Human Computer Interface Technology

Digital Signal Processing for HCI
October 14, 2002

*Copyright 1999-2002,
Perry R. Cook,
Princeton University

I. What's (a) DSP?

  • DSP stands for Digital Signal Processing (or Processor)
  • It's a branch of engineering mathematics which deals with the processing (filtering, detecting, etc.) of sampled data signals.
  • If we make some assumptions, like the signals are sampled uniformly in time, the systems we wish to study don't change (much) with time, and the systems are linear (definition to come), then the math gets very nice indeed.

  • We care about DSP for HCI because we get lots of signals from our sensors and it's assumed that we want to do intelligent things with them in efficient ways. DSP gives us a set of tools that make certain things easier than the alternative: scratch our heads, try lots of hacks and heuristics, write lots of code, and fail.

II. Linear Time Invariant Systems

  • A linear system is one that exhibits two properties:
    • Homogeneity: If x -> y, then kx -> ky for any constant k
    • Superposition: If x1 -> y1, and x2 -> y2, then (x1+x2) -> (y1+y2)

    • (NOTE: it's a convention that x is the input to a system and y is the output)

  • Examples of Linear Systems:
    • Room acoustics is linear, because sounds simply add up, and nothing changes except volume if you make the sounds louder or softer.
    • The outer and middle ear are essentially linear.
    • Your stereo amplifier is linear (as linear as you can afford), and any measure of distortion is a measure of the non-linearity.

  • One more property that's nice is Time Invariance, which just basically means that the system behaves the same independent of what time a signal is put into it:

    • if x(n) -> y(n), then x(n+T) -> y(n+T), for any constant T

  • OK, it's true that technically no system is time invariant, because physical parameters change (you amplifier ages, you change the volume and tone controls, etc.), but if the system is time invariant over a suitable observation period, or only changes slightly, or very slowly (relatively speaking), then we can still call it linear for analysis purposes.

III. Convolution

  • Linear, Time Invariant (LTI) systems can be dealt with by using a mathematical operation called convolution:

    y(n) = SUM[ x(n-T)h(T) ]

    where h(T) is the "impulse response" of the system H. Another way to think about the impulse response, and convolution, is that h(T) is just the response y(n) for a special type of x(n). called an impulse. That special impulse signal has value 1.0 at time zero, and value 0.0 all other times. This then means that convolution is just the impulse responses of the system to a train of weighted impulses (the individual samples of an arbitrary x are viewed as just weighted impulses at different times). Since the system is assumed to be LTI, we know that the weighting (homogeneity) and delaying (time invariance) and adding up (superposition) is legal.

IV. Filters

  • Digital Filters act linearly on a signal by forming weighted sums of the signal, and also of past values of the output of the filter:

    y(n) = g[x(n) + a1 x(n-1) + a2 x(n-2) + ... + aN x(n-N)
    + b1 y(n-1) + b2 y(n-2) + ... + bM y(n-M)]

    Linearity is obvious above, and if the coefficients ai and bj are constant, then the filter is also time- invariant. The figure below shows a generic linear time invariant digital filter. The Z-1 blocks represent a unit sample of delay.

  • A filter that only operates on the input signal, but not past values of the output, is called a Finite Impulse Response (FIR) filter. Such a filter would only have non-zero ai coefficients in the above equation, but all bj coefficients would be zero. We could also note from our discussion on convolution that the ai coefficients could be the hi sequence of an impulse response of a LTI system. Thus, a digital FIR filter can implement the convolution operation. The figure below shows an FIR filter.

V. Special Simple Filters: One Zero

  • The first order FIR filter is also called a "one zero" filter:

    y(n) = g (x(n) + a1 x(n-1))

  • If we let g = 0.5 and a1 = 1.0, the one zero filter implements a two-point moving average, simply summing up the current sample with the one previous.
  • If we let g = 1/T (the time interval between samples) and a1 = -1.0, then the two-point moving average approximates a differentiator
  • The frequency response of these two filters is shown here:

    It's pretty easy to see that the moving average filter is a simple low-pass filter which blocks signals at 1/2 sampling rate entirely (that's the zero), and the differentiater is the complimentary high-pass filter. The signal processing block diagram of a one-zero filter is shown below.

