Transforms - Convolutions

 
   
     
       

Convolution

       

Understanding Convolution

       

          Convolution is a fundamental operation in signal and system analysis. It combines two functions to produce a third that represents how the shape of one is modified by the other.        

        Mathematical definition of convolution.      
     
       
         

Definition:

         
           

\[ (f * g)(x) = \int_{-\infty}^{\infty} f(u)g(x-u)du \]

           

\[ f * g = g * f \quad ; \quad f * (g * h) = (f * g) * h \]

           

\[ f(x)g(x) \iff F(s) * G(s) \]

         
         

Key Properties:

         
               
  • Commutative, associative, and distributive over addition.
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  • Convolution in time domain = multiplication in frequency domain.
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Applications:

         
               
  • System response analysis in LTI systems.
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  • Image blurring and sharpening.
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  • Audio signal filtering.
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