EI331 Signals and Systems Lecture 1 Bo Jiang John Hopcroft Center - PowerPoint PPT Presentation

EI331 Signals and Systems Lecture 1 Bo Jiang John Hopcroft Center for Computer Science Shanghai Jiao Tong University February 26, 2019

Contents 1. Definition of Signals 2. Transformations of Independent Variable 3. Some Properties of Signals 3.1 Energy and power 3.2 Periodicity 3.3 Even/Odd symmetry 1/29

Signals Definition [The American Heritage � Dictionary of the English Language] Examples • voltages or currents in circuits • images, videos Mathematical Representation Function of one or more independent variables x : I → X t �→ x ( t ) 2/29

Examples of Signals Electrical voltage V o : R → R t �→ V o ( t ) Daily temperature Daily high temperatures, Shanghai, January 2019 16 14 T : I → R Temperature (C) 12 10 n �→ T [ n ] 8 6 4 2 0 1 11 21 31 Date 3/29

Examples of Signals Speech signal x : R → R t �→ x ( t ) Color Image P : I × J → R × G × B ( i , j ) �→ ( r [ i , j ] , g [ i , j ] , b [ i , j ]) 4/29

Continuous-time vs. Discrete-time Signals Focus on signals of 1-D independent variable • x : I → X , with I ⊂ R , often X ⊂ R or C • independent variable often referred to as “time” Continuous-time (CT) signal: x ( t ) • defined for interval I ⊂ R , often I = R • called analog signal if X is also continuum • notation: parentheses for continuous time, e.g. ( t ) Discrete-time (DT) signal: x [ n ] • defined for discrete set I , often I = Z • called digital signal if X is also discrete • notation: square brackets for discrete time, e.g. [ n ] 5/29

Continuous-time vs. Discrete-time Signals CT signal DT signal x [ n ] x ( t ) 1 2 3 4 5 6 7 8 9 10 n 1 2 3 4 5 6 7 8 9 10 O O t Signals from physical systems often continuous-time • electrical current, car speed Signals from computation systems often discrete-time • mp3, digital image 6/29

Continuous-time vs. Discrete-time Signals Sampling: converts CT signals to DT signals CT signal DT signal x ( t ) x [ n ] n 0 T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 T 9 T 10 T 0 1 2 3 4 5 6 7 8 9 10 t T = sampling period Important for computer processing of physical signals • sampled data contains no information about T • uniform sampling most common 7/29

Continuous-time vs. Discrete-time Signals Reconstruction: converts DT signals to CT signals x [ n ] x ( t ) Zero-order hold n 0 1 2 3 4 5 6 7 8 9 10 0 T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 T 9 T 10 T t T = sampling period x ( t ) Linear interpolation Different T yields different reconstructed signals 0 T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 T 9 T 10 T t 8/29

Contents 1. Definition of Signals 2. Transformations of Independent Variable 3. Some Properties of Signals 3.1 Energy and power 3.2 Periodicity 3.3 Even/Odd symmetry 9/29

Time Shift Time shift (Translation) operator τ b : x �→ τ b x ( τ b x )( t ) = x ( t − b ) ( τ b x )[ n ] = x [ n − b ] b ∈ R b ∈ Z τ b x x τ b O t 1 t 2 t t 1 + b t 2 + b t O Example: Radar, sonar, radio propagation • b > 0 : delay by b , right shift • b < 0 : advance by | b | , left shift 10/29

Time Reversal Time reversal (Reflection) operator R : x �→ Rx ( Rx )( t ) = x ( − t ) ( Rx )[ n ] = x [ − n ] x Rx R − t 2 − t 1 t 1 t 2 t t Example: Tape recording played backward 11/29

Time Scaling Time scaling operator S a : x �→ S a x need more ( S a x )( t ) = x ( at ) ( S a x )[ n ] = x [ an ] work for a ∈ R + a ∈ Z + a ∈ R + \ Z + x S a x S a t 1 t 2 t t 1 t 2 t O O a a Example: Audio played back at different speed • a > 1 : fast forward, compressed • 0 < a < 1 : slow forward, stretched 12/29

