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Phase Modulation problem

  1. Sep 10, 2008 #1
    1. The problem statement, all variables and given/known data

    Consider a tone modulated PM signal of the form

    [tex]s(t) = A_{c}\cos\left(2\pi f_{c}t + \beta_{p}cos(2\pi f_{m}t))\right)[/tex]

    where [itex]\beta_{p} = k_{p}A_{m}[/itex]. This modulated signal is applied to an ideal BPF with unity gain, midband frequency [itex]f_{c}[/itex] and passband extending from [itex]f_{c}-1.5f_{m}[/itex] to [itex]f_{c}+1.5f_{m}[/itex]. Determine the envelope, phase and instantaneous frequency of the modulated signal at the filter output as functions of time.

    2. Relevant equations

    [tex]s(t) = A_{c}Re\{exp(j2\pi f_{c}t + j\beta_{p}\cos\left(2\pi f_{m}t\right)\} = Re\{\~{s}(t)exp(j2\pi f_{c}t)\}[/tex]


    [tex]\~{s}(t) = A_{c}exp(j\beta_{p}\cos(2\pi f_{m}t)[/tex]

    3. The attempt at a solution

    The complex envelope [itex]\~{s}(t)[/itex] can be Fourier expanded as

    [tex]\~{s}(t) = \sum_{n=-\infty}^{\infty}c_{n}e^{j2\pi nf_{m}t}[/tex]


    [tex]c_{n} = \frac{A_{c}}{1/f_{m}}\int_{-1/2f_{m}}^{1/2f_{m}}exp(j\beta_{p}\cos(2\pi f_{m}t))e^{-j2\pi n f_{m}t}dt = \frac{A_{c}}{2\pi}\int_{-\pi}^{\pi}exp\left[j(\beta_{p}\cos x - nx)\right]dx[/tex]


    [tex]s(t) = \sum_{n=-\infty}^{\infty}c_{n}\cos(2\pi(nf_{m}+f_{c})t)[/tex]

    so, the Fourier Transform of [itex]s(t)[/itex] is give by

    [tex]S(f) = \frac{1}{2}\sum_{n=-\infty}^{\infty}c_{n}\left[\delta(f-nf_{m}-f_{c}) + \delta(f+nf_{m}+f_{c})\right][/tex]

    Let h(t) and y(t) denote the impulse response of the filter and the output of the filter respectively. Then since [itex]Y(f) = H(f)S(f)[/itex], we have

    [tex]y(t) = c_{-1}\cos(2\pi(f_{m}-f_{c})t) + c_{0}\cos(2\pi f_{c}t) + c_{1}\cos(2\pi(f_{m}+f_{c})t)[/tex]


    1. How do I write [itex]c_{n}[/itex] in terms of [itex]J_{n}(\beta_{p})[/itex], the n-th order Bessel function of the first kind?

    2. Is the envelope equal to [itex]\sqrt{c_{0}^2 + c_{1}^2 + c_{-1}^2}[/itex]?

    3. How does one define the phase and instantaneous frequency of the filter output?

    Thanks in advance!
  2. jcsd
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