Contents
Fourier Transform and its basic Properties:
Fourier Transform:
- $ \ X(f) = \int_{-\infty}^{\infty} x(t)\ e^{- j 2 \pi f t}\,dt $
Inverse Fourier Transform:
- $ f(x) = \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(x) = a.f(x) + b.g(x) $ then $ \ H(f)= a.F(f)+b.G(f) $
Time Shifting:
- If $ \ f(x)=g(x-x_0) $ then $ \ F(f)=e^{-2\pi i f x_0 }G(f) $
Frequency Shifting:
- If $ \ f(x)= e^{2\pi i x f_0}g(x) $ then $ \ F(f)=G(f-f_0) $
Time Scaling:
- If $ \ f(x)=g(ax) $ then $ \ F(f)=\frac{1}{|a|} G(\frac{f}{a}) $
Convolution:
