In order to reduce the power consumption of tag nodes, unnecessary feedback must be avoided in the master synchronization. In our proposed scheme, the acknowledgements are just transmitted at the time of satisfying the timing condition in tag receiver. That means a little energy may be cost in feedback. Therefore, it may be very beneficial to low-power working of the tag receiver. However, there are some problems to be solved in the following aspects. First, the stability of the mast synchronization is influenced by the long-delay feedback. Second, the precise synchronization required a small loop gain
which leads to large acquisition time. Third, the detection probability
and the false-alarm probability
caused by the wireless channel increase the average acquisition time and the transmitted energy in tag nodes obviously.
4.1. Stable Acquisition
The intuitive reason of instability in master synchronization is that the feedback control error departs from the right phase in master node due to the large delay. A simple method to ensure the stability of pulse master synchronization is by modulating the phase information into corresponding phase signal transmitted to tag node. After the tag node acquires this phase signal, that is, correlation exceeds the preset threshold, the carried phase information is fed back by the acknowledgement. Accepting the acknowledgement, the master node adjusts the VCO or NCO according to the phase information. Consequently, the phase of received signal would just agree with the phase of local oscillator in tag receiver.
Figure 5 shows the signaling procedure mentioned above. When there is burst data to the tag node
, the master node transmits the synchronization signaling to the tag receiver and adjusts its phase step by step. When the tag receiver
acquires the signal at the phase
, it demodulates the carried phase information and feeds it back to the master node by acknowledgement, or directly relays it back. Acknowledged by the tag node
, the master adjusts transmitter to the phase
, and starts sending data.
Considering the multiple access problem in the WSN with a mass of nodes, we can substitute the simple pulse signal with a pseudorandom multiaccess coded PCM signal. Adopting the multi-access code with a certain length not only resolves the problems of multi-access communication to tag nodes, but also reduces the synchronization false alarm probability
. Figure 6 shows the block diagram of symbol-level master synchronization system. Here, the integral window
of the tag correlator, is the length of multi-access code, not the pulse width.
4.2. Performance Analysis
The master synchronization is similar to the process of single dwell serial acquisition. In the traditional serial acquisition, the receiver needs to adjust the phase of local correlation template until it reaches consensus with that of the received signal. Different from serial acquisition, the adjustment of signal phase is transferred to the master transmitter in WSNs and the feedback has to be sent in wireless link. If the master synchronization requires a precision in the interval of
, the searching stepsize requires to be less than
. Noting that the synchronization accuracy of ultra-wideband impulse is affected by many factors, such as the searching stepsize, impulse waveform, multipath environment, modulation scheme, and decision threshold [17, 18], it is out of the range of this paper and should be further studied for the proposed scheme.
The serial searching period, which is also the maximum searching time without considering the missed detections, is
. The initial phase difference between the local phase of the tag receiver and that of the received signal corresponds to the uniform distribution. The time delay in feedback is denoted by
. In an ideal communication channel, apparently, the average acquisition time of the pulse master synchronization is
In the case that there are error decisions of acquisition, the master synchronization has a problem of mistaking non-synchronous status as synchronous status, which is defined by false alarm probability. On the contrary, the synchronous status may be detected as non-synchronous status either. Its probability is
, where
is the detection probability. By repeatedly sending
-periodic synchronization signaling for confirmation, the false alarm is eliminated with a penalty of
delay. If the synchronization is not confirmed, the serial search continues. Missing synchronization, the feedback system has to achieve acquisition in the next period of the serial search, which results in a long acquisition time.
In the communication of wireless sensor networks, the acknowledgement has to be delivered through wireless links. Not only should the false alarms and missed detections caused by the wireless fading and noise be considered, but also the feedback delay and the loss of acknowledgement in wireless channel may increase the acquisition time. Suppose
is the probability that the acknowledgement is correctly received, the Markov chain acquisition model can be applied to analyze the average acquisition time [19]. Using standard signal flow graph reduction techniques, one arrives at the desired result, namely,
If in addition
, the variance of acquisition time is obtained as
In the ideal case, the pulse master synchronization needs only the acquisition and confirmation acknowledgements. However, the false alarms may cause unnecessary feedbacks transmitted by the tag node. Apparently, the power consumption will increase with the increasing feedback times in the acquisition. Therefore, the average feedback time is an important parameter in the proposed master synchronization. Similar to the analysis above, it can be analyzed by the probability methods. The average initial clock difference is
. When there is no missed detection and no lost synchronization acknowledgement in the first search period, the times of extra feedbacks are
. When there is a missed detection or a lost acknowledgement in the first search period and no false in the second period, the extra feedbacks are
. When the acquisition is arrived in the third search period, the extra feedbacks are
. The average feedback times can be deduced by analogy
Disregarding the lose of the synchronization acknowledgement, the average feedback times is inversely proportional to the detection probability
and directly proportional to the false alarm probability
. However, with the common threshold decision in acquisition, the relationship between
and
is given by [19]
where
is the signal bandwidth,
is the signal-to-noise ratio (SNR), and
is the inverse function of
function. According to the equation, the false alarm probability decreases when the detection probability decreases. Obviously, a minimum feedback times can be obtained with the small detection and false alarm probabilities. Unfortunately, the small detection probability would also greatly extend the acquisition time.
Under a certain
and channel environment, it can be seen that there is a minimum average acquisition time according to (8) and (11). Suppose that
,
,
, and an additive-white-Gaussian-noise (AWGN) channel with
, the analysis results are illustrated in Figure 7. At
, there exists a minimum value of average acquisition time. It is worth noting that this minimum point varies at different SNR and
.