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The Ideal Lowpass Filter

Consider the ideal lowpass filter, depicted in Fig.4.1.

Figure 4.1: Desired amplitude response (gain versus frequency) for an ideal lowpass filter.

An ideal lowpass may be characterized by a gain of 1 for all frequencies below some cut-off frequency $ f_c$ in Hz, and a gain of 0 for all higher frequencies.5.2 The impulse response of the ideal lowpass filter is easy to calculate:

$\displaystyle h_{\hbox{ideal}}(n)$ $\displaystyle \isdef$ DTFT$\displaystyle ^{-1}_n\left(H_{\mbox{ideal}}\right)$  
  $\displaystyle \isdef$ $\displaystyle \frac{1}{2\pi}\int_{-\pi}^{\pi} d\omega e^{j\omega n }
1, & \left\vert\omega\right\vert\leq\omega_c \\ [5pt]
0, & \mbox{otherwise} \\
\end{array} \right.$  
  $\displaystyle =$ $\displaystyle \frac{1}{2\pi}\int_{-\omega_c}^{\omega_c} e^{j\omega n }d\omega$  
  $\displaystyle =$ $\displaystyle \frac{1}{2\pi jn}\left[e^{j\omega_c n} - e^{-j\omega_c n}\right]$  
  $\displaystyle =$ $\displaystyle \frac{\sin(\omega_c n)}{\pi n}$  
  $\displaystyle \isdef$ $\displaystyle 2f_c$   sinc$\displaystyle (2f_c n ), \quad n \in {\bf Z}
\protect$ (5.3)

where $ \omega_c\isdef 2\pi f_c$ denotes the normalized cut-off frequency in radians per sample. Thus, the impulse response of an ideal lowpass filter is a sinc function.

Unfortunately, we cannot implement the ideal lowpass filter in practice because its impulse response is infinitely long in time. It is also noncausal; it cannot be shifted to make it causal because the impulse response extends all the way to time $ n=-\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 generally concerned with finding an optimal approximation to the desired frequency response by minimizing some norm of a prescribed error criterion with respect to a set of practical filter coefficients, perhaps subject also to some constraints (usually linear equality or inequality constraints) on the filter coefficients, as we saw for optimal window design in § audio applications, optimality is difficult to define precisely because perception is involved. It is therefore valuable to consider also suboptimal methods that are ``close enough'' to optimal, and which may have other advantages such as extreme simplicity and/or speed. We will examine some specific cases below.

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Lowpass Filter Design Specifications
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Window Design by Linear Programming