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# Remember e i 2 π ut cos 2 π ut i sin 2 π ut the

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Remember: e i 2 π ut = cos ( 2 π ut ) + i sin ( 2 π ut ) The real part is a cosine of frequency u . The imaginary part is a sine of frequency u .

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The Fourier Transform The Fourier Transform The Fourier Series For a finite set of basis frequencies { u k } : All Functions { e k ( t ) } Harmonics { e i 2 π ut } Transform a k = f · e k a k = f · e i 2 π u k t = -∞ f ( t ) e k ( t ) dt = -∞ f ( t ) e - i 2 π u k t dt Inverse f ( t ) = k a k e k ( t ) f ( t ) = k a k e i 2 π u k t
The Fourier Transform The Fourier Transform The Fourier Transform Many tasks need an infinite number of basis functions (frequencies), each with their own weight F

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