Signal Processing and Linear Systems-B.P.Lathi copy

Signal Processing and Linear Systems-B.P.Lathi copy

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Unformatted text preview: will be lost. So, the bandwidth of 7jJ(t) cannot be less than B Hz. Hence, it is exactly equal to B Hz. 2 t.F+X ~1"1/>'P + X 7 (4.89) where X is unknown. To d etermine X , recall t hat for t he c ase k ---> 0, we found t he b andwidth t o b e 2 B. B ut a s Eq. (4.89) indicates, t his b andwidth is X w hen k ---> O. T herefore, X = 2 B, a nd B EM = 2 (t.F + B ) Hz (4.90) A m ore rigorous derivation of this result a ppears i n reference 4. N ote t hat w hen k ---> 0, t .F ---> 0 a nd t .F « B . O n t he o ther h and w hen k is very large, t .F » B . T he former case is k nown as t he n arrowband a ngle m odulation a nd t he l atter is known as t he w ideband a ngle modulation. Recall t hat for FM, .b(t) = m (t), a nd 1/>~ = m p, w here m p is t he p eak a mplitude o f m (t). Similarly, for P M, 1/>(t) = m (t). Hence, 1/>~ = m~, w here m~ is t he p eak a mplitude of m(t). T hus a nd (4.91) We observe an interesting fact in angle modulation. T he b andwidth o f t he m odulated s ignal is a djustable by choosing s uitable value of t .F o r t he c onstant k ( kj in F M o r kp i n P M). A mplitude m odulation lacks t his f eature. T he b andwidth of each AM scheme is fixed. I t is a general principle in communication t heory tThis assertion implies an assumption >b(t)lmax = 1>b(t)lminl 298 4 Continuous-Time Signal Analysis: T he Fourier Transform t hat widening a signal bandwidth makes t he signal more immune t o noise during transmission. T hus, widening the transmission bandwidth makes angle modulated signals can be m ade more immune t o noise. Moreover, this very property allows us t o reduce t he signal power required to achieve t he same quality o f transmission. Thus, angle m odulation allows us t o exchange signal power for bandwidth. Also, because of its constant amplitude, angle modulation has a major advantage over a mplitude modulation. This feature makes angle modulation less susceptible to nonlinear distortion. We shall see in t he following section (Sec. 4.8-3) t hat n o distortion results when we pass an angle modulated signal through a nonlinear device whose o utput y(t) a nd t he i nput x (t) a re related by y(t) = x 2 (t) [in general y (t) = L ;anxn(t)J. Such a nonlinearity can be disastrous in amplitude modulated systems. This i s t he p rimary reason why angle modulation is used in microwave relay systems, where nonlinear operation of amplifiers and o ther devices has thus far been unavoidable a t t he required high power levels. In addition, t he c onstant amplitude of F M gives it a kind of immunity against rapid fading. T he effect of amplitude variations caused b y rapid fading can b e eliminated by using automatic gain control a nd b andpass limiting4. Angle modulation is also less vulnerable t han a mplitude m odulation t o small interference from adjacent channels. B ut t he price for all these advantages is p aid in terms of increased bandwidth. We c an demonstrate t hat for t he same bandwidth, t he pulse code modulation ( PCM), discussed in C hapter 5, is s uperior to angle modulation 4 . 4 .8-3 Generation and Demodulation o f Angle M...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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