# Why we do that in AM demodulation?

1. Feb 23, 2013

### idmond dantes

Hey all,

"In AM synchronous demodulation, Why we don't divide m(t)coswt by cos(wt)
instead of multiplying by cos(wt), since this can be easily implemented by a simple divider circuit?"

If it's all about thinking mathematically, then it seems like it's more intuitive and a whole lot easier if we just divide by coswt, why we go through all the trouble and multiply then we have to know the trig identity of (cos(a)cos(b)) and then put a LPF after the output...

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

"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."

"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 and this will cause amplitude spikes which leads to circuit saturation."

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

2. Feb 24, 2013

### jsgruszynski

The multiplication is taking a modulated signal down to baseband (centered at 0 Hz) in order to demodulate. Essentially you are taking advantage of the trig identity:

http://en.wikipedia.org/wiki/List_o...#Product-to-sum_and_sum-to-product_identities

The modulation frequency is cleanly removed by multiplying leaving on the side bands which are exactly the audio modulated by the AM.

Multipliers are far easier to implement. At RF frequencies a strict magnitude divider isn't really practical. You may see the term "divider" used in the context of phase locked loops and synthesizers in RF but this isn't the same thing as a divider as you are thinking of.

3. Feb 24, 2013

### idmond dantes

yes, jsgruszynski I know these dividers(used in PLLs and RF synthesizers) are called arithmetic dividers which divide by a number and my demodulation scheme need a divider that can divide by a function like the sinusoid (cosine function). Is such a "divide by function" divider possible to implement it in a circuit or not?

4. Feb 24, 2013

### jsgruszynski

Not at high frequency nor accurately. Not in analog. Maybe digitally but at that point the advantages of just "downconverting" by multiplying pretty much solve the problem already.

5. Feb 24, 2013

### idmond dantes

yes, you're right about that because I always get this answer that you cannot do an analog circuit that can divide by a function may be a digital one would be feasible but it would be a more complicated solution than the demodulation by multiplying scheme.

but why with digital, you're saying it won't work at high frequencies and if it did, it won't be accurate, since division can easily be implemented in Digital signal processing even when you divide by a function, you will just have to deal with samples. can you elaborate more on this.
I'm looking forward to hear from you, jsgruszynski.

Last edited: Feb 24, 2013
6. Feb 25, 2013

### Staff: Mentor

I like the sound of that. Very difficult to argue around it.

7. Feb 25, 2013

### idmond dantes

well, we can still go around the "divide by zero" issue if we did all the demodulation process with digital signal processing and add some piece of code that checks for the case when the cosine function is zero then output some finite nonzero value.

8. Feb 25, 2013

### Staff: Mentor

Possibly. But how about when the cosine is small. As your professor #ⵒ pointed out, dividing a noise blip by a small number will lead to a large noise spike. That's never desirable.

9. Feb 25, 2013

### idmond dantes

In this case, we could increase our signal power to compensate for the amplified noise caused by dividing by a small number.

I believe that this whole thing "demodulation by division" actually does more harm than good. After all, what we gained from this :

1- we complicated our AM demodulator design by shifting the whole demodulation process to the digital domain to avoid the divide by zero problem.

2- we had to increase our signal power to compensate for the amplified noise that our scheme was the reason that it is amplified, in the first place.

Last edited: Feb 25, 2013
10. Feb 26, 2013

### sophiecentaur

Interesting . . . .This question could only, possibly, have been raised by the 'new' generation.
Whatever the arguments of your teachers, the historical answer is just 'Implementation'. Synchronous detection was implemented with analogue circuitry and a multiplication function is easily achievable with a simple mixer circuit - even just a diode will do what you want.
Those answers you got are not contradictory, I think - they just looked at two aspects of your suggestion.
If you want to do the demodulation 'numerically', sample by sample, then you do need to avoid the 'divide by zero' situation but that is very easy to achieve by making sure that the Cos function you are generating never contain samples at the zero crossings - that is just a matter of what phase you choose for your 'local oscillator' synthesis. Once you have locked your LO to the received carrier, the effect of the noise bursts around the zero crossings will, I think, generate noise which is related to the sampling frequency, rather than the base band frequencies so, once filtered, it wouldn't appear in the demodulated signal. This will be true, even for any samples from the division process which are 'full amplitude' (limiting value). The higher the sampling frequency, the less the effect of this noise energy - and this is the same situation as when you use low bit digital sampling and very much over-sample; the quantisation noise power is spread over a bigger and bigger bandwidth, away from the base bandwidth.

11. Feb 26, 2013

### Staff: Mentor

You'd request the distant radio station to increase their transmitter's power so that your reception was better? How practical does this sound?

12. Feb 27, 2013

### sophiecentaur

It would be safe to say that the 'big disadvantage' concerning the noise at the zero crossings, is probably non existent - for the reasons I gave above. Which leaves us with 'possible advantages'.

Not being very familiar with DSP techniques, I would be grateful if someone could justify the view that Division is easier than Multiplication, before we go any further. That certainly doesn't apply in Analogue.

13. Feb 27, 2013

### rbj

why can't both be true?

