ft-properties.slides.printing.6

# ft-properties.slides.printing.6 - The Fourier Transform:...

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Unformatted text preview: The Fourier Transform: Examples, Properties, Common Pairs The Fourier Transform: Examples, Properties, Common Pairs CS 450: Introduction to Digital Signal and Image Processing Bryan Morse BYU Computer Science The Fourier Transform: Examples, Properties, Common Pairs Magnitude and Phase Remember: complex numbers can be thought of as (real,imaginary) or (magnitude,phase). Magnitude: | F | = < ( F ) 2 + = ( F ) 2 1 / 2 Phase: ( F ) = tan- 1 = ( F ) < ( F ) Real part How much of a cosine of that frequency you need Imaginary part How much of a sine of that frequency you need Magnitude Amplitude of combined cosine and sine Phase Relative proportions of sine and cosine The Fourier Transform: Examples, Properties, Common Pairs Example: Fourier Transform of a Cosine f ( t ) = cos ( 2 st ) F ( u ) = Z - f ( t ) e- i 2 ut dt = Z - cos ( 2 st ) e- i 2 ut dt = Z - cos ( 2 st ) [ cos (- 2 ut ) + i sin (- 2 ut )] dt = Z - cos ( 2 st ) cos (- 2 ut ) dt + i Z - cos ( 2 st ) sin (- 2 ut ) dt = Z - cos ( 2 st ) cos ( 2 ut ) dt- i Z - cos ( 2 st ) sin ( 2...
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ft-properties.slides.printing.6 - The Fourier Transform:...

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