# The Fourier Transform of a Sampled Sequence

July 29, 2019

Back to: Sampling & Reconstruction

According to Theorem 1, πΈΜ(π) is the discrete-time Fourier transform of the sample sequence π(π). What is the Fourier transform of that sample sequence?

First, note that the Fourier transform is defined for functions over β or β (real or complex) and not defined for sequences, which are functions over the set of integers β€.

However, in an attempt to compute such a transform, it is possible to use the infinity enabled Dirac delta πΏ(π‘) to define a comb function ππ (π‘) as:

(1)

Ignoring that real-world ADCs/DACs cannot implement infinite voltage handling, the following steps lead to the mathematical definition of a Fourier transform of the sampled sequence.

(2)

(3)

(4)

(5)

(6)

(7)

(8)

### In conclusion:

- π·Μ (π) repeats every π = 2ππ₯s radians/second, or every π₯s Hertz
- π·Μ (π) reaches 0 at π = 1 just as it did before sampling (sample rate π₯s does not affect the width of the triangle)
- The height of each triangle is 2ππ₯s; the higher the sample rate π₯s the taller the triangle

The Fourier transform result above is very similar in form to the DTFT result in Theorem 1.