Equations for high pass filter? [closed]

Question

I made my own low pass filter in matlab by taking a moving average of the signal data. But if a moving average creates a low pass filter, how exactly does one design an equation for a high pass filter? I understand the intuition regarding using an average for low pass (high frequencies will average out to zero but low frequencies will average out to a number close to the signal value).

But is there any equation used for high pass filter?

Answer 1

There are a lot of equations for that! Perhaps the simplest one is the one-sample delay difference function,

y[n] = x[n] - x[n-1]

or, taking its Z-transform

H(z) = 1 - z^-1

Where H(z) = Y(z) / X(z) is the system equation for the filter.

Using AudioLazy with MatPlotLib (Python), you can see a frequency response plot for this highpass filter by typing. (Disclosure: I am the author of AudioLazy)

from audiolazy import z
(1 - z ** -1).plot().show()

You can apply it to a signal, as well

from audiolazy import z, Stream
filt = 1 - z ** -1
sig = Stream(1, 3, 1, -1, -3, -1) # Periodic signal
filt(sig).take(7)

Resulting in the first 7 samples:

[1.0, 2, -2, -2, -2, 2, 2]

The same can be done in GNU Octave (or MatLab):

filter([1, -1], [1], [1, 3, 1, -1, -3, -1, 1])

Which returns

[1, 2, -2, -2, -2, 2, 2]

That's a FIR filter in a 6-sample-periodic signal that decays from [-3;3] amplitude range to [-2;2] range in this example. If you try with a 12-sample-signal (lower frequency):

filt = 1 - z ** -1
sig = Stream(1, 2, 3, 2, 1, 0, -1, -2, -3, -2, -1, 0)
filt(sig).take(13)

Now the result is another square wave, but in the [-1;1] range. You should try the same with sinusoids, which are meaningful for the frequency response and should keep another sinusoid as the output of the filter, with the same frequency.

You can also use a resonator at the Nyquist frequency, giving you an IIR filter. There are several other filters design that can do so (e.g. Butterworth, Chebyshev, Elliptical), for different needs. Minimum phase, linear phase, filter stability and minimizing the ripple amplitude are some possible design goals (or constraints) you can have while designing a filter.