## Abstract

Algorithms for digitally processing sampled spatiotemporal signals typically ignore the fact that those samples were obtained time-sequentially, rather than instantaneously, at the beginning of each frame period The effect of displacing time-sequentially obtained samples to a non-time-sequential lattice is analyzed for both deterministic and nondeterministic signals Sample displacement is shown to cause unavoidable frequency-domain aliasing The resulting artifacts are compared in test signals that have been sampled with a lexicographic and bit-reversed ordering of the sample points

© 1983 Optical Society of America

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### Equations (34)

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(1)
$$h(x,y,t)={X}^{2}B\text{\u2211}_{l}g({\alpha}_{l}X,{\beta}_{l}X,lT)\times \delta (x-{\alpha}_{l}X,y-{\beta}_{l}X,t-lT)$$
(2)
$$\begin{array}{ll}{\alpha}_{l}=[l/M],\hfill & {\beta}_{l}=lmodM,\hfill \end{array}$$
(3)
$$H(u,\upsilon ,f)=\text{\u2211}_{m}\text{\u2211}_{n}\text{\u2211}_{p}{Q}_{mnp}G(u-m/A,\upsilon -n/A,f-p/B),$$
(4)
$${Q}_{mnp}=\frac{1}{{M}^{2}}\text{\u2211}_{l=0}^{{M}^{2}-1}exp[-\iota 2\pi (m{\alpha}_{l}/M+n{\beta}_{l}/M+pl/{M}^{2})]$$
(5)
$$\begin{array}{ll}\hfill {Q}_{000}& =1,\hfill \\ \hfill {Q}_{mn0}& ={\delta}_{mmodM}{\delta}_{nmodM},\hfill \\ \hfill {Q}_{00p}& ={\delta}_{pmodM}2,\hfill \\ \hfill {Q}_{m+aM,n+bM,p+cM}2& ={Q}_{mnp},\hfill \end{array}$$
(6)
$$\begin{array}{ll}U\le 1/(2X),\hfill & W\le 1/(2B)\hfill \end{array}$$
(7)
$$\u0125(x,y,t)={X}^{2}B\text{\u2211}_{l}g({\alpha}_{l}X,{\beta}_{l}X,lT)\times \delta (x-{\alpha}_{l}X,y-{\beta}_{l}X,t-\lfloor l/{M}^{2}\rfloor B)$$
(8)
$$c(x,y,t)=\delta (x)\delta (y)\text{rect}\{[t+(B-T)/2]/B\},$$
(9)
$$\text{d}(x,y,t)=1\times 1\times \text{\u2211}_{l}\delta (t-lB)$$
(10)
$$\u0125(x,y,t)=\text{d}(x,y,t)[c(x,y,t)\u2605h(x,y,t)],$$
(11)
$$\u0124(u,\upsilon ,f)=\text{\u2211}_{k}sinc[B(f-k/B)]\times exp[\iota \pi (B-T)(f-k/B)]H(u,\upsilon ,f-k/B)$$
(12)
$$\u0124(u,\upsilon ,f)=\text{\u2211}_{m}\text{\u2211}_{n}\text{\u2211}_{p}{Q}_{mn}(f-p/B)G(u-m/A,\upsilon -n/A,f-p/B),$$
(13)
$${Q}_{mn}(f)=\text{\u2211}_{k}{Q}_{mnk}sinc[B(f+k/B)]\times exp[\iota \pi (B-T)(f+k/B)].$$
(14)
$${Q}_{00}(f)\cong sinc[Bf]exp[\iota \pi (B-T)f/B]$$
(15)
$${S}_{hh}(u,\upsilon ,f)=\text{\u2211}_{m}\text{\u2211}_{n}\text{\u2211}_{p}{|{Q}_{mnp}|}^{2}\times {S}_{gg}(u-m/A,\upsilon -nA,f-p/B)$$
(16)
$${S}_{\u0125\u0125}(u,\upsilon ,f)=\text{\u2211}_{k}{sinc}^{2}[B(f-k/B)]{S}_{hh}(u,\upsilon ,f-k/B)$$
(17)
$${S}_{\u0125\u0125}(u,\upsilon ,f)=\text{\u2211}_{m}\text{\u2211}_{n}\text{\u2211}_{p}{R}_{mn}(f-p/B)\times {S}_{gg}(u-m/A,\upsilon -n/A,f-p/B)$$
(18)
$${R}_{mn}(f)=\text{\u2211}_{k}{|{Q}_{mnk}|}^{2}{sinc}^{2}[B(f+k/B)]$$
(19)
