If we integrate the given modulating signal, m(t), we arrive at the following FM signal:

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2 Part b If w intgrat th givn odulating signal, (, w arriv at th following signal: ( Acos( πf t + β sin( πf W can us anothr trigonotric idntity hr. ( Acos( β sin( πf cos( πf Asin( β sin( πf sin( πf Now, sinc β sin ( πf is a sall, slowly ti varying signal (rlativ to f w can trat it as our s( signal. Using this notation, th abov Hilbrt transfor proprtis, and so trigonotric idntitis w arriv at H { ( } ˆ ( Acos( β sin( πf sin( πf + Asin( β sin( πf cos( πf A + A Asin + A sin Asin + A sin [ sin( β sin( πf πf + sin( β sin( πf + πf ] [ sin( πf t β sin( πf + sin( πf t + β sin( πf ] ( πf t + β sin( πf ( β sin( πf πf + sin( πf t β sin( πf ( πf t + β sin( πf W thus arriv at th final solution: ( β sin( πf πf sin( β sin( πf πf ( Asin( πf t + β sin( πf

3 Probl Part a This is idntical to th intgral in probl.44, part b: ( A ( f t + β sin( πwt π, β cos ha W Part b f ha Part c This was covrd in class. Th first BPF liinats any nois in th spctru. Th output of th hard liitr loos li a variabl frquncy squar wav. W can writ th output of th hard liitr as follows: HL sin( nπ jnθ ( t ( c c n n ( πf t + β sin( πwt and, n θ nπ Rcall that copl ponntials can b pandd using Eulr s forula: j cos + j sin j cos j sin Thus, if w valuat our Fourir su fro inus infinity to plus infinity, all of th jsin trs cancl whil all of th cosin trs add. W ar lft with HL sin( nπ jnθ ( t ( c n n 4 cos π n nπ cos 4 3π ( nθ ( ( πf t + β sin( πwt cos( 6πf t + β sin( πwt + L All of th high frquncy trs ar liinatd whn th hard-liitd signal passs through th scond BPF. Also not that th aiu aplitud of th hard liitr is. Thus, du to th nonlinar natur of th hard liitr non of our trs can hav an aplitud largr than. W ar lft with

4 ( cos( πf t + β sin( πwt Passing through th diffrntiator rsults in π d dt π sin ( ( sin( πf t + β sin( πwt ( πf + β πw cos( πwt ( πf t + β sin( πwt ( f + βw cos( πwt This loos siilar to our AM signal, with th HF part insid th sin tr and th right hand factor rprsnting th odulating signal. Thus, aftr w pass th abov signal through th nvlop dtctor w arriv at v nv ( f + β W cos( πwt, βw f Th final stp is to pass this signal through th DC blocing capacitor. W thus arriv at th final output signal: ( f cos( πwt

5 Part d AM ( A[ + s( ] cos( πf, s( B cos( πwt W travl through th hard liitr. This ti, du to th natur of AM, our signal is truly priodic and loos as follows: X HL ( - T /f t W can s that now our signal coing out of th hard liitr, HL, is truly priodic and loos li a squar wav with aplitud. Th scond BPF following th hard liitr liinats all frquncis abov f. If you rcall, a squar wav can b writtn as a su of sinusoids of incrasing frquncis. Sinc f is th fundantal of this squar wav it is allowd to pass whil all of th raining HF trs ar liinatd. Thus our squar wav is soothd and th raining signal is ( cos( πf W pass this signal through th diffrntiator. π d dt π ( ( πf sin( πf f sin( πf This signal is in turn passd through th nvlop dtctor producing th following signal. ( f v nv

6 Th DC blocing capacitor liinats this signal to produc th final rsult. ( Part AM Cas, SSB ( cos( πf Th first BPF has no ffct sinc it is cntrd at our carrir frquncy, f. Multiplying by th cosin squars our cosin tr. W can us a trigonotric proprty. ( π f [ + cos( π ( f ] cos Th final LPF liinats th HF cosin tr. W ar lft with a constant. ( AM Cas, DSB Th only diffrnc hr is th BW of th first BPF. Sinc our signal is siply a spi in th frquncy doain th diffrnt BW has no ffct on th signal. Thus w obtain th sa answr as w did in th SSB cas. (

7 Cas, SSB ( cos( πf t + β sin( πwt W now fro our class nots that this signal can b writtn as { } jπf t jβ sin( πwt ( R And fro hr w now that thin of our incoing signal as jβ sin( πwt can b writtn as a Fourir Sris. Thus, w can ( J ( β cos( π ( f + nw n n Again, onc w pass this signal through th first BPF it liinats th HF trs whr n, laving us with ( J ( β cos( π ( f W BPF + Not that sinc our BPF only tnds to th right of our cntr frquncy w only pass th tr f + W and w liinat th tr f W. W ultiply b th carrir cosin and us a trigonotric proprty. BPF ( cos( πf J ( β cos( π ( f + W cos( πf J ( β cos( π ( f + W + J ( β cos( πwt Our final LPF liinats th HF tr and w ar lft with th solution. ( J ( β cos( πwt

8 Cas, DSB ( cos( πf t + β sin( πwt For this final cas w can again prss our input signal using th Fourir Sris prssion. ( J ( β cos( π ( f + nw n n W pass this signal through th BPF. Howvr, this ti notic that th filtr tnds lft by W as wll as right. So now w not only pass th low frquncy tr f + W but w also pass th low frquncy tr f W. W arriv at BPF ( J ( β cos( π ( f + W J ( β cos( π ( f W Not that th Bssl function is an odd function, i J ( β J( β th inus sign in th iddl of our BPF ( tr. W ultiply by our carrir cosin and apply th sa trigonotric proprty. BPF. This accounts for ( cos( πf J ( β cos( π ( f + W cos( πf J ( β cos( π ( f W cos( πf J ( β [ cos( π ( f + W + cos( πwt ] J ( β [ cos( π ( f W + cos( πwt ] Th LPF liinats th HF trs as usual and w ar lft with th following. Not that cosin is also an odd function. ( J ( β cos( πwt J ( β cos( πwt J ( β cos( πwt J ( β cos( πwt ( t

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