EI331 Signals and Systems Lecture 3 Bo Jiang John Hopcroft Center - PowerPoint PPT Presentation

EI331 Signals and Systems Lecture 3 Bo Jiang John Hopcroft Center for Computer Science Shanghai Jiao Tong University March 5, 2019

Contents 1. Complex Function of a Real Variable 2. Exponential and Sinusoidal Signals 2.1 CT complex exponential and sinusoidal signals 2.2 DT complex exponential and sinusoidal signals 3. Unit Impulse and Unit Step Functions 3.1 DT unit impulse and unit step functions 3.2 CT unit impulse and unit step functions 1/34

Complex Function Complex function f : G ⊂ C → C z �→ f ( z ) Four kinds of complex functions 1. real function of a real variable ◮ studied in calculus. e.g. f ( t ) = e t , t ∈ R 2. complex function of a real variable ◮ current focus. e.g. f ( t ) = e j 2 π t , t ∈ R 3. real function of a complex variable ◮ e.g. f ( z ) = | z | , z ∈ C 4. complex function of a complex variable ◮ later, e.g. f ( z ) = e z , z ∈ C 2/34

Complex Function of a Real Variable f : G ⊂ R → C t �→ f ( t ) Equivalent to two real functions of a real variable � u ( t ) = Re f ( t ) f ( t ) = u ( t ) + jv ( t ) ⇐ ⇒ v ( t ) = Im f ( t ) e.g. � u ( t ) = cos( 2 π t ) f ( t ) = e j 2 π t ⇐ ⇒ v ( t ) = sin( 2 π t ) 3/34

Complex Function of a Real Variable Calculus of f ( t ) = u ( t ) + jv ( t ) Limit lim t → t 0 f ( t ) = lim t → t 0 u ( t ) + j lim t → t 0 v ( t ) Continuity f ( t ) continuous ⇐ ⇒ u ( t ) and v ( t ) continuous Differentiation f ′ ( t ) = u ′ ( t ) + jv ′ ( t ) Integration � � � f ( t ) dt = u ( t ) dt + j v ( t ) dt 4/34

Contents 1. Complex Function of a Real Variable 2. Exponential and Sinusoidal Signals 2.1 CT complex exponential and sinusoidal signals 2.2 DT complex exponential and sinusoidal signals 3. Unit Impulse and Unit Step Functions 3.1 DT unit impulse and unit step functions 3.2 CT unit impulse and unit step functions 5/34

CT Complex Exponential Signals x ( t ) = Ce at , where C ∈ C , a ∈ C Real exponential signals: C ∈ R , a ∈ R 1. a > 0 : growing exponential 2. a < 0 : decaying exponential 3. a = 0 : constant x x x a > 0 a < 0 a = 0 C C C O O O t t t 6/34

CT Complex Exponential Signals x ( t ) = Ce at , where C ∈ C , a ∈ R 1. a > 0 : diverges from t axis, | x ( t ) | ր ∞ as t → ∞ 2. a < 0 : converges to t axis, | x ( t ) | ց 0 as t → ∞ 3. a = 0 : constant Im x Im x Im x C C C Re x Re x Re x t t t a > 0 a < 0 a = 0 7/34

CT Complex Exponential Signals Periodic complex exponential signals x ( t ) = e j ω 0 t = cos( ω 0 t ) + j sin( ω 0 t ) , where ω 0 ∈ R Re x Im x − 4 π − 2 π O 2 π 4 π t Re x ω 0 ω 0 ω 0 ω 0 Im x t − 4 π − 2 π O 2 π 4 π t ω 0 > 0 ω 0 ω 0 ω 0 ω 0 (Angular) frequency: ω 0 radians/s Frequency: f 0 = ω 0 / ( 2 π ) cycles/s, Hz Fundamental period: T 0 = 2 π/ | ω 0 | = 1 / | f 0 | s (only if ω 0 � = 0 ) 8/34

