B Signals and Systems I Solutions to Midterm Test 2. xt ()
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1 34-33B Signals and Sysems I Soluions o Miderm es 34-33B Signals and Sysems I Soluions o Miderm es ednesday Marh 7, 7:PM-9:PM Examiner: Prof. Benoi Boule Deparmen of Elerial and Compuer Engineering MGill Universiy Problem (3 marks (a [ marks] Consider he periodi signal x ( depied below. Give a mahemaial expression for x (. Find is fundamenal frequeny. Compue is Fourier series oeffiiens a k. Express x ( as a Fourier series. x ( e Ans: his signal an be wrien as: + ( m ( ( ( 3 m [ ] x e u m u m Is fundamenal period and frequeny are,. he average value over one period is given by: e a x( d e d e he FS oeffiiens a k for k are given by /3
2 34-33B Signals and Sysems I Soluions o Miderm es jk ak x( e d 3 jk ( + jk e e d e ( + jk jk 3 (3+ jk 3 3 e e e + jk + jk he Fourier series represenaion of x ( is (3+ jk 3 + jk + e jk k k k + jk. x ( ae e (b [ marks] Compue he oal average power of he hird harmoni omponens of x (. Ans: P a + a a 3o (3+ j 9 e + j e os + e e os e (sin ( [ marks] Suppose ha x ( is he inpu o an LI sysem wih impulse response h ( as shown below: x ( h ( y ( Calulae he oupu y (. Compue he oal average power in he fundamenal omponen of y ( dk. ha is he raio of oal average power in he fundamenal omponens P o of he oupu signal o is oal average power P? FS /3
3 34-33B Signals and Sysems I Soluions o Miderm es Ans: o find y (, i is bes o ompue he onvoluion in he ime domain. Noe ha here is no overlap on eiher side of he nonzero par of he period saring a zero. hus, we an onenrae on ha par of he signal only whih will generae one full period of he oupu signal. [ ] x ( e u( u( 3 3 h ( u ( u ( Le's ime-reverse and shif he impulse response. he inervals of ineres are: < : no overlap, so y (. < : overlap for τ < [ ] x ( e u( u( 3 3 τ h ( τ hen τ y ( h( τ x ( τ dτ e dτ e ( e τ τ 3/3
4 34-33B Signals and Sysems I Soluions o Miderm es < 3: overlap for < τ < hen τ y ( h( τ x ( τ dτ e dτ e ( e e ( e e 3 < 4: overlap for τ < 3 hen 3 3 τ y ( h( τ x ( τ dτ e dτ e 3 ( e e 4 : no overlap hen y ( hus τ 3 τ < e < y ( ( e e < 3 3 ( e e 3 < 4 4 and he periodi oupu signal is y ( y( m m o ompue he power raio, we have o ompue he oal average power in he oupu signal. A ime domain soluion is adoped: 4/3
5 34-33B Signals and Sysems I Soluions o Miderm es P y ( d ( (( ( e d+ e e d + e e d ( ( + 6 ( ( ( e + e d + e e d + e e + e d ( ( + 6 ( + e e + ( e ( e + ( e + e + e 3 + e e + + ( e e+ e + e + e + e + 4e + e e 3e ( e e ( e 4e e 4e e ( e e e e e e ( ( ( he frequeny response of he sysem is given by: + j j j H ( j h( e d e d e j, j j ( e e j j hen we obain he Fourier series oeffiiens of he oupu: k j (3+ jk 3 k ( e e dk H( j ak, k k + jk j. j 3 3 ( e e e d H( j a j Finally, he power in he fundamenal omponens is given by /3
6 34-33B Signals and Sysems I Soluions o Miderm es o + P d d d 6 8 j (3+ j 3 j (3 + j (3 + j ( e e ( e e + e + j j 4 j os e os e os sin e sin e sin ( [ ] P o P.94 and he raio of powers is raio.6.6%.8 Problem ( marks Consider he following sysem where he sampling frequeny is s. x ( y ( x Hbp ( j w ( z ( x H ( lp j xr ( + + g ( v ( os( + p ( δ ( k k Hlp ( j he inpu signal is x ( sin (, and. (a [ marks] Skeh he inpu signal x (. Find and skeh X( j, he Fourier ransform of he inpu signal. Assuming ha bp < bp are he uoff frequenies of he uni-magniude ideal bandpass filer Hbp ( j, find values for he uoff frequenies suh ha bp bp is minimized, and he oal average power of w ( is maximized. Skeh he Fourier ransform ( j of w (. 6/3
