence Fourier Transform

Fourier Transform Pairs Sequence Fourier Transform