New frequency generation in AM signal

[semc]

TL;DR Summary

Does new frequencies get generated when the signal is modulated?

Suppose I have a pure sine wave. Upon Fourier transforming (FT) the time signal, I obtain a delta function in the frequency domain. If I subsequently modulate this sine wave with another function, for instance, a Gaussian, the delta function in the frequency domain will broaden. I'm curious whether this broadening is purely a mathematical artifact or if new frequencies genuinely emerge during the modulation process. If new frequencies are actually generated, what is the physical mechanism that generates these new frequencies?

 

[Baluncore]

Modulation of an RF carrier, with an audio signal, will generate an upper and a lower sideband, each contains the audio spectrum. The audio is being "up-converted" to the carrier frequency.
https://en.wikipedia.org/wiki/Amplitude_modulation#Spectrum

 

[tech99]

TL;DR Summary: Does new frequencies get generated when the signal is modulated?

I'm curious whether this broadening is purely a mathematical artifact or if new frequencies genuinely emerge during the modulation process. If new frequencies are actually generated, what is the physical mechanism that generates these new frequencies?

It was thought by most people until about 1920 that the side frequencies were just a mathematical artifact - Professor G W O Howe was of this opinion, for instance. However, in the quest for more capacity on the trans Atlantic long wave radio telephone service, Carson in the USA experimented with narrow quartz crystal filters and found that the side frequencies do exist. This allowed the development of single sideband transmission, which doubled the capacity across the Atlantic. https://www.radschool.org.au/magazines/Vol74/pdf/SSB Theory.pdf
When we modulate a carrier with a modulating signal, we utilise interaction between the two - they are not just mixed. In one system, the amplitude of the carrier is multiplied by the amplitude of the modulation. It can be done by using the modulating signal to control the gain of the carrier amplifier. This produces the well known envelope, and it is apparent that additional frequencies are now present.

 

[Baluncore]

If new frequencies are actually generated, what is the physical mechanism that generates these new frequencies?

Modulation is a multiplication function, which is non-linear, so energy appears at new frequencies, being the sum and difference frequencies of the multiplied signals.

 

[DaveE]

The normal approach to think about the AM spectrum is to modulate the carrier with a single frequency, i.e. multiply two sine waves at different frequencies. This will give you 3 spikes in the spectrum as others have described. The you can extend this with Fourier transforms to more complex waveforms.

The net result is that the baseband spectrum is copied twice in the transmitted spectrum, one being a mirror image, around the carrier. The sum and difference.

Yes, spectrum analyzers and Fourier transforms represent real signals. The only thing that you might think of as an artifact is negative frequencies for some transforms. That's a more complex discussion that deserves it's own thread; one with a specific question.

https://en.wikipedia.org/wiki/Amplitude_modulation

 

[berkeman]

If new frequencies are actually generated, what is the physical mechanism that generates these new frequencies?

It's a consequence of the math of multiplying two sinusoidal signals of different frequencies:
$$cos(2 \pi f_1 t) cos(2 \pi f_2 t) = \frac{cos(2 \pi (f_1 + f_2) t) + cos(2 \pi (f_1 - f_2) t)}{2}$$
You can look up the standard equations for multiplying trig functions on Hyperphysics, for example:

http://hyperphysics.phy-astr.gsu.edu/hbase/trid.html

 

[DaveE]

what is the physical mechanism that generates these new frequencies?

Mixers/Modulators

 

[sophiecentaur]

TL;DR Summary: Does new frequencies get generated when the signal is modulated?

I'm curious whether this broadening is purely a mathematical artifact or if new frequencies genuinely emerge

As with all Science and Maths, you can;t really ask whether something that the Maths predicts is 'really happening'.

A way to think of what happens (and it does) in the modulation process is that the side bands carry information and information costs Energy so the signal spectrum always has a finite bandwidth. Depending on how the modulation is performed, the modulator circuit may add extra power of take power from the carrier and push it into the sidebands.

