I am trying to make a succinct summary of how algebraic operations in the time domain translate in the frequency domain (via the use of Fourrier/Laplace transforms). Here is what I have derived so far:

Time Domain                       Frequency Domain       
Summation/superposition   Summation/superposition
Time Delay                        Frequency shift                    
Integration/Differentiation  Multiplication/Division      
Multiplication/Division        Addition/Substraction        

However, I have noticed that if summation/superposition is the same in time and frequency domain, how can multiplication and division in time domain yield addition and subtraction in frequency domain. I also noticed that convolution in time domain results in multiplication in frequency domain so line 4 of the table looks wrong also...

Can someone clear this up or maybe provide a better summary of the different operations in time domain and how they translate in frequency domain?

Somewhat-related question: link.

  • 3
    $\begingroup$ How did you come up with that table? Lines 2 and 4 are wrong, and line 3 is unclear (multiplication/division by what?). I would believe that the correct versions of these correspondences can be found in most transform tables. Also note that duality helps, i.e., once you got one correspondence right, it also works the other way. $\endgroup$
    – Matt L.
    Apr 10, 2017 at 13:50
  • $\begingroup$ Could you point me to a correct table? $\endgroup$
    – Psi
    Apr 10, 2017 at 16:31
  • 2
    $\begingroup$ Googling Fourier transform properties table gives you more useable hits than you might want to check. $\endgroup$
    – Matt L.
    Apr 10, 2017 at 16:41


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