Nyquist sampling a bandlimited function. Sampling: The connection from CT to DT. Impulse Train sampling. Interpolation: Signal Reconstruction

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1 Sampling: he connection rom C to D Nyquist sampling a bandlimited unction ( Consider as an example sampling the bandlimited signal sinc 2 (5t F 1 5 Λ 5 An ininite number o waveorms could exactly pass through and thus share a set o samples. So how can we use a sampled representation to uniquely represent a waveorm? Sampling heorem A unction whose F is zero or > c (or with bandwidth B is ully and uniquely speciied by sampled values at equal intervals not exceeding s < 1 2 c = 1 save or any harmonic terms with zeros at the sampling points. hus the original signal can be exactly recovered rom the samples Minimum sampling rate s = is Nyquist rate. sampling interval = 1/ is Nyquist interval Kelvin Wagner, University o Colorado s=20hz =.05s t(sec s=10hz =.1s practical ilter ideal ilter t(sec.2.4 s=5hz =.2s t(sec.2.4 Oversampled Nyquist rate Undersampled Kelvin Wagner, University o Colorado 249 Impulse rain sampling Multiply a waveorm by a comb to extract samples x(t x(n x s (t = x(tcomb (t = x(n δ(t n Now use comb (t F 1 comb A 1 ( = comb ( We can represent the -20 F -10 o the sampled waveorm as the convolution 1/ comb( X s ( = X( comb ( his replicates the spectrum at intervals o 1/ in the requency domainideal ilter Recover original spectrum by spectral windowing ( A X s (Π = X( 2 c A/ x(t xs(t LPF xr(t H( comb(t x(t x(n A / comb( A/ ideal ilter A Kelvin Wagner, University o Colorado 250 Interpolation: Signal Reconstruction Ideal lowpass ilter recreates C signal by passing sampled signal through X r ( = X s (H( ( ( ( where H( = Π 2 c = Π = Π s = Π ( But such an ideal LPF has ininitely long impulse response and is not causal so diicult to implement exactly. Instead use approximate LPF { 1 < c H( 0 > c zero order hold simplest reconstruction impulse response ( t h(t = Π or ( t /2 h(t = Π 0th order hold 1st order hold x(t x(n x(n x(n Kelvin Wagner, University o Colorado 251 sinc( ideal LPF

2 Signal Reconstruction ( x r (t = x s (t h(t = x(n δ(t n h(t = For the ideal LPF H( = Π [ ] Assume we are sampling at the Nyquist rate 1 = s = x r (t = x(n h(t n h(t = sinc(t = 2 B s sinc(t h(t = sinc(t = sinc( s t [ ] t n x(n sinc[(t n ] = x(n sinc Exact reconstruction since sinc is 1 at sample point and identically 0 at all other samples Kelvin Wagner, University o Colorado time domain interpretation o Aliasing ω 0 = ω s /6 < ω s /2 x r (t = cos ω 0 t = x(t ω 0 = ω s /3 < ω s /2 x r (t = cos ω 0 t = x(t Kelvin Wagner, University o Colorado 254 Aliasing 1 = s < replicated spectra will old over and leak into LPF bandwidth producing erroneous spetra and also reconstructed time domain signal Consider sinusoidalsampled signal sinusoid For -the case o 2 0 = s x(t = sin(2π 0 t x(t = reconstructed sin(π s t x(n signal = sin(nπ x r (t = 0 x(t = cos(π s t x(n = cos(nπ = ( 1 n x r (t = cos(π s t = s Ideal LPF Adequately Sampled Critical Sampling Undersampled - Kelvin- Wagner, University o Colorado time domain interpretation o Aliasing ω 0 = 4ω s /6 > ω s /2 x r (t = cos(ω s ω 0 t x(t = cos 1 3 ω st cos 2 3 ω st ω 0 = 5ω s /6 > ω s /2 x r (t = cos(ω s ω 0 t x(t = cos 1 6 ω st cos 5 6 ω st Kelvin Wagner, University o Colorado 255