$${\xea}_{0}(x,y,t)=\u0125(x,y,t)-g(x,y,t+(B-T)/2).$$
(20)
$${{\sigma}_{\xea}}^{2}={\mathit{\iint}}_{\mathrm{\Omega}}\mathit{\int}{S}_{{\xea}_{0}{\xea}_{0}}(u,\upsilon ,f)\text{d}u\text{d}\upsilon \text{d}f$$
(21)
$${S}_{{\xea}_{0}{\xea}_{0}}(u,\upsilon ,f)={S}_{\u0125\u0125}(u,\upsilon ,f)+{S}_{gg}(u,\upsilon ,f)-2Re\{{S}_{\u0125g}(u,\upsilon ,f)exp[\iota \pi (B-T)f]\},$$
(22)
$${S}_{\u0125g}(u,\upsilon ,f)=sinc(Bf)exp[-\iota \pi (B-T)f]{S}_{gg}(u,\upsilon ,f)$$
(23)
$$\hfill {{\sigma}_{\xea}}^{2}& ={\iint}_{\mathrm{\Omega}}\mathit{\int}\{{R}_{00}(f)-2sinc(Bf)+1\}{S}_{gg}(u,\upsilon ,f)\text{d}u\text{d}\upsilon \text{d}f\hfill & +\text{\u2211}_{m}\text{\u2211}_{n}\text{\u2211}_{p}{\iint}_{\mathrm{\Omega}}\mathit{\int}{R}_{mn}(f-p/B){S}_{gg}(u-m/A,\upsilon -n/A,\hfill & f-p/B)\text{d}u\text{d}\upsilon \text{d}f,\hspace{0.17em}(m,n,p)\ne (0,0,0).\hfill $$
(24)
$$\begin{array}{ll}\hfill {S}_{gg}(u,\upsilon ,f)& =\{\begin{array}{ll}1/(8{U}^{2}W),\hfill & (u,\upsilon ,f)\in \mathrm{\Omega}\hfill \\ 0,\hfill & \text{else}\hfill \end{array},\hfill \\ \hfill \mathrm{\Omega}& =\{(u,\upsilon ,f):max[|u|/U,|\upsilon |/U,|f|/W]<1\},\hfill \end{array}$$
(25)
$$g(x,y,t)=\frac{1}{2}(1+cos\{2\pi [{x}^{2}/(8AX)-{f}_{0}t]\}).$$
(26)
$$g(r,t)=\frac{1}{2}(1+cos\{2\pi [{r}^{2}/(2AX)-{f}_{0}t]\}),$$
(27)
$${[{{\sigma}_{\xea}}^{2}]}_{al}=\text{\u2211}_{m}\text{\u2211}_{n}\text{\u2211}_{p}{\iint}_{\mathrm{\Omega}}\mathit{\int}{R}_{mn}(f-p/B){S}_{gg}(u-m/A,\upsilon -n/A,f-p/B)\text{d}u\text{d}\upsilon \text{d}f,(m,n,p)\ne (0,0,0).$$
(28)
$${[{{\sigma}_{\xea}}^{2}]}_{al}=\frac{1}{8}\text{\u2211}_{m}\text{\u2211}_{n}\text{\u2211}_{p=0}^{\infty}{\u220a}_{mnp}{\left\{[2-|m|/AU)]\times [2-|n|/AU)]{\mathit{\int}}_{-1}^{1-|p|/(BW)}{R}_{mn}(Wf)\text{d}f\right\}}^{+},$$
(29)
$${\u220a}_{mnp}=\{\begin{array}{ll}0,\hfill & (m,n,p)=(0,0,0),\hfill \\ 1,\hfill & \begin{array}{ll}(m,n)\ne (0,0)\hfill & p=0,\hfill \end{array}\hfill \\ 2,\hfill & p\ge 1,\hfill \end{array}$$
(30)
$${R}_{mn}(f)={sinc}^{2}(Bf)\u2605\text{rep}\frac{1}{T}\left[\text{\u2211}_{l=0}^{{M}^{2}-1}{|{Q}_{mnl}|}^{2}\delta (f+l/B)\right],$$
(31)
$${r}_{mn}(t)=\frac{1}{{M}^{2}}\text{\u2211}_{k=-{M}^{2}+1}^{{M}^{2}-1}{r}_{mnk}\delta (t+kT),$$
(32)
$${r}_{mnk}=[1-|k|/{M}^{2}]\text{\u2211}_{l=0}^{{M}^{2}-1}{|{Q}_{mnl}|}^{2}exp[-\iota 2\pi kl/{M}^{2}]$$
(33)
$${R}_{mn}(f)=\frac{1}{{M}^{2}}+\frac{2}{{M}^{2}}\text{\u2211}_{k=1}^{{M}^{2}-1}|{r}_{mnk}|cos(2\pi kBf/{M}^{2}+\underset{\_}{|{r}_{mnk}})$$
(34)
$$\begin{array}{l}{\mathit{\int}}_{-1}^{1-|p|/(BW)}{R}_{mn}(Wf)\text{d}f=\frac{1}{{M}^{2}}[2-|p|/(BW)]\hfill \\ +\frac{1}{\pi BW}\text{\u2211}_{k=1}^{{M}^{2}-1}\frac{|{r}_{mnk}|}{k}(sin\{2\pi kBW[1-|p|/(BW)]/{M}^{2}\hfill \\ +\underset{\_}{|{r}_{mnk}}\}-sin[-2\pi kBW/{M}^{2}+\underset{\_}{|{r}_{mnk}}])\hfill \end{array}$$