7.2. Signal Acquisition#
The objective of signal acquisition is twofold:
To identify whether the captured signal is a signal of interest.
To synchronize to the beginning of the packet. If a DLL is to run after acquisition, then only coarse synchronization to an accuracy of \(\pm \frac{T}{2}\) needs to be achieved. On the other hand, if no DLL is to run after acquisition, one may also perform one-shot fine symbol synchronization together with acquisition.
A special sequence of known symbols, called the acquisition/signature/prefix code sequence, is usually added in between the preamble (for the AGC to settle) and the data payload of each packet to allow the RX to perform acquisition. In particular, we will consider the simple case that the signature signal is a linearly modulated signal
(7.6)#\[\begin{equation} x(t) = \sum_{n=0}^{N-1} x[n] p(t-nT) \end{equation}\]where \(x[n]\) is the signature sequence and \(p(t)\) is the TX pulse shape.
7.2.1. Generalized likelihood acquisition#
Detection of the signature signal is the same as the signal capture problem that we discussed before, except that the whole acquisition signal \(x(t)\) is known to the RX and hence may be used for acquiring the RX signal.
In particular, we may model the RX signal w.r.t. the transmission of \(x(t)\) as
\[\begin{equation*} r(t) = x(t-\tau) e^{j\theta} + n(t) \end{equation*}\]where \(\theta\) is uniformly distributed over \([-\pi,\pi]\) and \(n(t)\) is an AWGN process as before. The delay \(\tau\) is deterministic but unknown. The generalized LRT may be used to solve this acquisition problem:
The ML estimate of \(\tau\) is given by (7.1).
The generalized LRT is equivalent to compare the statistic
(7.7)#\[\begin{equation} \max_{\tau} \left| r(t) * x^*(-t) \big|_{t=\tau} \right| = \max_{\tau} \left| \sum_{n} x^*[n] \tilde r(nT+\tau) \right| \end{equation}\]to a properly chosen decision threshold.
If the threshold is exceeded, the signature signal is acquired and the ML estimate \(\hat\tau\) gives us the timing of the beginning of the packet.
7.2.2. USRP implementation#
In practice, we often approximate the generalized LRT by continuously comparing the magnitude (or squared magnitude) of the MF output, i.e., \(r(t)*x^*(-t)\) in (7.7), to a threshold until the threshold is exceeded. The ML estimate \(\hat\tau\) is then searched over a small range beyond the delay at which the threshold is exceeded. Depending on the ratio of the sampling rate to the symbol rate (i.e., \(f_sT\)) and whether fine symbol synchronization is required in the acquisition step, we may need to implement the MF with interpolation.
If short packets are employed, the clocks in the USRP radios are accurate enough that there is usually no need to run a DLL throughout the whole packet at the RX. Instead, we may perform fine symbol synchronization only once, together with acquisition, using the signature signal directly. In this case, a larger interpolation factor is needed for the MF in (7.7) to obtain a fine resolution for the ML estimate \(\hat\tau\).