EI331 Signals and Systems Lecture 10 Bo Jiang John Hopcroft Center - PowerPoint PPT Presentation

EI331 Signals and Systems Lecture 10 Bo Jiang John Hopcroft Center for Computer Science Shanghai Jiao Tong University March 28, 2019

Contents 1. Properties of CT Fourier Series 1.1 Linearity 1.2 Time and Frequency Shifting 1.3 Time Reversal and Scaling 1.4 Differentiation and Integration 1.5 Multiplication and Periodic Convolution 2. Convergence of Fourier Series 2.1 Mean-square Convergence 1/31

Fourier Series Fourier series for x with period T and ω 0 = 2 π T , � ∞ x [ k ] e jk ω 0 t x ( t ) = ˆ k = −∞ Correspondence between periodic functions and doubly infinite sequences; time domain vs. frequency domain FS FS ← − − → ˆ or x ( t ) ← − − → ˆ x [ k ] x x • ˆ x consists of expansion coefficients of x in “basis” { e jk ω 0 t } • ˆ x alone does not uniquely determine x , • also need to know basis functions, or, equivalently, period T or fundamental frequency ω 0 • same coefficients with different bases (periods) give different functions (more later) 2/31

Linearity If x , y have same period T , so does their linear combination ax + by , and � ax + by = a ˆ x + b ˆ y Proof. ( � ax + by )[ k ] = � ax + by , e jk ω 0 t � = a � x , e jk ω 0 t � + b � y , e jk ω 0 t � = a ˆ x [ k ] + b ˆ y [ k ] Alternative proof. �  x [ k ] e jk ω 0 t  x ( t ) = ˆ   � k y [ k ]) e jk ω 0 t � = ⇒ ax ( t ) + by ( t ) = ( a ˆ x [ k ] + b ˆ  y [ k ] e jk ω 0 t y ( t ) = ˆ   k k 3/31

Time Shifting If x has period T and ω 0 = 2 π T , FS → e − jk ω 0 t 0 ˆ τ t 0 x = E − ω 0 t 0 ˆ � or x ( t − t 0 ) ← − − x [ k ] x where ( E a ˆ x )[ k ] = e jak ˆ x [ k ] Time shift ⇐ ⇒ linear phase change in frequency Proof. τ t 0 x )[ k ] = � τ t 0 x , e jk ω 0 t � = � x , τ − t 0 e jk ω 0 t � ( � = � x , e jk ω 0 ( t + t 0 ) � = � x , e jk ω 0 t 0 e jk ω 0 t � = e − jk ω 0 t 0 � x , e jk ω 0 t � = e − jk ω 0 t 0 ˆ x [ k ] = ( E − ω 0 t 0 ˆ x )[ k ] 2 e jt + 1 2 e jt + j 2 e − jt , sin( t ) = cos( t − π 2 ) = − j Example. cos( t ) = 1 2 e − jt 4/31

Frequency Shifting If x has period T and ω 0 = 2 π T , � FS e jm ω 0 t x ( t ) E m ω 0 x = τ m ˆ or ← − − → ˆ x [ k − m ] x where ( E a x )( t ) = e jat x ( t ) Modulation by harmonic exp. ⇐ ⇒ frequency shift Proof. ( � E m ω 0 x )[ k ] = � E m ω 0 x , e jk ω 0 t � = � x , E − m ω 0 e jk ω 0 t � = � x , e j ( k − m ) ω 0 t � = ˆ x [ k − m ] NB. Modulation by inharmonic exponential may change fundamental frequency or even result in aperiodic signal 5/31

Frequency Shifting 2 e j ω 0 t + 1 Example. x ( t ) = 1 + cos( ω 0 t ) = 1 + 1 2 e − j ω 0 t y ( t ) = e j 3 ω 0 t x ( t ) = e j 3 ω 0 t + 1 2 e j 4 ω 0 t + 1 2 e j 2 ω 0 t x ( t ) Re y ( t ) t t − T T 2 2 1 1 1 1 1 1 2 2 2 2 ω ω ω 0 − ω 0 2 ω 0 3 ω 4 ω 0 0 0 0 6/31

