# Continuous Time Fourier Transform

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

*
F*[

*x*(

*t*)] =

*X*(

*j*Ï‰) =

Inverse Fourier Transform is given by

*F*^{â€“1} [*X*(*j*Ï‰)] = *x*(*t*)

*x*(*t*) = *F*^{â€“1}[*X*(*j*Ï‰)] =

**Properties of Continuous Time Fourier Transform are given below:**

**Linearity :**If and then where*a*and*b*are constants.**Time-scaling :**If*x*(*t*) â†”*X*(Ï‰) then, for any real constant*a*,*x*(*at*) â†”*X*(Ï‰/*a*)

Linear scaling in time domain by a factor of â€˜*a*â€™ becomes linear scaling in frequency by a factor of â€˜1/*a*â€™ in frequency domain.**Duality or Symmetry :**

then**Time-shift :**

A linear phase shift in the signalâ€™s specturm is caused by the time delay in that signal.**Frequency-shift or Modulation :**

Due to modulation, the signal spectrum spreads to higher frequencies.**Differentiation in time :**

then**Frequency differentiation :**

then**Time Convolution Theorem :**, , then**Frequency convolution theorem :**

and then**Integration in time :**then

**Parsevalâ€™s Theorem for Energy Signals**

*E* = =

Parsevalâ€™s theorem for power signals

*P* =