AM is one of the easiest modulation forms to explain with hand waving but the same thing applies with FM (even). Also there are forms of modulation for which the output signal has no 'carrier power' at all.

 

[semc]

Thank you all for the insights shared. It appears that indeed new frequencies are generated through modulation, but the precise mechanism remains elusive. Some responses mention mixers, yet explanations about the actual generation mechanism are sparse. Basically all explanation just says that mixing generates sideband. This seems more like a result of Fourier transform than a explanation.

Other mentions point to nonlinearity, but again, what exactly is the mechanism? Drawing from my knowledge in nonlinear optics, nonlinearity arises from light interacting with asymmetric crystals. Is there a similar mechanism at play here? I'd greatly appreciate any resources that delve into this aspect.

Initially, my inquiry was slightly different, but I redirected it to draw parallels with AM signals, given their prevalence. If new frequencies truly stem from nonlinearity, I'm particularly curious about my original question. So my question was suppose I have a plane wave and I modulate with an optical chopper such that the pure sine wave is modulated by a comb function convolved with a rect function/Gaussian. In this scenario, where does the nonlinearity come into play to give rise to these new frequencies?

 

[berkeman]

So it seems that new frequencies are indeed generated. But what is the actual mechanism? Some replies mention mixer but I couldn't find any explanation on what is the generation mechanism.

You did understand my post #6 about the math, right? Many of your other threads are quite technical, so I'm guessing that you are fine with the math, but just looking for more intuition on how the mathematics of the multiplication/mixing happens...

I'll see if I can find a good explanation of how a diode mixer works; that should hopefully give you an intuitive understanding of how the non-linearity between the two input ports results in the multiplication/mixing at the output port. Wait one...

Update -- this is a pretty good introductory article: https://en.wikipedia.org/wiki/Frequency_mixer

 

[semc]

Yes I believe I understand your previous post. Thanks for the link. I guess the frequency mixer explains the nonlinearity in this case.

 

[Baluncore]

The amplitude distortion of a signal, by a non-linear device, generates harmonics of that signal.

An audio-mixer is an adder, it is linear. Distortion and new frequencies must be avoided.

A frequency-mixer is a multiplier, it is non-linear. New frequencies are required.

 

[semc]

Hi, I believe my primary concern is the linear case. I am concern with the modulation of light signal which can be achieved through a linear process (I believe). In that case the time signal is changed so does that mean the new frequency calculated using FT is a mathematical artifact?

 

[DaveE]

modulation of light signal which can be achieved through a linear process

Nope. A linear process is not a mixer. It can't be. The math is simple. If we define that a network has an input x, and an output y, then a linear network can be described as $y = a_1x$. Note that some will include a constant term $y =a_0 +a_1x$. That's not the usual definition of linear IMO, but it doesn't matter in this context.

So, if you input two sine waves into this linear network, $x = sin( \omega_1 t) + sin( \omega_2 t)$, the output is $y = a_1 sin( \omega_1 t) +a_1 sin( \omega_2 t)$, essentially the same spectral signature as the input.

In the general case, $y = f(x)$ can be a non-linear function. Any well behaved function can also be expressed as a polynomial expansion (Taylor series): $y = f(x) = a_0 + a_1 x + a_2 x^2 +a_3 x^3 +...$. In the real world, the first few terms are dominant, with decreasing effect as the exponent increases. So let's just look at the quadratic response; I'll leave a similar analysis of the higher order terms to others.

So, what if your network has this non-linear response: $y = a_0 + a_1 x + a_2 x^2$, and you input two sine waves $x = sin( \omega_1 t) + sin( \omega_2 t)$?

