Unit Step Function

1

p238 : 6.3.2 The Unit Step Function

The unit step function U(t-a) is defined :

at

atatU

,1

0,0

2

When a function f is multiplied by U(t-a), the unit step function " turns OFF " the portion of the graph of f from at 0 .

Unit Step Function

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Example :

Given 1ttf , then tftU 5 means that the ONLY portion of tf from 50 t is " turn OFF ".

Unit Step Function

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Another very useful unit step function is

U(t-a ) - U(t-b).

otherwise,0

,1 btabtUatU

Unit Step Function

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When a function f is multiplied by [U(t-a) - U(t-b)], only the portion of the graph of f from bta is

" turn ON ".

Unit Step Function

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Example : Given 1ttf , then tftUtU 51 means that ONLY the portion of tf from 50 t is " turn ON ".

Unit Step Function

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A piecewise function, f (t) is defined by different sub-functions for different intervals of t.

Example 1 :

otherwise

7442

20

0

,1,112

,5

tt

t

tt

t

tf sub-function

Unit Step Function

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Any piecewise function can be expressed as a single equation by :

1. applying unit step function on each sub-function

2. add each result obtained from 1.

Unit Step Function

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Example 2 : Express piecewise function tf from example 1 using unit step function.

1. applying unit step function on each sub-function

For 20 t , 555 tUtUtt

42112112 tUtUtt For 42 t ,

7411 tUtUtt For 74 t ,

For 7t , 0

otherwise

7442

20

0

,1,112

,5

tt

t

tt

t

tf

Unit Step Function

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2. Add each result obtained from 1.

tf

42112 tUtUt

741 tUtUt

Unit Step Function

25 tUtUt