Unit Impulse and Unit Step functions. Switch On the lower panel to multiply f(x-τ) by u or δ. Click to select/activate gray text.
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> [list] [*]δ(t) has an area of 1 [/list] For example, assume the spike of δ(t) is a narrow rectangle of height [i]h[/i] and width Δt. The area is [i]h[/i] Δt, which must have the constant value 1. Hence, [i]h[/i] = 1/Δt. The rectangle vanishes to a line as Δt → 0, and [i]h[/i] → ∞. 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: [url]http://www.youtube.com/watch?v=gZNm7L96pfY[/url] [list] [*]Part I 1. Sine or cosine from two points: [url]http://www.geogebratube.org/material/show/id/49208[/url] [b]→2. Unit Step, Unit Impulse[/b] 3. Triangle function λ: [url]http://www.geogebratube.org/material/show/id/50926[/url] 4. Sum λ1+ λ2 is a scalene triangle: [url]http://www.geogebratube.org/material/show/id/50004[/url] [/list]