General Affine Transformation of Time Affine transformation A a , b : x �→ A a , b x ( A a , b x )( t ) = x ( at + b ) ( A a , b x )[ n ] = x [ an + b ] a ∈ R \ { 0 } , b ∈ R a ∈ Z \ { 0 } , b ∈ Z A a , b x x A a , b a = 1 2 , b = 1 t 1 t 2 t t 1 − b t 2 − b t a a Can decompose as product of shift, reversal, scaling • a > 0 : A a , b = S a ◦ τ − b not unique • a < 0 : A a , b = S | a | ◦ R ◦ τ − b easier if shift first 13/29

Example of Affine Transformation Affine transformation A 1 2 , 1 = S 1 2 ◦ τ − 1 = τ − 2 ◦ S 1 2 x ( t ) x ( t + 1 ) τ − 1 t t − 2 − 1 1 2 3 4 5 6 7 8 − 2 − 1 1 2 3 4 5 6 7 8 A 1 2 , 1 S 1 S 1 2 2 x ( 1 x ( 1 2 t ) 2 t + 1 ) τ − 2 t t − 2 − 1 − 2 − 1 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 14/29

More Identities S a ◦ τ − b = τ − b a ◦ S a R ◦ τ − b = τ b ◦ R τ − b τ − b x ( t ) x ( t + b ) x ( t ) x ( t + b ) S a S a R R τ − b τ b a x ( at ) x ( at + b ) x ( − t ) x ( − t + b ) S a ◦ R = R ◦ S a R x ( t ) x ( − t ) S a S a R x ( at ) x ( − at ) 15/29

Contents 1. Definition of Signals 2. Transformations of Independent Variable 3. Some Properties of Signals 3.1 Energy and power 3.2 Periodicity 3.3 Even/Odd symmetry 16/29

Signal Energy and Power v ( t ) : voltage across 1 Ω resistor p ( t ) = | v ( t ) | 2 Instantaneous power � t 2 Energy over [ t 1 , t 2 ] E ( t 1 , t 2 ) = t 1 | v ( t ) | 2 dx � t 2 1 t 1 | v ( t ) | 2 dx Average power over [ t 1 , t 2 ] P ( t 1 , t 2 ) = t 2 − t 1 Total energy � ∞ ∞ � | x ( t ) | 2 dt | x [ n ] | 2 E ( x ) = E ( x ) = −∞ n = −∞ Average power � T N 1 1 � | x ( t ) | 2 dt | x [ n ] | 2 P ( x ) = lim P ( x ) = lim 2 N + 1 2 T T →∞ N →∞ − T n = − N 17/29

Finite-energy and Finite-power Signals Finite-energy signal E ( x ) < ∞ e.g. x ( t ) = 1 for t ∈ [ 0 , 1 ] and x ( t ) = 0 elsewhere Finite-power signal P ( x ) < ∞ e.g. x ( t ) = sin t for t ∈ ( −∞ , ∞ ) Some implications • E ( x ) < ∞ = ⇒ P ( x ) = 0 • P ( x ) > 0 = ⇒ E ( x ) = ∞ Caution • P ( x ) = 0 does not imply E ( x ) < ∞ • E ( x ) = ∞ does not imply P ( x ) > 0 x ( t ) = t − 1 / 2 for t ≥ 1 and x ( t ) = 0 elsewhere e.g. 18/29

Periodicity: Continuous-time Signal CT signal is periodic with period T ∈ R iff τ T x = x , i.e. x ( t + T ) = x ( t ) , ∀ t ∈ R Example: x ( t ) = sin t has period T = 2 π x O t − 4 π − 2 π 2 π 4 π Example: sawtooth signal x ( t ) = t − ⌊ t ⌋ has period T = 1 x O t − 4 − 3 − 2 − 1 1 2 3 4 19/29

Periodicity: Continuous-time Signal Fundamental period: smallest positive period (if exists) T 0 = min { T > 0 : x = τ T x } Example: x ( t ) = sin t has fundamental period T 0 = 2 π x O t − 4 π − 2 π 2 π 4 π Example: constant signal x ( t ) = 1 has no well-defined fundamental period, { T > 0 : x = τ T x } = R + x O t 20/29

