ADDITIONAL PROBLEMS (a) Find the Fourier transform of the half-cosine pulse shown in Fig. 2.40(a). Additional Problems 91
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1 ddiional Problems 9 n inverse relaionship exiss beween he ime-domain and freuency-domain descripions of a signal. Whenever an operaion is performed on he waveform of a signal in he ime domain, a corresponding modificaion is applied o he specrum of he signal in he freuency domain. n imporan conseuence of his inverse relaionship is he fac ha he ime-bandwidh produc of an energy signal is a consan; he definiions of signal duraion and bandwidh merely affec he value of he consan. n imporan signal processing operaion freuenly encounered in communicaion sysems is ha of linear filering. his operaion involves he convoluion of he inpu signal wih he impulse response of he filer or, euivalenly, he muliplicaion of he Fourier ransform of he inpu signal by he ransfer funcion (i.e., Fourier ransform of he impulse response) of he filer. Noe, however, ha he maerial presened in he chaper on linear filering assumes ha he filer is ime-invarian (i.e., he shape of he impulse response of he filer is invarian wih respec o when he uni impulse or dela funcion is applied o he filer). noher imporan signal processing operaion encounered in communicaion sysems is ha of correlaion. his operaion may provide a measure of similariy beween a signal and a delayed version of iself, in which case we speak of he auocorrelaion funcion. When he measure of similariy involves a pair of differen signals, however, we speak of he cross-correlaion funcion. he Fourier ransform of he auocorrelaion funcion is called he specral densiy. he Fourier ransform of he cross-correlaion funcion is called he cross-specral densiy. Discussions of correlaion and specral densiy presened in he chaper were confined o energy signals and power signals exemplified by pulse-like signals and periodic signals respecively; he reamen of noise (anoher realizaion of power signal) is deferred o Chaper 8. he final par of he chaper was concerned wih he discree Fourier ransform and is numerical compuaion. Basically, he discree Fourier ransform is obained from he sandard Fourier ransform by uniformly sampling boh he inpu signal and he oupu specrum. he fas Fourier ransform algorihm provides a pracical means for he efficien implemenaion of he discree Fourier ransform on a digial compuer. his makes he fas Fourier ransform algorihm a powerful compuaional ool for specral analysis and linear filering. DDIION PROBEMS.9 Find he Fourier ransform of he half-cosine pulse shown in Fig..4. g() g() g() g() (d) FIGURE.4 Problem.9
2 9 CHPER FOURIER REPRESENION OF SIGNS ND SYSEMS pply he ime-shifing propery o he resul obained in par o evaluae he specrum of he half-sine pulse shown in Fig..4. Wha is he specrum of a half-sine pulse having a duraion eual o a? (d) Wha is he specrum of he negaive half-sine pulse shown in Fig..4? (e) Find he specrum of he single sine pulse shown in Fig..4(d).. ny funcion g can be spli umambiguously ino an even par and an odd par, as shown by he even par is defined by and he odd par is defined by g o 3g g 4 Evaluae he even and odd pars of a recangular pulse defined by Wha are he Fourier ransforms of hese wo pars of he pulse?. he following expression may be viewed as an approximae represenaion of a pulse wih finie rise ime: where i is assumed ha W. Deermine he Fourier ransform of g. Wha happens o his ransform when we allow o become zero?. he Fourier ransform of a signal g is denoed by Gf. Prove he following properies of he Fourier ransform: If a real signal g is an even funcion of ime, he Fourier ransform Gf is purely real. If he real signal g is an odd funcion of ime, he Fourier ransform Gf is purely imaginary. n g a j n where G n f is he nh derivaive of Gf wih respec o f. p b G n f, (d).3 he Fourier ransform Gf of a signal g is bounded by he following hree ineualiies: ƒgfƒ ƒgƒ d n g d a j p b n G n g g d G fg f df ƒjpfgfƒ ƒjpf Gfƒ dg ` ` d d g d g ` ` d d g g e g o g e 3g g 4 g reca b exp pu du I is assumed ha he firs and second derivaives of g exis.
