1. Nature of Impulse Response - Pole on Real Axis. z y(n) = r n. z r

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1 . Nature of Impulse Respose - Pole o Real Axis Causal system trasfer fuctio: Hz) = z yz) = z r z z r y) = r r > : the respose grows mootoically > r > : y decays to zero mootoically r > : oscillatory, decayig expoetial r < : the output grows with oscillatios CL 69 Digital Cotrol, IIT Bombay c Kaa M. Moudgalya, Autum 6. Nature of Impulse Respose - Complex Cojugate Poles Hz) = z z re jω )z re jω ) A = α e jθ y) = α r [ e jω+θ) + e jω+θ) ] = α r cos ω + θ), r ω Calculate impulse respose Y z): Y z) z = = z z re jω )z re jω ) A z re + A jω z re jω A is complex, A is cojugate {u)} = y) = Ar e jω + A r e jω Arbitrary iput = sum of impulse fuctios Correspodig respose = sum of above siusoids CL 69 Digital Cotrol, IIT Bombay c Kaa M. Moudgalya, Autum 6 {δ k)}uk) = δ) u)

2 3. Properties of Cotiuous Siusoidal Sigals u a t) = A cos Ωt + θ), < t < A: amplitude Ω: frequecy i rad/s θ: phase i rad F : frequecy i cycles/s or Hertz Ω = πf u a t) = A cos πf t + θ). Cotiuous sigals with differet frequecies are differet u = cos π t ), u = cos π 7t ) coswt T p = F F fixed: periodic with period T p u a [t + T p ] = A cos πf t + F ) + θ) = A cos π + πf t + θ) = A cos πf t + θ) = u a [t] t 3. Frequecy of u rate of oscillatio of sigal - t is a cotiuous variable CL 69 Digital Cotrol, IIT Bombay 3 c Kaa M. Moudgalya, Autum 6 4. Complex Siusoids u a [t] = Ae jωt+θ) = A[cos Ωt + θ) + j si Ωt + θ)] For coveiece, egative frequecy Positive frequecy = couter clockwise rotatio Negative frequecy = clockwise rotatio u a [t] = A cos Ωt + θ) = A [e jωt+θ) + e jωt+θ) ] ] = Re [Ae jωt+θ) Re - real part. Im - imagiary ] u a [t] = A si Ωt + θ) = Im [Ae jωt+θ) CL 69 Digital Cotrol, IIT Bombay 4 c Kaa M. Moudgalya, Autum 6

3 5. Properties of Discrete Siusoidal Sigals - Periodicity u) = A cos w + θ), < < w = πf u) = cos πf + θ) u + N) = cos πf + N) + θ) iteger variable, sample umber A amplitude of the siusoid w frequecy i radias per sample θ phase i radias. f ormalized frequecy, cycles/sample u) is periodic with period N, N >, if ad oly if iff) u + N) = u) The smallest ozero N = fudametal period Equal iff there exists a iteger k: πf N = kπ f is ratioal f = k N N obtaied after cacellig the commo factors i k f is kow as fudametal period A discrete time siusoid is periodic oly if its frequecy f is a ratioal umber CL 69 Digital Cotrol, IIT Bombay 5 c Kaa M. Moudgalya, Autum 6 6. Properties of Discrete Siusoids - Idetical Sigals u) = A cos w + θ), < < w = πf iteger variable, sample umber A amplitude of the siusoid w frequecy i radias per sample θ phase i radias. f ormalized frequecy, cycles/sample All siusoidal sequeces u k ) u k ) = A cos w k + θ), w k = w + kπ, π < w < π are idistiguishable or idetical Oly the siusoids i rage π < w < π are differet Discrete time siusoids whose frequecies are separated by iteger multiple of π are idetical, i.e., cos w + π) + θ) = cos w + θ), π < w < π or < f < This property is differet from the previous oe Now fixed & varyig f Previously fixed f & varyig CL 69 Digital Cotrol, IIT Bombay 6 c Kaa M. Moudgalya, Autum 6

4 7. Properties of Discrete Siusoids - Highest Frequecy u) = A cos w + θ), < < w = πf iteger variable, sample umber A amplitude of the siusoid w frequecy i radias per sample θ phase i radias. f ormalized frequecy cycles/sample) cos w has bee plotted for cos*pi/).5.5 w values of π, π 4, π, π iteger 5 5 cos*pi/4) Highest oscillatio cos*pi/).5 cos*pi).5 w = π or w = π f = or f = CL 69 Digital Cotrol, IIT Bombay 7 c Kaa M. Moudgalya, Autum 6. Samplig a Cotiuous Sigal - Prelimiaries Let aalog sigal u a [t] have a frequecy of F Hz u a [t] = A cosπf t + θ) Uiform samplig rate T s s) or frequecy F s Hz) t = T s = F s u) = u a [T s ] < < = A cos πf T s + θ) = A cos π F ) + θ F s I our stadard otatio, u) = A cos πf + θ) It follows that f = F F s w = Ω F s = ΩT s Apply the uiqueess coditio for sampled sigals < f < F max = F s π < w < π Ω max = πf max = πf s = π T s CL 69 Digital Cotrol, IIT Bombay c Kaa M. Moudgalya, Autum 6

