This is the continuation of the PREVIOUS TUTORIAL.

**Steps for Graphical Convolution**

- First of all re-write the signals as functions of τ:
*x*(τ) and*h*(τ)

- Flip one of the signals around
*t*= 0 to get either*x*(-τ) or*h*(-τ)

- Best practice is to flip the signal with shorter interval
- We will flip
*h*(τ) to get*h*(-τ) throughout the steps

- Determine Edges of the flipped signal

- Determine the left-hand edge τ-value of
*h*(-τ): say τ_{L}_{,0} - Determine the right-hand edge τ-value of
*h*(-τ): say τ_{R}_{,0}

- Shifting
*h*(-τ) by a random value of*t*to obtain*h*(*t*-τ) and get its edges

- Determine the left-hand edge τ-value of
*h*(*t*-τ) as a function of*t*: say τ_{L,t}

Noteworthy: It will *forever *be…τ* _{L,t}* = t + τ

_{L}_{,0}

- Determine the right-hand edge τ-value of
*h*(*t*-τ) as a function of*t*: say τ_{R,t}

Noteworthy: It will *forever *be…τ* _{R,t}* = t + τ

_{R}_{,0}

- Find out Regions of τ-Overlap

- Determine intervals of
*t*over which the product*x*(τ)*h*(*t*-τ) possesses a single unique mathematical form in terms of τ

- For Each Particular Region: develop the Product
*x*(τ)*h*(*t*-τ) and Integrate

- Develop the product
*x*(τ)*h*(*t*-τ)

- Determine the boundaries of Integration by determining the interval of τ over which the mathematical product is nonzero

- Determine by discovering where the bounds of
*x*(τ) and*h*(*t*-τ) lie down - Call back that the bounds of
*h*(*t*-τ) are τand τ_{L,t}, which frequently depend upon the value of_{R,t}*t*

- Integrate the product
*x*(τ)*h*(*t*-τ) over the boundaries determined in 6b

- “Put Together” the output from the output time-sections for each of the region important:

- DO NOT add together all the sections

- Specify the output in “piecewise” manner

This example is provided in collaboration with Prof. Mark L. Fowler, Binghamton University.

- You May Also Read: Continuous-Time Convolution Properties