why we have to multiply in AM demodulation?

[idmond]

“In AM synchronous demodulation, Why we don’t divide the received signal, m(t)coswt, by cos(wt) using a simple divider circuit instead of multiplying by cos(wt)?”

I asked this question to two professors and I got different answers:

#Professor 1 Reply:
"For your scheme to work you must know exactly what the frequency w is that the transmitter is using, which tends to drift. So try your scheme with dividing
by cos (w+delta)t and see whether you can recover the signal."



#Professor 2 reply was:
"we don't use the division scheme for two reasons:

a- In the real world, noise is added to the received signal and it is going to look like this, m(t)coswt+n(t). If you then divide by coswt, you will get, m(t)+n(t)/coswt, and since the cosine function ranges from -1 to 1, thus for values of cosine less than 1, the noise term will increase much more, so you will get poor SNR.

b- When the cosine function goes to zero, you will divide by zero in this case, so what?"



I am now confused more than ever:-?. which one is true?

 

[babakta]

Hi
both answers seems to be correct but I agree with the second one more than the first answer.
Also there is another issue, what do you mean by a simple divider? which analog circuit can do the simple division that you suggest?
Looking forward to hear from you.

 

[idmond]

Hi
both answers seems to be correct but I agree with the second one more than the first answer.
Also there is another issue, what do you mean by a simple divider? which analog circuit can do the simple division that you suggest?
Looking forward to hear from you.

Hi, babakta. Thanks for your reply.

I didn't mean the divider to be simple literally, just any divider circuit that can divide by a function (not the one that divide by a number such as the dividers used in PLLs and RF synthesizers) like the cosine function (or any other sinusoid).

I too understand Professor 2 reasons, but Professor 1 reason seems a little bit vague and irreverent to me.

because, frankly i don't understand what the frequency shift has anything to do with choosing the multiplication scheme over the division scheme. After he told me that, I started analyzing the problem mathematically just like he told me, but again it was a dead end:

1- if we used the usual multiplication scheme for demodulation:

if the received AM modulated, frequency-shifted signal was something like this:

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then the demodulated signal would be:

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2- if we used the division scheme for demodulation:

if the received AM modulated, frequency-shifted signal was something like this:

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and if we pass through a low pass filter, we get the output signal:


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In both schemes, the spectrum of the message signal m(t) is shifted by an amount of . So if we had a PLL, it would track the frequency of the received carrier as it drifts and the frequency shift would be zero in both schemes, so we get:




1- for the division scheme :

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2- for the multiplication scheme:


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so in both schemes the problem is solved, the message signal is received.
what's the problem then?

 

[godfreyl]

I agree Professor 1 seems to have missed the point a little bit. With either multiplication or division frequency shift would be a problem, so for either scheme you need the PLL to keep the local oscillator synchronized with the incoming signal.