Ax1(t)+Bx2(t)Ay1(t)+By2(t)Linearityscaling & superpositioninputoutputTime invariancex(t-τ) y(t-τ)Linear TimeIn variantSystemy(t)x(t)Characteristic Functionsests=a+jbcomplex exponentialsLinear Time Invariant Systemsests=a+jbe-σtexponential decaye±jωts=σsinusoidss=±jωe-σ±jωtexponential sinusoidss=-σ ±jωLinear TimeInvariantSystemx(t)=sin(ωt)y(t)=A sin(ωt+φ)characteristic functions of LTI systemsoutput has same frequency as inputbut is scaled and phase shifted€ cosθ=ejθ+ e−jθ2Complex Exponential Signalssine oddy(θ)=y(θ +2πn)θ(rad) y1=sin(θ)0π/6π /4π /3Periodicsin(-θ)=sin(θ)cosine evencos(-θ)=-cos(θ)π /2π3π/22π0.50.7070.86610−10y2=cos(θ)10.8660.7070.50−101Sinusoidsθ=θ(t)ω: radian frequency (radians/sec)f: frequency (cycles/sec-Hz)ω=2πfT:period (sec/cycle)T = 1/f= 2π/ωrad/sec= (2π rad/cycle)*cycle/secsec/cycle= 1/ (cycle/sec)= (2π rad/cycle)/(rad/sec)A: amplitudeφ: phase (radians)orParameters:Relations:y(t)=A sin(ωt+φ)y(t)=A sin(2πft+φ)y(t)=y(t+T)Continuous sinusoidsContinuous sinusoidsθ=θ(t)ω: radian frequency (radians/sec)f: frequency (cycles/sec-Hz)A: amplitudeφ: phase (radians)orParameters:y(t)=A sin(ωt+φ)y(t)=A sin(2πft+φ)Phase shiftIn: x(t)=1sin(t)Out: y(t)=0.5sin(t-π/3)x(0)=0y(π/3)=0(t- π/3)delay of π/3plot moves to the right€ y n[ ]= sin(2π⋅150⋅ n)0 10 20 30 40 50 60 70 80 90 100-1-0.8-0.6-0.4-0.200.20.40.60.81n - samplesy[n]=cos(2*pi*n/50)TextEndsampled continuous sinusoids€ y(t) = sin( 2πt)€ Ts=150sec€ t = nTs€ y n[ ]= sin(π25⋅ n)Discrete SinusoidContinuous SinusoidSample rate:n01234y[n]00.1250.2490.3680.4818Sampled Continuous Sinusoidθ=θ[n]y[n]=A sin(ωn+φ)n=0, 1, 2…y[n]=A sin(2πfn+φ)A: amplitudeω: radian frequency (radians/sample)φ: phase (radians)f: frequency (cycles/sample)ω=2πfN:period (samples/repeating cycle [integer])f = k/N (f: rational number -> k/N is ratio of integers)rad/sample= (2π rad/cycle)*cycle/sampleRelations:Smallest integer N such that y[n]=y[n+N]Find an integer k so N=k/f is also an integer € N ≠1fDiscrete sinusoids€ y n[ ]= sin(2π⋅150⋅ n)T=50 samples (integer)€ y n[ ]= y[n + 50]€ sin(0) = sin(2π)Period of a Discrete Sinusoid:ex1y[n]=cos(2π(3/16)n)y[n]=A cos(2πfn+φ)N:period (samples/repeating cycle [integer])f = k/N (rational number -> k/N is ratio of integers)Smallest integer N such that y[n]=y[n+N]f=3/16let k=3N=(3)*16/3=16Find an integer k so N=k/f is also an integer 0 5 10 15 20 25 30-1-0.8-0.6-0.4-0.200.20.40.60.81n - samplesy[n]=cos(2*pi*3/16*n)TextEndperiod of discrete sinusoidsfrequency: f=3/16 cycles/sampleWhat is the period N?ex:Period of discrete sinusoids: ex2€ y n[ ]= sin(2π⋅350⋅ n)N=?? samples N: integer € y n[ ]= y[n + N ]50/3 ≠ integer€ f =350cyclessample€ N ≠1fPeriod of discrete sinusoids: ex3.€ y n[ ]= sin(2π⋅350⋅ n)N=?? samples€ y n[ ]= y[n + N ]€ 350⋅ N = k€ Nk=503samplescycleratio of integersrational numberN=50 samples, k=3 cyclesperiodicdiscrete functionperiod:€ f =350cyclessecfrequency:€ f ⋅ N = kN, k:integersPeriod of discrete sinusoids: ex3.