Next, an analysis of the probabilities that ML decisions are correct or incorrect for various detected signal levels leads to the following equation for the symbol error rate (SER) - that is, the probability of incorrect symbol detection: The rule turns out to be equivalent to the intuitive conjecture that the symbol most likely to have been transmitted, given the actual signal values detected during a symbol frame, is the one represented by 1 in each of the n time slots for which the detected signal values were the largest. This analysis leads to a maximum-likelihood (ML) decision rule to be used by the receiver. It is assumed that L( y) is finite and, as is true for many channels of practical interest, that L( y) increases monotonically with y. The analysis includes consideration of L( y) = p 1( y)/ p 0( y), defined as the likelihood ratio for receiving value y during the time slot.
For purposes of the analysis, the signal-propagation channel is assumed to be memoryless. The method is based partly on an analysis of the conditional probability, p 1( y) or p 0( y), that the actual value, y, of the noisy signal detected in a receiver during a given time slot represents a transmitted 1 or a transmitted 0, respectively. If the number of pulses per symbol is n, then the number of symbols in the alphabet is given by the binomial coefficient Multipulse PPM is a generalization of PPM in which pulses are transmitted during two or more of the M time slots. A symbol is represented by transmitting a pulse (representing “1”) during one of the time slots and no pulse (representing “0”) during the other M – 1 time slots. In conventional M-ary PPM, each symbol is transmitted in a time frame that is divided into M time slots (where M is an integer >1), defining an M-symbol alphabet. SER Values were calculated for 16-ary PPM using several different values n and two different noise levels in a Poisson channel. The method makes it possible, when designing an optical PPM communication system, to determine whether and under what conditions a given multipulse PPM scheme would be more or less advantageous, relative to other candidate modulation schemes. However, I cannot find any reference actually using any of these or at least discussing these topics for my case.Īny answer substantiated with a peer-reviewed paper is greatly appreciated.NASA’s Jet Propulsion Laboratory, Pasadena, CaliforniaĪ method of computing channel capacities and error rates in multipulse pulse-position modulation (multipulse PPM) has been developed.
This other question: How to calculate relative error? is somewhat related but I do not think it really applies. I have seen: How to calculate relative error when the true value is zero?īut since I do not have access to a measurement and its true value I am not sure how to apply the discussion there to this case, and I could not find any reference in the literature. What would you recommend to use under such situations? However, when the value of the quantity to measure tends to zero, the relative error can become very large, although the measurement itself might be very reliable, because for example the quantity to measure is zero within the instrumentation error. The relative error has many useful application in error propagation and so on, and it is frequently used to determine the quality of a measurement. Let's consider the case of a measurement $x$ and its correspondent error $\Delta x$ (which are both always positive).