Linear Time Invariant Systems. Ay 1 (t)+by 2 (t) s=a+jb complex exponentials

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1 Linear Time Invariant Systems x(t) Linear Time Invariant System y(t) Linearity input output Ax (t)+bx (t) Ay (t)+by (t) scaling & superposition Time invariance x(t-τ) y(t-τ) Characteristic Functions e st s=a+jb complex exponentials

2 Complex Exponential Signals s=a+jb s=σ s=±jω s=-σ ±jω e st e -σt e ±jωt e -σ±jωt exponential decay sinusoids exponential sinusoids cosθ = ejθ + e jθ characteristic functions of LTI systems x(t)=sin(ωt) Linear Time Invariant System y(t)=a sin(ωt+φ) output has same frequency as input but is scaled and phase shifted

3 Sinusoids θ(rad) y =sin(θ) π/6 π /4 π /3 π / π 3π/ π y =cos(θ) Periodic y(θ)=y(θ +πn) sine odd sin(-θ)=sin(θ) cosine even cos(-θ)=-cos(θ)

4 Continuous sinusoids θ=θ(t) y(t)=a sin(ωt+φ) y(t)=a sin(πft+φ) Parameters: A: amplitude φ: phase (radians) ω: radian frequency (radians/sec) or f: frequency (cycles/sec-hz) Relations: ω=πf rad/sec= (π rad/cycle)*cycle/sec T:period (sec/cycle) y(t)=y(t+t) T = /f= π/ω sec/cycle= / (cycle/sec)= (π rad/cycle)/(rad/sec)

5 Continuous sinusoids θ=θ(t) y(t)=a sin(ωt+φ) y(t)=a sin(πft+φ) Parameters: A: amplitude φ: phase (radians) ω: radian frequency (radians/sec) or f: frequency (cycles/sec-hz) Phase shift In: x(t)=sin(t) Out: y(t)=.5sin(t-π/3) x()= y(π/3)= (t- π/3) delay of π/3 plot moves to the right

6 Sampled Continuous Sinusoid Continuous Sinusoid y(t) = sin(πt) sampled continuous sinusoids Sample rate: t = nt s T s = 5 sec y[n]=cos(*pi*n/5) TextEnd Discrete Sinusoid [ ] = sin(π 5 [ ] = sin( 5 π n) y n y n n) n 3 4 y[n] n - samples

7 Discrete sinusoids θ=θ[n] n=,, y[n]=a sin(ωn+φ) y[n]=a sin(πfn+φ) A: amplitude φ: phase (radians) ω: radian frequency (radians/sample) f: frequency (cycles/sample) Relations: ω=πf rad/sample= (π rad/cycle)*cycle/sample N:period (samples/repeating cycle [integer]) Smallest integer N such that y[n]=y[n+n] Find an integer k so N=k/f is also an integer N f f = k/n (f: rational number -> k/n is ratio of integers)

8 Period of a Discrete Sinusoid:ex y[ n] = sin(π n) 5 T=5 samples (integer) y[ n] = y[n + 5] sin() = sin(π)

9 Period of discrete sinusoids: ex ex: y[n]=cos(π(3/6)n) What is the period N? period of discrete sinusoids y[n]=a cos(πfn+φ) frequency: f=3/6 cycles/sample y[n]=cos(*pi*3/6*n) TextEnd N:period (samples/repeating cycle [integer]) Smallest integer N such that y[n]=y[n+n] Find an integer k so N=k/f is also an integer f=3/6 let k=3 N=(3)*6/3= n - samples f = k/n (rational number -> k/n is ratio of integers)

10 y[ n] = sin(π 3 n) 5 Period of discrete sinusoids: ex3. f = 3 5 cycles sample y[ n] = y[n + N] N=?? samples N: integer N f 5/3 integer

11 discrete function y[ n] = sin(π 3 n) 5 Period of discrete sinusoids: ex3. frequency: f = 3 5 cycles sec period: N=?? samples y[ n] = y[n + N] f N = k N, k:integers 3 5 N = k N k = 5 3 samples cycle ratio of integers rational number periodic N=5 samples, k=3 cycles