€ y n[ ]= sin(2π⋅225⋅ n)N=?? samples (integer) € y n[ ]= y[n + N ]€ Nk=25 22irrational numbersampled discrete sinusoid aperiodic0 0.5 1 1.5 2 2.5 3 3.5 4-1-0.8-0.6-0.4-0.200.20.40.60.81time (sec)y=sin(2*pi*sqrt(2)/25*n)TextEndirrational frequency€ y t( )= sin(2π⋅ 2 ⋅ t)€ T =12secperiodic€ f =225€ f ⋅ N = knot a ratio of integers€ Ts=125seccontinuous functionsample€ t = nTsdiscrete functionperiod?N,k: integersAperiodic discrete sinusoidsT:period (sec/cycle)arbitrary continuous signaly(t+Τ)=y(t)After what interval doesthe signal repeat itself? PeriodicityPeriod of Sum of Sinusoids€ y t( )= y t + T( )0 1 2 3 4 5 6-1-0.500.510 1 2 3 4 5 6-2-1012time (sec)Tsum=? secondsT1=0.2 seconds, T2=0.75 secondsEx: Period of sum of sinusoidsTsum=3 secondsT1=0.2s=1/5 seconds1/5*k=3/4*lk/l=15/44/20s, 8/20s, 12/20s,16/20s, 20/20s, 24/20s,28/20s, 32/20s, 36/20s,40/20s, 44/20s, 48/20s,52/20s, 56/20s, 60/20s15/20s. 30/20s,45/20s, 60/20s 15 cycles4 cycles1/5s, 2/5s, 3/5s …T2=0.75s=3/4 seconds3/4s, 6/4s, …Tsum=15*T1=15/5=3 secondsTsum=4*T2=3/4*4=3 secondsseconds to complete cyclesseconds to complete cyclesrational numberLeast common multipley(t)=A sin(ωt+φ)θ=θ(t)y(θ)=sin(θ)instantaneous frequencyω=dθ/dtθ=ωt+φdθ/dt=ωsinusoid constant frequencychirp linearly swept frequencyω=dθ/dtω =((ω1-ω0)/T)t +ω0θ =(ω1-ω0)/2T t2 +ω0t + Cychirp(t)=A sin((ω1-ω0) /(2T) t2 +ω0t + φ)time varying argumentt ω0 ω0 Τ ω1integrateInstantaneous frequencyy(t)=A cos(ωt+φ)X= Aejφcomplex amplitude (constant)€ y(t) = Aejφejωt+ e− jωt[ ]2y(t)=Re{Aejφej(ωt)}trig functioncomplex conjugatesreal part ofcomplex exponentialrotating phasorejθ=cos(θ)+jsin(θ)Euler’s relations€ cosθ=ejθ+ e−jθ2€ sinθ=ejθ− e−jθ2 jy(t)=Re{Xej(ωt)}€ ejθ( )= ejωt+φ( )= ejφejωtRepresentations of a sinusoidAdd spectrumAddition cartesianPowerspolarRootspolarpolarcartesian polar cartesians=a+jb s=rejθ€ s = a2+ b2ej⋅a tanba( )€ s = r cosθ+ jr sinθ€ a1+ jb1( )+ a2+ jb2( )= a1+ a2( )+ j b1+ b2( )Subtraction cartesianMultiplicationpolarDivisionpolar€ r1ejθ1⋅ r2ejθ2= r1r2ejθ1+θ2( )€ a1+ jb1( )− a2+ jb2( )= a1− a2( )+ j b1− b2( )€ r1ejθ1r2ejθ2=r1r2ejθ1−θ2( )€ rejθ( )n= rnejnθ€ zn= s = rejθ€ z = s1/ n= r1/ nejθ/ n +2πk / n( ) € k = 1,2Kn −1Complex ArithmeticComplex ConversionsTrigonometric manipulations -> algebraic operations on exponentsWhy use complex exponentials?Vector representation (graphical)Trigonometric identities€ cos(x)cos(y) =12cos x − y( )− cos x + y( )[ ][ ]€ cos2(x) =1+ cos 2x( )2rexey= rex+y(rex)n= rnenxProperties of exponentials€ xn= x1/ n€ 1x= x−1Complex ExponentialsComplex ExponentialsTrigonometric manipulations -> algebraic operations on exponentsWhy use complex exponentials?Adding sinusoids of same frequency but multiple amplitudes and phases A cos(ωt+φ1)+ B cos(ωt+φ2)= C cos(ωt+φ3)A [cos(ωt)cos(φ1 )- sin(ωt)sin(φ1 )] + B [cos(ωt)cos(φ2 )- sin(ωt)sin(φ2 )] -[Asin(φ1)+Bsin(φ2)] sin(ωt) + [Acos(φ1)+Bcos(φ2)] cos(ωt)or.A2...2 A B cos φ1 φ2 B2sin.ω t atan.A cos φ1.B cos φ2.A sin φ1.B sin φ2.A cos ω φ1.B cos ω φ2.Re..A ejφ1e..j ω t..B e.j φ2e..j ω t.Re..A ejφ1.B e.j φ2e..j ω t.Re..A cos φ1.j sin φ1.B cos φ2.j sin φ2 e..j ω t.Re..A cos φ1.B cos φ2.j.A sin φ1.B sin φ2 cos.ω t.j sin.ω t.Re..A