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ft-properties.slides.printing

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

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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

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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: φ ( 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 π st ) F ( u ) = Z -∞ f ( t ) e - i 2 π ut dt = Z -∞ cos ( 2 π st ) e - i 2 π ut dt = Z -∞ cos ( 2 π st ) [ cos ( - 2 π ut ) + i sin ( - 2 π ut )] dt = Z -∞ cos ( 2 π st ) cos ( - 2 π ut ) dt + i Z -∞ cos ( 2 π st ) sin ( - 2 π ut ) dt = Z -∞ cos ( 2 π st ) cos ( 2 π ut ) dt - i Z -∞ cos ( 2 π st ) sin ( 2 π ut ) dt 0 except when u = ± s 0 for all u = 1 2 δ ( u - s ) + 1 2 δ ( u + s )

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The Fourier Transform:
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