# Unit Impulse, Unit Step

- Author:
- Ryan Hirst

Unit Impulse and Unit Step functions.
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I give δ(t) a height of 1 at the peak.
Say I vary the position δ with the parameter t. According to Newton, to represent the figure f(x-τ) by the (convolution) integral f(t)*δ(t), I must maintain a ratio of equality between the areas of the two figures, in the limit as Δt→ 0.
I am making up δ, so I will satisfy this condition by definition:<br>

- δ(t) has an area of 1

*h*and width Δt. The area is*h*Δt, which must have the constant value 1. Hence,*h*= 1/Δt. The rectangle vanishes to a line as Δt → 0, and*h*→ ∞. I may, then, represent the peak of δ(t) by infinite height and zero width. I will use an operational definition instead: δ(t) is any function which satisfies f(t) =f(t)*δ(t) In other words, δ(t) retrieves the original function. Voilà. Synthetic, piecewise integration. To DO: (Clumsy interface, needs work.) _______ Worksheets to accompany The Fourier Transform and its Applications; Prof. Osgood, Stanford University: http://www.youtube.com/watch?v=gZNm7L96pfY- Part I
1. Sine or cosine from two points: http://www.geogebratube.org/material/show/id/49208
**→2. Unit Step, Unit Impulse**3. Triangle function λ: http://www.geogebratube.org/material/show/id/50926 4. Sum λ1+ λ2 is a scalene triangle: http://www.geogebratube.org/material/show/id/50004