(New page: ==Fourier Transform and its basic Properties:== ===Fourier Transform:=== :<math>\ X(f) = \int_{-\infty}^{\infty} x(t)\ e^{- j 2 \pi f t}\,dt, </math> ===Inverse Fourier Transform:...) |
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===Fourier Transform:=== | ===Fourier Transform:=== | ||
| − | :<math>\ X(f) = \int_{-\infty}^{\infty} x(t)\ e^{- j 2 \pi f t}\,dt | + | :<math>\ X(f) = \int_{-\infty}^{\infty} x(t)\ e^{- j 2 \pi f t}\,dt </math> |
===Inverse Fourier Transform:=== | ===Inverse Fourier Transform:=== | ||
| − | :<math>f(x) = \int_{-\infty}^{\infty} X(f)\ e^{j 2 \pi f t}\,df | + | :<math>f(x) = \int_{-\infty}^{\infty} X(f)\ e^{j 2 \pi f t}\,df </math> |
for every real number ''f & x''. | for every real number ''f & x''. | ||
==Basic Properties of Fourier Transforms:== | ==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 | |
| − | Time Shifting: | + | Linearity: |
| + | :If <math>\ h(x) = a.f(x) + b.g(x)</math> then <math>\ H(f)= a.F(f)+b.G(f)</math> | ||
| + | |||
| + | Time Shifting: | ||
| + | |||
| + | :If <math>\ f(x)=g(x-x_0) </math> then <math>\ F(f)=e^{-2\pi i f x_0 }G(f)</math> | ||
Frequency Shifting: | Frequency Shifting: | ||
| + | |||
| + | :If <math>\ f(x)= e^{2\pi i x f_0}g(x) </math> then <math>\ F(f)=G(f-f_0)</math> | ||
Time Scaling: | Time Scaling: | ||
| + | |||
| + | :If <math>\ f(x)=g(ax) </math> then <math>\ F(f)=\frac{1}{|a|} G(\frac{f}{a})</math> | ||
Convolution: | Convolution: | ||
Revision as of 10:55, 23 September 2009
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:
