Amplitude Modulation with Phase-Offset

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 the following receiver front-end.

The lowpass filters are assumed to be ideal and have cut-off frequencies f_m. Find the Fourier transform of the signals y₁(t) and y₂(t).

3.

Draw the detailed block-diagram of a system which

• takes inputs y₁(t) and y₂(t),
• produces a signal proportional to A+m(t),
• does not assume knowledge of the phase $\phi$,
• but works reliable for all possible values of $\phi$.

Explain why your design will meet the above criteria.

4.

Give an alternate receiver block-diagram, which does not involve any multipliers and demodulates the AM-signal reliably for all values of the unknown phase $\phi$.

Hint: The following identities may be useful:

$$\cos(a+b) = \cos a \cos b - \sin a \sin b$$

$$\cos a \cos b = \frac{1}{2} (\cos(a-b) + \cos(a+b))$$

$$\cos a \sin b = \frac{1}{2} (\sin(a+b) - \sin(a-b))$$