3 High requencies old down to baseband Apparent Frequency o s Kelvin Wagner, University o Colorado 256 Frequencies close to Nyquist: just below s / Kelvin Wagner, University o Colorado 258 sinusoid o requency 0 will have samples identical to samples o sinusoid 0 + m s cos[2π 0 n ] cos[2π( 0 + m s n ] = cos[2π( 0 n + mn] = cos[2π 0 n ] a s/2 -s/2 undamental band a - - s/ Apparent requency o s n=s/2 sampled 3s/2 sinusoid 5s/2 phase change s 2s 3s n=s/2 s 3s/2 2s 5s/2 3s Sinusoid o requency 0 sampled at rate s will have an apparent requency a = 0 m s s 2 a < s m an integer 2 note since cos( ω a t + θ = cos(ω a t θ there is also phase change in regions a < 0 - Kelvin Wagner, University o Colorado- 257 Frequencies close to Nyquist: just above s / Kelvin Wagner, University o Colorado 259

4 Right at Nyquist N = s /2 Kelvin Wagner, University o Colorado Spectral reconstructed Components signal Anti-aliasing Filter Ideal LP ilter Practical Low Pass Filter - - Anti Aliasing ilter -s Not strictly bandlimited Spectrum o Input signal Aliased Lost Spectrum Lost Spectrum Kelvin Wagner, University o Colorado 262 s 2s Right at Nyquist N = s /2 Kelvin Wagner, University o Colorado 261 Practical Sampling with rectangular wave Rectangular pulse train ( with pulses o width τ t r(t = Π comb s (t τ With CFS representation ω s τ ( r(t = 2π sinc nωs τ e i2π s nt 2π }{{} So F o r(t is R(ω = 2π Sampled waveorm x s (t = r(tx(t = x(t with F R n ω s τ ( 2π sinc nωs τ δ(ω nω s 2π [ ( ] t Π comb s (t τ X s (ω = 1 X(ω R(ω = 2π r(t r(tx(t R n X(ω nω s Which just weights higher order copies at reduced amplitude X(ω R(ω Xs(ω Kelvin Wagner, University o Colorado 263 t t ω ω ω

5 D processing o C signals D processing o C signals 2π x(t x(t comb(t π xs(t Xc( A 1/ comb( Xs( A/ iω Hd(e 0 π Convert to Sequence x(n x[n] *h[n] y[n] Convert to ys(t yc(t H(e iω Impulse LPF train π 2 2π Ω=2π π 0 2π π 2 π ideal ilter 0 π A/ A Kelvin Wagner, University o Colorado 264 2π Express the sampled signal and its CF in terms o samples o the C signal x c (n x p (t = x c (n δ(t n X F p (jω = x c (n e jωn he D samples are just given by the sampled values x d [n] = x c (n whose DF is X d (Ω X d (e jω = x d [n]e jωn = x c (n e jωn So the CF and DF are related by X d (e jω = X p (jω/ Now since the CF o the sampled signal is repetitive X p (jω = 1 X c (j(ω kω s k= he DF o the samples is also repetitive and is just a requency scaled version o X p (jω and is periodic in Ω with period 2π X d (e jω = 1 X c (j(ω 2πk/ k= Kelvin Wagner, University o Colorado 265 ime shit o C signal using D processing ime shit o C signal using D processing Consider band limited signal x c (t sampled ast enough to avoid aliasing y c (t = x c (t Easy i = N is integer multiple o samples, otherwise use CF time-shit property Y c (jω = e jω X c (jω So or a bandlimited system with cuto ω c the transer unction is ( ω H c (jω = Π e jω 2ω c Choosing sampling req at Nyquist limit ω s = 2ω c D req response is H d (e jω = e jω / Ω < π yields D output y d [n] = x d [n / ] or integer /, but otherwise need to do bandlimited interpolation Consider an input x c (t = sinc(t/ at the limit o bandwidth allowed by Nyquist. his is sampled as x d [n] = 1 δ[n]. Now the desired C delated signal output is ( t y c (t = x c (t = sinc and since there is no aliasing the D sequence that will produce this is given by yielding So the delay system impulse response is y d [n] = 1 sinc(n / h[n] = sinc(n / Kelvin Wagner, University o Colorado 266 Kelvin Wagner, University o Colorado 267