Frequency Shifting 2 e j ω 0 t + 1 Example. x ( t ) = 1 + cos( ω 0 t ) = 1 + 1 2 e − j ω 0 t z ( t ) = cos( 3 ω 0 t ) x ( t ) = Re y ( t ) = 1 2 e j 3 ω 0 t + 1 4 e j 4 ω 0 t + 1 4 e j 2 ω 0 t + 1 2 e − j 3 ω 0 t + 1 4 e − j 4 ω 0 t + 1 4 e − j 2 ω 0 t x ( t ) z ( t ) t t − T T 2 2 1 1 1 1 1 2 2 2 2 1 1 1 1 4 4 4 4 ω ω ω 0 − ω 0 0 0 0 − 4 ω 0 − 3 ω 0 − 2 ω 0 2 ω 0 3 ω 0 4 ω 0 7/31

Time Reversal If x has period T , � FS Rx = R ˆ or x ( − t ) ← − − → ˆ x [ − k ] x Time reversal commutes with Fourier series FS x ( t ) x [ k ] ˆ R R FS x ( − t ) ˆ x [ − k ] Proof. ( � Rx )[ k ] = � Rx , e jk ω 0 t � = � x , Re jk ω 0 t � = � x , e − jk ω 0 t � = ˆ x [ − k ] Corollary. Fourier series preserves even/odd symmetry, i.e. x even ⇐ ⇒ ˆ x even, x odd ⇐ ⇒ ˆ x odd 8/31

Time Scaling If x has period T , then S a x has period T / a � FS S a x = ˆ x ( at ) ← − − → ˆ x [ k ] x or Time scaling preserves Fourier coefficients but changes fundamental frequency � ∞ x [ k ] e jk ω 0 t x ( t ) = ˆ k = −∞ different Fourier series � ∞ x [ k ] e jk ( a ω 0 ) t x ( at ) = ˆ k = −∞ 9/31

Time Scaling � ∞ � ∞ x [ k ] e jk ω 0 t x [ k ] e jk ( a ω 0 ) t x ( t ) = ˆ vs . x ( at ) = ˆ k = −∞ k = −∞ x ( t ) x ( 3 2 t ) t − T T t − T T 2 2 3 3 1 1 2 2 ω ω − 6 π − 2 π 0 2 π 6 π − 9 π − 3 π 0 3 π 9 π T T T T T T T T compression in time ⇐ ⇒ expansion in frequency expansion in time ⇐ ⇒ compression in frequency 10/31

Differentiation If x has period T , so does its derivative x ′ , and x ′ = M ω 0 ˆ � FS x ′ ( t ) or ← − − → jk ω 0 ˆ x [ k ] x where ( M a ˆ x )[ k ] = jak ˆ x [ k ] differentiation in time ⇐ ⇒ multiplication by jk ω 0 in frequency Proof. � x ′ [ k ] = 1 � x ′ ( t ) e − jk ω 0 t dt T T � 1 � � � = 1 � T T [ x ( t ) e − jk ω 0 t ] x ( t ) e − jk ω 0 t dt � 0 + jk ω 0 = jk ω 0 ˆ x [ k ] T T Alternatively, differentiate term by term � � x [ k ] e jk ω 0 t = ⇒ x ′ ( t ) = x [ k ] e − jk ω 0 t x ( t ) = ˆ jk ω 0 ˆ k k 11/31

Integration If x has period T and Fourier series � ∞ x [ k ] e jk ω 0 t x ( t ) = ˆ k = −∞ Integrating term by term � t x [ k ] e jk ω 0 t − 1 � y ( t ) � x ( s ) ds = ˆ x [ 0 ] t + ˆ jk ω 0 0 k � = 0 � � ˆ x [ k ] 1 x [ k ] e jk ω 0 t = ˆ ˆ x [ 0 ] t − + jk ω 0 jk ω 0 k � = 0 k � = 0 • y periodic iff ˆ x [ 0 ] = 0 , i.e. x has no DC component • if ˆ x [ 0 ] = 0 , y has period T , 1 ˆ y [ k ] = ˆ x [ k ] for k � = 0 jk ω 0 12/31

Example: Triangle Wave x ( t ) x ′ = x 1 1 4 sin( k π/ 2 ) 1 e jk π/ 2 , k � = 0 t ˆ x [ k ] = − T T jk ω 0 k π T 2 2 x 1 ( t ) = x ′ ( t ) x 1 = τ − T / 4 x 2 2 T x 1 [ k ] = 4 sin( k π/ 2 ) − 2 e jk π/ 2 − 2 t ˆ T δ [ k ] T k π T x 2 = 4 T x 3 − 2 x 2 ( t ) = x ′ ( t − T 4 ) 2 / T T x 2 [ k ] = 4 sin( k π/ 2 ) − 2 t ˆ T δ [ k ] k π T x 3 ( t ) 1 x 3 [ k ] = sin( k π/ 2 ) t ˆ k π 13/31

