AM modulation essentially consists of the TD multiplication of the modulating signal
by a cosine carrier.
In the FD, AM modulation effected a frequency shift of the spectrum of a band limited
modulating signal from being centered on zero to being centered on the carrier
frequency.
This is only half the story of using an AM modulated signal.
To recover the information in the AM signal, we have to “demodulate” it to reproduce
the original TD signal.
A) Why can’t we just low pass filter the modulated signal to get back the low frequency
modulating signal?
B) Show, by examining what happens in the FD, that we can effect demodulation in the
TD by i) multiplying the am modulated signal by the same cosine carrier that we
used to perform the modulation in the first place and then ii) passing the resulting
signal through a low pass filter.
C) What is the amplitude of the demodulated signal?
This is how AM
modulation is done.
In the time domain start
with the audio signal,
add a DC shift to make
it all positive and
multiply by the radio
frequency carrier to
make the modulated
signal.
Figs from Smith
Figure from Smith
This is how AM
modulation is done.
In the frequency domain
this corresponds to the
convolution of a shifted
delta with the audio
spectrum (note there
are 2 sidebands, taking
into account those
funny negative
frequencies - they really
exist!!).
Figure from Smith
We know multiplication in the TD is the same as convolution in the
FD. The top plot is the FT of the low-pass filter kernel. The middle
plot is the FT of the max non-aliased frequency we multiplied by - it
goes into a single point in the FD. The convolution of the two
sequences shifts the first (top) to be centered on d in the second
(middle). Resulting in the high pass filter kernel ( bottom)!
(Notice that we’re
making lots of use
of the
“periodicity” of
things in the
discrete domain
and negative
frequencies – for
both the LP filter
and the delta.
They’re REAL!)
Figure from Smith
AM modulation is done by multiplying two signals in the TD, which is
convolution in the FD
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When one of the signals is a single frequency, this gives a delta in the
FD. Convolution with a delta just shifts the FD of the second sequence
such that it becomes centered on the delta.
Notice we are including the negative frequencies in this analysis (it will
not work if we dont!).
So we have 2 copies of the signal in the FD, each has an amplitude of ½.
a) Why can’t we just low pass filter the modulated signal to get back the
low frequency modulating signal?
After modulation, there is no signal in the FD inside the LP filter limits,
so doing a LP filter does not result in any output.
Now we multiply the AM modulated TD signal (the whole thing in the
FD, with both positive and negative frequencies) by the “carrier”
frequency again. This causes another shift in the FD.
Now we get 4 copies of the spectrum, each at ¼ amplitude, but the two
in the middle add together to make one signal of half the amplitude.
Finally, now that we have a signal back in the low frequency range, we
can low pass filter it.
1
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1
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amplitude
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1 amplitude
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amplitude