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1.7. ANALYSIS OF LINEAR SYSTEMS WITH THE HELP OF FOURIER TRANSFORMATION

Lecture



The Fourier transform convolution theorem (1.6.14) turns out to be a very useful tool in analyzing linear systems. Consider the function   1.7.  ANALYSIS OF LINEAR SYSTEMS WITH THE HELP OF FOURIER TRANSFORMATION describing the image at the input of the linear system by the simplicity response   1.7.  ANALYSIS OF LINEAR SYSTEMS WITH THE HELP OF FOURIER TRANSFORMATION . The output image is described by the function   1.7.  ANALYSIS OF LINEAR SYSTEMS WITH THE HELP OF FOURIER TRANSFORMATION resulting from the convolution:

  1.7.  ANALYSIS OF LINEAR SYSTEMS WITH THE HELP OF FOURIER TRANSFORMATION (1.7.1)

Performing the Fourier transform of both parts of this equality and changing the order of integration in its right side, we get

  1.7.  ANALYSIS OF LINEAR SYSTEMS WITH THE HELP OF FOURIER TRANSFORMATION (1.7.2)

According to the shift theorem (1.6.13), the inner integral is equal to the product of the spectrum of the function   1.7.  ANALYSIS OF LINEAR SYSTEMS WITH THE HELP OF FOURIER TRANSFORMATION and the exponential phase shift factor. therefore

  1.7.  ANALYSIS OF LINEAR SYSTEMS WITH THE HELP OF FOURIER TRANSFORMATION (1.7.3)

Having executed Fourier transforms, we get

  1.7.  ANALYSIS OF LINEAR SYSTEMS WITH THE HELP OF FOURIER TRANSFORMATION (1.7.4)

Finally, the inverse Fourier transform gives a function that describes the output image:

  1.7.  ANALYSIS OF LINEAR SYSTEMS WITH THE HELP OF FOURIER TRANSFORMATION (1.7.5)

Expressions (1.7.1) and (1.7.5) represent two alternative ways of determining the output image of a linear spatially invariant system. The choice of one or another approach depends on the problem to be solved.


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Digital image processing

Terms: Digital image processing