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86 CHAPTER THREE
plus a remainder. This formula provides a standard way to approximate and com-
pute functions like sine and cosine. It’s the way compilers set up the computation.
It involves several multiply and accumulate steps. Each term in the equation is
another MAC. Generally, the remainder can be made arbitrarily small by carrying
out more terms (making n larger). A tutorial on the Taylor series can be found at
www.wikipedia.com/wiki/Taylors_theorem.
Finite Impulse Response (FIR) filters These are generally used for filtering a
continuous stream of information that represents audio or video. Consider the
reception of an audio signal in the presence of a strong 1 kHz interfering noise
source. We would like to remove the 1 kHz noise from our signal (as best we can).
If the audio signal is digitized, it can be fed into a FIR filter specifically designed
to filter out 1 kHz signals. The FIR filter method gives us a way to do this in as
precise a manner as required, governed only by cost.
Suppose we want to filter the signal x(t) to produce signal y(t). The generalized
formula for an n-stage FIR filter is given by
y1t2 h0 x1t2 h1 x1t 12 h2 x1t 22 ... hn x1t n2

where h1 ...hn are the coefficients of the filter. We’ll explain the math in a later
chapter, but we can see that this formula is also a series of MACs. A web site on
FIR filters can be found at www.wwc.edu/ frohro/qex/sidebar.html.
Fourier Transforms Fourier Transforms were developed, as we might guess, by
Joseph Fourier (see Figure 3-7) in the early 1800s. The transforms are a way of
representing any function, within certain bounds, as the superposition of a series
of pure sine waves. In this way, a function is broken down into a series of pure fre-

FIGURE 3-7 Joseph Fourier