a) Sketch the DSB-SC signal corresponding to the message signal m(t) = cos*2*pi*t.

Solution:

As the message signal is: m(t) = cos*2*pi*t.

While the frequency is = 1 Hz. and Time Period = 1 sec.

b) The DSB-SC signal of part (a) (above question) is applied at the input of an envelope detector. Show that the output of the envelope detector is not m(t), but |m(t)|. Show that, in general, if an AM signal [A + m(t)] cos(w_ct) is envelope-detected, the output is |A + m(t)|. Hence, show that the condition for recovering m(t) from the envelope detector is A + m(t) > 0 for all ‘t’.

Solution

First, the DSB-SC signal generated in part (a) is provided to the envelope detector e.g., given in the following figure:

The function of the envelope detector is to identify the envelope of the given (input) signal.

Envelope:

It is the waveform created by joining all the positive as well as negative peaks of the waveform e.g.,

Explanation:

As the diode conducts just the upper half of the AM signal furthermore due to charging as well as discharging action the output waveform will be |A + m(t)|

As shown in the figure above, the output of the envelope detector for [A + m(t)]cos(w_ct) (or [A + m(t)]cos(2*pi*f_ct) is |A +m(t)|.

Condition for retrieving the message signal is:

1/f_c < R_LC < 1/f_m

where,

f_c = Carrier Signal Frequency.
R_L = Load Resistance.
C = Capacitance.

So, if R_LC > 1/f_m then:

if R_LC < 1/f_c

Here output waveform fastly discharge therefore, the output is not capable to follow the message signal

So, the superior condition is:

1/f_c < R_LC < 1/f_m

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