Types of Jitter:
In the field of research focusing on errors created by measuring a signal two key components
are studied: Time base distortion and jitter. Time base distortion is deterministic (also called deterministic jitter) and the sum of the timing error. This type of jitter may appear random but it is systematic so we can attribute specific signals to this type of distortion. Random timing error is called jitter (or random jitter).
How do we get to the point where a numeric value can be assigned to jitter? We need to start with the signal. The signal of interest is represented as a continuous function of time containing components of jitter, systematic time base distortion and error.
\[m(t)=g\left(T(t)+h(t)+\tau (t)\right)+\epsilon (t)\]
In a pulse code modulation system the reference signal is generated by integrating $m(t)$ this result follows:
\[m_s\left[t\right]=\ \sum^{\infty }_{i=-\infty }{\triangle \left(\left(i-1\right)T_s\right)\int^{t_f}_{t_o}{\delta \left(\beta \left[i\right]-(\left[i-1\right]T_s+h\left[i\right]+\tau \left[i\right])+\epsilon\left[i\right]) d\beta }}\] sample: $y\left[i\right]$
the systematic time base distortion: $h\left[i\right]$ , $T\left[i\right]=(i-1)T_s$
is the target time of each sample, $T_s$ is the target time interval between samples, the additive noise: $h\left[i\right]$
we assume that the additive noise is zero-mean random variables with variance: ${\sigma }^2_{\$
then the random jitter error is: $\tau \left[i\right]$
Assume that the jitter is zero-mean random variables with variance: ${\sigma }^2_{\tau }$ \[\tau \left[i\right]\approx {\sigma }^2_{\tau }\]
Model for variance of the signal of interest: $var\left(y\left[i\right]\right)\ \approx \ {\sigma }^2_{\tau }{\left(g'\left(t[i]\right)\right)}^2+\ {\sigma }^2_\epsilon$
Obtain the signal variance of independent repeated measurements and use this model \[var\left(y\left[i\right]\right)\ \approx \ {\sigma }^2_{\tau }{\left(g'\left(t[i]\right)\right)}^2+\ {\sigma }^2_\epsilon\] $g'\left(t[i]\right)$ is the derivative of $g\left(t[i]\right)$ evaluated at $t\left[i\right] = T\left[i\right]+h\left[i\right]$ To solve for ${\sigma }^2_{\tau }$ \[{\sigma }^2_{\tau }\approx \ \frac{var\left(y\left[i\right]\right)-\ {\sigma }^2_\epsilon\ }{{\left(g'\left(t\left[i\right]\right)\right)}^2}=ct\] where \[c=\ \frac{1}{T}\int^T_0{v^T_1(\alpha )B(x_s\left(\alpha \right))B^T(x_s\left(\alpha \right))v_1(\alpha )d\alpha }\]
Three causes of jitter (there are more):
Three ways to correct for jitter effects:
\[m(t)=g\left(T(t)+h(t)+\tau (t)\right)+\epsilon (t)\]
In a pulse code modulation system the reference signal is generated by integrating $m(t)$ this result follows:
\[m_s\left[t\right]=\ \sum^{\infty }_{i=-\infty }{\triangle \left(\left(i-1\right)T_s\right)\int^{t_f}_{t_o}{\delta \left(\beta \left[i\right]-(\left[i-1\right]T_s+h\left[i\right]+\tau \left[i\right])+\epsilon\left[i\right]) d\beta }}\] sample: $y\left[i\right]$
the systematic time base distortion: $h\left[i\right]$ , $T\left[i\right]=(i-1)T_s$
is the target time of each sample, $T_s$ is the target time interval between samples, the additive noise: $h\left[i\right]$
we assume that the additive noise is zero-mean random variables with variance: ${\sigma }^2_{\$
then the random jitter error is: $\tau \left[i\right]$
Assume that the jitter is zero-mean random variables with variance: ${\sigma }^2_{\tau }$ \[\tau \left[i\right]\approx {\sigma }^2_{\tau }\]
Model for variance of the signal of interest: $var\left(y\left[i\right]\right)\ \approx \ {\sigma }^2_{\tau }{\left(g'\left(t[i]\right)\right)}^2+\ {\sigma }^2_\epsilon$
Obtain the signal variance of independent repeated measurements and use this model \[var\left(y\left[i\right]\right)\ \approx \ {\sigma }^2_{\tau }{\left(g'\left(t[i]\right)\right)}^2+\ {\sigma }^2_\epsilon\] $g'\left(t[i]\right)$ is the derivative of $g\left(t[i]\right)$ evaluated at $t\left[i\right] = T\left[i\right]+h\left[i\right]$ To solve for ${\sigma }^2_{\tau }$ \[{\sigma }^2_{\tau }\approx \ \frac{var\left(y\left[i\right]\right)-\ {\sigma }^2_\epsilon\ }{{\left(g'\left(t\left[i\right]\right)\right)}^2}=ct\] where \[c=\ \frac{1}{T}\int^T_0{v^T_1(\alpha )B(x_s\left(\alpha \right))B^T(x_s\left(\alpha \right))v_1(\alpha )d\alpha }\]
Three causes of jitter (there are more):
- Crystal Impurities
- Thermal variations
- Crystal boundaries with imperfect valence electron mapping.
Three ways to correct for jitter effects:
- Deconvolution
- Averaging
- Cross-correlation
No comments:
Post a Comment