# Finding an Intuitive Explanation of Mixer Theory

I don't know about you guys, but I've always been somewhat confused by fundamental mixer action. I've always been left unsatisfied by the standard trig identity explanation as one that's a tad too abstract. It has a kind of "it just is" feel about it and that always makes me want to dig deeper.

So my question is, how does a mixer create a carrier frequency and sidebands?

I've answered this question with my own intuition about how mixing works and how sidebands are formed and it would be good to get some feedback from more informed members. I'd also love to hear other peoples intuition as I find this topic very confusing.

• I suggest you replace the terms "modulating frequency" and "carrier frequency" with the terms "modulating signal" and "carrier signal," respectively. "Frequency" is a characteristic of a signal. As you point out, it is the "signals" that are being multiplied, not their "frequencies." Oct 12, 2018 at 15:21
• Great feedback Brian. I've updated my "note" and the text in question. Oct 12, 2018 at 16:03
• Hi Buck. If you find a comment helpful, it's best to upvote it by clicking on the triangle to its left. :-) Same thing applies to answers. Oct 12, 2018 at 17:06
• "At all other frequencies, the mis-timed rotation of the stick and the consequent distribution of the squiggles will produce zero vector averages" — what about the sum and difference frequencies? Can you explain how they fall out of the multiplication? [Also I agree w/what Phil said — recommend splitting this into a question about how mixing intuitively works, and then providing your writeup as an answer.] Oct 12, 2018 at 19:39

An intuitive picture of mixing can be created by understanding musical beat notes.

What is a beat note? If you play two equal amplitude sinewave-like notes slightly out of tune, instead of hearing two notes, you will usually hear something called a beat note or a periodically amplitude modulated note. If one gets out their handy dandy audio spectrum analyzer (or strobe tuner), one will find that the pitch of this new note is no longer at either of the two original note frequencies, but halfway in between.

However if you now plot this new additive waveform on your computer, you might note the phase of the new modulated sinewave changes to the opposite phase between every "hump". But a simple way to get rid of this phase change is to add a large enough sinewave of that new halfway-between center frequency.

Now all you've done is added 3 sinewaves of slightly different frequencies (two equal sidebands and a carrier halfway between them), but a plot show you've ended up with the exact same waveform as an envelope modulated carrier that an AM modulator would create via signal multiplication.

So an assumption that these two different domain representations are the same thing would be reasonable under various conditions (linearity, time-invariance, etc.)

If adding 3 sinusoids gives the same result as multiplying 2 sinusoids, then the inverse should also hold true, and a multiplying mixer should produce a signal with a carrier and two side bands (in an FFT plot of the 3 sinusoids).

• A rotation visualization might work better for an IQ mixer (e.g. the complex multiplication of two complex waveforms). Oct 13, 2018 at 16:08
• Is a beat note really the same thing? Two instruments playing out of tune notes add linearly, whereas a mixer multiplies. Oct 15, 2018 at 13:50
• Yes, linear addition of sines or cosines gives the same result as a multiplication of two other sinusoids. It’s a result of the trigonometric product-sum identity. Oct 15, 2018 at 16:49
• Also called the trigonometric sum-product identity. Oct 15, 2018 at 16:57
• I see, so two instruments don't mix, but the produce the same result as mixing the beat frequency and the average of the two frequencies together would Oct 15, 2018 at 17:33

Understanding how mixing actually works can be hard to grasp in an intuitive way. This explanation aims to shed some light on how mixing works in a way that doesn’t involve complicated maths. It doesn’t trot out the usual trig identity explanation, which I find falls short of a comprehensive explanation. Instead, the explanation rests on two major insights: an intuitive way of understanding Fourier analysis and the idea of multiplication as a scaling operator that scales the presence of the modulating signal and its effect on the carrier.

Let’s get the multiplication aspect out of the way first, since it’s the easier of the two to understand. Mixing occurs when two signals are multiplied by each other. Not added or subtracted – multiplied, this is very important. Let’s assume the carrier signal in a 1kHz sine wave and the modulating signal is a 100 Hz square wave of 50% duty cycle and is either on or off. If you “add” these two frequencies, you would get energy at the points in the cycle where the modulating signal is zero simply because some value + zero = some value.

On the other hand, if you multiply the two frequencies, you get zero energy where the modulating signal is zero and a scaled output of the carrier signal (scaled according to the magnitude of the modulation signal) where the modulation signal is non-zero.

This example, where the modulating signal is either on or off, makes it easier to understand the multiplicative effect and indeed many mixers operate using this exact method. However, you can see that the same principal applies when the modulating signal is a sine wave and you have points in the cycle where the amplitude falls to zero. You could say that modulation introduces a rate of change to the carrier and this idea of a changing carrier gives rise to the side bands, as explained below.

