# Phase Modulation problem

1. Sep 10, 2008

### maverick280857

1. The problem statement, all variables and given/known data

Consider a tone modulated PM signal of the form

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

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

2. Relevant equations

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

where

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

3. The attempt at a solution

The complex envelope $\~{s}(t)$ can be Fourier expanded as

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

where

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

Now,

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

so, the Fourier Transform of $s(t)$ is give by

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

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

$$y(t) = c_{-1}\cos(2\pi(f_{m}-f_{c})t) + c_{0}\cos(2\pi f_{c}t) + c_{1}\cos(2\pi(f_{m}+f_{c})t)$$

Questions

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

2. Is the envelope equal to $\sqrt{c_{0}^2 + c_{1}^2 + c_{-1}^2}$?

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

Cheers.
Vivek