In filter design, why isn't it possible to have a frequency response with phase 0? A linear phase filter will cause a delay in the output signal of the filter.

If I'm not mistaken, a filter with phase equal 0 would cause no delays in the output signal.

Can someone explain me why isn't it possibly to generate a 0 phase response filter? I mean, theoretically...

I'm studying for a test and I can't find the answer in textbooks.
A zero-phase frequency response is real-valued, and, by a property of the Fourier transform, the corresponding impulse response is symmetric. Since it is symmetric (i.e. it is non-zero for $t&lt;0$) it can't be causal, and consequently, it can't be implemented (without adding delay, which defeats the purpose).