In my recent post Is there a perfect synthesizer?, I of course talked about frequency modulation synthesis, and about the fact that what Yamaha implemented e.g. in their DX7 is in fact phase modulation – which is in that context “essentially the same”.
Some people have challenged that statement, or rather demanded a more thorough explanation, which extends beyond that mentioned in the literature section of the aforementioned post. So here it goes: an entire article about phase modulation, frequency modulation and how they are connected.
Frequency and Phase
Before we can start modulating them, we first need to get an understanding what “frequency” and “phase” is, and how they are related to each other.
First, assume a function  of an angle, y(φ). It may also depend on other variables, but we start with considering that angle for the time being. The angle is not measured in degrees, but in so-called radians, with 2π=360°(=”full circle”). Using those radians makes the discussion a little easier. The function y is periodic with 2π, which means that when the function argument reaches 2π (the period), the values start to repeat, or y(φ+2kπ)=y(φ), with k any integer. Apart from that, the function can be anything you’d like it to be. The function’s argument, the angle φ, is called the phase.
Next, we’ll introduce the angular frequency ω=2πf (unit: radian angles per second), and followingly, we’ll use frequency as short for “angular frequency”. The frequency here defines how an angle (in that case, our phase angle) increases per time amount, much like the speed of something describes how many metres per second (or kilometers per hour) it moves.
So that frequency is somehow related to the phase, but how? The simple case is that the frequency is constant, in that case the phase is φ(t)=ωt. In other words, if the angle progresses at the frequency ω, and it progresses for the time t, then after that time, the angle has increased by φ.
Things get trickier if the frequency changes with time, however. Then we can’t just use multiplication, we need to do an integral so the equation changes to
People versed in math will now start to point out that this equation misses an additive constant. Yes it does, but in the following discussion, we’ll just ignore additive constants and also constant scaling factors. We just assume we can work with them with a simple parameter in our synthesis engine. Please remember that last sentence, especially in the section on modulation, before you start to write to me to point out errors.
A simple analogy: running on a track
Assume you’re running on a track with a constant speed of 5 m/s. Which distance will you have covered after five seconds?
It’s simple: the distance s is your speed times the time you’re running at that speed, so s=vt=5m/s*5s=25m. And of course, if your speed changes during that run, it’s not so simple anymore – you need to consider for which time amount you’ve been running at which speed or how you’ve been accelerating – mathematically speaking: you need to do an integral.
The analogy here is plain: replace the distance s with our phase φ, the speed v with our frequency ω, and you’ll see it’s exactly the same.
Summary: Frequency and Phase
We have a phase φ and a frequency ω, which are related by the equation
or, in the special case of a constant frequency, by the simple equation
The phase is the argument of a function y(φ) which, with the equivalence (1) above, can be rewritten as
Now we’ll get on topic quickly, because that function y will later become the output of our synthesizer’s oscillator, and φ and ω are the quantities we want to modulate in phase or frequency modulation, respectively.
Modulation is a term from communication technology and describes varying a property (or several ones) of a periodic waveform, called the carrier signal, by another waveform, called the modulating signal. Now we’ve already got one waveform from the discussion before, y, and that will be our carrier signal. This carrier signal has properties, namely phase, frequency, and time, which all somehow depend on each other. As we can’t modulate time, we can look at modulating phase or frequency, and this is just what this article is all about .
We still leave the properties of our carrier signal undefined, but we’re gonna pick a simple example for our modulating signal: sine or cosine waveforms . In the discussion, we’ll switch freely between sine and cosine, mostly because in the scope of this discussion, they’re very similar – in fact, with cos(x)=sin(x+π/2), we can quickly transform one into the other.
In the chapter before, we’ve already considered a frequency varying with time. This can be seen as the sum of the constant part ωC (with C for “carrier”) and the variable, modulated part ω(t). This time dependency now surfaces in the form of the frequency being modulated by a cosine wave. So our frequency becomes a cosine function of time with in this case a constant frequency for the modulator ωM=const . Thusly, our signal frequency is
Putting that into equation (1) and using the integral of a cosine being a sine, we get
(3) -> (1): φFM(t)=∫ωC+cos(ωM*τ)dt=ωC*t+sin(ωM*t) (4).
This time, our frequency only has a constant component, ωC. The variable, modulated component is now added to the phase. We use a sine wave for the carrier (which, as we’ve discussed, is about the same as the cosine we did use before), and keep our modulator frequency ωM constant again.
With that, we get for the phase:
φPM(t)=ωC*t+sin(ωM*t) = φFM(t) (4).
It’s all the same! Really?
The proof is above. The equations are the same, so it’s the same. However, I can just wait for you challenging that, mostly with:
- it is only valid for specific output functions y,
- it is only valid for specific (in that case: sine/cosine) functions for modulating,
- it is no longer valid if you modulate the modulator signal, just like you can do in a proper FM/PM synth,
- I got both a FM and PM synth, and they sound completely different!
ad 1: “it is only valid for specific output functions y”
This is not the case. As you can see, we didn’t even look at the output function y itself, only at its argument.
ad 2: “it is only valid for specific (in that case: sine/cosine) functions for modulating”
This seems to be true: only some functions (exponential functions, to be precise, to which sine/cosine are related somehow ) derive and integrate into itself or nearly itself. So if you would use something that is not a sine or cosine, it will no longer work. Now wait a minute – didn’t Fourier tell us that all signals with specific properties (in short: all the relevant signals in this context) can be written as a sum of sines or cosines, so we could just break down our arbitrary modulating signal into cosines, then integrate and stay the same?
