082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

24.4 Spectra 767
uF(v)u
2
–4p –3p –2p –p p 2p 3p 4p
v
Figure24.4 { 1 |t|1
Spectrum of f (t) =
0 |t|>1.
Engineeringapplication24.1
Amplitudemodulation
Amplitudemodulationisatechniquethatallowsaudiosignalstobetransmittedas
electromagnetic radio waves. The maximum frequency of audio signals is typically
10 kHz. If these signals were to be transmitted directly then it would be necessary
to use a very large antenna. This can be seen by calculating the wavelength of an
electromagnetic wave offrequency 10 kHzusingthe formula
c=fλ
Herecisthevelocityofanelectromagneticwaveinavacuum (3 ×10 8 ms −1 ), f is
the frequency of the wave and λ is its wavelength, and hence λ = c = 30000 m.
f
It can be shown that an antenna must have dimensions of at least one-quarter of the
wavelengthofthesignalbeingtransmittedifitistobereasonablyefficient.Clearlya
verylargeantennawouldbeneededtotransmita10kHzsignaldirectly.Thesolution
istohaveacarriersignalofamuchhigherfrequencythantheaudiosignalwhichis
usuallytermedthemodulationsignal.Thisallowstheantennatobeareasonablesize
as a higher frequency signal has a smaller wavelength. The arrangement for mixing
the two signals isshown inFigure 24.5.
x(t)
3
x(t)cosv c t
Antenna
cosv c t
Figure24.5
Amplitudemodulation.
Let us now derive an expression for the frequency spectrum of an amplitudemodulated
signalgiven by
φ(t) =x(t)cosω c
t
where ω c
is the angular frequency of the carrier signal and x(t) is the modulation
signal. Now φ(t) =x(t)cos ω c
t can bewritten as
φ(t) =x(t) ejω c t +e −jω c t
2
➔