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Unformatted text preview: le Modulation 297 frequency Wx , t he s pectrum is n ot c oncentrated a t Wx , b ut s preads o ut o n b oth sides
of w x , a s c an b e seen from Fig. 4.24d in E xample 4.12. I n a t ypical angle m odulated
signal, t he c arrier frequency is directly p roportional t o m (t), which changes w ith
t. Hence, t he i nstantaneous frequency will also change w ith t continuously. Such
continuous s hift i n frequency will cause t he s pectral s pread b eyond t he b and 2 t.w.
Clearly, t he b andwidth o f t he angle m odulated s ignal is somewhat larger t han 2 t.w
r ads/s. How much larger? T his missing link c an b e found by looking a t t he r esults
derived earlier for t he case of k > O. L et us first d etermine t .w.
F rom Eq. (4.80), i t follows t hat Witt) = We + k~(t) (4.87) if t he p eak a mplitude o f .b(t) is d enoted b y 1/>~, t hen t he c arrier frequency varies in
t he r ange from We  kt,b~ t o We + k1/>~t T herefore t.W = k1/>~ (4.88a) T he c arrier frequency deviation t .F i n Hz is t .F = t .w
271" = ~1/>'
21T p (4.88b) As d emonstrated e arlier, because of s pectral s preading, t he angle m odulated s ignal
b andwidth is s omewhat l arger t han 2 t.F. L et t he a ctual b andwidth B EM in Hz b e = A cos [wet + hp(t)]
= A cos wet cos [k1/>(t)]  A s in wet sin [k1/>(t)]
< ':j A cos wet  Ak1/>(t) sin wet k>O (4.86) B EM = = Comparison of t he r ighthand side expression w ith 'PAM(t) i n Eq. (4.73a) shows
t hat t he two expressions are very similar. T he first t erm is t he c arrier, a nd t he
s econd t erm, r epresenting t he s idebands, has t he s ame form as t he D SBSC signal
corresponding t o t he b aseband s ignal Ak1/>(t). T he o nly difference is t hat t he c arrier
is s ine instead o f cosine. T his is j ust a m atter o f carrier phase difference of 71" /2.
Hence, t he b and w idth o f t he angle m odulated s ignal is the same as t hat o f AM
signal c orresponding t o t he b aseband signal 1/>(t). I f m (t) is b andlimited t o B Hz,
t hen t he b andwidth o f 1/>(t) is also B H z.t Hence, t he b andwidth of 'PEM(t) is 2B
Hz, t he s ame as t hat o f AM. B ut t his t rue only when k > O. L et u s now consider
t he g eneral case.
In angle m odulation, t he c arrier frequency is varied from its quiescent value
We. L et t he m aximum d eviation of t he c arrier frequency be t .w. I n o ther words,
t he c arrier frequency varies in t he r ange from We  t..w t o We + t .w. B ecause t he
c arrier frequency always remains in t his b and o f w idth 2t.w r adians/s, c ould we say
t hat t he r esulting s pectrum also remains within t his b and a nd t he b andwidth of
t he angle m odulated signal is 2t.w? T his a ssertion implies t hat if a sinusoid takes
a n i nstantaneous frequency Wx , t he r esulting s pectrum is c oncentrated only a t Wx .
T his is t rue only if t he c arrier has infinite duration. For a finite d uration s inusoid of
tBecause 7jJ(t) is t he output of a linear system when the input is m (t), the bandwidth of 7jJ(t)
cannot be greater than B Hz. Moreover, the filter is invertible. Hence, the filter bandWidth
cannot be less than B Hz, or some of the components of m (t)...
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 Spring '13
 Bayliss
 Signal Processing, The Land

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