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Chapters

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Embedded SystemsFPGAElectronics

Introduction to Digital Filters
    Analysis of a Digital Comb Filter
       Impulse Response

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Impulse Response

Figure 3.8 plots the impulse response of the example filter, as computed by the matlab script shown in Fig.3.9. (The plot-related commands are also included for completeness.4.2) The impulse signal consists of a single sample at time 0 having amplitude 1, preceded and followed by zeros (an ideal ``click'' at time 0). Linear, time-invariant filters are fully characterized by their response to this simple signal, as we will show in Chapter 4.

Figure 3.8: Impulse response of the example filter $ y(n) = x(n) + 0.5^3 x(n-3) - 0.9^5 y(n-5)$.
\includegraphics[width=4in]{eps/eir}

Figure 3.9: Matlab script for computing and plotting the impulse response of the example digital filter.
  
% Example comb filter:
g1 = 0.5^3;  B = [1 0 0 g1];      % Feedforward coeffs
g2 = 0.9^5;  A = [1 0 0 0 0 g2];  % Feedback coefficients

h = filter(B,A,[1,zeros(1,50)]);  % Impulse response
% h = impz(B,A,50); % alternative in octave-forge or MSPTB

% Matlab-compatible plot:
clf; figure(1); stem([0:50],h,'-k'); axis([0 50 -0.8 1.1]);
ylabel('Amplitude'); xlabel('Time (samples)'); grid;
cmd = 'print -deps ../eps/eir.eps'; disp(cmd); eval(cmd);

The impulse response of this filter is also easy to compute by hand. Referring to the difference equation

$\displaystyle y(n) = x(n) + 0.5^3 x(n-3) - 0.9^5 y(n-5) $

and setting the input $ x(n)$ to $ \delta(n) = [1,0,0,\dots]$, we compute, for $ n=0,1,2,\dots$,

\begin{eqnarray*} y(0) &=& 1\\ y(1) &=& 0\\ y(2) &=& 0\\ y(3) &=& 0.5^3 = 0.1... ...0.9^5)^3 = -0.9^{15} = -0.205891132094649\\ \vdots && \vdots\\ \end{eqnarray*}

Previous: Software Implementation in Faust
Next: Transfer Function

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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