# What is the discrete analogue of the Fourier transform of the delta function, and in what sense does it hold?

It is well-known that on $$\mathbb{R}$$, the delta function $$\delta(x)$$ has the Fourier transform representation $$$$\delta(x)=\int_\mathbb{R} e^{i2\pi kx}dk$$$$

which holds in the sense of distributions.

Let us think of a finite lattice $$\{ -\frac{1}{2}, \cdots, 0 , \cdots, \frac{1}{2} \}$$ with the spacing $$\frac{1}{N}$$ for some large $$N \in \mathbb{N}$$, so that the lattice has $$2N+1$$ elements.

Then, what would be discrete Fourier representation of the Kronecker delta function $$\delta_{0, x}$$? I guess it would be a discrete sum with a factor $$\frac{1}{N}$$ multiplied, but cannot figure out an exact form.

Also, in what sense does this discrete Fourier transform hold?

Could anyone clarify in the case of discreteness?

## 1 Answer

For any $$n=0,\dots,N-1$$

$$\frac 1 N \sum_{k=0}^{N-1}e^{2i\pi \frac{kn}N} = \delta_{0,n}\tag{1}$$ This is the discrete equivalent to the Poisson summation formula or the formula you wrote. So the Kronecker delta is discrete equivalent of the Dirac delta and $$(1)$$ is saying that the Discrete Time Fourier Transform of the constant function is that delta (of course, you may define the DTFT with a different normalization factor).