# Deriving the impulse response of an ideal low-pass filter

The impulse response of an ideal low-pass filter can be determined by setting $$H(\omega)=1$$ in the Fourier-representation $$h(n) = \frac{1}{2\pi}\int_{-\omega_c}^{\omega_c} H(\omega)e^{j\omega n}d\omega$$

The solution will be a function of form $$\sin(n)/n$$. Now, the motivation behind the substitution above is the desire to amplify each frequency component before $$\omega_c$$ by an equal amount. Hence the term ideal response. Also, the fact that for some $$\omega_0$$ there is a corresponding amplitude $$\vert H(\omega_0)\vert$$ is not ( as far as I know ) interesting while performing spectral analysis. What matters are the amplitude relations between some $$\omega_0$$ and $$\omega_1$$, the frequency components of a signal perhaps.

Therefore, the substitution $$H(\omega) = C$$, for any constant $$C$$ should be equally valid right?

I do remember a case where a paper was submitted to a journal in which the author called a filter with a constant gain $$\neq 1$$ an "allpass filter". One of the reviewers, who is a very famous professor, known to everybody who has ever heard the term "filter bank", wrote in his review that an allpass filter only deserves its name when it has a gain of unity, and the author was strongly advised to make appropriate changes. So, some people take gain constants quite seriously.