# Explanation relevant to the filter response ($\mathrm{sinc}$ interpolation) - Equation $(2)$ in my question

At present, I am learning the theory of operation of resampling for bandlimited periodic discrete signals using $\mathrm{sinc}$ interpolation. I am developing a design flow and having difficulty in understanding a particular expression/equation, hence I am stuck! Here I go:

Suppose we have a samples $x(nT_s)$ of a continuous time signal $x(t)$, where $t$ is time in seconds and $n$ is sample points that ranges over integers and ofcourse $T_s$ is the sampling period. $x(t)$ is bandlimited to $\pm F_s/2$, where $F_s = 1/T_s$.

Now, we know that the original signal $x(t)$ can be reconstructed or synthesized by the samples of $x(nT_s)$ using the equation:

$$x'(t) = x(nT_s)+H_s(t-nT_s) = x(t), \quad \text{for}\quad n =\ldots,-3,-2,-1,0,1,2,3,\ldots\tag1$$

and

$$H_s(t) = \mathrm{sinc}(\pi F_s t)$$

To resample $x(t)$ for the new rate i.e. $F'_s = 1/T'_s$, we have to evaluate equation $(1)$ at integer multiples of $T'_s$ i.e. $nT'_s$.

We know that, when the new sampling rate $F'_s$ is less than the original sampling rate or frequency $F_s$, the lowpass cutoff must be placed below half the new lower sampling rate.

Doubt - Based on the highlighted statement made above, the ideal low pass is given by:

$$H_s(t) = \min\left\{1,F'_s/F_s\right\}\mathrm{sinc}\left(\pi\min\left\{F'_s,F_s\right\}t\right) \tag2$$

where the scale factor maintains unity gain in the passband.

What does the equation $(2)$ represent? I understand that, if $F'_s$ is lower than $F_s$, then the filters cut off frequency should be half the $F'_s$, however, I am having trouble understanding this equation $(2)$ and how to benefit from it in the design process. Any explanation regarding this would be appreciated.

Source/Reference:

$\scriptstyle{\textrm{Julius O.Smith & Phil Gosett "A Flexible Sampling Rate Conversion Method" IEEE, 1984}}$.

A Sinc function is both a reconstruction formula for a band-limited waveform and the time domain representation (impulse response) of an ideal "brick-wall" low-pass filter. Since it does both, if you "fatten" the Sinc and then use it as an interpolation kernel, you get the reconstruction of the result of low-pass filtering the original waveform.

• Which windowing technique would you think is more suitable for overcoming gibbs effect, hamming or blackman?. And is it mandatory to apply FFT to the ideal filter kernel to get ideal frequency response in order to window?, or I can directly apply windowing to ideal filter kernel i.e.Time domain?. I am kind of confused, any clarity regarding this would be great! – PsychedGuy Feb 16 '15 at 10:23

Eq$(1)$ doesn't look right. It should be:

$$x'(t) = \sum_{n=-\infty}^{+\infty} x(nT_s) \mathrm{sinc}(t-nT_s)$$

For your doubt, consider subsampling (lowering $F_s$) as equivalent to reconstructing $x(t)$ and then sampling it at a lower frequency.

Eq$(2)$ comes from the combination of $2$ lowpass filters:

1. The ideal reconstruction filter (from eq.$1$)
2. a new anti-alias filter previos to sampling at the lower $F_s$

These $2$ filters combined should result in eq$(2)$ (several lowpass filters in cascade is equivalent to one filter with the minimum bandwidth, and the product of the gains).

• In the Eq(1), Hs(t) is supposed to be the filter impulse response ideally. However, for sinc filtering,like you said it would be sinc(t-nTs). So,I am presuming both to be right.Anyways, I will analyse the Eq(2) corresponding to your answer and get back later. Thanks! – PsychedGuy Feb 12 '15 at 12:59