The output of a communication channel is given by:

$x(t) = \sum_n{a_n}h(t-nT)$,

where $\{a_n\}$ are BPSK symbols, $h(t)$ is the channel response, and $T$ is the symbol period. If there is an inherent unknown timing offset $\tau$ inside the channel response we will instead have:

$x(t) = \sum_n{a_n}h(t-nT-\tau)$.

Say we need to sample $x(t)$ at times $\{kT\}$ and we can detect the symbols using the shifted channel response $h(t-nT-\tau)$. In other words we do not wish to resample $x(t)$ to compensate for the timing offset but we can estimate $\tau$. An estimate of $\tau$ is given by $z_k = x_k \hat{a}_{k-1} - x_{k-1}\hat{a}_k$ such that $E\{z_k\} = \tau$, where $x_k = x(kT)$, and $\{\hat{a}_k\}$ are the decisions of a (Viterbi) detector. This is the $M \& M$ estimate.

The question is how to use this estimate to recursively update the estimates of $\tau$ in a 1-st order PLL sense since the following equation will diverge.

$\hat{\tau}_{k+1} = \hat{\tau_k} + stepsize \times z_k$

The output of a communication channel is given by:

$x(t) = \sum_n{a_n}h(t-nT)$,

where $\{a_n\}$ are BPSK symbols, $h(t)$ is the channel response, and $T$ is the symbol period. If there is an inherent unknown timing offset $\tau$ inside the channel response we will instead have:

$x(t) = \sum_n{a_n}h(t-nT-\tau)$.

Say we need to sample $x(t)$ at times $\{kT\}$ and we can detect the symbols using the shifted channel response $h(t-nT-\tau)$. In other words we do not wish to resample $x(t)$ to compensate for the timing offset but we can estimate $\tau$. An estimate of $\tau$ is given by $z_k = x_k \hat{a}_{k-1} - x_{k-1}\hat{a}_k$ such that $E\{z_k\} = \tau$, where $x_k = x(kT)$, and $\{\hat{a}_k\}$ are the decisions of a (Viterbi) detector. This is the $M \& M$ estimate.

The question is how to use this estimate to recursively update the estimates of $\tau$ in a 1-st order PLL sense since the following equation will diverge.

$\tau_{k+1} = \tau_k + stepsize \times z_k$

The output of a communication channel is given by:

$x(t) = \sum_n{a_n}h(t-nT)$,

where $\{a_n\}$ are BPSK symbols, $h(t)$ is the channel response, and $T$ is the symbol period. If there is an inherent unknown timing offset $\tau$ inside the channel response we will instead have:

$x(t) = \sum_n{a_n}h(t-nT-\tau)$.

Say we need to sample $x(t)$ at times $\{kT\}$ and we can detect the symbols using the shifted channel response $h(t-nT-\tau)$. In other words we do not wish to resample $x(t)$ to compensate for the timing offset but we can estimate $\tau$. An estimate of $\tau$ is given by $z_k = x_k \hat{a}_{k-1} - x_{k-1}\hat{a}_k$ such that $E\{z_k\} = \tau$, where $x_k = x(kT)$ and $\{\hat{a}_k\}$ are the decisions of a (Viterbi) detector. This is the $M \& M$ estimate.

The question is how to use this estimate to recursively update the estimates of $\tau$ in a 1-st order PLL sense since the following equation will diverge.

$\hat{\tau}_{k+1} = \hat{\tau_k} + stepsize \times z_k$

The output of a communication channel is given by:

$x(t) = \sum_n{a_n}h(t-nT)$,

where $\{a_n\}$ are BPSK symbols, $h(t)$ is the channel response, and $T$ is the symbol period. If there is an inherent unknown timing offset $\tau$ inside the channel response we will instead have:

$x(t) = \sum_n{a_n}h(t-nT-\tau)$.

Say we need to sample $x(t)$ at times $\{kT\}$ and we can detect the symbols using the shifted channel response $h(t-nT-\tau)$. In other words we do not wish to resample $x(t)$ to compensate for the timing offset but we can estimate $\tau$. An estimate of $\tau$ is given by $z_k = x_k \hat{a}_{k-1} - x_{k-1}\hat{a}_k$ such that $E\{z_k\} = \tau$, where $x_k = x(kT)$, and $\{\hat{a}_k\}$ are the decisions of a (Viterbi) detector. This is the $M \& M$ estimate.

The question is how to use this estimate to recursively update the estimates of $\tau$ in a 1-st order PLL sense since the following equation will diverge.

$\hat{\tau}_{k+1} = \hat{\tau_k} + stepsize \times z_k$