x(at) 1 x(t t d ) e jωt d X( jω ) x(t)e jω 0t X( j(ω ω 0 )) READING ASSIGNMENTS Table of Easy FT Properties LECTURE OBJECTIVES

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1 ELEG- Signal Prossing and Communiations Ltur 4 Amplitud Modulation AM READING ASSIGNMENTS This Ltur: Chaptr, Stion - Nxt Lturs: Chaptr, Stion -3 Othr Itms: Tst #: Chaptrs 7-, Nxt Tusday Ritation: Rviw Chaptr 7- Closd book, - Z & Fourir Transform & Proprtis tabls givn Comprhnsiv Final, May 30, 0:30-:30 ECE- Signal Prossing First LECTURE OBJECTIVES Rviw of FT proprtis Convolution <--> multipliation Frquny shifting Sinwav Amplitud Modulation AM radio Frquny-division multiplxing FDM ECE- Signal Prossing First 3 Tabl of Easy FT Proprtis Linarity Proprty ax t + bx t a + b Dlay Proprty xt t d t d Frquny Shifting Saling 0t 0 xat a a ECE- Signal Prossing First 4

2 Tabl of FT Proprtis ht H Convolution Proprty yt = ht pt P Diffrntiation Proprty d dt 0t 0 ECE- Signal Prossing First 5 Convolution in th tim-domain yt = ht = hτ xt τ dτ Y = H orrsponds to MULTIPLICATION in th frquny-domain Y = H ECE- Signal Prossing First 6 Cosin Input to LTI Systm Y = H = H [πδ 0 +πδ + 0 ] = H 0 πδ 0 + H 0 πδ + 0 yt = H 0 0 t + H 0 0 t = H 0 0 t + H * 0 0 t = H 0 os 0 t + H 0 ECE- Signal Prossing First 7 Idal Lowpass Filtr H lp o o yt = if 0 < o yt = 0 if 0 > o ECE- Signal Prossing First 8

3 Th way ommuniation systms work From a Ntwork Prsptiv,,, Cox Cabl Stratgy: Broadband How do w shar bandwidth? Transport Ring Upstram 5 MHz 30 MHz 40 MHz 54 MHz I m p u l s Hadnd Voi & Rsrv Data Fibr Ring in Ring Analog TV Srvis Nod Downstram 550 MHz 750 MHz,000 HP Max 750 HP Avrag Digital TV Srvis 650 MHz Rsidntial vido, voi, data 750 MHz Voi & Rsrv Data Svs ECE- Signal Prossing First 9 Two-Way Spar "Broadast" Two-Way HDTV, VOD HFC Full Srvi Ntwork in Rsidntial Aras ECE- Signal Prossing First 0 Tabl of FT Proprtis ht H pt P Diffrntiation Proprty d dt 0t 0 ECE- Signal Prossing First Signal Multiplir Modulator pt yt = pt Y = P Multipliation in th tim-domain orrsponds to onvolution in th frquny-domain. Y = θ P θdθ ECE- Signal Prossing First

4 yt = pt Y = P = t < T / sint / = 0 t > T / / pt = os 0 t P = πδ 0 yt = os 0 t + πδ + 0 Y = [πδ 0 + πδ + 0 ] sin 7t yt = 0 t Y = 7 < < 0 0 Y = πt 0 lswhr ECE- Signal Prossing First 4 ECE- Signal Prossing First 3 Frquny Shifting Proprty 0 t xt t dt = 0 t dt 0t 0 = 0 Amplitud Modulator os t yt = os t Y = + + modulats th amplitud of th osin wav. Th rsult in th frqunydomain is two shiftd opis of. yt = os t Y = + + = t < T 0 t > T = sint yt = os t Y = sin + sin + + ECE- Signal Prossing First 5 ECE- Signal Prossing First 6

5 yt = os t Y = + + DSBAM Modulator yt = os t Y = os t ECE- Signal Prossing First 7 If =0 for > b and > b,th rsult in th frquny-domain is two shiftd and sald xat opis of. ECE- Signal Prossing First 8 DSBAM Wavform In th tim-domain, th nvlop of sin-wav paks follows Doubl Sidband AM DSBAM Typial bandlimitd input signal Frquny-shiftd opis Lowr sidband Uppr sidband ECE- Signal Prossing First 9 ECE- Signal Prossing First 0

6 DSBAM Dmodulator DSBAM Dmodulation yt = os t wt vt os t os t wt = [os t] = + os t W = V = H W ECE- Signal Prossing First H = < o 0 > o V ECE- = H W Signal = Prossing First if b < o < b Frquny-Division Multiplxing FDM FDM Blok Diagram mittr Shifting sptrum of signal to highr frquny: Prmits transmission of low-frquny signals with high-frquny EM wavs By alloating a frquny band to ah signal multipl bandlimitd signals an shar th sam hannl AM radio: khz 0 khz bands FM radio: MHz 00 khz bands Sptrum of inputs must b bandlimitd Nd > b os t os t ECE- Signal Prossing First 3 ECE- Signal Prossing First 4

7 Frquny-Division D-Mux Bandpass Filtrs for D-Mux os t os t ECE- Signal Prossing First 5 ECE- Signal Prossing First 6 Amplitud Modulator yt = os t + Y = os t Phas modulats th amplitud of th osin wav. Th rsult in th frqunydomain is two SHIFTED opis of. DSBAM: Frquny-Domain Typial bandlimitd input signal Frquny-shiftd opis + Lowr sidband Uppr sidband ECE- Signal Prossing First 7 ECE- Signal Prossing First 8

8 ECE- Signal Prossing First 9 DSBAM Dmod Phas Synh wt vt os t os + t os t t x t y = W =? what if os π = = V