Convolution_Deconvolution

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					Convolution

   A mathematical operator which computes the pointwise overlap
    between two functions.

   Discrete domain:



   Continuous domain:
Discrete domain

Basic steps
   1.   Flip (reverse) one of the functions.
   2.   Shift it along the time axis by one sample.
   3.   Pointwise multiply the corresponding values of the two digital
        functions.
   4.   Sum the products from step 3 to get one point of the digital
        convolution.
   5.   Repeat steps 1-4 to obtain the digital convolution at all times that the
        functions overlap.
Continuous domain example
Continuous domain example
LTI (Linear Time-Invariant) Systems

   Convolution can describe the effect of an LTI system on a signal

   Assume we have an LTI system H, and its impulse response h[n]

   Then if the input signal is x[n], the output signal is y[n] = x[n] * h[n]




            x[n]                 H              y[n] = x[n]*h[n]
Fourier Series

   Most periodic functions can be expressed as a (infinite) linear
    combination of sines and cosines

     F(t) = a0 + a1cos (ωt) + b1sin(ωt) +
                  a2cos (2ωt) + b2sin(2ωt) + …

             
                
         =      n 0
                       (an cos(nt ) bn sin(nt ))


     F(t) is a periodic function with
                                                            2
                                                         
                                                            T
Most Functions?

                                                           2
   F(t) is a periodic function with
                                                        
                                                           T
   must satisfy certain other conditions:

     –   finite number of discontinuities within T
     –   finite average within T
     –   finite number of minima and maxima
Calculate Coefficients


  F (t )  n 0 (an cos(nt ) bn sin(nt ))
                     




         T

          f (t )dt
  a0    0
                 T
             T                               T
       2                                 2
  ak 
       T      f (t ) cos(kt )dt
             0
                                    bk 
                                         T    f (t ) sin(kt )dt
                                             0
Example

                                                                2
   F(t) = square wave, with T=1.0s (           )
                                                             
                                                                1.0


                                                        Series1



            1




            0
                0   1    2     3        4   5       6             7
Example


                                4
                 F (t )            sin t
                            
                                                     Series1
                                                     Series2


     1




     0
         0   1     2        3         4      5   6             7
Example


              4        4           4
    F (t )  sin t     sin 3t  sin 5t
                     3          5

                                      Series1
                                      Series3


      1




      0
          0   1   2   3   4   5   6         7
Why Frequency Domain?

   Allows efficient representation of a good approximation to the
    original function

   Makes filtering easy

   And a whole host of other reasons….
But One Really Important One


   Note that convolution in the time domain is equivalent to
    multiplication in the frequency domain (and vice versa)!
Fourier Family
Fourier Transform

   Definitions:




   Can be difficult to compute => Often rely upon table of transforms
Delta function

   Definition:




   Often, the result of the Fourier Transform needs to be expressed in
    terms of the delta function
Fourier Transform pairs

                               

                               



   There is a duality in all transform pairs




                   

				
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