# Chapter 2 Summary: Continuous-Wave Modulation. Belkacem Derras

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1 ECEN 44 Communicaion Theory Chaper Summary: Coninuous-Wave Modulaion.1 Modulaion Modulaion is a process in which a parameer of a carrier waveform is varied in accordance wih a given message (baseband) signal. In coninuous-wave (CW) modulaion, he modulaed carrier is a sinusoidal signal of he form c() = cos(πf c +φ), where f c is he carrier frequency. I is clear ha c() is characerized by 3 parameers: he ampliude, he frequency f c, and he phase φ. Varying each of hese 3 parameers proporionally o a meassage signal m() will resul in a cerain ype of modulaion. Thus we can have ampliude modulaion (AM), frequency modulaion (FM), and phase modulaion (PM). FM and PM are relaed and someimes hey are called angle modulaion. Wihin AM we have several varians which are: AM, DSB-SC, SSB, and VSB. SSB and VSB can be wih or wihou carrier. The modulaion process is performed on baseband signal for several reasons: (1) o adap he signal bandwidh o he ransmission channel bandwidh, () o be able o frequency-muliplex many message signals and ransmi hem alogeher across he same channel, and (3) o be able o use a small size receiving anennas.. Ampliude Modulaion For he differen ampliude modulaion mehods he modulaed signals have he following forms AM signal: s() = [1 + k a m()]cos(πf DSB-SC signal: s() = m()cos(πf SSB signal: VSB signal: s () s () = = -----m() cos( π f -----m() cos( π f ± ± -----mˆ () sin( π f -----m () sin( π f where mˆ () = m () is he Hilber ransform of m() and m () is he response of VSB filer o an inpu m(). π In commercial TV broadcasing a VSB plus a carrier is used for he modulaion of video signals. This makes he receiver much simpler since in he presence of a carrier, an envelope deecor is used o demodulae he ransmied signal..3 Angle Modulaion This includes FM and PM. In FM, he insananeous frequency of a carrier wave is proporional o he message signal m(), i.e. f i () =f c + k f m() and in PM he insananeous phase is proporional o m(), i.e. φ() =πf c + k p m(). Therefore an FM generaor is simply an inegraor followed by a phase modulaor and a PM generaor is simply a differeniaor followed by a frequency modulaor. FM signal: s () = cos π f c + πk f m( τ) dτ PM signal: s () = cos( π f c + k p m ()) In fac FM and PM signals may be wrien using a single expression s () = cos π f c + πk f m( τ)h ( τ) dτ, where h() =δ() for PM and h() =u() (uni sep funcion) for FM. In commercial FM broadcasing h() is chosen as he impulse response of a preemphasis filer used before he modulaion. Sricly speaking, such a modulaion is in fac neiher FM nor PM bu a mixure of he wo. The preemphasis filer is used o boos high frequency componens of he message signal m() because in FM modulaion noise affecs more high frequency componens of m(). A he receiver and afer demodulaion, a filer called deemphasis filer is used o undo he effec of he preemphasis filer used a he ransmier. Universiy of Colorado a Boulder, ECE Deparmen 1 1/6/1

2 ECEN 44 Communicaion Theory.4 Generaion of Ampliude-Modulaed Signals The generaion of he differen ampliude-modulaed signals can be performed using 3 differen ypes of modulaors: 1. Muliplier Modulaors: in his case he modulaion is performed in a sraighforward manner by muliplying m() by cos(πf.. Nonlinear Modulaors: in his case modulaion is obained by passing he message signal m() hrough a nonlinear device, hen by using an appropriae bandpass filer we can exrac he desired componen. 3. Swiching Modulaors: In his case he muliplicaion operaion required for modulaion is replaced by a simple swiching operaion. This is done in he following way. If we muliply he message signal m() by any periodic signal q() (no necessarily a pure sinusoid) ha has a fundamenal frequency f c. Such a signal q() can be expressed in erms of Fourier series as q () = C n cos( π f c + φ). This shows ha he frequency conen n = (specrum) of m()q() is he specrum M(f) shifed o ± f c, ± f c,..., ± nf c,... Wih a bandpass filer of bandwidh W and cenered a f c, we can ge he desired AM signal, where W is he bandwidh of m(). In his mehod, an appropriae and simple swiching signal q() should be employed. For SSB and VSB signals, in addiion o he seps of hese hree mehods, one more sep is required which is a bandpass filering o exrac one side band for SSB or o exrac one side band and a vesige of he oher side band for VSB..5 Generaion of Angle-Modulaed Signals There are wo mehods for generaing FM signals: direc mehod and indirec mehod. 1. Direc mehod is based on a volage-conrolled oscillaor (VCO) o which he message signal m() is applied as inpu. And since he oupu frequency of a VCO varies proporionally o he inpu signal, he resul is an FM signal a he VCO oupu. This mehod is no used for high precision applicaions since i is based on varying one of he reacive parameers (C or L) of a resonnan circui of an oscillaor.. Indirec mehod uses a narrowband frequency modulaor (NBFM) followed by a frequency muliplier. The NBFM uses m() as inpu and produces s NBFM () = cos(πf - π k f sin(πf a() as oupu, where a () = m( τ) dτ. The frequency muliplier is a nonlinear device followed by a bandpass filer o exrac he desired signal. Noice here ha he generaion of an NBFM is obained from he implemenaion of he expression s NBFM () using a DSB-SC modulaor. The generaion of PM signals is similar o ha FM signals since FM and PM are relaed. Narrowband PM (NBPM) can be generaed is a similar way as he NBFM, excep ha he NBPM signal is given by s NBPM () = cos(πf - k p sin(πf m(). For wideband PM (WBPM) signals, heir generaion is accomplished using direc or indirec mehod as in he case of wideband FM (WBFM). This is simply because a PM modulaor is jus an FM modulaor whose inpu is dm()/d. Therefore a PM modulaion consiss of a differeniaor followed by an FM modulaor..6 AM Demodulaion (Deecion) In general an AM signal is demodulaed using an envelope deecor or a coheren deecor (synchronous demodulaor). Envelope deecor is used when he modulaed signal includes he carrier, whereas coheren deecor is used when he modulaed signal expression doesn include any carrier waveform. In erms of complexiy, envelope deecion is a very simple mehod when compared o coheren deecion mehod. The laer mehod employs a phase-locked loop (PLL) circui which synchronizes he frequency and phase of a locally-generaed carrier wih hose of he received signal. As menioned previously, he use of an envelope deecor in a receiver requires he ransmission of he carrier waveform which consumes mos of he ransmission power. Therefore such a demodulaion (deecion) mehod is used when we have one ransmier and many receivers as in broadcasing radio and TV sysems. The coheren deecion mehod, on he oher hand, is used when no carrier is ransmied which saves an imporan amoun of ransmission power bu a he cos of some addiional complexiy o he receiver. Universiy of Colorado a Boulder, ECE Deparmen 1/6/1

