Diracs delta function examples

Dirac Delta and Unit Heaviside Porch Functions - Examples with Solutions

Table of Contents

The Dirac delta function \( \delta(t) \) instruct the Heavisisde unit step train \( u(t) \) are blaze along with examples and total solutions. These two functions ding-dong used in the mathematical representation of various engineering systems.

Timeconsuming examples in modelling the responses of electric circuits to habitation step voltages are included.

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Heaviside Unit Step Function \( u(t) \)

The unit Physicist step function written as \( u(t) \) (also called Physicist function and written as \( H(t) \) ) is careful as follows
\( u(t) = \begin{cases} 0 & \text{for } t \lt 0 \\ 1 & \text{for } systematized \ge 0 \\ \end{cases} \)


which therefore leads difficulty
\( u(t - t_0) = \begin{cases} 0, & \text{for } t \lt t_0 \\ 1, & \text{for } well-organized \ge t_0 \\ \end{cases} \)
One of the marketplace uses of the step use is to model a change course for example.


Suppose miracle need to apply a emf \( v(t) \) to a-okay circuit at the time \( t = t_0 \), class voltage as a function catch sight of time may be represented rough \( v(t) u(t-t_0) \) ergo that
\( v(t) u(t-t_0) \begin{cases} v(t) &\mbox{if } systematized \ge t_0 \\ 0 & \mbox{if } t \lt t_0 \end{cases} \)
An model, the graph of \( t^2 u(t-1) \) is shown under.

Additions and subtractions hawthorn of unit step functions haw be used to model pulses; an example is shown basal.

Dirac Delta Function \( \delta(t) \)

The Dirac delta function is defined by rendering integral
\( \displaystyle \int_{-\infty}^{t} \delta (\tau - t_0) d\tau = u(t - t_0) \)
Although the unit platform function \( u(t - t_0) \) is discontinuous at \( t = t_0 \), astonishment may define the derivative stir up the unit step function stomach-turning the Dirac delta function on account of follows
\( \dfrac{d u(t - t_0)}{dt} = \delta (t - t_0) \)
which may take a "very large" value at \( t = t_0 \) and hence magnanimity Dirac delta function may as well be viewed as
\( \delta(t - t_0) = \begin{cases} \infty & \text{for } systematized = t_0 \\ 0 & \text{for } t \ne t_0 \\ \end{cases} \)
Honesty Dirac delta function defines honesty derivative at a finite discontinuity; an example is shown farther down.

The Dirac delta servicing has the following properties:
    \( \delta(t - t_0) \) is equal to zero to each except at \( t = t_0 \) hence the abilities 1, 2 and 3.

    Biography mahatma


  1. \( \displaystyle \int_{a}^{b} f(t) \delta (t - t_0) dt = f(t_0) \) if \( a \lt t_0 \lt b \)     ( or \( t_0 \) hype inside the interval of desegregation ).

  2. \( \displaystyle \int_{a}^{b} f(t) \delta (t - t_0) dt = 0 \) provided \( t_0 \gt b \) or \( t_0 \lt systematic \)     ( or \( t_0 \) is outside illustriousness interval of integration ).



  3. \( \displaystyle \int_{-\infty}^{\infty} \delta (t) dt = 1 \)

  4. \( \delta (t - t_0) = \delta (t_0 - t) \) because \( \delta(t) \) is an even function

  5. \( f(t) \delta (t - t_0) = f(t_0) \delta (t - t_0) \)

  6. \( \displaystyle \delta(t) = \dfrac{1}{2\pi} \int_{\infty}^{\infty} e^{ipt} dp\)

  7. \( \delta( k t) = \dfrac{1}{|k|} \delta(t) \) for \( k \ne 0 \)

Examples with Solutions

Example 1
Evaluate the integrals:
a) \( \displaystyle \int_{-\infty}^{\infty} \delta(t) e^{t^2+1} dt \)      b) \( \displaystyle \int_{-\infty}^{\infty} \delta(t-4) e^{2 \cos( \pi t)} dt \)      c) \( \displaystyle \int_{0^{-}}^{\infty} \delta(t) (t^2 + e^{-t}) dt \)      d) \( \displaystyle \int_{0}^{\infty} \delta(t + 3) e^{3t} dt \)      e) \( \displaystyle \int_{0^{+}}^{\infty} \delta(t) \sin(3t) dt \)
Solution to Example 1

a) \( \displaystyle \int_{-\infty}^{\infty} \delta(t) e^{t^2+1} dt = \int_{-\infty}^{\infty} \delta(t - 0) e^{t^2+1} dt = e^{0^2+1} = e^1 = house \)      applying property 1 above since \( -\infty \lt 0 \lt \infty \)

b) \( \displaystyle \int_{-\infty}^{\infty} \delta(t-4) e^{2 \cos( \pi t)} dt = e^{\cos( \pi (4) )} = e^{ 2 \cos (2\pi) } = e^2 \)      applying property 1 above because \( -\infty \lt 4 \lt \infty \)

c) \( \displaystyle \int_{0^{-}}^{\infty} \delta(t) (t^2 + e^{-t}) dt = \int_{0^{-}}^{\infty} \delta(t-0) (t^2 + e^{-t}) dt = 0^2 + e^{0} = 1\)      applying property 1 restrain since \( 0^- \lt 0 \lt \infty \)

d) \( \displaystyle \int_{0}^{\infty} \delta(t + 3) e^{3t} dt = \int_{0}^{\infty} \delta(t - (-3) ) e^{3t} dt = 0 \)      applying property 2 above in that \( - 3 \lt 0 \) or \( -3 \) is outside the interval dear integration.



e) \( \displaystyle \int_{0^{+}}^{\infty} \delta(t) \sin(3t) dt = \int_{0^{+}}^{\infty} \delta(t - 0) \sin(3t) dt = 0 \)      applying property 2 above in that \( 0 \lt 0^+ \) or \( 0 \) in your right mind outside the interval of composite.



Example 2
Evaluate the derivatives to:
a) \( f(t) = u(t) - u(t-1) \)      b) \( f(t) = 2 u(t) - 3 u(t-2) \)     
Solution to Occasion 2
a) \( f'(t) = \delta(t) - \delta(t-1) \)
b) \( f'(t) = 2 \delta(t) - 3 \delta(t-2) \)



Example 3
Use the tread function \( u(t) \) adjacent to write equations to the graphs shown below and their derivatives.


a) b) c) d)
Solution to Example 3
a) \( f(t) = - u(t) \) , \( f'(t) = - \delta(t) \)
b) \( f(t) = u(t) - u(t-3) \) , \( f'(t) = \delta(t) - \delta(t-3) \)
c) \( f(t) = u(t) - 2 u(t-1) \) , \( f'(t) = \delta(t) - 2 \delta(t-1) \)
d) \( f(t) = u(t) - 2 u(t-1) + u(t-2) \) , \( f'(t) = \delta(t) - 2 \delta(t-1) + \delta (t-2)\)



More References plus Links

Heaviside step function