[sdiy] A question about phasers/notch filters

[Tom Wiltshire]

On 9 Oct 2011, at 09:30, ASSI wrote:

>> 
>> The bit I don't get is why adding that shifted signal back to the
>> original would give *moving* notches. I suppose it's to do with the beat
>> frequencies between the two signals, in the same way that two close
>> frequencies sound like one frequency modulated by an LFO (which isn't
>> really there). I'll have to think about that a bit more until it becomes
>> clearer.
> 
> Richie Burnett already answered that question briefly in response to how the 
> Buchla 297 might work.  Frequency is a measure of how fast the (unwrapped) 
> phase changes, so a constant frequency difference between two signals 
> imparts a linearly increasing phase shift (aka delay) between them.  Mapped 
> back to the concept of a traditional phaser that means you'd need to 
> implement notches that continually move to infinity and you'd have to 
> implement an infinite number of them.  That of course isn't realizable, 
> which is exactly why in a traditional phaser you will have to reverse the 
> direction of modulation at some point.

The key here is:
"Frequency is a measure of how fast the (unwrapped) 
phase changes, so a constant frequency difference between two signals 
imparts a linearly increasing phase shift (aka delay) between them"

Which I guess is what Neil meant about the fundamental relationship between frequency and phase.

What I was missing was that the original signal also needs to be provided in quadrature - I didn't realise that. Richie pointed me at the following site with a list of handy identities:

http://www.sosmath.com/trig/Trig5/trig5/trig5.html

The key ones for the frequency shifter are the Product-to-Sum formulas at the bottom, sin(u)sin(v) and cos(u)cos(v).
The one I was referring to with my 'beat frequency' example is actually the same thing in reverse, the Sum-to-Product formula:

sin(u)+sin(v) = 2sin(u+v/2)cos(u-v/2)

..so if you add two sine waves together, what you get is one sinewave at half the sum of the frequencies, modulated at half the difference.


Thanks to everyone for taking the time to explain this to me. I've really learned something, and now I *do* get how a frequency shifter works, and how it produces infinite phasing.

Tom