Convolution Sum and Convolution Integral

Learning - Signals and Systems, Uncategorized

summaryOfConvolution

One of the most important operation in signals processing is convolution. For a typical LTI system, the output can be produced by convolving the input with the given impulse responses. In general, we have 2 types of convolutions:

  • Convolution sum for Discrete LTI system
  • Convolution Integral for Continuos LTI system

convolutionSum.PNG

convolutionIntegral.PNG

 

Here, we will see how to compute convolution integral. Basically, the convolution integral involves 4 steps as shown below:

convolutionIntegralStep.PNG

 

Let’s us look at the simple example, x(t) and h(t) is given below:

ci_Eg

So follow step 1 to use a dummy variable τ to replace t

ci_step1

then flip h(τ) as indicated in step 2

ci_step2

Repeating step 3 & 4 until obtaining the final answer

ci_step3-4

ci_step3-4(2)

 

ci_step3-4(3)

ci_step3-4(4)

so, the final answer is

finalAns

 

Similarly, we can also compute the convolution sum for Discrete LTI system by using the following 4 steps:

convolutionSumStep.PNG

we will look at some related examples on computing convolution sum in the next post. Stay tuned.

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