6 Bandpass Sampling Bandpass Sampling: Undersampled ω2 ω1 ω1 B ω2 ω2 ω1 -s B ω1 ω2 s< 2s 3s -B 0 4B -B 0 4B Multiple Orders but not overlapping Multiple overlapping Orders Downconverted Sampling BPF BPF Downconverted Sampling Kelvin Wagner, University o Colorado 268 Kelvin Wagner, University o Colorado 269 Sampling Oscilloscope or periodic repetitive signals Sampling Oscilloscope Simulation Ultrahigh speed periodic waveorm Ultrahigh speed periodic waveorm Impulse train or sampling Impulse train or sampling Allow impulse train samples to drit through repetitive periodic waveorm = k + δ For repetitive sampling δ = /M should be chosen as integer raction o period Consider B =100GHz bandwidth signal repeating every =1ns choose = 10001ns 1000 samples will drit through periodic waveorm, but require repetitions. requires precise timing (psec and low SNR will require averaging (1000 in 1sec Kelvin Wagner, University o Colorado 270 i Slight spacing dierence between periodic comb comb r (t and sampling comb comb (t Product in time domain gives convolution in requency domain Kelvin Wagner, University o Colorado 271

7 2-D combs Recovery o Image rom sampled version comb (x, y = comb (xcomb (y = m= m= δ(x m 2-D grid o deltas spaced by X in x and by Y in y comb X,Y (x, y = 1 ( x XY comb X, y = δ(x mx Y scaling gives each δ(x, y unit area Sampling s (x, y = (x, y 1 XY comb ( x X, y Y δ(y n δ(y ny = (mx, ny δ(x mx, y my m,n Fourier ransorming F s (u, v = 1 XY F (u, v XY comb (Xu, Y v = 1 (u XY F m X, v n Y m,n which is an array o replicas o spectrum F (u, v separated by 1/X and 1/Y Kelvin Wagner, University o Colorado 272 CCD/CMOS detector arrays and inite sized pixel image sampling D. Brady, Optical Imaging and Spectroscopy, Wiley, 2008 Convolution o object with PSF image is detected on X Y array with MN pixels o(x, y = (x, yh(x x, y ydxdy = h Pixel sampling unction p(x, y describes integrating pixel sensitivity proile g mn = ˆ X 2 X 2 ˆ Y 2 Y 2 o(x, y p(x m x, y n y dx dy continuous smoothed unction by convolving with p(x, y, then sample g(x, y = o(x, y p(x x, y y dx dy g mn = g(x, y δ(x m x, y n y dx dy g(x, y is bandlimited since it is sampling an optical image limited by OF H(u, v with bandwidth 1 λf/# G(u, v = F (u, vh(u, vp (u, v P (u, v is the Pixel ranser Function (PF or also reerred to as detector MF. Kelvin Wagner, University o Colorado 274 * * X Y When (x, y is bandlimited, then replicas can be made non-overlapping i the Nyquist criteria is satisied 1 2X > 1 x 2Y > y Isolate one order rom non-overlapping repeated spectra o sampled image using aperture in Fourier domain ( ( u v H(u, v = Π Π x y G(u, v = F s (u, vh(u, v In real domain, this gives sinc interpolator h(x, y = x sinc(2 x x y sinc(2 y y ( x ( y g(x, y = [comb comb (x, y] h(x, y X Y = 4B x B y XY g(nx, my sinc[ x (x nx]sinc[ y (y ny ] n m Sampling heorem Why does this work? At each other sample point, the zeroes o the sinc cancel any crosstalk. Works in 2-D with Cartesian sampling grid. Kelvin Wagner, University o Colorado 273 Image samples in terms o spectra F(u,v g mn = e i2πum x e i2πvn y F (u, vh(u, vp (u, vdudv Discrete Fourier transorm o image samples G pq = 1 M/2 N/2 e iπmp/m e iπnq/n g N 2 mn = e iπp/m u x e iπq/n v y M 2 +1 N 2 +1 sin[π(p um x ] sin[π(q vn y ] F (u, vh(u, vp (u, vdudv sin[π(p/m u x ] sin[π(q/n v y ] F ( p M x, q N x H ( p M x, q N x P Shannon scaling unction approximation ( p q M x, N x Bx By F(u,v 1/X v v 1/Y When aliasing avoided sin[π(p um x ] sin[π(p um x /M] = ( 1 m Msinc(uM x p mm m G(u, v is periodically replicated in spectra o periodically sampled spectra ( p + mm G pq = G, a + nn M x N y m= Kelvin Wagner, University o Colorado 275 u u

