B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

212 ANGLE MODULATION AND DEMODULATION
that m(t) has all the information of m(t), the cell width in m(t) must be no greater than the
Nyquist interval of 1/2B second according to the sampling theorem (Chapter 6).
It is relatively easier to analyze FM corresponding to m(t) because its constant amplitude
pulses (cells) of width T = 1/2B second. Consider a typical cell starting at t = t k . This
cell has a constant amplitude m(t k )- Hence, the FM signal corresponding to this cell is a
sinusoid of frequency W e + k t
m(t k ) and duration T = I/2B, as shown in Fig. 5.6b. The FM
signal for m(t) consists of a sequence of such constant frequency sinusoidal pulses of duration
T = 1/2B corresponding to various cells of m(t). The FM spectrum for m(t) consists of
the sum of the Fourier transforms of these sinusoidal pulses corresponding to all the cells.
The Fourier transform of a sinusoidal pulse in Fig. 5.6b (corresponding to the kth cell) is a
sine function shown shaded in Fig. 5.6c see Eq. (3.27a) with r = 1/2B and Eq. (3.26) with
Jo =f e + k t
m(t k )/2n :
Note that the spectrum of this pulse is spread out on either side of its center frequency W e +
k_t m(t k ) by 4n Bas the main lobe of the sine function. Figure 5 .6c shows the spectra of sinusoidal
pulses corresponding to various cells. The minimum and the maximum amplitudes of the cells
are - and m p m p
, respectively. Hence, the minimum and maximum center frequencies of
the short sinusoidal pulses corresponding to the FM signal for all the cells are W e - k t m p
and w, + kt mp , respectively. Consider the sine main lobe of these frequency responses as
significant contribution to the FM bandwidth, as shown in Fig. 5.6c. Hence, the maximum and
the minimum significant frequencies in this spectrum are W e +kt mp +4n B and W e -kt mp -4JT B,
respectively. The FM spectrum bandwidth is approximately
I
2n
BFM = - (2k t
m p
+ 8nB) = 2 -- + 2B
( k t m P
2n
)
Hz
We can now understand the fallacy in the reasoning of the pioneers. The maximum and
minimum carrier frequencies are W e + kt m p
and W e - kt m p
, respectively. Hence, it was reasoned
that the spectral components must also lie in this range, resulting in the FM bandwidth of2k r
m p
.
The implicit assumption was that a sinusoid of frequency w has its entire spectrum concentrated
at w. Unfortunately, this is true only of the everlasting sinusoid with T = oo (because it turns
the sine function into an impulse). For a sinusoid of finite duration T seconds, the spectrum is
spread out by the sine on either side of w by at least the main lobe width of 2n /T. The pioneers
had missed this spreading effect.
For notational convenience, given the deviation of the carrier frequency (in radians per
second) by ±k t m p
, we shall denote the peak frequency deviation in he1tz by !1f . Thus,
m max - m mm · m
!1f = k
f = f = k _!!_
2 - 2JT 1 2n
The estimated FM bandwidth (in hertz) can then be expressed as
BFM '.::::'. 2(!1f + 2B) (5. 12)
The bandwidth estimate thus obtained is somewhat higher than the actual value because this
is the bandwidth corresponding to the staircase approximation of m(t), not the actual m(t),
which is considerably smoother. Hence, the actual FM bandwidth is somewhat smaller than