Multiplication If x and y have same period T , so does their product xy , and � ∞ FS xy = ˆ � x ∗ ˆ or x ( t ) y ( t ) ← − − → ˆ x [ m ]ˆ y [ k − m ] y m = −∞ multiplication in time ⇐ ⇒ convolution in frequency Proof. � � � � � ∞ � ∞ x [ m ] e jm ω 0 t y [ ℓ ] e j ℓω 0 t x ( t ) y ( t ) = ˆ ˆ m = −∞ ℓ = −∞ � ∞ � ∞ y [ ℓ ] e j ( m + ℓ ) ω 0 t = ˆ x [ m ]ˆ m = −∞ ℓ = −∞ � � � ∞ � ∞ e jk ω 0 t = ˆ x [ m ]ˆ y [ k − m ] ( k = m + ℓ ) k = −∞ m = −∞ 14/31

Periodic Convolution Periodic convolution x ∗ y of x and y with same period T � ( x ∗ y )( t ) = x ( τ ) y ( t − τ ) d τ T Properties • Commutativity x ∗ y = y ∗ x • Associativity ( x ∗ y ) ∗ z = x ∗ ( y ∗ z ) • Bilinearity �� � �� � � ∗ = a i b j ( x i ∗ y j ) a i x i b j y j i , j i j 15/31

Periodic Convolution Fourier coefficients satisfy FS � x ∗ y = T ˆ x ˆ or ( x ∗ y )( t ) ← − − → T ˆ x [ k ]ˆ y [ k ] y convolution in time ⇐ ⇒ multiplication in frequency Proof. � x ∗ y )[ k ] = 1 ( x ∗ y )( t ) e − jk ω 0 t dt ( � T T � �� � = 1 e − jk ω 0 t dt x ( τ ) y ( t − τ ) d τ T T T � 1 � � � y ( t − τ ) e − jk ω 0 t dt = x ( τ ) d τ T T T � � � e − jk ω 0 τ ˆ = x ( τ ) y [ k ] d τ = T ˆ x [ k ]ˆ y [ k ] T 16/31

Conjugation and Symmetry x = x ∗ , and If x has period T , so does its complex conjugate ¯ x ∗ = R ˆ FS � x ∗ x ∗ ( t ) x [ − k ]) ∗ or ← − − → (ˆ Proof. � x ∗ [ k ] = � x ∗ , e jk ω 0 t � = � x , e − jk ω 0 t � = ˆ x [ − k ] Corollary. If x is real, ˆ x is conjugate symmetric i.e. ˆ x [ − k ] = ˆ x [ k ] Corollary. If x is real and even, ˆ x is also real and even ˆ x [ k ] = ˆ x [ − k ] = ˆ x [ k ] Corollary. If x is real and odd, ˆ x is purely imaginary and odd − ˆ x [ k ] = ˆ x [ − k ] = ˆ x [ k ] 17/31

Contents 1. Properties of CT Fourier Series 1.1 Linearity 1.2 Time and Frequency Shifting 1.3 Time Reversal and Scaling 1.4 Differentiation and Integration 1.5 Multiplication and Periodic Convolution 2. Convergence of Fourier Series 2.1 Mean-square Convergence 18/31

Vector Space Vector space V over field F ( R or C ) has two operations • addition: x , y ∈ V = ⇒ x + y ∈ V • scalar multiplication: λ ∈ F , x ∈ V = ⇒ λ x ∈ V satisfying following axioms 1. commutativity of addition: x + y = y + x , ∀ x , y ∈ V 2. associativity of addition: ( x + y ) + z = x + ( y + z ) , ∀ x , y , z ∈ V 3. identity element of addition: ∃ 0 ∈ V s.t. x + 0 = x , ∀ x ∈ V 4. inverse elements of addition: ∀ x ∈ V , ∃ y ∈ V s.t. x + y = 0 5. identity element of scalar multiplication: 1 x = x , ∀ x ∈ V 6. compatibility of scalar and field multiplications λ 1 ( λ 2 x ) = ( λ 1 λ 2 ) x , ∀ λ 1 , λ 2 ∈ F , x ∈ V 7. distributivity: λ ( x + y ) = λ x + λ y , ∀ λ ∈ F , x , y ∈ V 8. distributivity: ( λ 1 + λ 2 ) x = λ 1 x + λ 2 x , ∀ λ 1 , λ 2 ∈ F , x ∈ V 19/31