Periodicity: Continuous-time Signal Question. What’s period of x ( t ) = x 1 ( t ) + x 2 ( t ) if x i has period T i ? Answer. Sufficient condition for x to have period T is T = m 1 T 1 = m 2 T 2 for integers m 1 and m 2 . This requires T 1 / T 2 = m 2 / m 1 ∈ Q Examples • x ( t ) = sin( t ) + sin( 2 t ) has T 0 = 2 π ◮ T 1 = 2 π , T 2 = π , T 1 / T 2 = 2 ; take m 1 = 1 , m 2 = 2 . • x ( t ) = sin( t ) + sin( 3 t / 2 ) has T 0 = 4 π ◮ T 1 = 2 π , T 2 = 4 π/ 3 , T 1 / T 2 = 3 / 2 ; take m 1 = 2 , m 2 = 3 . • x ( t ) = sin( t ) + sin( π t ) is aperiodic! x O t − 4 π − 2 π 2 π 4 π 21/29

Periodicity: Continuous-time Signal How to prove x ( t ) = sin( t ) + sin( π t ) is aperiodic? Proof. By contradiction. Suppose x ( t ) has period T > 0 . 1. sin( t + T ) + sin( π ( t + T )) = sin( t ) + sin( π t ) 2. t = 1 = ⇒ sin( 1 + T ) − sin( π T ) = sin( 1 ) 3. t = − 1 = ⇒ sin( − 1 + T ) − sin( π T ) = − sin( 1 ) 4. subtract 3. from 2. 2 sin( 1 ) cos( T ) = sin( T + 1 ) − sin( T − 1 ) = 2 sin( 1 ) 5. 4. = ⇒ cos( T ) = 1 = ⇒ T = 2 k π for k ∈ Z + 6. substitute T = 2 k π into 2. sin( 1 ) − sin( 2 k π 2 ) = sin( 1 ) = ⇒ sin( 2 k π 2 ) = 0 ⇒ π = m / ( 2 k ) ∈ Q � ⇒ 2 k π 2 = m π for m ∈ Z = 7. 6. = 22/29

Periodicity: Discrete-time Signal DT signal is periodic with period N ∈ Z iff τ N x = x , i.e. x [ n + N ] = x [ n ] , ∀ n ∈ Z Example: x [ n ] = sin( π 5 n ) has period N = 10 x − 4 π 2 π n O − 2 π 4 π Example: x [ n ] = sin n is aperiodic! x n O − 4 π − 2 π 2 π 4 π 23/29

Periodicity: Discrete-time Signal Question. When is x [ n ] = sin( ω n ) periodic? Solution. Suppose x [ n ] has period N > 0 . sin( ω ( n + N )) = sin( ω n ) ⇐ ⇒ ω N = 2 k π for k ∈ Z . Necessary condition for periodicity ω 2 π = k N ∈ Q . Also sufficient (check!) sin( ω n ) periodic ⇐ ⇒ ω is rational multiple of 2 π 24/29

Periodicity: Discrete-time Signal Fundamental period: smallest positive period N 0 = min { N > 0 : x = τ N x } Example: x [ n ] = sin( π 5 n ) has N 0 = 10 x − 4 π 2 π n O − 2 π 4 π Example: constant signal x [ n ] = 1 has N 0 = 1 (cf. x ( t ) = 1 !) x O t 25/29

Periodicity: Discrete-time Signal Question. What’s fundamental period of x [ n ] = sin( ω n ) ? Solution. Periodic iff ω = 2 π k N for k ∈ Z , N ∈ Z + . In this case, N is a period. Fundamental period is N / gcd( N , k ) . Proof. Clearly true for k = 0 . Consider k � = 0 . WLOG, assume gcd( N , k ) = 1 and show N is fundamental period. Proof by contradiction. 1. Suppose 0 < N 1 < N also a period 2. ω N 1 = 2 π m for m ∈ Z \ { 0 } = ⇒ kN 1 = mN 3. 2. = ⇒ N divides kN 1 ⇒ N divides N 1 � 4. gcd( N , k ) = 1 = 26/29