# Very basic question about filters in signal reconstruction

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?

## 1 Answer

Neither a filter with a rectangular impulse response, nor one with a triangular impulse response is a very good interpolation / reconstruction filter. If you use a filter with a rectangular impulse response, the resulting output is just a piecewise constant signal. Since you want the constant pieces to have exactly the duration between two samples, the width of the rectangle must equal $$T_s$$. You could call that a zeroth order interpolation.

A first-order or linear interpolation between the sample values is achieved with a triangular filter kernel. If you look at it in the time domain you can see that the triangle with its apex aligned with a sampling instant $$k$$ must extend from the previous sample $$k-1$$ to the next sample $$k+1$$ in order to provide linear interpolation when all those triangles are summed up. So for linear interpolation, the base of the triangle must have a width of $$2T_s$$.

Of course, that triangular function is the convolution of two rectangles of width $$T_s$$, so its Fourier transform is the square of the Fourier transform of the rectangular pulse.

• Thanks for the detailed answer! That pretty much confirms what I had understood. So we choose those sizes to get a zeroth and first order interpolation, and looking at what happens in the frequency doesn’t really make sense, is that right? (The Sinc function doesn’t really have a “cutoff” frequency, perhaps?) – yggdrasil Dec 23 '18 at 4:23
• @A.S.: Yes, with those simple interpolators it appears most natural to consider in the time-domain. As I've pointed out in my answer, those filters don't have an especially good low pass characteristic. – Matt L. Dec 23 '18 at 17:13