VI. Special Simple Filters: One Pole

  • A filter that operates on past values of the output signal (possibly in addition to past input samples) is called an Infinite Impulse Response (IIR) filter. Other names for this type of filter include "Recursive", "Feedback", and "Auto-Regressive." The first order IIR filter is also called a "one pole" filter:

    y(n) = (g x(n)) + (b1 y(n-1))

  • If we let g = 1.0 and b1 = 1.0, the one pole filter implements a "digital integrator", summing up all past input samples. This is not a very stable setting of the b coefficient, because feeding the filter a constant (DC) signal would result in the filter growing to infinity. Note also that any value of b which is greater than 1.0 in absolute value yields an unstable filter, which would grow to infinity for any input other than identically zero.
  • Some useful common settings of the one-pole filter are the
    one-pole low-pass with 0 < b1 < 1.0, g = (1.0 - b1), and the
    one-pole high-pass with -1.0 < b1 < 0, g = (1.0 - |b1|).
  • The frequency response of these two filters is shown here, with b1 = 0.9 on the left, and b1 = -0.9 on the right:

  • The signal processing block diagram of a one-pole filter is shown below.

VII. Non-linear One Pole Application:
The Envelope Follower

  • A very useful algorithm for HCI and other signal detection applications is the Envelope Follower.
  • First the signal is rectified (absolute value) or squared (energy vs. power), then a special one pole low-pass smoothing filter is applied. The idea is to track low frequency amplitude peaks accurately while filtering out high frequencies. The way this is accomplished is with a signal dependent filter pole and gain. The figure below shows this:
  • If we choose bup = 0.5 or so, and bdown = 0.99, we'll get a fast rising response and a slow falling response. The slow falling response is what filters out the high frequency information we don't want. The result of envelope following on a quasi-periodic impulsive signal (the sound of footsteps) is shown here:
  • This is similar in software to the hardware circuit shown in the signal conditioning notes. Noisy EMG signals from the BioMuse are typical of signals that would benefit from envelope following. The EMG envelope signal is a good estimate of overall muscle tension.

  • VIII. Two Pole and Two Zero Filters

  • The second order feedforward filter is called a "Two Zero" filter. The second order feedback filter is called a "Two Pole" filter. The generic 2nd order filter with both feedback and feedforward is called a "Two-pole Two-zero" filter, or a "BiQuad" (for two quadratic equations, one in y(n) and one in x(n)). With a second order building block, a complex resonance (poles) or anti-resonance (zeroes) can be implemented. The way to determine the coefficients of the filter is:

    a/b_1 = 2 r cos(2 * PI * freq / SRATE)

    a/b_2 = - r*r

    where 0 < r < 1.0 is a resonance parameter
    freq is the resonant or antiresonant frequency in Hz.
    and SRATE is the sampling rate in Hz.

  • Setting r to 1.0 makes the two-pole filter into an oscillator. Setting r to less than one gives the two- pole filter an impulse response that is an exponentially decaying sinusoidal oscillation.

  • IX. Two Pole/Zero Applications: Tone detection and 60Hz. hum elimination

  • A two-pole resonator can be used for detecting sinusoids in a complex or noisy signal. For example, if we designed one resonantor for each of the tones corresponding to the rows and columns of the touch-tone phone dialer, we could place an envelope follower on the output of each of those, and use simple AND gates on pairs of those outputs to indicate which number was dialed.
  • A common use of a two-zero filter is to reject a certain sinusoidal frequency, such as 60Hz hum elimination. The figure below shows a sound of speaking which also includes so much 60 cycle hum that the speech signal is barely visible. Listen to it here Below that is the spectrum of that signal. Below that is the same signal (and resulting spectrum) after processing through the filter:

    y(n) = x(n) - 2*0.9*cos(60.0*2PI/8000)*x(n-1) + 0.90*0.90*x(n-2)

  • Listen to the processed signal here

    X. The Fast Fourier Transform

  • The Fourier Transform represents any signal as a linear decomposition of orthogonal sinusoids.
  • The Discrete Fourier Transform does this on sampled data. The Fast Fourier Transform is a fast way of computing the DFT for data sets of lengths which are powers of two.
  • Convolution in the Time Domain is multiplication in the Frequency Domain, so filtering can be done using the FFT.
  • See the Sound Synthesis Notes for more information about Fourier analysis, spectrum graphing, and synthesis.

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