7 34-33B Signals and Sysems I Soluions o Miderm es Ans: x ( 4 j Fourier ransform X( j e, <, > X( j X( j - - he uoff frequenies suh ha bp bp is minimized, and he oal average power of w ( is maximized are: bp, bp 3. ( j / /3
8 34-33B Signals and Sysems I Soluions o Miderm es ( j (b [ marks] Assume ha he ideal uni-magniude lowpass filer Hlp( j has a uoff frequeny lp. Find he sampling frequeny s ha will allow reovery of he sperum of he inpu signal. Does i mee he requiremen of he sampling heorem? Skeh he spera Z( j and X ( j. Give an expression for X ( j. r he sperum an be reovered if he sampling frequeny is r (sampling period. his frequeny does no saisfy he sampling heorem, bu i allows us o bring he sperum bak around DC. s Z( j Z( j /3
9 34-33B Signals and Sysems I Soluions o Miderm es Xr ( j Xr ( j j Fourier ransform of xr ( is X ( ( e, r j X j <., > ( [ marks] Assume ha he seond ideal uni-magniude lowpass filer Hlp ( j has a uoff frequeny lp. Skeh he sperum of he oupu signal G( j and ompue is oal energy. G( j G( j /3
10 34-33B Signals and Sysems I Soluions o Miderm es oal energy in g ( is ompued as: E G( j d Problem 3 ( marks Suppose ha x ( is periodi of period and x ( a k. Prove Parseval's relaion: z P x( d ak. Ans: k S F j( k m ak am e d k m, k m, k m jk jm m m P x( d x( x ( d ake a e d k aa k k k ak k Problem 4 ( marks he differenial equaion of a seond-order lowpass Buerworh filer has he form: d y( dy( + ( ( + y x. d d (a [3 marks] ha is? I is he uoff frequeny of he filer, for whih H( j, also alled he -3dB uoff. (b [6 marks] Find he frequeny response of his filer, hen ompue and skeh is magniude. Frequeny response is /3
11 34-33B Signals and Sysems I Soluions o Miderm es Magniude: H B H ( j B Y( j ( j X( j ( j j L NM + + j j j + ( + ( + ( L NM ( + ( 4 O QP + O QP + ( [6 marks] Design a highpass filer saring wih he above Buerworh filer, i.e., give is frequeny response and is differenial equaion. A highpass filer an be obained as follows: HBhp ( j H ( j ( B j + j + ( j + j + j + ( j Differenial equaion of highpass filer: d y ( ( ( ( + dy ( y d x dx d + d + d d Problem ( marks he inpu signal of he LI differenial sysem shown below (iniially a res is x ( u ( + u ( + u ( and is orresponding oupu is y ( ( + u ( + u ( + ( u (. x ( H( j y ( (a [6 marks] Skeh he inpu signal x ( and he oupu signal y (. /3
12 34-33B Signals and Sysems I Soluions o Miderm es x ( y ( (b [7 marks] Compue he Fourier ransform of he oupu Y( j. Mehod : (dire + j Y( j y( e d j j ( + e d+ ( e d jτ ( jτ τe dτ + ( τ e dτ jτ j jτ e dτ + ( e τe dτ j j ( e j jτ ( e e e j j + dτ j j j ( e j ( e ( e + e + j j j j j j j ( e ( e e ( e + + j j ( j os( 4sin ( / sin ( / ( ( ( τ + in firs in., τ in seond in. Mehod : onvoluion of wo pulses o ge y(. he oupu signal an be represened as he onvoluion of wo pulses of widh.: y ( [ u ( +. u (. ] [ u ( +. u (. ] ( + u( + u( + ( u( hus, is F is a "sin" squared: 4sin ( / Y( j ( sin ( sin ( ( ( [7 marks] Compue he Frequeny response H( j of he sysem. /3
13 34-33B Signals and Sysems I Soluions o Miderm es Noiing ha y ( is simply he running inegral of x (, we find ha he FR of he sysem is H( j. j Y( j e an also ompue he F of he inpu and ake he raio H( j : X( j j j X( j ( sin( e ( sin( e. j. j sin( e sin( e jsin( sin( jsin ( sin ( Y( j H( j X( j j jsin ( END OF SOLUIONS 3/3
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