Lecture 3.7: Fourier transforms Matthew Macauley Department of - PowerPoint PPT Presentation

Lecture 3.7: Fourier transforms Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4340, Advanced Engineering Mathematics M. Macauley (Clemson) Lecture 3.7: Fourier transforms Advanced Engineering Mathematics 1 / 7

What is a Fourier transform? Definition Suppose f : R → C vanishes outside some finite interval. Its Fourier transform is defined by � ∞ � π L f ( x ) e − i ω x dx = lim f ( x ) e − i ω x dx . F ( f ) = � f ( ω ) = L →∞ − π L −∞ Suppose f vanishes outside [ − π L , π L ]. Extend this function to be 2 π L -periodic. Note that � ∞ � π L 1 1 1 f ( x ) e − inx / L dx = f ( x ) e − inx / L dx = c n . � f ( n / L ) = 2 π L 2 π L 2 π L − π L −∞ Thus, we can write f ( x ) as � � ∞ ∞ � n � 1 c n e i π n L x = e i π n � L x . f ( x ) = f L 2 π L n = −∞ n = −∞ Let ω n = n L and ∆ ω = 1 L . Taking the limit as ∆ ω → 0 yields � ∞ � ∞ � ∞ L x = 1 f ( ω n ) e i πω n x ∆ ω = 1 c n e i π n � � f ( ω ) e i πω x d ω. f ( x ) = lim lim 2 π 2 π L →∞ ∆ ω → 0 −∞ n = −∞ n = −∞ This is called the inverse Fourier transform of � f ( ω ), also denoted F − 1 ( � f ). M. Macauley (Clemson) Lecture 3.7: Fourier transforms Advanced Engineering Mathematics 2 / 7

Example: a rectangular pulse   1 − 0 . 5 < x < 0 . 5  Consider a 2 L -periodic function defined by f ( x ) = 0 . 5 x = 0 . 5   0 0 . 5 < | x | < L . If L = 1, compute its complex Fourier series. How does this compare to L = 2? To L = 200? What is its Fourier transform? M. Macauley (Clemson) Lecture 3.7: Fourier transforms Advanced Engineering Mathematics 3 / 7

A “continuous” version of a Fourier series Every continuous function f : [ − π, π ] → C can be decomposed into a discrete sum of complex exponentials: � π � ∞ c n = 1 f ( x ) e − inx dx , c n e inx , f ( x ) = let ω = 1 . 2 π − π n = −∞ Every continuous function f : [ − 2 π, 2 π ] → C can be decomposed into a discrete sum of complex exponentials: � 2 π � ∞ c n = 1 f ( x ) e − inx / 2 dx , c n e inx , f ( x ) = let ω = 1 / 2 . 4 π − 2 π n = −∞ Every continuous function f : [ − 200 π, 200 π ] → C can be decomposed into a discrete sum of complex exponentials: � 200 π � ∞ 1 f ( x ) e − inx / 200 dx , c n e inx , f ( x ) = c n = let ω = 1 / 200 . 400 π − 200 π n = −∞ Now take the limit as L → ∞ . . . Every continuous function f : ( −∞ , ∞ ) → C can be decomposed into a discrete sum integral of complex exponentials: � ∞ � ∞ c ω = 1 f ( x ) e − i ω x dx = 1 c ω e i ω x d ω, � f ( x ) = f ( ω ) . 2 π 2 π −∞ −∞ M. Macauley (Clemson) Lecture 3.7: Fourier transforms Advanced Engineering Mathematics 4 / 7

The sine cardinal (sinc) function The Fourier transform of the “rectangle function” in the previous example is � 1 x = 0 sinc( x ) = sin x x � = 0 x This is called the “sampling function” in signal processing. M. Macauley (Clemson) Lecture 3.7: Fourier transforms Advanced Engineering Mathematics 5 / 7

“Evil twins” of the Fourier transform Our Fourier transform and inverse transform: � ∞ � ∞ f ( x ) = 1 � f ( x ) e − i ω x dx , � f ( x ) e i ω x d ω f ( ω ) := and 2 π −∞ −∞ The opposite Fourier transform and its inverse: � ∞ � ∞ f ( ω ) := 1 ˇ f ( x ) e − i ω x dx , ˇ f ( x ) e i ω x d ω and f ( x ) = 2 π −∞ −∞ The symmetric Fourier transform and its inverse: � ∞ � ∞ 1 1 ⌢ f ( x ) e − i ω x dx , ⌢ ( x ) e i ω x d ω f ( ω ) := √ and f ( x ) = √ f 2 π 2 π −∞ −∞ The canonical Fourier transform and its inverse: � ∞ � ∞ � f ( x ) e − 2 π i ξ x dx , � f ( x ) e 2 π i ξ x d ξ f ( ξ ) := and f ( x ) = −∞ −∞ This last definition is motivated by of the relation ω = 2 πξ between angular frequency ω (radians per second) and oscillation frequency ξ (cycles per second, or “Hertz”). It is easy to go between these definitions: √ � ω � � f ( ω ) = 2 π ˇ ⌢ ( ω ) = � = � f ( ω ) = 2 π f f f ( ξ ) . 2 π M. Macauley (Clemson) Lecture 3.7: Fourier transforms Advanced Engineering Mathematics 6 / 7

Recall that the Laplace transform of a function f ( t ) is � ∞ f ( t ) e − st . F ( s ) = −∞ To get its Fourier transform, just plug in s = i ω : � ∞ � � f ( t ) e − i ω t = F ( s ) F ( i ω ) = s = i ω . � −∞ Because of this, these transforms share many similar properties: Property time-domain frequency domain c 1 � f 1 ( ω ) + c 2 � Linearity c 1 f 1 ( t ) + c 2 f 2 ( t ) f 2 ( ω ) e − i ω t 0 � Time / phase-shift f ( t − t 0 ) f ( ω ) � e i ν t f ( t ) Multiplication by exponential f ( ω − ν ) c � 1 Dilation by c > 0 f ( ct ) f ( ω/ c ) df ( t ) i ω � Differentiation f ( ω ) dt − d ˆ Multiplication by t tf ( t ) f ( ω ) d ω f 2 ( ω ) · � � f 2 ( ω ) = ( � f 1 ∗ � Convolution f 1 ( t ) ∗ f 2 ( t ) f 2 )( ω ) M. Macauley (Clemson) Lecture 3.7: Fourier transforms Advanced Engineering Mathematics 7 / 7