Filter math

drewfx1 recently pointed out that FIR and IIR filters both have benefits, and that each can be applied to suit different priorities.
 
I went to study up and found this video helpful:
 

mike_mccue
 
I went to study up and found this video helpful:
 

Mike, if you truly understood this video tutorial, then I'm duly impressed.  When I was studying engineering in college, I frequently encountered the kind of math involved in DSP calculations (although I never directly studied DSP itself).  However, that ship sailed a long time ago.  This stuff isn't for the faint of mind.
 

dmbaer

mike_mccue
 
I went to study up and found this video helpful:
 

Mike, if you truly understood this video tutorial, then I'm duly impressed.  When I was studying engineering in college, I frequently encountered the kind of math involved in DSP calculations (although I never directly studied DSP itself).  However, that ship sailed a long time ago.  This stuff isn't for the faint of mind.
 

Hi David,
 I don't think "truly understand" applies for me, but I did find that I was able to follow the explanation as the overview it seemed it was meant to be, and I felt like I understood more at the end than I did at the beginning.
 
 I was especially interested in the idea that FIR refers to a data table, that so called Linear Phase EQs are FIR, and that the reference to a large data table can cause lots of loading. It made me think of the infamous LP64 EQ and all its glitches.
 
 I'd like to follow some examples of IIR where the variables are replaced with actual numbers so I can get a better understanding of what goes on.
 
 best regards,
mike

mike_mccue
It seems like seeing first hand how a series of samples go in one side and come out the other with a 1 octave wide boost at 5kHz would be something that would help with furthering an understanding.
 

That's exactly what the impulse response shows you.
 
A single sample impulse input to the filter contains all frequencies with flat response and the impulse response output by the filter contains the filtered response. You have a series of filter coefficients that produce a given impulse response. The impulse response completely defines what the filter does in the time domain - it equates to the frequency and phase response of the filter in the frequency domain. 
 
The filter coefficients define either an FIR or an IIR filter and you just have a series of single sample delays, multiply the delayed samples by each associated coefficient and add each result back to your signal (with an IIR some of the coefficients feed back as well). IOW, the basic structure of the filter is either FIR or IIR, but is otherwise the same regardless of what the filter is actually doing - what changes is the coefficients.
 
The tricky part is getting the filter coefficients that produce the desired results. 

What you're going to see if you do that Mike is exactly what you see now if you just look at any waveform before and after you put it through a filter. Do you really want to go through multiplying each sample by each coefficient and adding them together? IMO, it's going to be very tedious and not very enlightening.
 
 
But you can think about it this way - filters work by "remembering" the previous samples (the capacitor does this in the analog world) and using phase shift.
 
Consider that at a given sampling frequency there is a phase difference between adjacent samples based on the relationship between the frequency of the signal and the sampling frequency. At low frequencies the phase shift is very small - 0° at 0Hz (aka DC) and 180° at the Nyquist frequency (exactly 1/2 the sampling rate = exactly 2 samples per cycle = 180° phase difference).
 
So basically what is happening is you are phase shifting your original signal with the single sample delays, multiplying the signal that has been phase shifted by the appropriate coefficients, and then adding the result back into the original signal. This will either add or cancel at a given frequency, and to varying degrees depending on the amount of phase shift at a given frequency and the value of the coefficient (including whether it's positive or negative).

bitflipper
 
BTW, for those reading this thread with a big cartoon question-mark hovering over their head, one of the best simplified explanations I've come across is in Mr Aldrich's book, required reading for anyone dipping their toes into the technical side of things generally.

Er ... thanks, Bit ... I guess.  Damn Amazon and it's insidious "buy with one click"!

You're so good with explanations, Drew! Maybe next you could address what makes a compressor "musical".