I am studying digital signal processing and I would like to ask a question about sampled signal -> original signal reconstruction filters.
Most textbooks use a box filter or a tent filter to show how a discrete-time sampled signal can be reconstructed to (an approximation in this case) of the original signal. However my question is:
Given a discrete signal result from sampling a signal with a sampling interval "Ts", the spectrum will consist of infinite replicas spaced at 1/Ts and scaled by 1/Ts. Now I assume 1/Ts is more than double the original signal maximum frequency, so there is no aliasing.
Our intent with the box (rect) or tent filter is try to "cut off" the spectrum replica in the middle with convolution. What most textbooks do not explain, is what size we should choose the rect or tent to be. Most examples I've seen use a box filter which has a base of Ts, or a tent filter with a base of 2*Ts. Why the base of the tent filter is double of that of the box filter? Is this an arbitrary choice? Looking at the fourier transform of a tent function (squared sinc) compared to that of a rect function (sinc) we shouldn't need to double the base to get the cutoff frequency near the (ideal) 1/2*Ts. With a tent with base 2*Ts we actually hit 0 amplitude at 1/4*Ts. Aren't we cutting out more frequency than we should?