AM Stereo Signals

The use of quadrature carrier multiplexing provides the basis for the generation of AM stereo signals. One particular form of such a signal is described by

$$s(t) = A_c (m_l(t) \cos(2 \pi f_0 t - \frac{\pi}{4}) + m_r(t) \cos(2 \pi f_0 t + \frac{\pi}{4})),$$

where f_c is the carrier frequency and m_l(t) and m_r(t) are the left-hand and right-hand output signals of the loudspeakers, respectively. You may assume throughout that m_l(t) and m_r(t) are lowpass signals with bandwidth f_m much smaller than f₀.

1.

Draw a block diagram of a system with inputs m_l(t) and m_r(t) that generates the signal s(t).

2.

Compute the Fourier transform of the signal s(t).

3.

A coherent mono receiver can be implemented as in the following block diagram:

Derive an expression for the signal c(t) and then compute the Fourier transform of the signal c(t).

4.

Assume the lowpass filter to be ideal with cut-off frequency f_c satisfying $f_m < f_c \ll f_0$, find an expression for the output signal $\hat{m}(t)$. Explain how the mono receiver processes the received signal containing stereo signals.

5.

Now a stereo receiver as shown in the following block diagram is employed:

Proceeding as in parts (c) and (d), compute expressions for the two signals c₁(t) and c₂(t). Then determine how the coefficients a_i, i=1,2,3,4 must be chosen such that the original left-hand and right-hand signals m_l(t) and m_r(t) are recovered.

Hint:

$$\begin{array} {l} \cos(x \pm y) = \cos x \cos y \mp \sin x \... ...\ \cos x \sin y = \frac{1}{2}(\sin(x+y) - \sin(x-y))\end{array}$$