Difference between revisions of "Convolution Shortcuts"
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\(\begin{align}
\delta(t)*f(t)&=f(t)\\
\delta(t-a)*f(t)&=f(t-a)\\
\delta(t)*f(t-b)&=f(t-b)\\
\delta(t-a)*f(t-b)&=f(t-a-b)\\
\end{align}\)
\(\begin{align}
u(t)*f(t)&=\int_{-\infty}^{t}f(\tau)~d\tau\\
r(t)*f(t)=u(t)*u(t)*f(t)&=\int_{-\infty}^{t}\int_{-\infty}^{\gamma}f(\tau)~d\tau~d\gamma\\
\end{align}\)
\(\begin{align}
u(t)*u(t)&=r(t)=tu(t)\\
u(t)*r(t)=u(t)*u(t)*u(t)&=q(t)=\frac{1}{2}t^2u(t)\\
u(t)*q(t)=r(t)*r(t)=u(t)*u(t)*u(t)*u(t)&=\frac{1}{6}t^3u(t)\\
\mbox{equivalent of }n\mbox{ steps convolved together}&=\frac{1}{(n-1)!}t^{n-1}u(t)
\end{align}\)
\(\begin{align}
(e^{-at}\,u(t))*(e^{-at}\,u(t))&=\int_{-\infty}^{\infty} e^{-a\tau}\,u(\tau)\,e^{-a(t-\tau)}\,u(t-\tau)\,d\tau\\
&=u(t)\int_{0}^t e^{-a\tau}\,e^{-a(t-\tau)}\,d\tau=u(t)\int_{0}^t e^{-a\tau}\,e^{-at}\,e^{a\tau}\,d\tau\\
&=e^{-at}u(t)\int_{0}^t e^{-a\tau}\,e^{a\tau}\,d\tau=e^{-at}u(t)\int_{0}^t d\tau\\
&=e^{-at}u(t)\left[ \tau \right]_0^t=e^{-at}u(t)\left[t-0\right]\\
&=t\,e^{-at}\,u(t)
\end{align}\)
\(\begin{align}
(e^{-at}\,u(t))*(e^{-bt}\,u(t))&=\int_{-\infty}^{\infty} e^{-a\tau}\,u(\tau)\,e^{-b(t-\tau)}\,u(t-\tau)\,d\tau\\
&=u(t)\int_{0}^t e^{-a\tau}\,e^{-b(t-\tau)}\,d\tau=u(t)\int_{0}^t e^{-a\tau}\,e^{-bt}\,e^{b\tau}\,d\tau\\
&=e^{-bt}u(t)\int_{0}^t e^{-a\tau}\,e^{b\tau}\,d\tau=e^{-bt}u(t)\int_{0}^t e^{(b-a)\tau}\,d\tau\\
&=e^{-bt}u(t)\left[ \frac{e^{(b-a)\tau}}{b-a}\right]_0^t=e^{-bt}u(t)\left[\frac{e^{(b-a)t}}{b-a}-\frac{1}{b-a}\right]\\
&=\left[\frac{e^{-at}-e^{-bt}}{b-a}\right]\,u(t)
\end{align}\)
\(\begin{align}
(e^{-at}\,u(t))*(u(t))&=(e^{-at}\,u(t))*(e^{-0t}\,u(t))\\
&=\left[\frac{e^{-at}-e^{-0t}}{0-a}\right]\,u(t)=\left[\frac{1-e^{-at}}{a}\right]\,u(t)
\end{align}\)
\(\begin{align}
y(t)&=x(t)*h(t)\\
~&=(u(t-a)) * (e^{-2t}u(t))\\
~&=\delta(t-a) * u(t) * e^{-2t}u(t)\\
~&=\delta(t-a) * \int_{-\infty}^{t} e^{-2\tau} u(\tau)~d\tau = \delta(t-a) * u(t)\int_{0}^{t} e^{-2\tau} ~d\tau\\
~&=\delta(t-a) * \left( \frac{1-e^{-2t}}{2} \right)u(t)\\
~&=\left( \frac{1-e^{-2(t-a)}}{2}\right) u(t-a)
\end{align}\)
(→Convolution Between Exponentials) |
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\end{align}</math></center> | \end{align}</math></center> | ||
| − | ==Convolution with | + | ==Convolution with Steps and Ramps== |
<center><math>\begin{align} | <center><math>\begin{align} | ||
u(t)*f(t)&=\int_{-\infty}^{t}f(\tau)~d\tau\\ | u(t)*f(t)&=\int_{-\infty}^{t}f(\tau)~d\tau\\ | ||
| Line 23: | Line 23: | ||
==Convolution Between Exponentials== | ==Convolution Between Exponentials== | ||
