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From (5.54) the average energy per symbol for MPAM is 1 Es = M in Software Encoding barcode code39 in Software From (5.54) the average energy per symbol for MPAM is 1 Es = M ANSI/AIM Code 39 for vb




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From (5.54) the average energy per symbol for MPAM is 1 Es = M generate, create none none in none projectscreate code 39 vb.net A2 i Microsoft Office Word Website 1 = M. M i=1 1 (2i 1 M )2 d2 = (M 2 1)d2 . 3 (6.20). Thus we can wr ite Ps in terms of the average energy E s as Ps = 2(M 1) Q M 6 s M2 1 . (6.21).

Consider now M none none QAM modulation with a square signal constellation of size M = L 2 . This system can be viewed as two MPAM systems with signal constellations of size L transmitted over the in-phase and quadrature signal components, each with half the energy of the original MQAM system. The constellation points in the insymbol error probability phase and quadrature branches take values A i = (2i 1 L)d, i = 1, 2, .

. . , L.

The for each branch of the MQAM system is thus given by (6.21) with M replaced by L = M and s equal to the average energy per symbol in the MQAM constellation: 2( M 1) Q Ps = M 3 s M 1 . (6.

22). Note that s is multiplied by a factor of 3 in (6.22) instead of the factor of 6 in (6.21) since the MQAM constellation splits its total average energy s between its in-phase and quadrature branches.

The probability of symbol error for the MQAM system is then 2 2( M 1) 3 s Q . (6.23) Ps = 1 1 M 1 M The nearest neighbor approximation to probability of symbol error depends on whether the constellation point is an inner or outer point.

If we average the nearest neighbor approximation over all inner and outer points, we obtain the MPAM probability of error associated with each branch: 2( M 1) Ps Q M 3 s M 1 . (6.24).

For nonrectang none none ular constellations, it is relatively straightforward to show that the probability of symbol error is upper bounded as Ps 1 1 2Q 3 s M 1. 4Q. 3 s M 1 (6.25). The nearest ne none for none ighbor approximation for nonrectangular constellations is Ps Mdmin Q d min 2N0 , (6.26). where Mdmin is the largest number of nearest neighbors for any constellation point in the constellation and d min is the minimum distance in the constellation. Example 6.3: For 16QAM with b = 15 dB ( s = log2 M b ), compare the exact probability of symbol error (6.

23) with the nearest neighbor approximation (6.24), and with the symbol error probability for 16PSK with the same b that was obtained in the previous example..

Solution: The none none average symbol energy s = 4 101.5 = 126.49.

The exact Ps is then given by Ps = 1 1 2(4 1) Q 4 3 126.49 15. = 7.37 10 7 ..

The nearest ne none for none ighbor approximation is given by Ps 2(4 1) Q 4 3 126.49 15 = 3.68 10 7 ,.

which differs by roughly a factor of 2 from the exact value. The symbol error probability for 16PSK in the previous example is Ps 1.916 10 3 , which is roughly four orders of magnitude larger than the exact P s for 16QAM.

The larger Ps for MPSK versus MQAM with the same M and same b is due to the fact that MQAM uses both amplitude and phase to encode data, whereas MPSK uses just the phase. Thus, for the same energy per symbol or bit, MQAM makes more ef cient use of energy and thus has better performance..

The MQAM demod none none ulator requires both amplitude and phase estimates of the channel so that the decision regions used in detection to estimate the transmitted bit are not skewed in amplitude or phase. The analysis of the performance degradation due to phase estimation error is similar to the case of MPSK discussed above. The channel amplitude is used to scale the decision regions to correspond to the transmitted symbol: this scaling is called Automatic Gain Control (AGC).

If the channel gain is estimated in error then the AGC improperly scales the received signal, which can lead to incorrect demodulation even in the absence of noise. The channel gain is typically obtained using pilot symbols to estimate the channel gain at the receiver. However, pilot symbols do not lead to perfect channel estimates, and the estimation error can lead to bit errors.

More details on the impact of amplitude and phase estimation errors on the performance of MQAM modulation can be found in [15, 10.3][16]..

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