3 ddiional Problems 93 g() FIGURE.4 Problem.3 Consruc hese hree bounds for he riangular pulse shown in Fig..4 and compare your resuls wih he acual ampliude specrum of he pulse..4 Consider he convoluion of wo signals g and g. Show ha d d 3g g 4 c d d g d g 3g g 4 d c g d d g.5 signal x of finie energy is applied o a suare-law device whose oupu y is defined by he specrum of x is limied o he freuency inerval W f W. Hence, show ha he specrum of y is limied o W f W. Hin: Express y as x muliplied by iself..6 Evaluae he Fourier ransform of he dela funcion by considering i as he limiing form of recangular pulse of uni area, and sinc pulse of uni area..7 he Fourier ransform Gf of a signal g is defined by Deermine he signal g..8 Consider a pulselike funcion g ha consiss of a small number of sraigh-line segmens. Suppose ha his funcion is differeniaed wih respec o ime wice so as o generae a seuence of weighed dela funcions, as shown by d g d ai k i d i where he k i are relaed o he slopes of he sraigh-line segmens. Given he values of he and i, show ha he Fourier ransform of g is given by k i y x Gf d Gf 4p f a i, f, f, f k i exp jpf i Using his procedure, show ha he Fourier ransform of he rapezoidal pulse shown in Fig..4 is given by Gf p f b a sin3pf b a 4 sin3pf b a 4 g() b a a b FIGURE.4 Problem.8
4 94 CHPER FOURIER REPRESENION OF SIGNS ND SYSEMS.9 recangular pulse of ampliude and duraion a may be viewed as he limiing case of he rapezoidal pulse shown in Fig..4 as approaches a. Saring wih he resul given in par of Problem.8, show ha as b approaches a, his resul approaches a sinc funcion. Reconcile he resul derived in par wih he Fourier-ransform pair of E. (.)..3 e x and y be he inpu and oupu signals of a linear ime-invarian filer. Using Rayleigh s energy heorem, show ha if he filer is sable and he inpu signal x has finie energy, hen he oupu signal y also has finie energy. ha is, if hen ƒyƒ d.3 Deermine he overall ampliude response of he cascade connecion shown in Fig..43 consising of N idenical sages, each wih a ime consan RC eual o. Show ha as N approaches infiniy, he ampliude response of he cascade connecion approaches he Gaussian funcion expa where for each value of N, he ime f b, consan is seleced so ha he condiion is saisfied. b ƒxƒ d 4p N R R R C Buffer amplifier C Buffer amplifier C Buffer amplifier FIGURE.43 Problem.3.3 Suppose ha, for a given signal x, he inegraed value of he signal over an inerval is reuired, as shown by y x d Show ha y can be obained by ransmiing he signal x hrough a filer wih is ransfer funcion given by Hf sincf exp jpf n adeuae approximaion o his ransfer funcion is obained by using a low-pass filer wih a bandwidh eual o >, passband ampliude response, and delay >. ssuming his low-pass filer o be ideal, deermine he filer oupu a ime due o a uni sep funcion applied o he filer a, and compare he resul wih he corresponding oupu of he ideal inegraor. Noe ha Sip.85 and Si p>..33 Show ha he wo differen pulses defined in pars and of Fig..44 have he same energy specral densiy: g f 4 cos pf p 4 f
5 ddiional Problems 95 g() g() FIGURE.44 Problem Deermine and skech he auocorrelaion funcions of he following exponenial pulses: g exp au g exp aƒƒ g exp au expau where u() is he uni sep funcion, and u is is ime-reversed version..35 Deermine and skech he auocorrelaion funcion of a Gaussian pulse defined by.36 he Fourier ransform of a signal is defined by sincf. Show ha he auocorrelaion funcion of his signal is riangular in form..37 Specify wo differen pulse signals ha have exacly he same auocorrelaion funcion..38 Consider a sinusoidal signal g defined by g exp p g cospf u cospf u Deermine he auocorrelaion funcion R g of his signal. Wha is he value of R g? Has any informaion abou g been los in obaining he auocorrelaion funcion? Explain..39 Deermine he auocorrelaion funcion of he riple pulse shown in Fig..45. g() 3 3 FIGURE.45 Problem.39.4 e Gf denoe he Fourier ransform of a real-valued energy signal g, and R g denoe is auocorrelaion funcion. Show ha B dr g d R d 4p.4 Deermine he cross-correlaion funcion R of he recangular pulse g and riple pulse g shown in Fig..46, and skech i. Wha is R? re hese wo signals orhogonal o each oher? Why? f ƒgfƒ 4 df
6 96 CHPER FOURIER REPRESENION OF SIGNS ND SYSEMS g () g () FIGURE.46 Problem.4.4 Consider wo energy signals g and g. hese wo signals are delayed by amouns eual o and seconds, respecively. Show ha he ime delays are addiive in convolving he pair of delayed signals, whereas hey are subracive in cross-correlaing hem..43 n energy signal x, is Fourier ransform Xf, auocorrelaion funcion R x, and energy specral densiy x f are all relaed, direcly or indirecly. Consruc a flow-graph ha porrays all he possible direc relaionships beween hem. If you are given he freuency-domain descripion Xf, he auocorrelaion funcion R x can be calculaed from Xf. Ouline wo ways in which his calculaion can be performed..44 Find he auocorrelaion funcion of a power signal g whose power specral densiy is depiced in Fig..47. Wha is he value of his auocorrelaion funcion a he origin? S g (f ) f FIGURE.47 Problem Consider he suare wave g shown in Fig..48. Find he power specral densiy, average power, and auocorrelaion funcion of his suare wave. Does he wave have dc power? Explain your answer. g() (seconds) FIGURE.48 Problem Consider wo periodic signals g p and g p ha have he following complex Fourier series represenaions: g p a c,n exp jpn
7 ddiional Problems 97 and g p he wo signals have a common period eual o. Using he following definiion of cross-correlaion for a pair of periodic signals, R a > > c,n exp jpn g p g p d show ha he prescribed pair of periodic signals saisfies he Fourier-ransform pair.47 periodic signal g p of period is represened by he complex Fourier series where he c n are he complex Fourier coefficiens. he auocorrelaion funcion of g p is defined by Consider he sinusoidal wave Deermine he auocorrelaion funcion R gp and plo is waveform. Show ha R gp >. R g p a c n expjpn> R gp > g p g p d > g p cospf c u.48 Repea pars and of Problem.47 for he suare wave: a c, n c, n d f n g p c, 4 4, for he remainder of period.49 Deermine he power specral densiy of he sinusoidal wave of Problem.47, and he suare wave of Problem Following a procedure similar o ha described in Secion. ha led o he flow graph of Fig..36 for he 8-poin FF algorihm based on decimaion-in-freuency, do he following: Develop he corresponding flow graph for he 8-poin FF algorihm based on decimaionin-ime. Compare he flow graph obained in par wih ha described in Fig..36, sressing he similariies and differences beween hese wo basic mehods for deriving he FF algorihm.
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