# What are the advantages of using the Dirichlet kernel for interpolation?

Suppose we define the Dirichlet kernel as:

$$\frac{\sin(\pi N x /2)}{\pi N\sin(x /2)}$$

(Note: I'm not entirely sure what the standard definition of the Dirichlet kernel is; mine is slightly different than the Wikipedia one, for example. Comments on this are appreciated as well.)

I belive this can be shown to be an aliased version of a sinc function ($$\frac{\sin(\pi x)}{\pi x}$$). The Dirichlet kernel can be used as a lowpass interpolation filter, but as shown here, it has poorer frequency characteristics than a truncated windowed sinc.

My question is, in spite of its drawbacks, are there any advantages to using the Dirichlet kernel over a windowed sinc for band limited interpolation? For example, I've heard that it produces better results than a windowed sinc near the edges of a signal (where you get errors when the sinc filter runs over the edge of the data; thus it would be advantageous especially for short signals). I think this is related to treating the signal being interpolated as if it's been replicated (or mirrored) at the ends, but I can't find it worked out anywhere.

• Is this question about using the advantages of Dirichlet vs. No interpolation or vs. some other interpolation? Commented May 18 at 19:02
• @robertbristow-johnson the main alternative I have in mind is a windowed sinc filter for bandlimited interpolation, but there could be others. I've edited the question a bit. Commented May 18 at 21:25
• I think that's better, Gillespie, because, especially if $N$ is large, you can have far fewer terms to add up. I would suggest a Kaiser winder. For perfect bandlimited interpolation, it's the Dirichlet thingie for odd $N$, but it's slightly different for even $N$. Commented May 18 at 21:29

• Dan, is it always? What if the period $N$ is even? Then it's $$\sum_{n=-\infty}^{\infty} x[n] \, \operatorname{sinc}(t-n) = \sum_{n=0}^{N-1} x[n] \frac{\sin\big(\pi (t-n)\big)}{N \tan\big(\pi (t-n)/N\big)}$$ with $\tan(\cdot)$ in the denomimator. Commented May 18 at 21:24