Obviously not. This is the moment where the scaling factors we so hastily decided to ignore above start to come back into discussion. Before, we could just rescale the entire thing and be fine. But now, due to the scaling factor, each summand gets a different scaling factor, so this no longer works.
So, if you want to use an arbitrary signal for modulating, and can’t create its integral (or derivative) – both is easy, you just need a lowpass or highpass, respectively – then there is indeed a big difference between modulating the phase and the frequency.
ad 3: “it is no longer valid if you modulate the modulator signal, just like you can do in a proper FM/PM synth”
Also, this is true. As a simple example, if you modulate the modulator frequency in phase modulation with another sine, then you’d need a sum of two multiplied sine/cosine functions to get the same result. It still can be done, only it’s more complicated.
ad 4: “I got both a FM and PM synth, and they sound completely different!”
In addition to what we explained above, here’s yet another reason, taken from my article about the perfect synth: Even if it’s the same in theory, different implementations of the same synthesis architecture will still sound different. And that’s a good thing.
Summarizing – do we need both?
We’ve seen, in the first chapter, that in theory and for some special cases, frequency modulation and phase modulation are exactly the same. And in general, you can get the same kinds of sounds with both approaches, only one specific sound might be easier to obtain one way or the other.
So why do we need it in the first place?
The vast majority of analogue synthesizers of the 70s worked with subtractive synthesis. One or several oscillator signals rich in overtones (saw, triangle, square, pulse) were sent through one or several filters which would then remove the upper or sometimes lower frequency content, or cut out a segment in the middle. The resulting sounds analogue synths are known (and to this day, highly regarded) for are fat leads, warm strings or throbbing bass sounds. And those adjectives say it all. You usually don’t hear someone raving about an analogue synth doing metallic leads, crystal-clear strings or clangy basses. They just can’t do it – at least not with a number of oscillators and filters that makes sense .
The reason has to do with the frequency spectrum of different instruments, especially of idiophonic percussion instruments which are not based on a harmonic frequency spectrum. And so, while the frequency spectrum of those subtractive synthesis examples can look convincingly like the sustained part of a string sound, a church bell sound is completely different. And this is where FM – or PM – comes in. It simply allows us to generate sounds that can’t be done with subtractive synthesis.
But there’s more. While with subtractive synthesis, the result was always very stable with regard to changes in the synth’s parameters (i.e. slowly lowering the lowpass cutoff frequency would make the sound equally slowly more dull), modulating a key parameter in FM – like the modulator’s frequency, the resulting sound will change quite drastically, if sometimes not very intuitively. So for some expressive playing of sounds, this can also greatly add to your sonic repertoire.
The latter (and here I’m coming back to my line of argument from the last article) is also the reason why FM synthesis cannot be simply replaced by using sample-based synthesis: even if you can imitate that electric piano even better with samples than with FM synthesis, you can’t morph the sound as crazily as you can e.g. on a Yamaha FS1R.
Why the confusion with the names?
FM synthesis was first introduced as a mass product by Yamaha in 1983 in their DX7 synthesizer. This is a well-known fact – but is it true?
The synthesis implemented here was licensed by Yamaha from a patent by John Chowning at Stanford University for a “method of synthesizing a musical sound” . While this application talks of frequency modulation, what the author describes (even though he doesn’t exactly go out of his way to keep his physical units in check) clearly indicates PM.
So did Yamaha simply license that faulty wording with the patent? I’d rather guess that they simply stuck with it because “frequency modulation” and FM were already known household language – FM radio stood for “better, clearer, and more modern” in comparison to AM radio, so it made sense to exploit that existing analogy for marketing this new technology.
Why PM was used for the actual implementation instead of FM is really open for some guessing. Some sources  claim that for low frequency (as opposed to radio frequency) applications, PM is easier to implement, especially in a digital synthesizer. Hordijk also points out that PM allows for selfmodulation of an oscillator without changing the fundamental frequency – and that is an important property for doing musical implementations of the concept .
- For the math geeks: this needn’t be a function necessarily. However, sticking to a function makes life easier, and also does not limit our choices because all of the signals we’ll be looking at will be functions.
- Yes, in theory it would be possible to modulate time as well, but no one has ever done this for communcation engineering or use in synthesizers – prove me wrong, please!
- Incidentially, Yamaha used sine waves for their FM synthesis products, such as the DX7.
- But yes, this frequency can also vary with time by being modulated by yet another modulator, and this is also done in FM synthesizers.
- The details here: cosine and sine are the real and imaginary part of the result of a complex exponential function with only an imaginary argument (i.e. exp(jx)=jsinx+cosx). With exp(jx)’=jexp(jx), by looking only at the real part, we get cosx’=-sinx, and looking at the imaginary part, sinx’=cosx.
- Rob Hordijk: FM workshop (from: Nord Modular Tips & Tricks).
- If you move away from the “number of oscillators that makes sense” limitation, you can of course create those metallic, crystal-clear or clangy sounds by adding signals from a large number of (sine) oscillators. This is called additive synthesis, as used by synthesizers like the Kurzweil K150 or the Kawai K5000.
- John M. Chowning “Method of synthesizing a musical sound“, U.S. patent 4018121.
- Unfortunately, I don’t remember those sources – anyone?