12 Aperiodic discrete sinusoids continuous function ( ) = sin(π t) y t T = sec sample t = nt s periodic T s = 5 sec y=sin(*pi*sqrt()/5*n) TextEnd irrational frequency discrete function [ ] = sin(π y n period? [ ] = y[n + N] y n N=?? samples (integer) n) time (sec) f = 5 f N = k N,k: integers N k = 5 not a ratio of integers irrational number sampled discrete sinusoid aperiodic

13 Periodicity arbitrary continuous signal y(t+τ)=y(t) After what interval does the signal repeat itself? T:period (sec/cycle)

14 Period of Sum of Sinusoids y( t) = y( t + T)

15 Ex: Period of sum of sinusoids T=. seconds, T=.75 seconds time (sec) Tsum=? seconds

16 seconds to complete cycles T=.s=/5 seconds /5s, /5s, 3/5s 4/s, 8/s, /s, 6/s, /s, 4/s, 8/s, 3/s, 36/s, 4/s, 44/s, 48/s, 5/s, 56/s, 6/s 5 cycles Least common multiple /5*k=3/4*l k/l=5/4 seconds to complete cycles T=.75s=3/4 seconds 3/4s, 6/4s, 5/s. 3/s, 45/s, 6/s rational number 4 cycles Tsum=5*T=5/5=3 seconds Tsum=4*T=3/4*4=3 seconds Tsum=3 seconds

17 Instantaneous frequency y(θ)=sin(θ) time varying argument θ=θ(t) instantaneous frequency ω=dθ/dt sinusoid constant frequency y(t)=a sin(ωt+φ) θ=ωt+φ dθ/dt=ω chirp linearly swept frequency ω=dθ/dt ω =((ω -ω )/T)t +ω integrate θ =(ω -ω )/T t +ω t + C t ω ω Τ ω y chirp (t)=a sin((ω -ω ) /(T) t +ω t + φ)

18 Representations of a sinusoid y(t)=a cos(ωt+φ) y(t) = Ae jφ e jωt + e jωt [ ] y(t)=re{ae jφ e j(ωt) } trig function complex conjugates real part of complex exponential e j( θ) = e j( ωt+φ) = e jφ e jωt X= Ae jφ complex amplitude (constant) y(t)=re{xe j(ωt) } rotating phasor Add spectrum Euler s relations e jθ =cos(θ)+jsin(θ) cosθ = ejθ + e jθ sinθ = ejθ e jθ j

19 cartesian polar polar cartesian s=a+jb s=re jθ s = Complex Conversions a + b e j a tan( b ) a s = rcosθ + jrsinθ Complex Arithmetic Addition Subtraction cartesian cartesian ( a + jb ) + ( a + jb ) = ( a + a ) + j( b + b ) ( a + jb ) ( a + jb ) = ( a a ) + j( b b ) Multiplication polar r e jθ r e jθ = r r e j ( θ +θ ) Division polar r e jθ r e jθ = r e j ( θ θ ) r Powers polar ( re jθ ) n = r n e jnθ Roots polar z n = s = re jθ z = s / n = r / n e j θ / n +πk / n ( ) k =,Kn

20 Complex Exponentials Why use complex exponentials? Trigonometric manipulations -> algebraic operations on exponents Trigonometric identities cos(x)cos(y) = [ cos ( x y ) [ cos( x + y) ]] ( ) cos + cos x (x) = Vector representation (graphical) Properties of exponentials re x e y = re x+y (re x ) n = r n e nx n x x = x = x / n

21 Complex Exponentials Why use complex exponentials? Trigonometric manipulations -> algebraic operations on exponents Adding sinusoids of same frequency but multiple amplitudes and phases or A cos(ωt+φ )+ B cos(ωt+φ )= C cos(ωt+φ 3 ) A [cos(ωt)cos(φ )- sin(ωt)sin(φ )] + B [cos(ωt)cos(φ )- sin(ωt)sin(φ )] -[Asin(φ )+Bsin(φ )] sin(ωt) + [Acos(φ )+Bcos(φ )] cos(ωt) A. A. B. cos φ φ B. sin ω. t A. cos ω φ B. cos ω φ Re. A. e jφ... e j ω t B. e j. φ... e j ω t Re. A. e jφ B. e j. φ... e j ω t atan A. cos φ B. cos φ A. sin φ B. sin φ Re. A. cos φ j. sin φ B. cos φ j. sin φ... e j ω t Re. A. cos φ B. cos φ j. A. sin φ B. sin φ. cos ω. t j. sin ω. t Re. A. sin φ B. sin φ. sin ω. t A. cos φ B. cos φ. cos ω. t A. cos φ B. cos φ. sin ω. t A. sin φ B. sin φ. cos ω. t A. sin φ B. sin φ. sin ω. t A. cos φ B. cos φ. cos ω. t. j *sum formula for sin & cos trig id Re( )=cos Adding complex amplitudes cos + jsin

