10/08 - Mechanics of Convolution & Fourier Transform

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?

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