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

  • Low Pass

  • 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 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

  • FIR 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..1

  • Filter 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 reinforce

  • This 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 sample

  • Now the cost is greater, and the latency is higher, but the quality can be very good

  • Where 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 inputs

  • Convolution 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, Hanning

  • Note 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"

Last modified: Thursday, 11 April 2019, 3:49 PM