| + | Note - the following work if $$a$$ and/or $$b$$ is 0. | ||
| + | * Same exponent | ||
<center><math>\begin{align} | <center><math>\begin{align} | ||
| − | (e^{-at}\,u(t))*(e^{-bt}\,u(t))&=\int_{-\infty}^\infty} e^{-a\tau}\,u(\tau)\,e^{-b(t-\tau)}\,u(t-\tau)\,d\tau\\ | + | (e^{-at}\,u(t))*(e^{-at}\,u(t))&=\int_{-\infty}^{\infty} e^{-a\tau}\,u(\tau)\,e^{-a(t-\tau)}\,u(t-\tau)\,d\tau\\ |
| − | &=u(t)\int_{0}^t e^{-a\tau}\,e^{-b(t-\tau)}\,d\tau=u(t)\int_{0}^t e^{-a\tau}\,e^{-bt}\,e^{ | + | &=u(t)\int_{0}^t e^{-a\tau}\,e^{-a(t-\tau)}\,d\tau=u(t)\int_{0}^t e^{-a\tau}\,e^{-at}\,e^{a\tau}\,d\tau\\ |
| + | &=e^{-at}u(t)\int_{0}^t e^{-a\tau}\,e^{a\tau}\,d\tau=e^{-at}u(t)\int_{0}^t d\tau\\ | ||
| + | &=e^{-at}u(t)\left[ \tau \right]_0^t=e^{-at}u(t)\left[t-0\right]\\ | ||
| + | &=t\,e^{-at}\,u(t) | ||
| + | \end{align}</math></center> | ||
| + | |||
| + | * Different exponents | ||
| + | <center><math>\begin{align} | ||
| + | (e^{-at}\,u(t))*(e^{-bt}\,u(t))&=\int_{-\infty}^{\infty} e^{-a\tau}\,u(\tau)\,e^{-b(t-\tau)}\,u(t-\tau)\,d\tau\\ | ||
| + | &=u(t)\int_{0}^t e^{-a\tau}\,e^{-b(t-\tau)}\,d\tau=u(t)\int_{0}^t e^{-a\tau}\,e^{-bt}\,e^{b\tau}\,d\tau\\ | ||
&=e^{-bt}u(t)\int_{0}^t e^{-a\tau}\,e^{b\tau}\,d\tau=e^{-bt}u(t)\int_{0}^t e^{(b-a)\tau}\,d\tau\\ | &=e^{-bt}u(t)\int_{0}^t e^{-a\tau}\,e^{b\tau}\,d\tau=e^{-bt}u(t)\int_{0}^t e^{(b-a)\tau}\,d\tau\\ | ||
| − | &=e^{-bt}u(t)\left[ \frac{e^{(b-a)\tau | + | &=e^{-bt}u(t)\left[ \frac{e^{(b-a)\tau}}{b-a}\right]_0^t=e^{-bt}u(t)\left[\frac{e^{(b-a)t}}{b-a}-\frac{1}{b-a}\right]\\ |
&=\left[\frac{e^{-at}-e^{-bt}}{b-a}\right]\,u(t) | &=\left[\frac{e^{-at}-e^{-bt}}{b-a}\right]\,u(t) | ||
| + | \end{align}</math></center> | ||
| + | |||
| + | * "Single" exponent | ||
| + | <center><math>\begin{align} | ||
| + | (e^{-at}\,u(t))*(u(t))&=(e^{-at}\,u(t))*(e^{-0t}\,u(t))\\ | ||
| + | &=\left[\frac{e^{-at}-e^{-0t}}{0-a}\right]\,u(t)=\left[\frac{1-e^{-at}}{a}\right]\,u(t) | ||
\end{align}</math></center> | \end{align}</math></center> | ||
Latest revision as of 01:43, 18 September 2023
The following is a list of convolutions that are good to know. In each case, \(f(t)\) represents an arbitrary function while \(a\) and \(b\) represent constants.
Contents
Convolution with Impulses
Convolution with Steps and Ramps
Convolution Between Singularity Functions
Convolution Between Exponentials
Note - the following work if $$a$$ and/or $$b$$ is 0.
- Same exponent
- Different exponents
- "Single" exponent
Examples
Exponential and Shifted Step
Find \(y(t)\) if \(x(t)=u(t-a)\) and \(h(t)=e^{-2t}u(t)\):
Questions
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