教育部網路通訊人才培育先導型計畫 Wireless Communication Technologies 1 Acquisition time (simple example) The first problem is how long it takes to acquire the code. That is, align the receiver code generator to within a fraction (1/2 or 1/4) of a chip. Assume the code has N PN symbols, the probability of detection is unity ( ), the probability of false alarm is zero ( ), and the dwell time (integration time) is second. Then assuming no Doppler shifts and no oscillator instabilities, the time to search all N chips in half chip increments (2N cells) is and the mean acquisition time is just When the detection probability is not unity and false alarm probability is not zero, the calculation is no longer quite so simple

教育部網路通訊人才培育先導型計畫 Wireless Communication Technologies 2 Example Mean acquisition time calculation Show that if and, then. Note that when the result agree with 【 Sol. 】

教育部網路通訊人才培育先導型計畫 Wireless Communication Technologies 3 Acquisition time for the single dwell time search Consider the simplified filter, square, and integrate detector acquisition circuit shown in Figure Assume that there are q cells to be searched. Now q may be equal to the length of the PN code to be searched or some multiple of it. For example, if the update size is one-half chip, q will be twice the code length to be searched

教育部網路通訊人才培育先導型計畫 Wireless Communication Technologies 4 Acquisition time for the single dwell time search Further assume that if a ‘hit’ (output is above threshold) is detected by the threshold detector, the system goes into a verification mode that may include both an extended duration dwell time and an entry into a code loop tracking mode. In any event, we model the ‘penalty’ of obtaining a false alarm as sec, and the dwell time itself as sec. If a true hit is observed, the system has acquired the signal, and the search is completed. Assume the false alarm probability and probability of detection are given. Clearly the time to acquire, that is, to obtain a true hit (not a false alarm) is a random variable. Also we assume that is constant (time invariant) and as a consequence, the analysis of the model follows discrete time invariant Markov processes and flow graphs theory

教育部網路通訊人才培育先導型計畫 Wireless Communication Technologies 5 Analysis of acquisition time Let each cell be numbered from left to right so that the kth cell has an a priori probability of having the signal present, given that it was not present in cells 1 through k-1, of The generating function flow diagram is given in Figure using the rule that at each node the sum of the probability emanating from the node equals unity. Consider node1. The probability a priori of having the signal present is, and the probability of it not being present in the cell is. Suppose the signal were not present. Then we advance to the next node (node 1a); since it corresponds to a probabilistic decision and not a unit time delay, no z multiplies the branch going to it

教育部網路通訊人才培育先導型計畫 Wireless Communication Technologies 6 Generating function flow graph for acquisition time 2.4.6

教育部網路通訊人才培育先導型計畫 Wireless Communication Technologies 7 Analysis of acquisition time At node 1a either a false alarm occurs, with probability, which requires one unit of time to determine ( sec) and then K unit of time ( sec) are needed to determine that there is no false alarm. Or there is no false alarm with probability ( ), which takes one dwell time to determine, which requires the branch going to node 2. Now consider the situation at node 1 when the signal does occur there. If a hit occurs, then acquisition occurs and the process is terminated, hence the node F denoting ‘finish’. If there was no hit at node 1 (the integrator output was below the threshold), which occurs with probability, one unit of time would be consumed. This is represented by the branch, leading to node

教育部網路通訊人才培育先導型計畫 Wireless Communication Technologies 8 Analysis of acquisition time At node 2 in the upper left part of the diagram either a false alarm occurs with probability and delay (K+1) or a false alarm does not occur with a delay of 1 unit. The remaining portion of the generating function flow graph is a repetition of the portion just discussed with the appropriate node changes. In order to obtain the transfer function of the flow graph we shall reduce it by combining sections at a time. Let Then the flow graph can be drawn as shown in following Figure

教育部網路通訊人才培育先導型計畫 Wireless Communication Technologies 9 Reduced flow graph diagram 2.4.6

教育部網路通訊人才培育先導型計畫 Wireless Communication Technologies 10 Analysis of acquisition time Letting and we can reduce the flow graph to that of following Figure; since takes into account the feedback loops

教育部網路通訊人才培育先導型計畫 Wireless Communication Technologies 11 Analysis of acquisition time Letting and, the generating function flow graph is redrawn in a slightly simpler form

教育部網路通訊人才培育先導型計畫 Wireless Communication Technologies 12 Analysis of acquisition time By inspection the flow graph is given by Writing and in terms of q, we have so 2.4.6

教育部網路通訊人才培育先導型計畫 Wireless Communication Technologies 13 Analysis of acquisition time As a check should be unity. (Hint: ) The mean acquisition time is given by (after some algebra) with being included in the formula to translate from our unit time scale

教育部網路通訊人才培育先導型計畫 Wireless Communication Technologies 14 Analysis of acquisition time As a partial check on U(z), let and. Then we have This result can be obtained by direct calculation by noting that the mean time to acquire is given by (the a priori probability is 1/q ) For the usual case when, is given by 2.4.6

教育部網路通訊人才培育先導型計畫 Wireless Communication Technologies 15 Analysis of acquisition time The variance of the acquisition time is given by It can be shown that (when and ) As a partial check on the variance result, let and. Then we have which is the variance of a uniformly distributed random variable, as one would expect for the limiting case