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Sampling

Sampling. COS 323. Signal Processing. Sampling a continuous function. . Signal Processing. Convolve with reconstruction filter to re-create signal. . =. How to Sample?. Reconstructed signal might be very different from original: “aliasing”. . . Why Does Aliasing Happen?.

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Sampling

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  1. Sampling COS 323

  2. Signal Processing • Sampling a continuous function 

  3. Signal Processing • Convolve with reconstruction filter tore-create signal  =

  4. How to Sample? • Reconstructed signal might be very different from original: “aliasing”  

  5. Why Does Aliasing Happen? • Sampling = multiplication by shah function III(x) (also known as impulse train)  = III(x)

  6. Fourier Analysis • Multiplication in primal space = convolution in frequency space • Fourier transform of III is III sampling frequency F(III(x)) III(x)

  7. Fourier Analysis • Result: high frequencies can “alias”into low frequencies  =

  8. Fourier Analysis • Convolution with reconstruction filter = multiplication in frequency space  =

  9. Aliasing in Frequency Space • Conclusions: • High frequencies can alias into low frequencies • Can’t be cured by a different reconstruction filter • Nyquist limit: can capture all frequencies iff signal has maximum frequency  ½ sampling rate = 

  10. Filters for Sampling • Solution: insert filter before sampling • “Sampling” or “bandlimiting” or “antialiasing” filter • Low-pass filter • Eliminate frequency content above Nyquist limit • Result: aliasing replaced by blur OriginalSignal Prefilter Sample ReconstructionFilter ReconstructedSignal

  11. Ideal Sampling Filter • “Brick wall” filter:box in frequency • In space: sinc function • sinc(x) = sin(x) / x • Infinite support • Possibility of “ringing”

  12. Cheap Sampling Filter • Box in space • Cheap to evaluate • Finite support • In frequency: sinc • Imperfect bandlimiting

  13. Gaussian Sampling Filter • Fourier transform of Gaussian = Gaussian • Good compromise as sampling filter: • Well approximated by function w. finite support • Good bandlimiting performance

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