1. Sampling COS 323, Spring 2005

2. Signal Processing Sampling a continuous functionSampling a continuous function 

3. Signal Processing Convolve with reconstruction filter to re-create signalConvolve with reconstruction filter to re-create signal  =

4. How to Sample? Reconstructed signal might be very different from original: “aliasing”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)Sampling = multiplication by shah function III (x) (also known as impulse train)  = III (x)

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

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

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

9. Aliasing in Frequency Space Conclusions: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 samplingSolution: insert filter before sampling – “Sampling” or “bandlimiting” or “antialiasing” filter – Low-pass filter – Eliminate frequency content above Nyquist limit – Result: aliasing replaced by blur Original Signal PrefilterPrefilterSampleSample Reconstruction Filter Reconstructed Signal

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

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

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