Phase-Offset in AM Modulation

Consider an amplitude modulated signal x(t) in which the carrier's phase $\phi$ is not equal to zero, i.e.,

$$x(t) = (A + m(t)) \cos(2 \pi f_c t + \phi).$$

You may assume throughout the problem that the highest frequency f_m in the message signal is much smaller than the carrier frequency f_c and that the constant A has been chosen such that (A+m(t)) > 0 for all t.

1.

Find the Fourier transform X(f) of the transmitted signal x(t) in terms of of the Fourier transform M(f) of the message signal m(t).

2.

Consider first the coherent demodulator in the following block diagram

Notice, that the demodulator is ``unaware'' of the phase-offset $\phi$ in the modulator! Find an expression for the Fourier transform Y(f) of the signal y(t).

3.

Assuming that the lowpass is ideal with cut-off frequency f_m, find an expression for the output signal z(t). How does the phase-offset affect coherent demodulation?

4.

For the remainder of the problem, consider a non-coherent demodulator of the following form

The subsystems labeled $(\cdot)^2$ and $\sqrt{\cdot}$ take the square and square-root of their respective inputs.
Find an expression for the Fourier transform of the signal x²(t) in terms of the Fourier transform M(f) of the message signal m(t).

5.

Assuming that the Fourier transform M(f) of the message signal m(t) is given by $M(f) = \Pi(\frac{f}{2f_m})$, sketch the magnitude spectrum of the signal x²(t).

6.

Assuming that the lowpass is ideal with cut-off frequency f_m, find an expression for the ouput signal y(t) in terms of m(t). What effect does the phase-offset have on the non-coherent demodulator?

Hint: The following identities may be useful

$$\cos(x) \cdot \cos(y) = \frac{1}{2} ( \cos(x-y) + \cos(x+y)).$$

$$\cos^2(x) = \frac{1}{2} (1 + \cos(2x))$$