Difference between revisions of "Fourier Transforms and its Properties" - Rhea

Revision as of 11:58, 23 September 2009

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

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

                                                                                 for every real number f & x.

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.

@@ Line 2: / Line 2: @@
 ===Fourier Transform:===
-:<math>\ X(f) = \int_{-\infty}^{\infty} x(t)\ e^{- j 2 \pi f t}\,dt </math>
+:<math>\ F(f) = \int_{-\infty}^{\infty} x(t)\ e^{- j 2 \pi f t}\,dt </math>
 ===Inverse Fourier Transform:===
-:<math>f(x) = \int_{-\infty}^{\infty} X(f)\ e^{j 2 \pi f t}\,df </math>
+:<math>\ f(t) = \int_{-\infty}^{\infty} X(f)\ e^{j 2 \pi f t}\,df </math>
                                                                                    for every real number ''f & x''.
@@ Line 13: / Line 13: @@
 Linearity:
-:If <math>\  h(x) = a.f(x) + b.g(x)</math> then <math>\ H(f)= a.F(f)+b.G(f)</math>
+:If <math>\  h(t) = a.f(t) + b.g(t)</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>
+: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:
-:If <math>\ f(x)= e^{2\pi i x f_0}g(x) </math> then <math>\ F(f)=G(f-f_0)</math>
+: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:
-:If <math>\ f(x)=g(ax) </math> then <math>\ F(f)=\frac{1}{|a|} G(\frac{f}{a})</math>
+:If <math>\ f(t)=g(at) </math> then <math>\ F(f)=\frac{1}{|a|} G(\frac{f}{a})</math>
 Convolution:
+Convolution in Time domain corresponds to multiplication in Frequency domain.