CT Complex Exponential Signals Periodic complex exponential signals x ( t ) = e j ω 0 t = cos( | ω 0 | t ) − j sin( | ω 0 | t ) , where ω 0 < 0 Re x Im x − 4 π − 2 π O 2 π 4 π t Re x | ω 0 | | ω 0 | | ω 0 | | ω 0 | Im x t ω 0 < 0 − 4 π − 2 π O 2 π 4 π t | ω 0 | | ω 0 | | ω 0 | | ω 0 | Fundamental frequency: | ω 0 | , | f 0 | 9/34

CT Complex Exponential Signals Periodic complex exponential signals x ( t ) = Ce j ω 0 t = | C | cos( ω 0 t + φ )+ j sin( ω 0 t + φ ) , where C = | C | e j φ Re x Im x − 4 π − 2 π 2 π 4 π O t Re x ω 0 ω 0 ω 0 ω 0 Im x − 4 π − 2 π 2 π 4 π t ω 0 ω 0 ω 0 ω 0 O t ω 0 > 0 10/34

CT Complex Exponential Signals Sinusoidal signals x ( t ) = A cos( ω 0 t + φ ) , A ∈ R Conversion between exponentials and sinusoids Ae j ( ω 0 t + φ ) = A cos( ω 0 t + φ ) + jA sin( ω 0 t + φ ) A cos( ω 0 t + φ ) = A 2 e j φ e j ω 0 t + A 2 e − j φ e − j ω 0 t = A · Re e j ( ω 0 t + φ ) A sin( ω 0 t + φ ) = A · Im e j ( ω 0 t + φ ) Same periodicity • always periodic with fundamental frequency | ω 0 | • larger | ω 0 | , faster oscillation 11/34

CT Complex Exponential Signals General Complex Exponential Signals where C = | C | e j φ , a = r + j ω 0 x ( t ) = Ce at , ⇓ x ( t ) = | C | e rt e j ( ω 0 t + φ ) = | C | e rt cos( ω 0 t + φ )+ j | C | e rt sin( ω 0 t + φ ) Re x Re x envelop | C | e rt | C | e rt O t O t r > 0 r < 0 damped sinusoids 12/34

DT Complex Exponential Signals x [ n ] = C α n = Ce β n , where C ∈ C , α = e β ∈ C Real exponential signals: C ∈ R , α ∈ R (but β ∈ C !) 1. α > 1 : monotonically growing 2. 0 < α < 1 : monotonically decaying 3. α = 1 : constant x x a > 1 0 < a < 1 n n O O 13/34

DT Complex Exponential Signals x [ n ] = C α n = Ce β n , where C ∈ C , α = e β ∈ C Real exponential signals: C ∈ R , α ∈ R (but β ∈ C !) 4. α < − 1 : growing magnitude, alternating sign 5. − 1 < α < 0 : decaying magnitude, alternating sign 6. α = − 1 : constant magnitude, alternating sign ( β = j π ) x x x n n n α < − 1 − 1 < α < 0 α = − 1 14/34

DT Complex Exponential Signals General Complex Exponential Signals where C = | C | e j φ , α = | α | e j ω 0 x [ n ] = C α n , ⇓ x [ n ] = | C |·| α | n e j ( ω 0 n + φ ) = | C |·| α | n cos( ω 0 n + φ )+ j | C |·| α | n sin( ω 0 n + φ ) Re x Re x envelop | C | · | α | n | C | · | α | n n n | α | > 1 0 < | α | < 1 15/34

DT Complex Exponential Signals Sinusoidal Signals x [ n ] = | C | e j ( ω 0 n + φ ) = | C | cos( ω 0 n + φ ) + j | C | sin( ω 0 n + φ ) Periodicity • periodic ⇐ ⇒ ω 0 = 2 π k N for k ∈ Z , N ∈ Z + • fundamental period N 0 = N / gcd( N , k ) Fundamental frequency • zero if N 0 = 1 • 2 π/ N 0 if N 0 > 1 Example. x [ n ] = e j 3 π n has N 0 = 2 , fundamental frequency π , not 3 π ! Note e j 3 π n = e j π n . 16/34

DT Complex Exponential Signals Aliasing • e j ω 1 t = e j ω 2 t , ∀ t ∈ R ⇐ ⇒ ω 1 = ω 2 • e j ω 1 n = e j ω 2 n , ∀ n ∈ N ⇐ ⇒ ω 1 = ω 2 + 2 k π, k ∈ Z frequencies differing by 2 k π yields same discrete sinusoid Example. ω 1 = 1 , ω 2 = 1 + 2 π x t n For DT signals, suffices to consider frequencies on an interval of length 2 π , e.g. [ 0 , 2 π ) or ( − π, π ] 17/34

