The Fourier Transform of a Sampled Sequence

July 29, 2019

According to Theorem 1, 𝖸̃(πœ”) 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 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 𝛿(𝑑) to define a comb function 𝗑𝑑 (𝑑) as:

(1)   \begin{equation*} x_{d}(t)=\sum_{n\in\mathbb{Z}}x(n\tau)\delta(t-n\tau) \end{equation*}

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)   \begin{equation*} \tilde{Y}(\omega)=\int_{t\in\mathbb{R}}\text{x}_{d}(t)e^{-i\omega t}dt\text{ by definition of FT} \end{equation*}

(3)   \begin{equation*} =\int_{t\in\mathbb{R}}\left[\sum_{n\in Z}x(t)\delta(t-n\tau)\right]e^{-i\omega t}dt\text{ by definition of }x_{d}(t) \end{equation*}

(4)   \begin{equation*} =\sum_{n\in Z}\int_{t\in\mathbb{R}}x(t)\delta(t-n\tau)e^{-i\omega t}dt\text{ by linearity of }\int_{t}dt \end{equation*}

(5)   \begin{equation*} =\sum_{n\in Z}x(n\tau)e^{-i\omega n\tau}\text{ by definition of }\delta(t) \end{equation*}

(6)   \begin{equation*} =\frac{2\pi}{\tau}\sum_{n\in Z}\tilde{X}\left(\omega+\frac{2\pi}{\tau}n\right)\text{ by IPSF} \end{equation*}

(7)   \begin{equation*} =\frac{2\pi}{\tau}\sum_{n\in Z}\tilde{X}\left(\omega-\frac{2\pi}{\tau}n\right)\text{ by absolute summability property} \end{equation*}

(8)   \begin{equation*} =2\pi \text{F}_{s}\sum_{n\in Z}\tilde{X}(\omega-2\pi \text{F}_{s}n)\text{ by definition of sample rate }\text{F}_{s} \end{equation*}

In conclusion:

  • 𝖷̃ (πœ”) repeats every πœ” = 2πœ‹π–₯s radians/second, or every π–₯s Hertz
  • 𝖷̃ (πœ”) reaches 0 at πœ” = 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 2πœ‹π–₯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.