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Embedded SystemsFPGAElectronics

Introduction to Digital Filters
    Time Domain Digital Filter Representations
       Finite Impulse Response Digital Filters
          Convolution Representation of FIR Filters

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Convolution Representation of FIR Filters

Notice that the output of the $ k$th delay element in Fig.5.5 is $ x(n-k)$, $ k=0,1,2,\ldots,M$, where $ x(n)$ is the input signal amplitude at time $ n$. The output signal $ y(n)$ is therefore

$\displaystyle y(n)$ $\displaystyle =$ $\displaystyle b_0 x(n) + b_1 x(n-1) + b_2 x(n-2) + \cdots + b_M x(n-M)$  
  $\displaystyle =$ $\displaystyle \sum_{m=0}^M b_m x(n-m)$  
  $\displaystyle =$ $\displaystyle \sum_{m=-\infty}^{\infty} h(m) x(n-m)$  
  $\displaystyle \isdef$ $\displaystyle (h\ast x)(n)$ (6.7)

where we have used the convolution operator ``$ \ast $'' to denote the convolution of $ h$ and $ x$, as defined in Eq.$ \,$(5.4). An FIR filter thus operates by convolving the input signal $ x$ with the filter's impulse response $ h$.

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Next: The ``Finite'' in FIR

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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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