Let’s have a look at the product of two sine waves, one at 1kHz and one at 100Hz produced by Maxima:

Here you can clearly see the multiplication operation at work. Notice how the amplitude of the carrier signal drops to zero at intervals coinciding with the drop in the amplitude of the modulating 100Hz signal. Notice also, how the carrier is changing up and down at a particular rate determined by the modulation frequency.

The next thing to understand is a beautifully intuitive way to visualize how Fourier analysis “detects” the presence of energies at specific frequencies. Similar intuition is available at several sources. My favorites are:

Stuart Riffle's bead and stick intuition

3Blue1Brown's Fourier Visualized

In either case, the essential intuition is the idea of spinning the signal at different frequencies and measuring the "balance" of the resulting plot. If the signal is spun at a frequency where energy is present, the correspondence of the spin frequency and the peaks in the signal will skew the plot off-balance.

It's relatively easy to imagine spinning the signal at the carrier frequency and consider the resulting plot as a series of loops off to one side. The loop magnitudes will change according to the modulation but the average will still be an off-centre vector indicating energy at the carrier frequency.

However, the sideband's are trickier to understand. My intuition is related to the rate of change of the carrier. If you imagine the signal is spun at the carrier frequency as before but now you want to match the rate of increase (or decrease) at every instant as the carrier ramps upwards (or downwards) according to the modulation. Then you will need to spin faster (or slower) to catch that ever rising (or falling) slope and you will understand that the sideband frequency is that frequency that produces a non-destructive increase (or decrease) to the carrier.

The example chosen here is the ideal scenario of multiplying two pure sine waves to simplify the explanation. In reality, a practical mixer will exhibit an exponential or chopping type action. In these cases, the multiplication is inherent to the process alongside many other mathematical terms (power terms) and whilst you get the “multiplying action” described above, you also get other types of signal operations that will produce additional intermodulation products.

• "If you imagine the signal is spun at the carrier frequency as before but now you want to match the rate of increase (or decrease) at every instant […]" — this is interesting but if I imagine a single sideways circle/disk tracing the pattern it either has to change size or has to rotate back and forth which is a much bigger range of speeds (=frequencies) than I would expect. [No idea what this means or if it's even on the right track, just a random observation!] Feb 22, 2022 at 19:00
• @natevw-AF7TB For sidebands, I like to think of the bead and stick or you could go with just visualizing am signal as given in my post and simply imagine spaced dots along the time axis representing the "spin" or "test" frequency. It's easy to see in either case, the loops (or dots) lining up at the carrier frequency. But, if you slow the "spin" frequency down a bit (or speed it up a bit), you could imagine the loops spreading out around the stick axis with varying amplitudes. The thing is, they will average out to some positive value (much lower in amplitude). These are your sidebands. Feb 24, 2022 at 12:51

I found the discrete case of sampling easier to initially grasp. Imagine you have a sine wave with frequency = 1, that is it completes a full cycle in each unit on the x axis:

Now if you sample that with frequency = 1 (meaning, on each integer of the x axis), you are sampling at the same phase each time, so you just get a straight line.

This is because the sampling frequency and signal frequency are equal, so their difference is zero. A frequency of zero is a straight line.

If we change the sampling frequency to 1.1 (meaning, we sample at every 1/1.1 or about every 0.90909090 units along the x axis), then we get something more interesting:

A sine wave with frequency 0.1 appears because the sampling frequency (1.1) minus the signal frequency (1) is 0.1.

• Nice visual explanation of aliasing.
– wbg
Mar 2, 2022 at 22:52

One can look at a signal in the time domain and, if it is not a perfect sine wave, Fourier analysis tells us that more than one frequency exists. For instance, a square wave contains the fundamental frequency plus many odd harmonics which can be seen on a spectrum analyzer - a device that effects a Fast Fourier Transform (FFT) on the signal from the time domain to the frequency domain.

Here is a plot of an AM signal in the time domain and in the frequency domain. It is obvious from the time domain graph that the carrier signal Amplitude is being Modulated (AM). The frequency domain graph makes it clear that the modulation frequency is a single sine wave that creates the two sidebands characteristic of an amplitude modulated signal.

http://www.ni.com/cms/images/devzone/tut/am1.JPG

Seems to me that mixer products are intuitively obvious when displayed in the frequency domain, i.e. seeing is believing. \$25 SDR dongles plus free software running on a PC provide a great visual learning experience about carriers and sidebands from DC to 1+ GHz.