10/08 - Mechanics of Convolution & Fourier Transform

From Class Wiki
Jump to: navigation, search

Mechanics of the Convolution

Remember from the game:

Input LTI System Output Reason
 \delta (t)\,\!  \Longrightarrow  h(t) \,\! Given
 \delta (t-\lambda)\,\!  \Longrightarrow  h(t-\lambda) \,\! Time Invarience
 x(\lambda) \delta (t-\lambda)\,\!  \Longrightarrow  x(\lambda)h(t-\lambda) \,\! Proportionality
 x(t) = \int_{-\infty}^{\infty} x(\lambda) \delta (t-\lambda)\, dx  \Longrightarrow  \underbrace{\int_{-\infty}^{\infty} x(\lambda)h(t-\lambda)\, dx}_{Convolution Integral} Superposition

We will also denote the convolution as  x(t) * h(t) \equiv \int_{-\infty}^{\infty} x(\lambda)h(t-\lambda)\, dx

Communative Property

x(t) * h(t) \,\! =\int_{-\infty}^{\infty} x(\lambda)h(t-\lambda)\, d\lambda Let  t-\lambda = u \,\! thus  du = -d\lambda \,\!
=-\int_{\infty}^{-\infty} x(t-u)h(u)\, du The order of integration switched due to changing from -\lambda = u\,\!
=\int_{-\infty}^{\infty} h(u)x(t-u)\, du
=h(t)*x(t) \,\!

Example 1

\delta(t)*x(t)\,\! =\int_{-\infty}^{\infty} \delta(\lambda)x(t-\lambda)\,d\lambda
=x(t)\int_{-\infty}^{\infty} \,d\lambda
=x(t)\,\!

Example 2

 \overbrace{[u(t)-u(t-1)]}^{x(t)}*\overbrace{[u(t-1)-u(t-3)]}^{h(t)} =
\begin{cases}
0, & t \le 1 \\
\int_{0}^{t-1} 1 \cdot 2 \,d\lambda, & 1 \le t \le 2\\
\int_{0}^{1} 1 \cdot 2 \,d\lambda, & 2 \le t \le 3\\
\int_{t-3}^{1} 1 \cdot 2 \,d\lambda, & 3 \le t \le 4\\
0, & t > 4 \\
\end{cases} = \begin{cases}
0, & t \le 1 \\
2\,t - 2, & 1 \le t \le 2\\
2, & 2 \le t \le 3\\
-2\,t - 4, & 3 \le t \le 4\\
0, & t > 4 \\
\end{cases}

  • In this case, we are doing the FSMI to  h(t) \,\!
  • If  u(t) \,\! isn't involved, then you can plug n chug with the integral. The u(t) will change the limits, which can be impractical to evaulate if you have more than 2.
  • ? Does it matter which one you FSMI?

Oct8.PNG

Convolution: A visual approach

  • Flip: Flip one about the dependant axis
  • Shift: The initial flipped function is at  t = 0 \,\!. Shift this function for the multiply & integrate
  • Multiply: Multiply the two functions
  • Add/Integrate: You may need to make multiple equations for different intersections

Convolution3.PNG