# The Fourier Transform of a Sampled Sequence

July 29, 2019

According to Theorem 1, 𝖸̃(𝜔) is the discrete-time Fourier transform of the sample sequence 𝗒(𝑛). What is the Fourier transform of that sample sequence?

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

However, in an attempt to compute such a transform, it is possible to use the infinity enabled Dirac delta 𝛿(𝑡) to define a comb function 𝗑𝑑 (𝑡) 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:

• 𝖷̃ (𝜔) repeats every 𝜔 = 2𝜋𝖥s radians/second, or every 𝖥s Hertz
• 𝖷̃ (𝜔) reaches 0 at 𝜔 = 1 just as it did before sampling (sample rate 𝖥s does not affect the width of the triangle)
• The height of each triangle is 2𝜋𝖥s; the higher the sample rate 𝖥s the taller the triangle

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