[sdiy] Phase shifts and instantaneous frequency

[Aaron Lanterman]

I had to add my two cents in here.

In the sophomore "Introduction to Signal Processing" class I teach, we  
talk about "instantaneous frequency." If you have a cosine with a  
generic function in its argument, like

x(t) = cos(psi(t)),

we define the instantaneous frequency (in Hertz) as

f(t) = (1/(2pi)) (d/dt) psi(t).

Clearly, if you apply this idea to cos(2pi f t), you get an  
instantaneous frequency of f, which is what you expect. You can then  
extend this notion to all sorts of things. If you want to think about  
adding a phase that varies with time, you get

x(t) = cos(2 pi f t + theta(t)), and you get an instantaneous  
frequency of

f(t) = f + ((d/dt) theta(t))/(2pi)

So, say theta(t) = A*2*pi*cos(t) would give you an instantaneous  
frequency of f(t) = f - A sin(t).

So you can think of your time-varying phase as imparting a time- 
varying frequency (or vice versa).

This should be familiar to those who dig into the distinctions of  
"phase modulation" vs. "frequency modulation."

- Aaron