DT Complex Exponential Signals High frequencies around ( 2 k + 1 ) π , low frequencies around 2 k π x x x n n n x [ n ] = cos( 0 · n ) = 1 x [ n ] = cos( π n / 4 ) x [ n ] = cos( π n / 2 ) x x x n n n x [ n ] = cos( 3 π n / 4 ) x [ n ] = cos( π n ) x [ n ] = cos( 5 π n / 4 ) x x x n n n x [ n ] = cos( 3 π n / 2 ) x [ n ] = cos( 7 π n / 4 ) x [ n ] = cos( 2 π n ) = 1 18/34

Contents 1. Complex Function of a Real Variable 2. Exponential and Sinusoidal Signals 2.1 CT complex exponential and sinusoidal signals 2.2 DT complex exponential and sinusoidal signals 3. Unit Impulse and Unit Step Functions 3.1 DT unit impulse and unit step functions 3.2 CT unit impulse and unit step functions 19/34

DT Unit Impulse (Unit Sample) � 1 , n = 0 δ [ n ] = δ 0 n = 0 , n � = 0 1 n 0 Shifted version τ k δ � 1 , n = k τ k δ [ n ] = δ [ n − k ] = δ kn = 0 , n � = k 1 n 0 k 20/34

DT Unit Impulse (Unit Sample) Sampling property x δ = x [ 0 ] δ, or ( x δ )[ n ] = x [ n ] δ [ n ] = x [ 0 ] δ [ n ] , ∀ n ∈ Z More generally, x τ k δ = x [ k ] τ k δ, or x [ n ] δ [ n − k ] = x [ k ] δ [ n − k ] , ∀ n ∈ Z x [ n ] n 0 1 δ [ n − k ] n 0 k k x [ n ] δ [ n − k ] n 0 21/34

DT Unit Impulse (Unit Sample) Recall from linear algebra, , 0 , . . . , 0 ) T , e k = ( 0 , . . . , 0 , 1 k = 1 , 2 , . . . , n ↑ k -th form basis of R n (and C n ) n x = ( x 1 , x 2 , . . . , x n ) n = � x k e k k = 1 Similarly, { τ k δ : k ∈ Z } is a basis of C Z , space of doubly infinite sequences of complex numbers ∞ ∞ � � x = x [ k ] τ k δ, or x [ n ] = x [ k ] δ [ n − k ] , ∀ n ∈ Z k = −∞ k = −∞ 22/34

DT Unit Step � 0 , n < 0 u [ n ] = 1 , n ≥ 0 1 n 0 Relation to unit impulse ∞ n � � u [ n ] = δ [ n − k ] = δ [ m ] running sum m = −∞ k = 0 δ [ n ] = u [ n ] − u [ n − 1 ] first (backward) difference 23/34

CT Unit Step Function Also called Heaviside (step) function � 0 , t < 0 u ( t ) = 1 , t > 0 • undefined at t = 0 • sometimes u ( 0 ) = 0 , 1 , 1 / 2 Oliver Heaviside u ( t ) (from Wikipedia) 1 t O 24/34

CT Unit Impulse Function Also called Dirac delta function or δ function δ ( t ) = du dt ( t ) = lim ∆ → 0 δ ∆ ( t ) where δ ∆ ( t ) = u ( t + ∆ 2 ) − u ( t − ∆ 2 ) ∆ Physical models Paul Dirac (from Wikipedia) • density of point mass/charge • impulse force By calculus � 0 , t � = 0 du dt ( t ) = + ∞ , t = 0 t 25/34

Specification of Function by Action Classically, function is defined by specifying value at each point in its domain, e.g. x : R → R t �→ x ( t ) Idea. Define function by “action” on “test functions” Example in physics • impulse force specified by change of momentum Example in linear algebra • Matrix A ∈ R n × n ◮ specify each entry A ij ◮ specify action A x for all vector x ∈ R n 26/34