Trapezoid as the Convolution of Two Rectangles
This is relevant to calculating uncertainties. If you have two inputs
that exhibit rectangular distributions, the output will exhibit a
Suppose we have a rectangular distribution R1 where the half-width
and half-maximum (HWHM) is h1, and similarly R2 where the HWHM
is h2. We assume for convenience, without loss of generality, that
h2 ≥ h1.
We now calculate y := x1 + x2, where x1 is drawn from R1 and
x2 is drawn from R2. This situation is shown in figure 1.
: Trapezoid as the Convolution of Two
Here are some of the key relationships:
| || ||R1|| ||R2|| ||Trapezoid|| ||Lebesgue||Remarks |
|center:|| ||c1|| ||c2|| ||c1 + c2 || |
|HWHM:|| ||h1|| ||h2|| ||h2|| ||L∞||independent of h1 |
|HWtop:|| ||h1|| ||h2|| ||h2 − h1 || |
|HWbase:|| ||h1|| ||h2|| ||h2 + h1|| ||L1||worst-case deviation |
|stdev:|| ||σ1|| ||σ2|| ||√(σ12 + σ22)|| ||L2||Euclidean norm |
|stdev:|| || || || || |
| HWHM/stdev:|| || || || || |
| height:|| || || || || ||normalized to unit area |
The HWHM of the trapezoid is independent of the width of R1, so
long as R1 is narrower than R2.
The ratio of HWHM to stdev varies quite a bit:
Ratio = √3 ≈ 1.73 (if the
skinnier rectangle is very skinny, so the trapezoid itself is very
- Ratio = √1.5 ≈ 1.22 (if the two rectangles
have equal width, so the trapezoid is triangular).