We get an output
$y = a_0 + a_1 [sin( \omega_1 t) + sin( \omega_2 t)]+ a_2 [sin( \omega_1 t) + sin( \omega_2 t)]^2$
$y = a_0 + a_1 [sin( \omega_1 t) + sin( \omega_2 t)] + a_2 [sin^2( \omega_1 t) + 2 sin( \omega_1 t) sin( \omega_2 t) + sin^2( \omega_2 t)]$

Using the trig identities you learned in high school (and then forgot because you knew you could look them up later when you need them):

$y = a_0 + a_1 [sin( \omega_1 t) + sin( \omega_2 t)] +$
$\frac{a_2}{2} [1 - cos( 2 \omega_1 t) + 2 cos( (\omega_1- \omega_2) t) - 2 cos( (\omega_1 + \omega_2) t) + 1 - cos( 2 \omega_2 t)]$

Here you can see how the harmonics and sum and different frequencies are generated by non-linear responses.

 

[Baluncore]

In that case the time signal is changed so does that mean the new frequency calculated using FT is a mathematical artifact?

The square of the FT tracks energy in the spectrum. It is NOT an artifact of the FT, it is real energy spreading.

Chopping a light source is not linear.
When you chop a narrowband light source, you are multiplying the light source by a square wave in the time domain, that switches between zero and one. You will see sidebands appear on both sides of the fundamental, offset by the chopping frequency.

 

[semc]

Nope. A linear process is not a mixer.

Upon reconsideration, I acknowledge my oversight in assuming linearity in modulation, especially given examples like AM that illustrate its nonlinear nature. This prompts me to question whether modulation using an optical chopper is also categorized as a nonlinear process. If it is indeed nonlinear, I'm curious to understand where this nonlinearity originates and what mechanisms contribute to the generation of new frequencies in this context.

 

[semc]

You will see sidebands appear on both sides of the fundamental, offset by the chopping frequency.

Ok great. So what is the physical process that generates these sidebands?

My main problem is that I fail to reconcile the spectra broadening effect due to modulation with any physical process that I know.

 

[Baluncore]

So what is the physical process that generates these sidebands

It is amplitude modulation, by the change in time of the multiplicative transmission coefficient.

If the light path has a varying transmission coefficient, say an on-off square wave, then odd harmonics of the chopper frequency will appear as sidebands. If the chopper was a rotating polarizer, or a "butterfly valve" screen, then only the rectified sinewave harmonics would appear in the sidebands.

 

[semc]

Pardon me but this explanation is saying amplitude modulation causes side bands or varying transmission causes sidebands while the question is how the sideband arises. It's akin to stating that sum frequency generation in a crystal produces new frequencies without exploring the specific interactions between light and crystal axes that give rise to these new frequencies through the susceptibility tensor. Or we can also explain at the microscopic level through energy level transition.

Of course it could be that the explanation is just flying over my head and I just don't get it...

 

[Baluncore]

If the character of a crystal is changed by one signal, then another signal can be modulated by that change. The same is true of any (non-linear) medium.
https://en.wikipedia.org/wiki/Luxemburg–Gorky_effect

 

[sophiecentaur]

This prompts me to question whether modulation using an optical chopper is also categorized as a nonlinear process

I think you are looking for some essential difference between the Maths and the Science. All Maths does is to model a Physical process. It describes what goes on. In the same way, the Fourier transform just transforms a variation in time of a signal (time domain) into a a variation in frequency (frequency domain). There are often practical limits which affect the choice domain that we choose to describe a signal or process.

Your optical chopper could be put in a black box with two inputs and one output and an entirely different mechanism could be used to mimic its function and produce an identical modulated signal.

AS for the difference between a linear and a non linear process, a linear process will produce an output signal that is identical to the input signal(s) except for a possible change in signal level. There will be no other spectral components generated. If other spectral frequencies are produced then the process is non-linear. Introducing specific ideas of particular forms of non linearity doesn't actually help the argument; there are no exceptions. The Trig Identities in post #6 are just mathematical identities and show that the output of a 'perfect' multiplier can be described as sum and difference frequencies. If you look on a scope at the result of adding together two waves of different frequencies you will see the familiar beat pattern.

