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The Simplest Lowpass Filter

Let's start with a very basic example of the generic problem at hand: understanding the effect of a digital filter on the spectrum of a digital signal. The purpose of this example is to provide motivation for the general theory discussed in later chapters.

Figure 1.1: Amplitude response (gain versus frequency) specification for the ideal low-pass filter.
\begin{figure}\input fig/kfigtpone.pstex_t

Our example is the simplest possible low-pass filter. A low-pass filter is one which does not affect low frequencies and rejects high frequencies. The function giving the gain of a filter at every frequency is called the amplitude response (or magnitude frequency response). The amplitude response of the ideal lowpass filter is shown in Fig.1.1. Its gain is 1 in the passband, which spans frequencies from 0 Hz to the cut-off frequency $ f_c$ Hz, and its gain is 0 in the stopband (all frequencies above $ f_c$). The output spectrum is obtained by multiplying the input spectrum by the amplitude response of the filter. In this way, signal components are eliminated (``stopped'') at all frequencies above the cut-off frequency, while lower-frequency components are ``passed'' unchanged to the output.

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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 (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 for details.


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