# Amplitude modulation by nikeborome

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• pg 1
```									    Periodogram of a
sinusoid + spike

• Single high value is sum of
sine curves all in phase at
time t0:

 (t  t0 )   cos  (t  t0 )d

Periodogram of a
sinusoid + white
noise
• White noise is sum of sine
curves with equal amplitude
but random phases:

w(t)   cost   ( )d


• NOTE THAT BOTH FLAT
NOISE AND A SPIKE HAVE
FLAT PERIODOGRAMS
Amplitude modulation
• Suppose the amplitude of the
oscillation changes
sinusoidally with time:

X(t)  (1  Asin t  Bcos t)sin t
 sin t
 (A / 2)[cos(   )t  cos(   )t]
 (B / 2)[sin(   )t  sin(   )t]

• This produces sidelobes at
.
Phase modulation
• Suppose the phase
increases at a rate that has
a sine variation superposed:
X(t)  sin(t  Asin t  Bcos t)sin t.
Note that sin( x  x)  sin x  x cos x :
X(t)  sin t  A cost sin t  Bcost cos t
 sin t
 (A / 2)[sin(   )t  sin(   )t]
 (B / 2)[cos(   )t  cos(   )t]
• This also gives sidelobes at
but with different
phases relative to those
produced by amplitude
modulation.
Phase relations for sidelobes
• Combined amplitude & phase modulation:
X(t)  (1  Asin t  Bcos t)sin(t   sin t   cos t)
B                 A
 sin t       sin(   )t      cos(   )t
2                  2
B                 A
      sin(   )t      cos(   )t
2                  2
• Frequency spectra:
A( )  C(   )  C(   )    S()
Amplitude
modulation
B( )  S(   )  S(   )
 ( )  S(   )  S(   )                       Phase
modulation
 ( )  C(   )  C(   )
C()

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