but that is just a linear superposition of two waves, observed on a wide band device that admits both frequencies (i.e. a scope). If you choose to look with two narrow band filters, you will see two independent waves with different frequencies and no other frequencies. The existence of sum and difference patterns is all in the 'eye of the observer'. Note: if you wanted to 'detect' that beat pattern you would need to introduce a non-linear detector / demodulator so the overall system wouldn't be linear. (No receiver circuit can be made without non linear elements and that's something that the Maths on its own tends to hide)

 

[sophiecentaur]

Upon reconsideration, I acknowledge my oversight in assuming linearity in modulation

I re-read this and I would point out that there are plenty of (analogue) modulation systems that produce a (virtually) identical version of the wanted programme signal at the other end of the chain, having mixed, multiplied and frequency-shifted several times (with many non-linear processes). The output can be made as near a linear / undistorted version of the input as you want or can pay for.

My hobby horse is that people should avoid trying to force classification at all cost; it can be useful but is not to be relied on in all circs.

 

[Baluncore]

AS for the difference between a linear and a non linear process, a linear process will produce an output signal that is identical to the input signal(s) except for a possible change in signal level. There will be no other spectral components generated. If other spectral frequencies are produced then the process is non-linear. Introducing specific ideas of particular forms of non linearity doesn't actually help the argument; there are no exceptions.

When we find unexpected new frequencies present, we must then search for the non-linearity.

Once the two sine waves have been linearly added to make one signal, if that signal is passed through a rectifier diode, it is obvious. But if it is passed through a resistor, the effect is more subtle. The beat difference frequency will appear, rectified and low-pass filtered, as the changing temperature of the resistor.

An on/off switch is a non-linear component.

 

[sophiecentaur]

I have to admit, you had me going for a while but I think we can get around that one as follows: The (instantaneous) temperature rise of the resistor will be a function of the signal voltage squared so that detector can, i think, be regarded as a square law detector with a low pass filter due to the thermal time constant. A very short time constant will display the 'whole' signal with both carrier frequencies but the temperature will still be proportional to V².
.

 

[Baluncore]

I have to admit, you had me going for a while ...

That was not my intention.
Mathematically, the instantaneous power dissipated in a resistor, is proportional to sin²(v).

A filament lamp, with an AC supply, at frequency f, has a small brightness component at 2f, but absolutely no component at f.
The biggest new elephant in the spectrum, is the huge brightness component at zero-frequency.

The mechanism that introduces the non-linearity, seems to be the insensitivity of the resistive filament, to the polarity of the bi-directional current flow.

It is not the squaring, so much as the rectification inherent in the squaring, that generates new frequency components. The Absolute(v) function would also generate the new components, and a few higher harmonics.

 

[sophiecentaur]

The mechanism that introduces the non-linearity, seems to be the insensitivity of the resistive filament, to the polarity of the bi-directional current flow.

Exactly. It's a square law and the thermal time constant becomes relevant once the frequency is high and the thermal mass is high. All I'm saying is that there is inherent non-linearity so the basic principle of mixing, rather than addition still applies and a thermal detector falls into the non linear camp.

 

[sophiecentaur]

It is not the squaring, so much as the rectification inherent in the squaring,

Both a small filament and an ideal diode will have even order non linearity. The filament will have just a squared term and the diode will have an infinity of even terms. The spectrum from the filament will have just second harmonic and the diode will have an infinity of even harmonics. But the time constant blurs it all out and raises a DC component.

 

[Baluncore]

We agree that;
Squaring a fundamental sinewave produces a DC offset with a pure sinewave at 2f.
Rectifying a sinewave produces DC and even harmonics, decreasing towards infinity.
The time constant attenuates the amplitude of the harmonics.

The spectrum from the filament will have just second harmonic and the diode will have an infinity of even harmonics. But the time constant blurs it all out and raises a DC component.

While I claim that the DC component is there, quite independent of the time constant.

 

[DaveE]

But the time constant blurs it all out and raises a DC component.

While I claim that the DC component is there, quite independent of the time constant.

I think he's saying that the LPF reduces the HF stuff, then the DC is more prominent, not that there's actually more of it.

