Digital Audio Filters
Idea: get signal into system as close to Nyquist as possible
Do filtering mostly in software (or digital hardware)
Can build much better filters
Aside: Number Representation
How shall we represent samples for this kind of processing?
Obvious choice: integers at sampling resolution
Can get weird for 24-bit, so promote to 32?
Math is tricky: overflow etc. Promote to next higher size?
What resolution to output? May have more or less precision than started with
Obvious choice: floating-point
Scale input to -1..1 or 0..1
32 or 64 bit? (32-bit conveniently has 24 bits of precision)
Issues of precision and resolution mostly go away (Inf and NaN).
Fast with HW support, slow otherwise especially on 8-bit hardware
Less obvious choice: "fixed-point"
Treat integer as having implicit fixed "binary point"
.1001011000000001 1.001011000000001 -.001011000000001 10010110.00000001
Fiddly, especially for languages that don't allow implementing a fixed-point type with normal arithmetic
Slightly slower than integer: must keep the decimal in the right place
Typical used on integer-only embedded systems, "DSP chips"
Strongly suggest 64-bit floating point for this course: just say no to a bunch of annoying bugs
Obvious approach: Convert to frequency domain, scale the frequencies you don't want, convert back
For real-time filter output, this in principle means doing a DFT and inverse DFT at every sample position, which seems…expensive to get one sample out
Can cheat by sliding the window more than one, but you will lose time information from your signal
Also, DFT has ripple: frequencies between bin centers will be slightly lower than they should be, since they are split between two bins and the sum of gaussians there isn't quite 1
Frequency resolution can be an issue: a 128-point FFT on a 24KHz sample will produce roughly 200Hz bins, so the passband is going to be something like 400Hz, which is significant
FIR and IIR Filters
We characterize filters in terms of impulse response: what if you have an input sample consisting of a single pulse of amplitude 1 and then zeros forever?
Taking a look at the DFT sum, our DFT filter will treat an impulse anywhere in its window identically (linear time-invariant). When the pulse leaves the window, the FFT will then say 0 forever
We call this Finite Impulse Response: an impulse presented to the filter will eventually go away
A trick that we will explore is to actually use past filter outputs as well as inputs to decide the next filter output
In this case, an impulse will make it into the outputs, which means that it will be looped back into the inputs: Infinite Impulse Response
Of course, the IIR filter should reduce the amplitude of the impulse over time, else badness. Such a filter is a stable filter
IIR filters have cheap implementation (analog or digital) per unit quality, but:
Are less flexible
Are harder to design
Have lots of issues with stability, noise, numerics