Contents
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} F(f)\ e^{j 2 \pi f t}\,df $
for every real number f & t.
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.
- If $ \ h(t)=f(t)*g(t) $ then $ \ H(f)=F(f).G(f) $
