13 Continuous-Time Modulation

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1 13 Continuous-Time Modulation In this lecture, we begin the discussion of modulation. This is an important concept in communication systems and, as we will see in Lecture 15, also provides the basis for converting between continuous-time and discrete-time signals. In its most general sense, modulation means using one signal to vary a parameter of another signal. In communication systems, for example, if a channel is particularly suited to transmission in a certain frequency range, the information to be transmitted may be embedded in a carrier signal matched to the channel. The mechanism by which the information is embedded is modulation; that is, the information to be transmitted is used to modulate some parameter of the carrier signal. In sinusoidal frequency modulation, for example, the information is used to modulate the carrier frequency. In sinusoidal amplitude modulation, the carrier is sinusoidal at a frequency that the channel can accommodate, and the information to be transmitted modulates the amplitude of this carrier. Furthermore, in communication systems, if many different signals are to be transmitted over the same channel, a technique referred to asfrequency division multiplexing is often used. In this method each signal is used to modulate a carrier of a different frequency so that in the composite signal the information for each of the separate signals occupies nonoverlapping frequency bands. The modulation property for Fourier transforms applies directly to amplitude modulation, that is, the interpretation in the frequency domain of the result of multiplying a carrier signal by a modulating signal. From the modulation property we know that for amplitude modulation the spectrum of the modulated output is the convolution of the spectra of the carrier and the modulating signal. When the carrier is either a complex exponential or a sinusoidal signal, the spectrum of the carrier is one or a pair of impulses and the result of the convolution is then to shift the spectrum of the modulating signal to a center frequency equal to the carrier frequency. Modulation with a single complex exponential and with a sinusoidal signal are closely related. With a complex exponential carrier, demodulation, i.e., recovery of the original modulating signal, is relatively straightforward, basically involving modulating a second time with the complex corjugate signal. With sinusoidal 13-1

Signals and Systems 13-2 amplitude modulation, demodulation consists of modulating again with a sinusoidal carrier followed by lowpass filtering to extract the original signal.

2 Signals and Systems 13-2 amplitude modulation, demodulation consists of modulating again with a sinusoidal carrier followed by lowpass filtering to extract the original signal. This form of demodulation is typically referred to as synchronous demodulation since it requires synchronization between the sinusoidal carrier signals in the modulator and demodulator. However, by adding a constant to the modulating signal or equivalently injecting some carrier signal into the modulated output, a simpler form of demodulator can be used. This is referred to as asynchronous demodulation and typically results in a less expensive demodulator. However, the fact that a carrier signal is injected into the modulated signal represents an inefficiency of power transmission. Suggested Reading Section 7.0, Introduction, pages Section 7.1, Continuous-Time Sinusoidal Amplitude Modulation, pages Section 7.2, Some Applications of Sinusoidal Amplitude Modulation, pages Section 7.3, Single-Sideband Amplitude Modulation, pages

3 Continuous-Time Modulation 13-3 SINUSOIDAL AMPLITUDE MODULATION 13.1 Sinusoidal amplitude and frequency modulation with a sinusoidal carrier. SINUSOIDAL FREQUENCY MODULATION A A t 13.2 Block diagram of amplitude modulation and some examples of commonly used carrier signals.

4 Signals and Systems 13-4 xt x(t) CMt 4 1- ff ( X(w) * CMI~ 13.3 Spectra associated with amplitude modulation with a complex exponential carrier. x(t) - YGw) C(w) t + OG 43. 2e0* Iz I I W 21 e-ws 0 -twot +.) t liwe Y~w) X(w) y(t) + 3M (WC - WM 4 1w. + "M)i ~WM WM 00s (w~t + 6,) Re jy(t)} xkt) X y(t) 13.4 Representation of amplitude modulation with a complex exponential carrier in terms of amplitude modulation with two sinusoidal carriers with a 90* phase difference. X() -- M WM y"w) IMl" o y(t) ly(c) + Y* (-w)j Re jy(t) (Y() - Y*(-w) jim jy(t)}

5 Continuous-Time Modulation 13-5 ejwct Ideal lowpass filter H(w) e jct 13.5 The use of amplitude modulation with a complex exponential carrier to implement bandpass filtering with a lowpass filter. X y(t) w(t) X f(t) WO CO X(o) Y(o) W(o) 13.6 Spectra associated with the system in Transparency F- - (JO I A- F(w) (-we - WO 0 ) (-W d 1 + WO)

6 Signals and Systems Equivalent frequency response associated with the system in Transparency (71 (-Wc - Wo 0 ) (-wc + wo) H, (W) 2 wo H2(F Representation of amplitude modulation with a complex exponential carrier in terms of amplitude modulation with two sinusoidal carriers with a 90* phase difference.