8 Interplay between sample spacing 1 : P (u, vh(u, v and OF cuto λf/# Aliased OF o Sampled Imager OF spectrum replicated at u = 1/ x and v = 1/ y When adequately sampled OF replicas do not overlap For unequal spacing replicas can overlap in one dimension not in the other A little overlap is tolerable due to the small OF at the edges Aberrations are usually signiicant and permit even more overlap o weak wings Kelvin Wagner, University o Colorado Pixel spacing and Aliasing o the Pixel ranser Function P (u, v CCD Sensor Organization: Frame ranser and Interline ranser Kelvin Wagner, University o Colorado Kelvin Wagner, University o Colorado 278 Kelvin Wagner, University o Colorado 279

9 Sampling o Images: InCoherent Imaging Sampling o Images: InCoherent Imaging Image sampling can alias, but partially mitigated with wide pixel sensitivity proiles More samples is usually better, but point sampling will still alias Kelvin Wagner, University o Colorado 280 Sampling o Images: Coherent Case or Digital Holography with intererometric zone plates Kelvin Wagner, University o Colorado 281 Sampling o Images: Coherent Case or Digital Holography with intererometric zone plates Aliasing highly prevalent due to no rollo o high requencies Large re beam angle can still yield centered zone plate due to aliasing Kelvin Wagner, University o Colorado 282 Kelvin Wagner, University o Colorado 283

10 Color interpolation aects in Bayer ilters vs Foveon or Monochrome CCDs Foveon CMOS detector arrays and image sampling Kelvin Wagner, University o Colorado 284 Kelvin Wagner, University o Colorado MF Limits using Monochrome CCD or color CCDs with Bayer Filters 2d 2d A sampled signal has a Fourier spectrum that is a periodically repeated version o the spectrum o the sampled signal. xs(t = x(tcomb (t = 22d 2d X Sampling arrays 285 Sampling Summary Unaliased Nyquist limited region associated with RGB Bayer ilter 22d F x(n δ(t n Xs( = X( comb ( Aliasing occurs i adjacent spectral orders overlap. he original signal can be recovered exactly rom the samples i there is no aliasing. Xs( Π = X( 2c 2d 4d d 2d I the signal is sampled at a rate more than twice its highest requency (or ull 2-sided bandwidth, s = 1/ >, there will be no aliasing v Fourier space 1/ 2d Kelvin Wagner, University o Colorado v 1/ 22d 1/2d u 1/4d 1/2d u 1/2d A signal can not be simultaneusly time limited and bandlimited. u 286 he sinc unction is the ideal interpolating unction in 1-D, but since it is ininite and noncausal it can not be used in practice X X t n xr (t = x(n sinc[(t n ] = x(n sinc Kelvin Wagner, University o Colorado 287