An ideal low-pass filter is a type of filter that passes all frequency components of a signal below a certain frequency, called the cutoff frequency ($\omega_{0}$), and blocks all frequency components above the cutoff frequency. This filter has the following [[frequency response]]: ![[low-pass-filter.excalidraw.svg]] $ H(e^{j\omega}) = \begin{cases} 1, &|\omega| \lt \omega_{0} \\ 0, &\omega_{0} \lt |\omega| \lt \pi \end{cases} $ Using the [[discrete-time Fourier transform (DTFT)]] to compute the [[impulse response]]: $ \begin{align} h(n) &= \frac{1}{2\pi} \int _{-\omega_{0}} ^{\omega_{0}} e^{j\omega n} \, d\omega \\ &= \frac{e^{j\omega_{0}n}-e^{-j\omega n}}{2\pi j n} \\ &= \frac{\sin(\omega_{0} n)}{\pi n} \end{align} $ The impulse response is thus the sampling of a sinc function: ![[sinc-function.png]] [[Huang 2001]] The ideal low-pass filters are not realizable because the impulse response has values from $n=-\infty$ to $n=+\infty$. An approximation can be computed by multiplying the ideal impulse response by a [[window function]] and transforming it into a [[finite impulse response (FIR) system]] filter.