7 Continuous-Time Modulation 13-7 c(t) = cos (WCt + Oe) ei(wt + 0) + e et COM X(C,) 13.9 Spectra associated with amplitude modulation with a sinusoidal carrier. we Ie C(o) I wejc 1 YC ) C-je l(w elo Cos (Wct + 0,) I Demodulation of an amplitude-modulated signal with a sinusoidal carrier. WC (- -M) (- + WpM)W C(W) tie twew W(W) i-2j9,i I 1 2jOC

8 Signals and Systems 13-8 MODULATOR x(t) y(t) Block diagram of sinusoidal amplitude modulation and demodulation. cos (w et + 0 C) DEMODU LATOR y (t) H (w) 2 x (t) -W w w Lowpass filter cos (oct + Oc) WM < w < (2oc m COS Wat Block diagram for frequency division multiplexing. Xa (t) COS ob t Ya (t) Xb (t) Yb (t) w(t) Cos Wc t X c(w

9 Continuous-Time Modulation 13-9 X, (W) Xb (W) X ( M -wm WM (A -wm W ( \M M L Spectra illustrating frequency division multiplexing. -Wb (A.) W(W) Ar7in K> K>r7~~A - Wa - C -Cb Wa Lb Mc L Demultiplexing - Demodulation Demultiplexing and demodulation of a frequency division multiplexed signal. Bandpass filter COS Wat Lowpass filter Xa (t)

10 Signals and Systems Demodulation of an amplitude-modulated signal with a sinusoidal carrier. [Transparency repeated] j2i Cos (WC t + 0c) Y (W) 1 el6-11 (W,- M) W. (W. + WM) W C(I ciw) re~li Weit W(W) -2jO I~O - w, I W M2 L cos (ct + 0c) Effect of loss of synchronization in phase in a synchronous sinusoidal amplitude modulation/demodulation system. (oe - +c)]x(t) Lowpass filter cos (oct + 4c) w(t) = x(t) Cos (Wmt + 6,) cos (ot + = [cos (0c ~ oc)]x(t) +-I- x(t) cos (2wct + Oc + #c) lowpass component

11 Continuous-Time Modulation A simple system and associated waveform for an asynchronous demodulation system. y(t) - IA + x(ti cos wct Cos ('t AT A A ~NA NA~ AAAW4-A- IY -V.N va,lv -yy V Modulator associated with asynchronous sinusoidal amplitude modulation. For such a system the carrier must be irjected into the output. This transparency shows time waveforms for the asynchronous modulation system.

12 Signals and Systems y(t) - A+ x (t)i cos oct Frequency spectra associated with an asynchronous modulation system. cos (", t ita 1 2 ira X(W) Single-sideband modulation in which only the upper sidebands are retained. ~W M W M Y(W) sideband sideband sideband sideband Y, (w) -we -C W

13 Continuous-Time Modulation Single-sideband modulation in which only the lower sidebands are retained. v (t) The use of a highpass filter to obtain a single-sideband signal. c W H(w) -Loc cw A 2

14 MIT OpenCourseWare Resource: Signals and Systems Professor Alan V. Oppenheim The following may not correspond to a particular course on MIT OpenCourseWare, but has been provided by the author as an individual learning resource. For information about citing these materials or our Terms of Use, visit:

13 Continuous-Time Modulation

13 Continuous-Time Modulation Recommended Problems P13.1 c(t) x(t) x ;y(t) Figure P13.1-1 In the amplitude modulation system in Figure P13.1-1, the input x(t) has the Fourier transform shown in Figure