22 Complex Exponentials Amplitude modulation (multiply two sinusoids of different frequencies) Acos(ω t) B cos(ω t+φ)= C(cos(ω 3 t +φ )+ cos(ω 4 t +φ )) A. cos ω. B. cos ω φ A. B. cos ω. cos ω. cos φ sin ω. sin φ A. B. cos ω. cos φ A. B. cos ω. sin ω. sin φ cos ω ω cos ω ω sin ω ω sin ω ω A. B.. cos φ A. B.. sin φ cos ωs cos ωd sin ωs sin ωd A. B.. cos φ A. B.. sin φ... A B cos φ. cos ωs cos φ. cos ωd sin φ. sin ωs sin φ. sin ωd. A. B. cos ωs φ sin ωd φ or A. cos ω. B. cos ω φ A. e j. ω. e j ω. B. e j. ω φ e j. ω φ cos = complex conj mult. exponentials=add exponents... 4 A B exp j. φ ω ω exp j. φ ω ω exp j. φ ω ω exp j. φ ω ω... 4 A B. cos φ ω ω. cos φ ω ω. A. B. cos φ ω ω cos φ ω ω *sum formula for sin & cos trig id *product formula for sin & cos trig id *sum formula for sin & cos trig id cos = complex conj

23 Ex. ( ) Re Ae Acos πkf t + φ k n j πft +φ { } { } = Re Ae jφ e j πft { } = Re Xe j πft ( ) ( ) Ae jφ e j πft +e j πft e = X j πft +e j πft A k cos( πft + φ k ) = Re A k e πft +φ k { } = Re A k e φ k eπft 3cos π 4t + π Representations of Sinusoids Sum multiple cosines same frequency n n = Re A k e φ k { } e πft ( ) cos( π 4t π 6 ) + cos( π 4t + π 3 ) Re 3e j π e jπ 4t e j π 6 e jπ 4t + e π 3 e Re 3e j π e j π 6 + e 3 π jπ 4t e { } Re 5.34e j.545 π 4t e 5.34cos( π 4t +.545) jπ 4t n { } sum complex amplitudes

24 Multiply cosines of different frequencies A cos( ω t) A cos( ω t + φ) A A A 4 A A 4 A A e jω t + e jω t e j ( ω t +φ ) + e j( ω t +φ) A e jωt e j ω ( ( t +φ) + e jωt e j( ω t +φ) + e jωt e j( ω t +φ) + e jωt e j( ω t +φ) ) e j ω t +ω ( ( t +φ) + e j( ω t ωt +φ) + e j( ω t ωt +φ) + e j( ωt +ω t +φ) ) ( cos( ( ω + ω )t + φ) + cos( ( ω ω )t + φ) )

25 Composite signals (waveform synthesis) x(t) = A + A k ( ) = X + Re cos πkf t + φ k e j πkf t synthesize a periodic signal x(t) from a sum of a series of sinusoids - the Fourier series. f k =kf, k integer f fundamental frequency f k harmonic frequencies Note: The sum of harmonic sinusoids is periodic with a period equal to the fundamental period. ex. = 8 k odd π k k even f = 5Hz

26 x(t) = A + Composite signals (waveform synthesis) A k ( ) = X + Re cos πkf t + φ k 8 k odd = π k k even X = 8/(π )=.85 f =kf = 5=5 x(t) =.85cos( π5t + π) e j πkf t 8 = π k e jπ k odd k even =

27 .8 x(t) = A + k=3 Composite signals (waveform synthesis) A k ( ) = X + Re cos πkf t + φ k 8 = π k e jπ k odd k even X 3 = 8/(3 π )=.85 f 3 =kf =3 5=75 e j πkf t x(t) =.85cos( π5t + π) +.9cos( π 75t + π) =

