Amplitude modulation by nikeborome

VIEWS: 13 PAGES: 5

									    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()

								
To top