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:
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.
- 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.