28 x(t) = A + k=5 Composite signals (waveform synthesis) A k ( ) = X + Re cos πkf t + φ k 8 = π k e jπ k odd k even X 5 = 8/(5 π )=.34 f 5 =kf =5 5=5 e j πkf t x(t) =.85cos( π5t + π) +.9cos( π 75t + π) +.34 cos( π5t + π) =

29 spectrum e jπ.453e jπ X e jπ.45e jπ.5 jπ.6e.6e jπ f x(t) =.85cos( π5t + π) +.9cos( π 75t + π) +.34cos( π5t + π) +... x(t) =.85 e π e j π 5 t + e j π 5 t ( ) +.9 ( ) +.34 e π e j π 75 t j π 75 t + e ( ) +... e π e j π5 t j π5 t + e

30 Fourier Series For a given signal, how do we find for each k? = A k e jφ k Fourier Analysis x(t) = A + where A k ( ) = X + Re cos πkf t + φ k e j πkf t X = T T x(t)dt f :fundamental frequency T =/ f = T T x(t)e j πkt T dt

31 Fourier Series x(t) = t t < T x(t) = A + A k ( ) = X + Re cos πkf t + φ k e j πkf t X = T T x(t)dt.4.4 t. T = T tdt = T t T = T T = T t.4 Mathematica: athena%add math athena%math In[]:=/T*Integrate[t,{t,,T}] Out[]:=T/ = T T x(t)e j πkt T dt

32 x(t) = t t < T x(t) = A + X = T = T = T = T = j T T A k te j πkt T dt Fourier Series ( jπk +) e jπk T π k π k ( jπk +) kπ π k + T π k T π k = j T πk = T πk e j π ( ) = X + Re cos πkf t + φ k T π k π k e j πkf t e jπk = ( e j π ) k = k =

33 Fourier Series Mathematica: In[]:= /T*Integrate[t*Exp[-I**Pi*k*t/T],{t,,T}] = T T te j πkt T dt ( I) k Pi -((- + E - ( I) k Pi) T) Out[]= ( I) k Pi E k Pi In[3]:= Simplify[%,Element[k,Integers]] e jπk = e jπk = k I T Out[3]= ---- k Pi = j T πk = T πk e j π

34 x(t) = t t < T x(t) = A + X = T = T πk e j π A k cos πkf t + φ k x(t) = T + T cos πkf t + π πk Fourier Series ( ) = X + Re ( ) e j πkf t x(t) = T + T π cos ( πf t + π ) + T π cos ( π f t + π ) +K f :fundamental frequency T =/ f.4 y t t t.4 f = 5Hz T =/ f =.4 7 terms

35 x(t) = t t < T x(t) = A + X = T = T πk e j π A k cos πkf t + φ k x(t) = T + T cos πkf t + π πk Fourier Series ( ) = X + Re ( ) e j πkf t x(t) = T + T π cos ( πf t + π ) + T π cos ( π f t + π ) +K.4.4 y t t t.4 Defined between <t<.4 Periodic with period.4

36 Fourier Series:Square Wave x(t) = t < T T t < T z t t.4 X = T T dt + dt T T T In[]:=/T*Integrate[,{t,,T/}]+ /T*Integrate[-,{t,T/,T}] Out[]:= X =

37 Fourier Series:Square Wave = T T e j πkt T dt + e j πkt T T dt T T In[]:=/T*Integrate[Exp[-I**Pi*k*t/T],{t,,T/}]+ /T*Integrate[- Exp[-I**Pi*k*t/T],{t,T/,T}] -I k Pi I k Pi -I ( - E ) I (- + E ) Out[]= k Pi ( I) k Pi E k Pi In[3]:= Simplify[%,Element[k,Integers]] k -I (- + (-) ) Out[7]= k Pi = j( + ( ) k ) kπ = j 4 k odd kπ k even = j ( ) kπ = j4 kπ = j ( + ) kπ = j ( ) kπ

38 x(t) = t < T T t < T X = = j 4 k odd kπ k even 4 = kπ e j π k odd k even x(t) = A + A k ( ) = X + Re cos πkf t + φ k e j πkf t. x(t) = 4 cos πf π ( t π ) + 4 cos π 3 f 3π ( t π ) +K.5 y t z t t.4