## Discrete Fourier transform

Posted by peeterjoot on October 11, 2013

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For decoupling trial solutions of lattice vibrations we have what appears to be a need for the use of a discrete Fourier transform. This is described briefly in [1], but there’s no proof of the inversion formula. Let’s work through that detail.

Let’s start given a periodic signal of the following form

and assume that there’s an inversion formula of the form

Direct substitution should show if such a transform pair is valid, and determine the proportionality constant. Let’s try this

Observe that the interior sum is just when . Let’s sum this for the values , writing

With we have

Observe that , and necessarily takes on just integer values. We have terms of the form , for integer in the numerator, always zero. In the denominator, the sine argument is in the range

We can visualize that range as all the points on a sine curve with the integer multiples of omitted, as in fig. 1.1.

Clearly the denominator is always non-zero when . This provides us with the desired inverse transformation relationship

**Summary**

Now that we know the relationship between the discrete set of values and this discrete transformation of those values, let’s write the transform pair in a form that explicitly expresses this relationship.

We have also shown that our discrete sum of exponentials has an delta function operator nature, a fact that will likely be handy in various manipulations.

# References

[1] S.S. Haykin. *Communication systems*. Wiley, 1994.

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