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Spectral Audio Signal Processing
FIR Digital Filter Design
The Ideal Lowpass Filter

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The Ideal Lowpass Filter

**Figure B.1:** Amplitude response (gain versus frequency) specification for the ideal low-pass filter.
$\begin{figure}\input fig/ilpf.pstex_t \end{figure}$

Consider the ideal lowpass filter, depicted in Fig.B.1. The ideal lowpass filter is characterized by a gain of 1 for all frequencies below some cut-off frequency in normalized Hz, and a gain of 0 for all higher frequencies.^B.1 The impulse response of the ideal lowpass filter is easy to calculate:

$\displaystyle h_{\hbox{ideal}}(n)$	$\displaystyle \isdef$	DTFT $\displaystyle ^{-1}_n\left(H_{\mbox{ideal}}\right)$
	$\displaystyle \isdef$	$\displaystyle \frac{1}{2\pi}\int_{-\pi}^{\pi} d\omega e^{j\omega n } \left\{\be... ...ga\right\vert\leq\omega_c \\ [5pt] 0, & \mbox{otherwise} \\ \end{array}\right.$
	$\displaystyle =$	$\displaystyle \frac{1}{2\pi}\int_{-\omega_c}^{\omega_c} e^{j\omega n }d\omega$
	$\displaystyle =$	$\displaystyle \frac{1}{2\pi jn}\left[e^{j\omega_c n} - e^{-j\omega_c n}\right]$
	$\displaystyle =$	$\displaystyle \frac{\sin(\omega_c n)}{\pi n}$
	$\displaystyle =$	$\displaystyle 2f_c$ sinc $\displaystyle (2f_c n ), \quad n \in {\bf Z} \protect$	(B.1)

where $\omega_c\isdef 2\pi f_c$ denotes the normalized cut-off frequency in radians per sample.

Unfortunately, we cannot implement the ideal lowpass filter in practice because its impulse response is infinitely long in time. Also note that it is noncausal. It cannot be shifted to make it causal because the impulse response extends all the way to time $-\infty$ . It is clear we will have to accept some sort of compromise in the design of any practical lowpass filter. The subject of digital filter design is concerned with finding optimal designs under prescribed error criteria. We will examine a few cases below and give pointers to more information.

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written by 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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