21.1.2 Rectangle function
The rectangle function Π is 0 everywhere except on [−1/2,1/2],
where it is 1; namely,
Π(x)=θ(x+1/2)−θ(x−1/2)
where θ is the
Heaviside function. The rectangle function is a special case of boxcar function
(see Section 21.1.1) for a=−1/2 and b=1/2.
The rect
command computes the rectangle function.
-
rect takes
x, an identifier or an expression.
- rect(x) returns the value of the rectangle function
at x.
Example
|
| |
θ | ⎛
⎜
⎜
⎝ | | + | | ⎞
⎟
⎟
⎠ | −θ | ⎛
⎜
⎜
⎝ | | − | | ⎞
⎟
⎟
⎠ |
| | | | | | | | | | |
|
To compute the convolution of the rectangle function with itself, you
can use the Convolution Theorem (see Section 21.4.2).
| R:=fourier(rect(x),x,s):; ifourier(R^2,s,x) |
|
| |
−2 x θ | ⎛
⎝ | x | ⎞
⎠ | +x θ | ⎛
⎝ | x+1 | ⎞
⎠ | +x θ | ⎛
⎝ | x−1 | ⎞
⎠ | +θ | ⎛
⎝ | x+1 | ⎞
⎠ | −θ | ⎛
⎝ | x−1 | ⎞
⎠ |
| | | | | | | | | | |
|
This result is the triangle function tri(x) (see section 21.1.3).