MATH1012 - Lecture 32

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MATH1012 - Lecture 31
Skeleton Slides

Heaviside Function

• The Heaviside function $H(t)$ is defined to be: $H(t)=\{\begin{array}{ll}0,& t<0,\\ 1,& t\ge 0.\end{array}$
  • Also called the unit step function and denoted $u(t)$

For any $a\in \mathbb{R}$ the graph of $H(t-a)$ is obtained from the graph of $H(t)$ by shifting $a$ units (to the right if $a>0$ and to the left if $a<0$)
$g(t)H(t-a)=\{\begin{array}{ll}0,& t<a,\\ g(t),& t\ge a.\end{array}$

• Multiplying $g(t)$ by $H(t-a)$ has the effect of suppressing the function until time $t=a$ and then activating it

The Pulse Function

$H(t-a)-H(t-b)=\{\begin{array}{ll}0,& t<a,\\ 1,& a\le t<b,\\ 0,& b\le t.\end{array}$

• This function is equivalent to turning on a switch at $t=a$ then turning it off again at a later $t=b$
  • For a certain region, the function is non-zero
  • Everywhere else, the function equals $0$
• Pulse functions are often used in engineering, signal processing, control systems, and physics to model short bursts of energy or signals

Note that: $g(t)H(t-a)-H(t-b)=\{\begin{array}{ll}0,& t<a,\\ g(t),& a\le t<b,\\ 0,& b\le t.\end{array}$

• $H(t-a)$ is sometimes denoted by $u(t-a)$ or $u_a(t)$

The Heaviside Shift Theorem

• The Heaviside shift theorem describes how a time-domain shift/delay affects a function’s Laplace transform
• If $F(s)=\mathcal{L}(f(t))$, then for any $a\ge 0$ we have: $\mathcal{L}[f(t-a)H(t-a)]=e^{-as}F(s)$
  • Proof on slides

$\mathcal{L}(H(t-a))=\frac{e^{-as}}{s}$
$\mathcal{L}((t-a)H(t-a))=\frac{e^{-as}}{s^2}$

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Annotated Slides
End of Unit!