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Continuous Time Fourier Transform


Fourier transform of non-periodic function x(t) denoted as

[x(t)] = X(jω) = 887.png


Inverse Fourier Transform is given by


F–1 [X(jω)] = x(t)

x(t) = F–1[X(jω)] = 892.png

Properties of Continuous Time Fourier Transform are given below:

  1. Linearity : If 897.png and 902.png then 907.png where a and b are constants. 
  2. Time-scaling : If x(t X(ω) then, for any real constant ax(at 912.png X(ω/a)
    Expansion  Compression
    Linear scaling in time domain by a factor of ‘a’ becomes linear scaling in frequency by a factor of ‘1/a’ in frequency domain.
  3. Duality or Symmetry : 917.png
    then 923.png
  4. Time-shift : 928.png
    A linear phase shift in the signal’s specturm is caused by the time delay in that signal.
  5. Frequency-shift or Modulation : 933.png 938.png
    Due to modulation, the signal spectrum spreads to higher frequencies.
  6. Differentiation in time : 943.png
    then 948.png
  7. Frequency differentiation : 953.png
    then 958.png 
  8. Time Convolution Theorem : 963.png968.png, then 974.png
  9. Frequency convolution theorem : 979.png
    and 984.png then 989.png 
  10. Integration in time : 994.png then

Parseval’s Theorem for Energy Signals

E = 1004.png = 1009.png

Parseval’s theorem for power signals

P = 1014.png

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