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DSP_Chapter02 CHAPTER 2 Discrete-Time Signals and Systems CHAPTER 2 Discrete-Time Signals and Systems YANG Jian nxryang@126.com School of Information Science and Technology Yunnan University 2008-03 ~ 2008-06 云南大学课程:数字信号处理 Discrete-Time Signals and Systems in ...

DSP_Chapter02
CHAPTER 2 Discrete-Time Signals and Systems CHAPTER 2 Discrete-Time Signals and Systems YANG Jian nxryang@126.com School of Information Science and Technology Yunnan University 2008-03 ~ 2008-06 云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 2 OutlineOutline • Discrete-Time Signals • Typical Sequences and Sequence Representation • Discrete-Time Systems • Time-Domain Characterization of LTI Discrete- Time Systems • Classification of LTI Discrete-Time Systems • Summary • Assignment and Experiment 云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 3 Discrete-Time SignalsDiscrete-Time Signals • Time-Domain Representation – Sequence of numbers: • — sequence • — samples • — sample value or nth samples, a real or complex value – An example of a discrete-time signal with real-valued samples: • is defined only for integer value of { }[ ]x n n [ ]x n { } { }[ ] , 0.3, 0.76, 0, 1, 2, 0.92, x n = − ↑ … … [ ]x n n 云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 4 Discrete-Time SignalsDiscrete-Time Signals • Length of a Discrete-time signal – Finite-length or infinite-length sequence – Length or duration of a finite-length sequence – Appending with zeros or zero-padding – Right-sided sequence, causal sequence – Left-sided sequence, anticasual sequence – Two-sided sequence 云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 5 Discrete-Time SignalsDiscrete-Time Signals • Operation on sequences – Basic operation • Product / modulation: • Addition: • Scalar multiplication ( gain / attenuation ): • Time-shifting, delay / advance: • Time-reversal / folding operation: – Combination of Basic Operations • Linear combination • Example 2.4, see p35 2[ ] [ ] [ ]w n x n y n= + 3[ ] [ ]w n Ax n= 4 5( ) [ 1] ( ) [ 1]w n x n w n x n= − = + 1[ ] [ ] [ ]w n x n y n= ⋅ 6 ( ) [ ]w n x n= − 云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 6 Discrete-Time SignalsDiscrete-Time Signals • Operation on sequences – Sampling Rate Alteration • Up-sampling, interpolation: • Down-sampling, decimation: [ / ], 0, , 2 , , [ ] 0, ,u x n L n L L x n otherwise = ± ±=  … [ ] [ ]y n x nM= 云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 7 Discrete-Time SignalsDiscrete-Time Signals • Classification of Sequences – The number of sequences: finite / infinite • Finite-length sequences: – Symmetry • conjugate-symmetric (even): • conjugate-antisymmetric (odd): • Any complex sequence x[n] can be expressed as a sum 1 2[ ] 0, x n n N and n N= < > [ ] [ ]x n x n∗= − [ ] [ ]x n x n∗= − − * * [ ] [ ] [ ] 1 1where, [ ] ( [ ] [ ]), [ ] ( [ ] [ ]) 2 2 cs ca cs ca x n x n x n x n x n x n x n x n x n = + = + − = − − 云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 8 Discrete-Time SignalsDiscrete-Time Signals • Classification of Sequences – Periodity: periodic / aperiodic • Periodic sequence: – Energy and Power Signals 2 2 : [ ] 1: lim [ ] 2 1 x n K x K n K energy x n power P x n K ε ∞ =−∞ →∞ =−  = = + ∑ ∑ : , 0 : , x x x x energy signals P power signals P ε ε − < ∞ = − = ∞ < ∞ [ ] [ ], for all n, k is any integer.x n x n kN= +� � 云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 9 Discrete-Time SignalsDiscrete-Time Signals • Classification of Sequences – Other types of Classification • Bounded: • Absolutely summable: • Square-summable: [ ] xx n B≤ < ∞ [ ] n x n ∞ =−∞ < ∞∑ 2[ ] n x n ∞ =−∞ < ∞∑ 云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 10 Typical Sequences and Sequence Representation Typical Sequences and Sequence Representation • Some Basic Sequences – Unit sample sequence: • An arbitrary sequence can be