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Understand all the needed ideas about envelope
detector in time and frequency domain
By : Oussama Gassab
Contact : [email protected]
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(( ))
282
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1- to understand how the envelope detector demodulate the
signal (( how it extract the original signal s(t) from the
modulated signal x(t) )) and why the condition 1
RC 0
must be achieved , we have to analyze the concepts in frequency
domain
When look to the phenomena of envelope detector in time
domain we will see only the detection of the envelope ,
when the capacitor is charged and discharged, and all the
other amazing concepts cannot be seen
Because when we try to construct and prove the relation
1
RC 0 we will only say that the period of the charging
capacitor must be very large than the period of the frequency 0
so 0 0 1
RC 0
But when we try to prove 1
RC we will not have clear
vision in time domain .
To take very clear understand we have to observe the
phenomena in both time and frequency domains.
Since the AM signal is in the
form
= 1 + 0
The diode is On for the
positive wave of 0 and
is Off in the negative wave
So the diode will follow the following graph
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The Fourier series of this signal
=1
2+
2
(1)
2 + 1
+
=0
cos(2 + 1)0
So the current i(t) is
= where = 1 + 0 so
= 1 + 1
20 +
2
1
2 + 1 0
+
=0
cos(2 + 1)0
We have
0 cos 2 + 1 0 =1
2 cos 2 + 2 0 + cos(2)0
With small simplification we got :
= 1 + 1
+
1
20
2
1
42 1 20
+
=1
We have obtained the Fourier series of the current through the
diode
When we drew the spectrum () of the current , we will
observe
If the spectrum of s(t) is given by
0 t
d(t)
1
-
Than the spectrum of () is given by
The resistor and the capacitor circuit
construct low pass filter with transfer
function
=
1+ the cutoff frequency is
1
and
Rits graph as shown
below . when the current passes
through this low pass filter
We will obtain the original signal as
-
Shown below
So to retain our signal that must 1
RC must be between 0
and and since 0 that is mean 1
RC must satisfy the
condition 1
RC 0
As an remark we have always to understand any phenomena in
both time and frequency domains in order to get complete
understanding about this phenomena
