# Digital Filters

## Filters

Change amplitude / phase of frequencies of sound

Many applications

"Tone" control, "Equalizer"

Effects, e.g. "wah"

Band limiting for antialiasing, resampling, etc

Etc etc

## Common Ideal Filter Shapes

Usually 0-1 with Passband, Stopband: goal is to block some range of frequencies while leaving others alone

High Pass, Bandpass, Band Notch

## Units and Normalization

Common to leave out sampling rate and gain in DSP

In time domain, samples are just numbered

In frequency domain, frequencies range from 0..1 where 1 is the Nyquist limit

Amplitude is normalized to -1..1 in time domain, 0..1 in frequency domain

We have already talked about omega, dB

- There are several dB scales floating around

## Filter "Quality" Measures

The ideal low pass filter is a "brick wall":

Gain in passband is exactly 1 for all frequencies

Gain in stopband is exactly 0 for all frequencies

Transition is instantaneous (vertical) at corner frequency

## Analog Filters

Made of electricity: resistors, capacitors, inductors, op-amps, etc.

Analog filters are simple, of necessity

Analog filters are kind of meh: typically use as few of them as possible when digital is available

Obvious example: anti-aliasing and DC removal "blocking" (typically a blocking capacitor)for DAC and ADC

## Aside: Linear Time-Invariant Systems

Normal filter definition / requirement

Output signal is a linear function of input signal ("no distortion")

- Preserves frequencies of input waves

Output signal does not depend on input time

- Signals are notionally infinite, so this is a hard constraint

Analog filters are LTI

## Digital 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

Forgot to talk about this earlier: 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

Fast

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

## DFT Filters

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 1Frequency 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 awayA trick that we will explore is to actually use past filter

*outputs*as well as inputs to decide the next filter outputIn 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*filterFIR 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

## Simple FIR Lowpass Filter

Let's design an FIR lowpass filter

First, some notation:

`x(n)`

is the nth sample of input,`y(n)`

is the nth sample of output. Amplitude of sample is assumed -1..1Filter equation:

`y(n) = (x(n) + x(n - 1)) / 2`

Why is this a low-pass filter? For higher frequencies if sample

`x(n)`

is positive sample`x(n-1)`

will tend to be negative, so they will tend to cancel. For lower frequencies the sample`x(n)`

will be close to`x(n-1)`

so they will reinforceThis filter is kind of bad: the frequency response doesn't have much of a "knee" at all

On the other hand, this filter is stupidly cheap to implement, and has very little latency: the output depends only on the current and previous samples

## Wider FIR Filters

Normally, you want a much sharper knee

To get that, you typically use more of the history

For standard FIR filters, it is common to use thousands of samples of history

General FIR filter:

`y(n) = (1/k) x(n-k … n) ∙ a(k … 0)`

So

*k*multiplications and additions per sampleNow the cost is greater, and the latency is higher, but the quality can be

*very*goodWhere do the coefficients

`a`

come from? More in a bit

## Inversion, Reversal, Superposition

Why the obsession with lowpass? Because we can get the other kinds "for free" from the lowpass

Inversion: Negate all coefficients and add 1 to the "center" coefficient — this flips the spectrum, so high-pass

Reversal: Reverse the order of coefficients — this reverses the spectrum, so high-pass

Superposition: Average the coefficients of two equal-length filters — this gives a spectrum that is the product of the filters. If one is low-pass and the other high-pass, this is band-notch. We can then invert to get bandpass.

## Convolution

A filter can be thought of as a

*convolution*of the input signal: sum of possibly delayed weighted inputsConvolution is probably out of scope for this course, but pretty cool

Interestingly,

**multiplication in the frequency domain is convolution in the time domain**. This means that we can use a DFT as a convolution operator if we like

## FIR "Windowing" Filters

In general, simplest low-pass filters: take a "window" of past samples, then "round off the corners" by multiplying by some symmetric transfer function

There are

*many*window functions, each with their own slightly different properties as filters: simple things like triangular, plausible things like cosine, and weird things like Blackman, Hamming, HanningNote that windowing is also how we deal with edge effects of DFT: we make the signal have period equal to the DFT size by applying a window, but this also low-passes and changes the signal

## FIR Chebyshev "Remez Exchange" Filters

There's a fancy mathematical trick for approximating a given desired filter shape with high accuracy for a given filter size

Involves treating filter coefficients as coefficients of a Chebyshev Polynomial, then adjusting the coefficients until maximum error is minimized

Probably not something you want to do yourself, but there are programs out there that will do it for you

## IIR Filters

Can get much better response per unit computation by feeding the filter output back into the filter (?!)

In some applications, a 12th-order IIR filter can replace a 1024th-order FIR filter

Design of these filters really wants a full understanding of complex analysis, outside the scope of this course

Fortunately, many standard filter designs exist: Chebyschev, Bessel, Butterworth, Biquad, etc

Basic operation is the same as FIR, except that you have to remember some output:

`y(n) = (1/(k+m)) (x(n-k … n) ∙ a(k … 0) + y(n-m-1 … n-1) ∙ a(0 … m))`

Always use floating point, as intermediate terms can get large / small

Really, just look up a filter design and implement it: probably too hard to "roll your own"