represented by unit sample sequence in time-domain. – Unit step sequence: 1, 0 [ ] 0, 0 n n n δ ==  ≠ 0 1, 0 [ ] 0, 0 [ ] [ ] , [ ] [ ] [ 1] n k n n n n k n n n µ µ δ δ µ µ = ≥=  < = = − −∑ 云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 11 Typical Sequences and Sequence Representation Typical Sequences and Sequence Representation • Sinusoidal and Exponential Sequences – The real sinusoidal sequence: – The exponential sequence: • The sinusoidal sequence are periodic of period N as long as is an integer multiple of , i.e., . The smallest possible N is the fundamental period of the sequence. 0[ ] cos( ), x n A n nω φ= + −∞ < < ∞ 0 0 0 0 0 0 ( ) 0 0 [ ] , , , [ ] cos( ) sin( ) jn j n j n n n x n A n e A A e x n A e e A e n j A e n σ ω φ σ ω φ σ σ α α ω φ ω φ + + = − ∞ < < +∞ = = = = + + + 0 Nω 2π 0 2N rω π= 云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 12 Typical Sequences and Sequence Representation Typical Sequences and Sequence Representation • Some Typical Sequences – Rectangular window sequence: – Real exponential sequence: • Representation of an Arbitrary Sequence – An arbitrary sequence can be represented as a weight sum of basic sequence and its delayed version. 1, 0 1 [ ] 0, R n N w n otherwise ≤ ≤ −=  [ ] [ ]nx n a nµ= [ ] [ ] [ ] k x n x k n kδ∞ =−∞ = −∑ 云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 13 Discrete-Time SystemDiscrete-Time System • Discrete-time system • Simple Discrete-Time Systems – The accumulator – The M-point moving-average filter • p52: Example 2.11, Program 2_4 – The factor-of-L interpolator – Median filter [ ] { [ ]} -y n T x n n= ∞ < < ∞ Discrete-time System Output y[n]Input x[n] 云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 14 Discrete-Time SystemDiscrete-Time System • Classification of Discrete-Time System – Linear system – Shift-Invariant System – LTI System The linear time-invariant discrete-time system satisfies both the linearity and the time-invariance properties. 1 1 2 2 1 2 1 2 [ ] [ ], [ ] [ ], [ ] [ ] [ ] [ ] if x n y n x n y n then x n x n y n y nα β α β → → + → + 0 0 [ ] [ ], [ ] [ ]if x n y n then x n n y n n→ − → − 云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 15 Discrete-Time SystemDiscrete-Time System • Classification of Discrete-Time System – Causal System In a causal discrete-time system, the th output sample depends only on input samples for and does not depend on input samples for . 1 1 2 2 1 2 1 2 [ ] [ ] [ ] [ ] { [ ] [ ], } { [ ] [ ], } if u n y n and u n y n then u n u n for n N implies also that y n y n for n N → → = < = < 0n 0[ ]y n [ ]x n 0n n≤ 0n n> 云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 16 Discrete-Time SystemDiscrete-Time System • Classification of Discrete-Time System – Stable System Definition of bounded-input, bounded-output (BIBO) stable. • Passive and Lossless Systems – The passive (无源的) system – The lossless system: the same energy x[ ] , [ ] , x y if n B n then y n B n < ∀ < ∀ 2 2[ ] [ ] n n y n x n ∞ ∞ =−∞ =−∞ ≤ < ∞∑ ∑ 云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 17 Discrete-Time SystemDiscrete-Time System • Impulse and Step Responses – Input sequence x[n] output sequence y[n] – Unit sample response or impulse response : – Unit step response or step response : [ ]h n [ ] [ ]n h nδ → [ ]s n [ ] [ ]n s nµ → → 云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 18 Time-Domain Characterization of LTI Discrete-Time Systems Time-Domain Characterization of LTI Discrete-Time Systems • Input-Output Relationship – The response y(n) of the LTI discrete-time system to x(n) will be given by the convolution sum: – The operation • Step 1, time-reverse: • Step 2, shift n sampling period: • Step 3, product: • Step 4, summing all samples: [ ] [ ] [ ] [ ] [ ] [ ] [ ] k k y n x k h n k x n k h k x n h n ∞ ∞ =−∞ =−∞ = − = − ∗∑ ∑ � [ ] [ ]h k h k→ − [ ] [ ]h k h n k→ − [ ] [ ] [ ]x k h n k v k− → [ ] [ ] [ ] k k v k x k h n k ∞ ∞ =−∞ =−∞ = −∑ ∑ 云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 19 Time-Domain Characterization of LTI Discrete-Time Systems Time-Domain Characterization of LTI Discrete-Time Systems • Some useful properties of the convolution operation – Commutative (交换率): – Associative (结合率) for stable and single-sided sequences: – Distributive (分配率): 1 2 2 1[ ] [ ] [ ] [ ]x n x n x n x n∗ = ∗ 1 2 3 1 2 3[ ] ( [ ] [ ]) ( [ ] [ ]) [ ]x n x n x n x n x n x n∗ ∗ = ∗ ∗ 1 2 3 1 2 1 3[ ] ( [ ] [ ]) [ ] [ ] [ ] [ ]x n x n x n x n x n x n x n∗ + = ∗ + ∗ 云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 20 • Simple Interconnection Schemes – Cascade Connection: – Parallel Connection: – Inverse System: Time-Domain Characterization of LTI Discrete-Time Systems Time-Domain Characterization of LTI Discrete-Time Systems 1 2( ) ( ) ( )h n h n h n= ∗ 1 2[ ] [ ] [ ]h n h n h n= + 1 2[ ] [ ] [ ]h n h n nδ∗ = 1 2 2 1 1 2[ ] [ ] [ ] [ ] [ ] [ ]h n h n h n h n h n h n→ → → ≡ → → → ≡ → ∗ → 1[ ]h n 2[ ]h n ⊕ ≡ 1 2[ ] [ ]h n h n+ → → → → →↓ ↑ 云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 21 Time-Domain Characterization of LTI Discrete-Time Systems Time-Domain Characterization of LTI Discrete-Time Systems • Stability Condition in Terms of the Impulse Response – An LTI digital filter is BIBO stable if only if its impulse response sequence is absolutely summable, i.e.: • Causality Condition in Terms of the Impulse Response – An LTI discrete-time system is causal if and only if its impulse response is a causal sequence satisfying the condition: [ ]h n [ ] n S h n ∞ =−∞ = < ∞∑ [ ] 0, 0h k for k= < 云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 22 Time-Domain Characterization of LTI Discrete-Time Systems Time-Domain Characterization of LTI Discrete-Time Systems • Linear Constant Coefficient Difference Equation – An important subclass of LTI discrete-time systems is characterized by a linear constant coefficient difference equation of the form: – The order of the system is given by max(N, M) 0 0 0 0 0 [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ], 1 N M k k k k N M k k k 1 k0 0 N M k k k 1 k0 0 d y n k p x n k d pthen y n y n k x n k d d d por y n y n k x n k if d d d = = = = = = − = − = − − + − = − − + − = ∑ ∑ ∑ ∑ ∑ ∑ 云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 23 Classification of LTI Discrete-Time Systems Classification of LTI Discrete-Time Systems • Classification based on impulse response length – Finite impulse response ( FIR ): – Infinite impulse response ( IIR ): 1 2 1 2[ ] 0,h n for n N and n N , with N N= < > < 2 1 [ ] [ ] [ ] N k N y n h k x n k = = −∑ 0 For a causal IIR system [ ] [ ] [ ] n k y n x k h n k = = −∑ 云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 24 Classification of LTI Discrete-Time Systems Classification of LTI Discrete-Time Systems • Classification Based on the output calculation process – Non-recursive system: If the output sample can be calculated sequentially (顺序地), knowing only the present and pass input samples. – Recursive system: If the computation of the output involves past output samples. – Remarks: • FIR — Non-recursive • IIR — Recursive 云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 25 SummarySummary • The LTI system has numerous applications in practice. • The LTI system can be described by an input- output relation composed of a linear constant coefficient difference equation. • The LTI discrete-time system is usually classified in terms of the length of its impulse response. 云南大学课程:数字信号处理 Discrete-Time Signals and Systems in the Time-Domain 26 Assignment and ExperimentAssignment and Experiment • Assignment(作业) – 第二章:2.6, 2.19, 2.21, 2.33。 – 3月19日上午10:40以前交。 • Experiment(实验) – 实验一:离散时间信号的时域分析 – 必做内容:Q1.29, Q1.30, Q1.31, Q1.32, Q1.33。 – 选做内容: Q1.34, Q1.35。 – 3月12日完成实验,3月19日下午2:30前交实验报告。
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