Actually I got confused about definition of frequency response of Ideal low-pass filters because in some books they mention that phase of H(f) should be linear in pass band and it's value is $ -2\pi t_{0}$ enter image description here

However in other books they mention that phase should be zero in pass band.

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So I want to know their differences and where does $ -2\pi t_{0}$ comes from?


An ideal low-pass filter has frequency response $H(f)$ equal to the brickwall function: $$H(f)=\mathbb I[|f|<f_c]$$ where $\mathbb I[s]$ is the indicator function, which is equal to 1 if the statement $s$ is true and 0 otherwise, and $f_c$ is the cutoff frequency. Since this frequency response is real, its phase is zero.

This ideal filter is not stable and non-causal. It also has zero delay; it corresponds to $t_0=0$ in the first definition you list above. Now consider the case where we want an ideal LPF, but with some delay $t_0>0$. If the filter input is a sinusoid of frequency $f_0<f_c$, then the filter output is delayed a time $$-\frac{-2\pi f_0t_0}{2\pi f_0}=t_0.$$

In other words, the first filter in your question is ideal with a delay $t_0$ that you can measure or specify; the second filter is ideal with delay equal to 0. As pointed out in another answer, this helps to make the filter "more causal" -- it allows you to implement a filter that approximates the ideal.

See here for another explanation of phase and delay.

  • $\begingroup$ Thanks Mbaz for your good explanation and your contribution $\endgroup$ – Ehsan Zakeri Nov 24 '16 at 5:49

This difference stems from the (approximated) fact of causality:

Looking at the time-domain response of an ideal lowpass with linear phase, it is a sinc function that has its peak at time $t=t_0$. The LP without the phase has the peak at $t=0$. Now, if you assume that you can neglect the relatively low values of the sinc at times $|t|>t_0$ the first filter (with time-shift) becomes causal. HOwever, this is just an approximation (because sinc is infinitely long in time).


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