4 ECEN 44 Communicaion Theory widh (including he audio signal and he differen gaps) is 6 MHz. ombinaion of FM and PM is used in commercial radio FM broadcasing. FM is also used o moduale he audio signal in TV sysems..11 Summary Table 1: Summary of he differen coninuous-wave (CW) modulaion schemes Modulaed Signals B T Deecion FOM AM signal s() = [1 + k a m()]cos(πf W Envelope 1 + DSB-SC signal s() = m()cos(πf W Coheren 1 s () SSB-SC signal W Coheren 1 = -----m() cos( π f ± -----mˆ () sin( π f s () SSB+C signal W Envelope = -----m() cos( π f ± -----mˆ () sin( π f + cos( π f 1 + s () VSB-SC W+f v Coheren 1 = -----m() cos( π f ± -----m () sin( π f s () VSB+C W+f v Envelope = -----m() cos( π f ± -----m () sin( π f + cos( π f FM signal ( f + W) Frequency discriminaor or s () = cos π f c + πk f m( τ) dτ PLL PM signal ( f + W) Frequency discriminaor or s () = cos( π f c + k p m ()) PLL 1 + 3k f Pm W k p Pm The FOM is used o measure he receiver performance agains he noise for each ype of demodulaion. The FOM for demodulaion using envelope deecion is obained for he small noise case, where he carrier-o-noise raio is much greaer han uniy. I is clear from he expressions of FOM ha for ampliude modualion, coheren deecion performs beer han envelope deecion and in general angle modulaion (FM and PM) performs beer han ampliude modula- Universiy of Colorado a Boulder, ECE Deparmen 4 1/6/1

5 ECEN 44 Communicaion Theory ion (AM, DSB-SC, SSB, VSB). The comparison beween FM and PM can be made by aking he raio of heir corresponding FOMs, which is pw , where k f = f/a m and k p = f/a m. f is he maximum frequency deviaion, A m is k he 3k f peak ampliude of m(), and A m is he peak ampliude of dm()/d. Therefore i is clear ha for some meassage signals m(), PM performs beer han FM and for oher message signals FM performs beer han PM. The superioriyof FM or PM depends on he ype of message signal m() and how he peak ampliudes of m() and is derivaive are relaed. For message signals wih A m >> A m, PM is beer han FM and if A m << A m, hen FM is beer han PM. The ransmission bandwidhs of FM and PM are no given in closed forms as in he case of ampliude modulaion. They are given in approximae forms using he well-known Carlson s rule B T =( f + W). Bu someimes his rule is used for narrowband cases and a more relaxed rule is used for wideband cases which is B T = ( f + W)..1 Noaions m(): Message signal A m : Peak ampliude of m() P m : Average power of m() dm()/d: Derivaive of message signal A m : Peak ampliude of dm()/d mˆ () : Hilber ransform of m() m (): Filered version of m() s(): Modulaed signal f c : Carrier frequency : Carrier ampliude f v : Vesigial bandwidh W: Message signal bandwidh f: Maximum frequency deviaion k f : Frequency sensiiviy k p : Phase sensiiviy k a : Ampliude sensiiviy β: Modulaion index for angle modulaion (PM and FM) µ: Modulaion index for ampliude modulaion B T : Transmission bandwidh (bandwidh of s()) FOM: Figure of meri Universiy of Colorado a Boulder, ECE Deparmen 5 1/6/1

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