ch7_parts5and6

ch7_parts5and6 - 7.5 Bit Error Probability We now know how...

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7.5 Bit Error Probability We now know how to design rules for deciding which of M signals (or symbols) has been sent, and how to estimate the performance of these decision rules. Sending one of M signals conveys m = log 2 M bits, so that a hard decision on one of these signals actually corresponds to hard decisions on m bits. In this section, we discuss how to estimate the bit error probability, or the bit error rate (BER), as it is often called. 01 c N s d 00 10 11 N Figure 7.32: QPSK with Gray coding. QPSK with Gray coding: We begin with the example of QPSK, with the bit mapping shown in Figure 7.32. This bit mapping is an example of a Gray code, in which the bits corresponding to neighboring symbols di ff er by exactly one bit (since symbol errors are most likely going to occur by decoding into neighboring decision regions, this reduces the number of bit errors). Let us denote the symbol labels as b [1] b [2] for the transmitted symbol, where b [1] and b [2] each take values 0 and 1. Letting ˆ b [1] ˆ b [2] denote the label for the ML symbol decision, the probabilities of bit error are given by p 1 = P [ ˆ b [1] = b [1]] and p 2 = P [ ˆ b [2] = b [2]]. The average probability of bit error, which we wish to estimate, is given by p b = 1 2 ( p 1 + p 2 ). Conditioned on 00 being sent, the probability of making an error on b [1] is as follows: P [ ˆ b [1] = 1 | 00 sent] = P [ML decision is 10 or 11 | 00 sent] = P [ N c < - d 2 ] = Q ( d 2 σ ) = Q 2 E b N 0 where, as before, we have expressed the result in terms of E b /N 0 using the power e ffi ciency d 2 E b = 4. We also note, by the symmetry of the constellation and the bit map, that the conditional probability of error of b [1] is the same, regardless of which symbol we condition on. Moreover, exactly the same analysis holds for b [2], except that errors are caused by the noise random variable N s . We therefore obtain that p b = p 1 = p 2 = Q 2 E b N 0 (7.90) 63
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The fact that this expression is identical to the bit error probability for binary antipodal signaling is not a coincidence. QPSK with Gray coding can be thought of as two independent BPSK systems, one signaling along the I component, and the other along the Q component. Gray coding is particularly useful at low SNR (e.g., for heavily coded systems), where symbol errors happen more often. For example, in a coded system, we would pass up fewer bit errors to the decoder for the same number of symbol errors. We define it in general as follows. Gray Coding: Consider a 2 n -ary constellation in which each point is represented by a binary string b = ( b 1 , ..., b n ). The bit assigment is said to be Gray coded if, for any two constellation points b and b which are nearest neighbors, the bit representations b and b di ff er in exactly one bit location. Nearest neighbors approximation for BER with Gray coded constellation: Consider the i th bit b i in an n -bit Gray code for a regular constellation with minimum distance d min . For a Gray code, there is at most one nearest neighbor which di ff ers in the i th bit, and the pairwise error probability of decoding to that neighbor is Q ( d min 2 σ ) . We therefore have P (bit error) Q η P
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