Because, the phrase "time constant" implies a linear response, which can't make more DC than whatever it gets.

 

[Baluncore]

I think he's saying that the LPF reduces the HF stuff, then the DC is more prominent, not that there's actually more of it.

And now you have done it.
The DC component is neither more prominent, nor raised.

The DC component is exactly what it is according to the FT.
Any harmonics present form a symmetrical ± ripple on the DC output.
The amplitude of those sinusoids may be attenuated.

 

[DaveE]

And now you have done it.

What's the point here? You do know that the 3 of us all understand this, right? It's not "rocket science" for analog EEs after all. My apologies for getting involved. I have been duly chastised.

 

[Swamp Thing]

It sometimes happens that members find it hard to see "where the OP is coming from", as they say these days. This could be because the experts don't remember grappling with the question from that particular perspective -- a perspective which, in the big picture, may not be not very useful. Or it could be because their specific gifts made it easy for them as students to sail through easily. So then you have threads like this one where the OP remains dissatisfied with the replies until the thread fizzles out or gets locked.

In this particular case I remember asking questions along nearly the same lines as the OP, which is why I found this pretty interesting:

It was thought by most people until about 1920 that the side frequencies were just a mathematical artifact ... Carson in the USA experimented with narrow quartz crystal filters and found that the side frequencies do exist.

As a student I wanted to know exactly why and how a filter (like one of Carson's filters in the quote) would produce an output when the input is an AM carrier centered somewhere far away from the filter's bandwidth. The answer that I finally figured out was something like this...

It's easier to start with a suppressed carrier AM signal that contains only the sidebands and no component at the carrier frequency. The thing about this signal is that its phase is zero degrees for one half of the modulation cycle and 180 degrees for the other half of the modulation cycle. If you apply this to a filter that is centered on the carrier frequency, the filter will build up an output during one half cycle, only to ramp down its output during the second half cycle as the opposing phase input drives it down. Think of a swing that is subjected to little pushes at its resonant frequency for a certain time, followed by the same number of pulls for the same time, and so on. So a filter of infinite Q will not build up significant output over many modulation cycles.

Now, what if the filter is centered around the carrier frequency plus the modulation frequency? Imagine that the filter is already "ringing" ("swinging") at its own resonance frequency. This ringing output is sometimes be in phase with the carrier frequency, and sometimes out of phase. It alternates between 0 and 180 according to the modulation half-cycle, if we take the carrier as our phase reference. But if we look at the suppressed carrier AM, its phase is also alternating between 0 and 180. Thus the input can keep on driving the filter's resonances to higher and higher levels -- theoretically infinite output if the Q is infinite.

Adding the carrier frequency back to get conventional AM won't affect this output, a la superposition.

QED, sort of, I think.

I am not sure if this will address the OP's concern, but there you go.

 

[Averagesupernova]

I think what the OP is looking for can be summed up most simply by saying that you cannot change a sine wave in any manner without generating new frequencies.

 

[sophiecentaur]

A snag is that the whole topic is more complicated than this thread is suggesting and we're trying to sum things up too simply.

I think what the OP is looking for can be summed up most simply by saying that you cannot change a sine wave in any manner without generating new frequencies.

"any form" is an overstatement. Phase and amplitude changes are linear.

Because, the phrase "time constant" implies a linear response, which can't make more DC than whatever it gets.

Another overstatement. A rectifier circuit will be operating with different charge and discharge time constants. A 'good' transformer with loads of copper and iron will produce a fast charge slope and a high resistance load will produce a slow discharge slope. The mean (DC) Voltage will be affected by both time constants. Without some regulation, audio amp psu's have to be over-engineered to reduce hum.

The best answer here is to appreciate how frequency and time domain descriptions can be interchanged by the FT and forget the nuts and bolts of the circuit involved.

 

[Averagesupernova]

"any form" is an overstatement. Phase and amplitude changes are linear.

Linear relative to what? Any change made to a sine wave will put energy in a different part of the spectrum than the original carrier during said change.