14. Feb 27, 2013

### sophiecentaur

Divide and conquer. A well known student trick!
"But the other teacher said x and you said y - you must be wrong sir"

15. Feb 27, 2013

### sophiecentaur

Actually, I could be cynical and say that your question may have taken them totally on the hop and they came out with the first thing that came into their heads. Your idea will almost certainly not come into the syllabus (it's very novel) and they (unlike PF!!!) probably would rather you hadn't asked it.

16. Feb 27, 2013

### jim hardy

I think the question has been answered several times but perhaps not in direct enough words.

Professor 2 is certainly right.

Since cos(anything) is never greater than 1, niether is ((cos(anything))^2. With multiplication, as input gets smaller so does output.

So a multiplier will never be asked to deliver a large result because its gain is always less than 1.
But a divider must be capable of quite high gain, hence it is capable of producing quite large numbers.

I have built analog dividers. To keep them stable with small denominators is well nigh impossible.

When designing any machine why would you intentionally make it inherently unstable?

old jim

17. Feb 27, 2013

### sophiecentaur

That is an excellent 'analogue' answer Jim and those ideas have coloured other responses.
However, when this is done numerically, you can do all sorts of arithmetic tricks and I'm not so sure that it is a complete no-no. Stability doesn't need to be a problem as with your ancient steam analogue systems. Though god knows why anyone would actually want to demodulate this way. The OP obviously came from way outfield amongst the theoretical daisies but there should always be a really good answer to the 'why not practical?' question.
You could limit the output amplitude of the division (even using non-linear quantising) and that will only give you short bursts of 'dodgy' (limiting value) samples. Some intelligent AGC could ensure that there are never more than a few of these and this corresponds to a short enough noise burst so that most of the spectral content is actually out of band. I'm assuming that the processing would need to be heavily over-sampled so I can't help thinking that the gut reaction may be mis-placed - as with bit slice coders and decoders which work at hundreds of times over-sampling.
It's such an innocent little question, though and the algebra is right enough. It reminds me of the rules for minimising errors in evaluating formulae with early calculators : multiply then divide then multiply then divide etc. to avoid too many zeros and too few sig figs building up.

18. Feb 27, 2013

### jim hardy

I have to think demodulation infers real time processing ergo not a lot of time for fancy arithmetic -

but i am absolutely inexperienced in DSP. I can only marvel at its capabilities and daydream about understanding the appnotes.

Usually fundamental principles apply to any operation whether it be done numerically or analog. We implemented a minimum denominator in one analog application..

Thanks sophie - that is food for thought.

old jim

19. Feb 27, 2013

### sophiecentaur

You would be amazed what they can do with numbers these days Jim! If you can do it in Excel or Basic, you can do it real-time at GHz speeds. Some of these young lads have never wound a coil in their lives!!!

20. Feb 27, 2013

### sophiecentaur

Actually, I just re-read the OP and I really don't know what a "simple divider circuit" consists of - surely he doesn't mean a flip-flop???? That's the simplest 'divider circuit' I could think of. DSP circuits are in no way "simple".
Could this all be a huge misunderstanding?

21. Feb 27, 2013

Some food for thought.

1) In the high frequency ranges you cannot sample your signal fast enough for your method to work and you cannot amplify at these frequencies, so you use a reference and a non-linear mixer to produce a signal in a range that can be handled more easily. Of course this is not really a problem in the ranges where AM is still used.

2) The signal from your method will look like **** on the short time scales. You substitute the multiplication with cos(x) by a multiplication with 1/cos(x). This signal has terrible spikes. The signal to noise in real space is the worst at the zero crossings, and this is the moment were you weight your signal the strongest. Any noise and non linear distortion will produce strong noise at the carrier frequency and its harmonics. Therefore you will definitely need a low pass filter.

3) Mathematically the multiplication with a sine wave and low pass filtering is a scalar product in Hilbert space, which is very similar to a division. It is very well understood in Fourier space. The stronger the signal is low pass filtered after the mixer, the more the noise bandwidth is reduced, I believe that this property cannot be retained using the division.

4) I think it is interesting that a multiplication and a division can result in the same signal I have to think about that some more.

22. Feb 27, 2013

### sophiecentaur

A mathematical identity is no guarantee of a working circuit, surely. The analytical world can do cancelling top and bottom and can assume ω and ω are the same. You can also 'prove' 1=0 with a bit of sleight of hand and a hidden divide by zero.

But the two functions do not produce the same signal aamof. You need to low-pass filter the produces of a multiplier to eliminate the other (RF) components, in practice.

23. Feb 27, 2013

### jim hardy

I took it to mean something akin to AD634, an impressive transconductance analog multiplier IC which can be wired to do division.
From its datasheet a caution about small denominators:

in other words, dont try gain>100.

My experience with time division approach for steam flow was similar.
While time division proved more precise than transconductance approach with small denominators, it just isn't feasible to resolve fluid flow signals(ΔP across an orifice) below about a 100::1 turndown.
I would imagine the same applies to a radio signal plucked from the air.

But i am respectful of DSP's capabilities.

old jim

btw = thanks...

24. Feb 27, 2013

### idmond dantes

by "them", you mean the members or the professors?

25. Feb 27, 2013

### idmond dantes

I didn't mean the divider to be simple, literally. It just occurred to mean that since the multiplication scheme requires multiplication and then filtering while the division scheme requires only one operation, i.e division, it would be much simpler in Implementation.
Clearly I was wrong about that.