9.1. Basic Model#

  • Suppose that the linearly modulated signal \(x(t) = \sum_n b[n] p(t-nT)\) is transmitted. Further assume that the TX pulse shape \(p(t)\) satisfies the ISI-free property, like the RRC pulse.

  • Assuming that timing synchronization is achieved, the RX signal can be modeled by

    \[\begin{equation*} r(t) = x(t) e^{j(\omega_o t + \theta)} + n(t) \end{equation*}\]

    where \(n(t)\) is AWGN as before. The deterministic unknown constants \(\omega_o\) and \(\theta\) model the carrier frequency and phase offsets of the LO w.r.t. the carrier in the RX signal, respectively. We also assume that the oscillators used in the TX and LO are sufficiently accurate so that \(|\omega_o| \ll \frac{2\pi}{T}\). We are to jointly estimate (or track) \(\omega_o\) and \(\theta\).

9.2. ML Estimation Using Pilot Symbols#

  • Suppose first that \(b[n]\) is known to the RX. That is, they are all pilot symbols. This can be the situation in which we reuse the signature sequence for acquisition and/or timing synchronization as pilots to perform carrier synchronization. Under this condition, it can be shown that the joint ML estimator for \((\omega_o, \theta)\) is

    (9.1)#\[\begin{split}\begin{align} (\hat \omega_o, \hat \theta) &= \arg\!\max_{\!\!\!\!\omega_o, \theta} \text{Re} \left\{ \int_{-\infty}^{\infty} r(t) x^*(t) e^{-j(\omega_o t + \theta)} dt \right\} \label{e:dd_ml_con} \\ & \approx \arg\!\max_{\!\!\!\!\omega_o, \theta} \text{Re} \left\{ \sum_n \hat r[n] e^{-j(\omega_o nT+\theta)} \right\} \end{align}\end{split}\]

    where

    \[\begin{equation*} \hat r[n] = b^*[n] \left[ r(t) * p^*(-t) \big|_{t=nT} \right]. \end{equation*}\]

    Note that the approximated ML estimator above is obtained based on the assumption that \(\omega_o \ll \frac{2\pi}{T}\).

  • It is clear that the (approximate) ML estimator in (9.1) can be rewritten as

    (9.2)#\[\begin{equation} (\hat \omega_o, \hat \theta) = \arg\!\max_{\!\!\!\!\omega_o, \theta} \text{Re} \left\{ e^{-j\theta} \hat R( e^{j\omega_o T} ) \right\} \end{equation}\]

    where

    \[\begin{equation*} \hat R( e^{j\hat\omega}) \triangleq \sum_n \hat r[n] e^{-j\hat\omega n} \end{equation*}\]

    is the DTFT of \(\hat r[n]\). The decision rule in (9.2) is equivalent to

    (9.3)#\[\begin{split}\begin{align} \hat \omega_o &= \frac{1}{T} \arg\!\!\max_{\!\!\!\!\hat \omega \in [-\pi, \pi)} \left| \hat R(e^{j\hat\omega}) \right| \notag \\ \hat \theta &= \angle \hat R(e^{j\hat \omega_o T}). \end{align}\end{split}\]

  • We may also periodically insert pilot symbols into the data symbol sequence. Because of the ISI-free property of the TX pulse, we may simply ignore the data symbols and consider only the pilot symbols as far as estimating the carrier frequency and phase goes. For example, if a pilot symbol is inserted every \(P\) symbol periods, the ML estimator is then:

    (9.4)#\[\begin{split}\begin{align} \hat \omega_o &= \frac{1}{PT} \arg\!\!\max_{\!\!\!\!\hat \omega \in [-\pi, \pi)} \left| \hat R_P(e^{j\hat\omega}) \right| \notag \\ \hat \theta &= \angle \hat R_P(e^{j\hat \omega_o T}). \end{align}\end{split}\]

    where

    \[\begin{equation*} \hat R_P( e^{j\hat\omega}) \triangleq \sum_n \hat r[nP] e^{-j\hat\omega n}. \end{equation*}\]

    The form of \(\hat \omega_o\) clearly requires \(|\omega_o| < \frac{\pi}{PT}\) for the joint ML estimator to work. Thus the rate of pilot symbols must also satisfy the Nyquist criterion corresponding to the maximum possible range of carrier frequency offset.

  • USRP implementation:

    • We may collect the symbol-rate samples of the MF (matched to the TX pulse shape) output corresponding to the pilot symbols, and then employ FFT to obtain a frequency-domain sampled version of the DTFT of the collected MF output to implement the joint ML estimator in (9.3) or (9.4).

    • If the packet is short such that \(\omega_o\) and \(\theta\) remain constant within the packet, then it suffices to perform a one-shot ML estimation of \(\omega_o\) and \(\theta\) using the joint ML estimator in (9.3) or (9.4). Then, we may correct the frequency and phase offsets by multiplying the properly timed MF output \(\tilde r[n] = r(t) * p^*(-t) \big|_{t=nT}\) with \(e^{-j(\hat\omega_o nT + \hat\theta)}\) before doing coherent demodulation for the data symbols.