5 9. Properties of Discrete Siusoids - Alias As w, freq. of oscillatio, reaches maxi- u [t] = cos π t ), u [t] = cos π 7t ), T s = mum at w = π u ) = cos π 7 ) = cos π ) What if w > π? = cos π π ) ) π = cos = u ) w = w w = π w u ) = A cos w = A cos w u ) = A cos w = A cosπ w ) = A cos w = u ) coswt w is a alias of w t CL 69 Digital Cotrol, IIT Bombay 9 c Kaa M. Moudgalya, Autum 6. Fourier Series of Cotiuous Periodic Sigals xt) is periodic with a fudametal period T p = F It has a Fourier Series: xt) = C k e jπkf t Wat to calculate C k Multiply both sides by e jπlf t Itegrate from to + T p, T p = F t +T p xt)e jπlf t dt = = t +T p e jπlf t t +T p C k e jπk l)ft dt ) C k e jπkf t dt CL 69 Digital Cotrol, IIT Bombay c Kaa M. Moudgalya, Autum 6

6 . Fourier Series of Cotiuous Periodic Sigals t +T p t xt)e jπlft +T p dt = C k e jπk l)ft dt If k l, let = k l: Hece t +T p = C l T p +,k l C k e jπk l)f t jπk l)f +T p e jπf t +T p = e jπf +T p ) e jπf = e jπf e jπ ) = xt)e jπlf t dt = C l T p C l = t +T p xt)e jπlft dt = xt)e jπlft dt T p T p T p Sice, periodic. xt), C l : Fourier Series Pair CL 69 Digital Cotrol, IIT Bombay c Kaa M. Moudgalya, Autum 6. Fourier Trasform of Cotiuous Aperiodic Sigals Aperiodic o Fourier series If xt) = outside T p, T p ), Costruct a periodic sigal x p [t] with a period T p. lim Tp x p [t] = xt) x p [t] = C k e jπkft, F = T p C k = Tp T p T p x p [t]e jπkft dt As x p = x over oe period, C k = Tp T p T p xt)e jπkft dt As x vaishes outside oe period, C k = xt)e jπkft dt T p Defie Fourier Trasform of xt) as XF ) = xt)e jπf t dt XF ) is a fuctio of the cotiuous variable F. C k are samples of XF ). C k = T p X[kF ] = F X[kF ] CL 69 Digital Cotrol, IIT Bombay c Kaa M. Moudgalya, Autum 6

7 3. Fourier Trasform of Aperiodic Sigals - Cotiued x p [t] = C k e jπkf t XF ) = x p [t] = F xt)e jπf t dt X[kF ]e jπkf t F = = F T p x p [t] = X[k F ]e jπk F t F xt) = lim T p x p[t] = lim F = X[k F ]e jπk F t F XF )e jπf t df If we let radia frequecy Ω = πf xt) = π X[Ω] = X[Ω]e jωt dω, xt)e jωt dt xt), X[Ω] are Fourier Trasform Pair CL 69 Digital Cotrol, IIT Bombay 3 c Kaa M. Moudgalya, Autum 6 4. Frequecy Respose Discrete Time Fourier Trasform Apply u) = e jw to g) ad obtai output y: y) = g) u) = gk)u k) = = e jw gk)e jwk gk)e jw k) Defie Discrete Time Fourier Trasform Ge jw ) = gk)e jwk = = Gz) z=e jw Provided, absolute covergece: gk)e jwk < For causal systems, BIBO stability gk)z k z=e jw gk) < CL 69 Digital Cotrol, IIT Bombay 4 c Kaa M. Moudgalya, Autum 6

8 5. Frequecy Respose - Cotiued y) = e jw gk)e jwk = e jw Ge jw ) Write i polar coordiates: Ge jw ) = Ge jw ) e jϕ - ϕ is phase agle: y) = e jw Ge jw ) e jϕ = Ge jw ) e jw+ϕ). Iput is siusoid output also is a siusoid with followig properties: Output amplitude gets multiplied by the magitude of Ge jw ) Output siusoid shifts by ϕ with respect to iput. At ω where Ge jw ) is large, the siusoid gets amplified ad at ω where it is small, the siusoid gets atteuated. The system with large gais at low frequecies ad small gais at high frequecies are called low pass filters Similarly high pass filters CL 69 Digital Cotrol, IIT Bombay 5 c Kaa M. Moudgalya, Autum 6