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===Fourier Transform:=== | ===Fourier Transform:=== | ||
| − | :<math>\ | + | :<math>\ F(f) = \int_{-\infty}^{\infty} x(t)\ e^{- j 2 \pi f t}\,dt </math> |
===Inverse Fourier Transform:=== | ===Inverse Fourier Transform:=== | ||
| − | :<math>f( | + | :<math>\ f(t) = \int_{-\infty}^{\infty} F(f)\ e^{j 2 \pi f t}\,df </math> |
| − | for every real number ''f & | + | for every real number ''f & t''. |
==Basic Properties of Fourier Transforms:== | ==Basic Properties of Fourier Transforms:== | ||
| Line 13: | Line 13: | ||
Linearity: | Linearity: | ||
| − | :If <math>\ h( | + | :If <math>\ h(t) = a.f(t) + b.g(t)</math> then <math>\ H(f)= a.F(f)+b.G(f)</math> |
Time Shifting: | Time Shifting: | ||
| − | :If <math>\ f( | + | :If <math>\ f(t)=g(t-t_0) </math> then <math>\ F(f)=e^{-2\pi i f t_0 }G(f)</math> |
Frequency Shifting: | Frequency Shifting: | ||
| − | :If <math>\ f( | + | :If <math>\ f(t)= e^{2\pi i t f_0}g(t) </math> then <math>\ F(f)=G(f-f_0)</math> |
Time Scaling: | Time Scaling: | ||
| − | :If <math>\ f( | + | :If <math>\ f(t)=g(at) </math> then <math>\ F(f)=\frac{1}{|a|} G(\frac{f}{a})</math> |
Convolution: | Convolution: | ||
| + | Convolution in Time domain corresponds to multiplication in Frequency domain. | ||
| + | |||
| + | :If <math>\ h(t)=f(t)*g(t)</math> then <math>\ H(f)=F(f).G(f)</math> | ||
| + | ---- | ||
| + | [[Hw3ECE438F09boutin|Back to HW3 ECE438 Fall 2009]] | ||
| + | |||
| + | [[ECE438|Back to ECE438]] | ||
Latest revision as of 06:34, 25 August 2010
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) $
