Fourier_1a

Fourier_1a - Introduction to Sampled Signals and Fourier...

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Physics116C, 4/28/06 D. Pellett Introduction to Sampled Signals and Fourier Transforms References: Essick, Advanced LabVIEW Labs Press et al., Numerical Recipes , Ch. 12 Brigham, The Fast Fourier Transform and its Applications Any original presentations copyright D. Pellett 2006 1
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Sampled Signals and Nyquist Frequency As we were saying at the end of Physics 116B: 2
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Sampling Theorem 3
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Aliasing of f > fc In general, will need to limit bandwidth with a low-pass filter before digitizing. For audio response to 17 kHz with 44 kHz sample frequency must use sharp cutoff filter to eliminate f > f c = 22 kHz (although little power may remain in this part of the spectrum). 4
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Fourier Transform To pursue this we need Fourier transforms • This is the convention used in Essick • It has a nice correspondence with complex phasors from 116A 5
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Some FT Theorems (again, same convention as Essick) • Convolution integral is important concept • Also useful in statistics (as is Fourier transform). More on this later 6
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Convolution Theorem – We will look at an example soon Again, we 7
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Power Spectral Density in Frequency Define 8
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Some Useful Transform Pairs • Spacing of δ fcns in t is T while spacing in f is 1/T 9
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Time Domain Frequency Domain h ( t ) = A - T 0 < t < T 0 = A 2 t = - T 0 , T 0 = 0 t < - T 0 , t > T 0 H ( f ) = 2 AT 0 sinc(2 π T 0 f ) h ( t ) = 2 Af 0 sinc(2 π f 0 t ) H ( f ) = A - f 0 < f < f 0 = A 2 f = - f 0 , f 0 = 0 f < - f 0 , f > f 0 h ( t ) = K H ( f ) = K δ ( f ) h ( t ) = K δ ( t ) H ( f ) = K h ( t ) = n = -∞ δ ( t - nT ) H ( f ) = n = -∞ 1 T δ
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Fourier_1a - Introduction to Sampled Signals and Fourier...

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