Wiki claims that this discrete process is the "same as" the low pass filter:
One proof strategy that I can think of:
function lowpass(real[0..n] x, real dt, real RC)
var real[0..n] y
var real α := dt / (RC + dt)
y := α * x
for i from 1 to n
y[i] := α * x[i] + (1-α) * y[i-1]
I'm now trying to understand why this doesn't "wrap around". I would naively
expect: (1) low frequency = minimal destructive interference (2) high frequency
(phase difference 180 degrees) = maximal destructive interference (3) EVEN
HIGHER frequency (phase difference greater than 180 degrees) = non-maximal
I believe the resolution is to remember that we only build signals upto what
Niquist lets us, so we build at max a sin wave of frequency
- Split signal into fourier components
- Notice that if the frequency is higher than some threshold, we will start introducing phase deltas larger than . This will cause cancellation of a wave.
- See that this makes sure that only low frequency signals live.
fs/2 where the same
fs. Thus, when we phase shift, we will get a phase of at max 180 degrees.
As to why the Niquist ratio is
1/2, imagine a circle. a
1/2 is how much
"ambiguity" we have in the phase. In that, if we jump by more that
1/2 of the circle), we will ????