Note for Principles of Communication Systems - PCS By vtu rangers

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PRlNCKPUi\:S OF COMMUl\UCA,TlON SYSTEM - 4[.SEM ECE MODULE 1: AMPLITUDE MODULATION Time domain representation of AM Amplitude Modulation may be defined as a process in which the amplitude of the carrier wave is varied to the instantaneous value (amplitude) ofthe modulating or base band signal. Let us consider a sinusoidal carrier wave CCt)given as : C(t) = Ac CDS Wet = Ac CDS 2nfet (1) Here Ac is the maximum amplitude of the carrier wave and Weis the carrier frequency. For simplicity here we have assumed that the phase of the carrier wave is zero in equation (1). Let M(t) denote the modulating or base band signal. Then according to amplitude modulation, the maximum amplitude Ac of the carrier will have to be made proportional M(t). to the instantaneous amplitude of the modulating signal The standard equation for amplitude modulated (AM) wave may be expressed as : Set) = Ac cos wet + KaM(t) Ac cos wet Or, Set) = Ac[l + KaM(t)] CDS Wet (2) •••••••••••••••••• (3) Figure 1 shows the modulating signal or base band signal, carrier signal and modulated signal. Mo'dulatihgrsignal; ".-." C(i:) Mr.RAVIKlRAN M.Tech, Dept OF ECE, CBIT KOLAR Page 1

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--PIUNCIPl"ES OF COMMUNKCAtflON SYSTEM - 4 SEM ECE Points about Amplitude 1. It may be observed of amplitude 2. Modulation that equation modulated It may be noted that carrier The modulating the time-domain behaviour signal. signal having frequency 3. (1) or (2) describe = signal [i.e C(t) Ae cos wet] is a fixed frequency We . or baseband signal M(t) contains the information to be is super imposed upon transmitted. 4. In the process the carrier of amplitude of amplitude The 6. variations resulting amplitude modulation 7. Equation from the information process whereas modulation, the amplitude modulated varies according to the with the value of the modulating amplitude unchanged and is equal to called as the envelope of AM wave consists signal. This means that the unique property carrier and phase of the carrier Ae [1 +K, M(t)] . This implies that the amplitude 1 shows that the envelope of the modulated is called wave. This wave has a constant of the AM signal remains The AM wave has time-varying baseband in the form signal. an amplitude cos wet and amplitude the frequency maximum value of the information signal M(t). The frequency Figure is transmitted signal. In of amplitude-modulation of the wave is changing around Ae in accordance 8. of the carrier signal of simply AM wave. (3) represents frequency the of amplitude constant instantaneous modulation, variations of the carrier signal. signal In the process remain this information signal in the form of amplitude other words, in amplitude 5. modulation, We of the AM wave. of the modulating or of AM is that the envelope has the same shape as the message signal or baseband signal. Mr.RAVIKIRAN MTech, Dept OF ECE, CBIT KOLAR Page 2

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-----------~ ------------------- PRINCIPLES OF COMMUN!CATlON SYSTEM - 4-SEM ECE Frequency - Domain Representation or Spectrum of AM Waves Frequency Mathematical Expression and Waveforms Let met) is the modulating signal and the carrier signal is given by the expression: C(t) = Accos 2nfct ...................(1) Then the equation for the AM will be set) = Accos Zrtf-t + Accos 2nfc t Ksmjt] (2) This equation describes the AM wave in time-domain. However, if we want to know the frequency description or frequencies present in AM wave, we will have to find its spectrum or frequency-domain representation. For this purpose, first we have to take the Fourier transform of AM wave. Let Set) or S( co) denote the Fourier transform of set), C(t) or C(w) denotes the Fourier transform of c(t) M(t) or M(w) denoted the Fourier transform of met). Let the modulating signal met) and its Fourier transform M(t) be shown in figure 1 (a) . f'!l(tJ o~------~------~----t --_1: -·\ro ----r----~ ~rn Let the modulating signal or message signal met) be band-limited to the interval: Mr.RAVIKIRAN M.Tech, Dept OF ECE, CBIT KOLAR Page 3

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PIUNClPLES OF COMMUNICATION SYSTEM - 4\-SEM IEeE This means that the modulating signal does not have any frequency component outside the interval (-Wrn, Wrn). It may be noted at this point that in figure 1 (a) the modulating signal frequency ranges extend from -Wrn to Wrn i.e. it includes negative frequencies also from -Wrn to 0 . Practically there is no meaning of negative frequency. In fact, the negative frequency is used for mathematical convenience only. Hence we can say that the modulating signal contains frequencies simply the bandwidth of modulating signal is of a cosine signal cos wet Wrn . from 0 to consists of two impulse at n[,B(w+ we) +o{w- wd] We and - We as : Ccs ')..rr-tc.-t .•............... (3) I ~ nA[o(w+ wc)+.S(lu-wdl ~i[6 Clt):: Ac. c.o~ Since the carrier signal is C(t)= Ac;(lS'Wct, therefore Acoswct or We know that the Fourier transform oR --' COSWct ~ Wrn ...•.•....•..... (4t [f+-tJ +& [of --t=J} ~lTtc. t AcCb'\:tITtc.{ ~ ~c{OC-t+-~c.)--\-o(-t-+e-)J Figure 1 (b) shows the carrier signal Aecos 2nfe t and its Fourier transform. C(w) Figure 1 (b) Now, the Am wave is given as: set) = Ac cos2nfc t + ,AcKa m(t)cos 2nfc t (5) To find the spectrum of AM wave, we take its Fourier transform. Mr.RAVIKIRAN M.Tech, Dept OF ECE, CBIT KOLAR Page 4