What is the definition of the rectangular pulse function used in the Fourier method?

Q: What is the definition of the rectangular pulse function used in the Fourier method?

The rectangular pulse function, denoted as $f(t)$, is defined piecewise. It takes the value 1 when the absolute value of $t$ is less than or equal to 1 (i.e., $|t| \le 1$), and it takes the value 0 for all other values of $t$ (i.e., $|t| > 1$). This function serves as the input for the Fourier transform calculation in the video.

The rectangular pulse function, denoted as

f(t)

, is defined piecewise. It takes the value 1 when the absolute value of

t

is less than or equal to 1 (i.e.,

|t| \le 1

), and it takes the value 0 for all other values of

t

(i.e.,

|t| > 1

). This function serves as the input for the Fourier transform calculation in the video.

Conditions

$t$ is a real number.
The function is defined on the entire real line.

Reasoning, step by step

Identify the variable $t$ as the independent variable.
Determine the condition for the non-zero value: $|t| \le 1$ .
Assign the value 1 to $f(t)$ under this condition.
Determine the condition for the zero value: $|t| > 1$ (or 'otherwise').
Assign the value 0 to $f(t)$ under this condition.
Write the piecewise definition: $f(t) = \begin{cases} 1 & (|t| \le 1) \\ 0 & (\text{other}) \end{cases}$ .
Visualize the graph as a rectangle of height 1 centered at the origin with width 2.

Example

The whiteboard displays: ' $f(t) = \begin{cases} 1 & (|t| \le 1) \\ 0 & (\text{other}) \end{cases}$ ' alongside a rectangular pulse graph.

Common misconceptions

Assuming the pulse is from 0 to 1; it is symmetric from -1 to 1.
Confusing the rectangular pulse with the Dirac delta function; the pulse has finite width and height.
Thinking the value is undefined at $t=\pm 1$ ; the condition $|t| \le 1$ includes the boundaries.

Watch the explanation

BilibiliFourier series, Fourier transform, Green's functions｜海浪晶灵aa

0:40 – 1:03Watch this moment ↗

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Answers are generated from source material and independently checked. Consult the original video or creator if something is unclear.