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Frequency spectrum of the modulated signal g(t)

  1. Feb 10, 2009 #1
    1. The problem statement, all variables and given/known data

    Let the baseband signal be
    s(t)=cos(2πfst+π3), where fs=5kHz. Radio carrier is
    c(t)=sin(2πfct), where fc=100MHz. Using the amplitude modulation of g(t)=(1+s(t))c(t), what is the frequency spectrum of the modulated signal g(t)?
    What are the amplitude and phase shift of each frequency component in g(t)?

    2. Relevant equations

    g(t)=(1+s(t))c(t)


    3. The attempt at a solution

    g(t)= (1+ cos(2πfst+π3)) sin(2πfct)
    g(t)= sin(2πfct)+ cos(2πfst+π3) sin(2πfct)
    g(t)=
     
  2. jcsd
  3. Feb 11, 2009 #2

    Redbelly98

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    Looks good so far. The first sin(...) term gives you one of the frequency components.

    You'll need to express "cos(2πfst+π3) sin(2πfct)" as the sum of distinct, single-frequency sin and/or cos terms. You can do that using these trig identities:

    sin(x + y) = sin(x)·cos(y) + cos(x)·sin(y)
    sin(x - y) = sin(x)·cos(y) - cos(x)·sin(y)

    p.s. welcome to PF :smile:
     
  4. Feb 11, 2009 #3
    I try this
    g(t)= sin(2πfct)+ cos(2πfst+π3) sin(2πfct)
    g(t)= sin(2πfct)+ (cos(2πfst)+cos(π3)) sin(2πfct)
    g(t)= sin(2πfct)+ (cos(2πfst)sin(2πfct)+cos(π3)sin(2πfct))
    but don't know how to continue
     
  5. Feb 12, 2009 #4

    Redbelly98

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    There's a problem there, because
    cos(2πfst+π3) and cos(2πfst)+cos(π3)​
    are not equivalent.

    If you add the two equations in my earlier post, you'll have an expression for
    sin(x) cos(y)​
    which will be useful here.
     
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