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Continuous Time Graphical Convolution Example

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This is the continuation of the PREVIOUS TUTORIAL.

Steps for Graphical Convolution

1. First of all re-write the signals as functions of τ: x(τ) and h(τ)
1. 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
1. Determine Edges of the flipped signal
• Determine the left-hand edge τ-value of h(-τ): say τL,0
• Determine the right-hand edge τ-value of h(-τ): say τR,0
1. 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 τL,t

Noteworthy: It will forever be…τL,t = t + τL,0

• Determine the right-hand edge τ-value of h(t-τ) as a function of t: say τR,t

Noteworthy: It will forever be…τR,t = t + τR,0

1. 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
1. 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 τL,t and τR,t , which frequently depend upon the value of t
• Integrate the product x(τ) h(t-τ) over the boundaries determined in 6b
1. “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.

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