Fourier Transforms and its Properties - Rhea

Fourier Transform and its basic Properties:

Fourier Transform:

$ \ F(f) = \int_{-\infty}^{\infty} x(t)\ e^{- j 2 \pi f t}\,dt $

Inverse Fourier Transform:

$ \ f(t) = \int_{-\infty}^{\infty} X(f)\ e^{j 2 \pi f t}\,df $

                                                                                 for every real number f & x.

Basic Properties of Fourier Transforms:

Suppose a and b are any complexn numbers, if h(x) ƒ(x) and g(x) Fourier Transform to H(f) F(f) and G(f) respectively, then

Linearity:

If $ \ h(t) = a.f(t) + b.g(t) $ then $ \ H(f)= a.F(f)+b.G(f) $

Time Shifting:

If $ \ f(t)=g(t-t_0) $ then $ \ F(f)=e^{-2\pi i f t_0 }G(f) $

Frequency Shifting:

If $ \ f(t)= e^{2\pi i t f_0}g(t) $ then $ \ F(f)=G(f-f_0) $

Time Scaling:

If $ \ f(t)=g(at) $ then $ \ F(f)=\frac{1}{|a|} G(\frac{f}{a}) $

Convolution: Convolution in Time domain corresponds to multiplication in Frequency domain.