Quadrature Amplitude ModulationQAM调制

2E P (−1 | 1) = P(3 | 1) = Q N 0 Hence the probability of error given that u=1 is 2E Pe|1 = 2Q N 0
Quadrature amplitude modulation.doc

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If we assume that grey coding of the 16 QAM modulation and assume that errors are only made to the neighboring lattice point then a single bit error will occur every time there is a symbol error. Also for the 16 QAM modulation Es=4Eb as there are 4 bits per symbol. Consequently 4 Eb Pb = 3Q 5N 0
-3
-1
1
3
u
Define E as an energy scaling factor such that the distance between a constellation point and the decision line is E . Consider the case where u=”1” was sent. The probability that an error occurs such that it is interpreted as u=”-1” or u=”3” is given by the conditional probability
Next we have to determine the average energy per PAM symbol as a function of E which is 3 2 + 12 + 12 + 3 2 E pam = E = 5E 4 Also the energy per QAM symbol is twice that of the energy per PAM modulation. Hence, denoting Es as the average QAM symbol energy we have
The exact probability of symbol error is determined in the same manner as we did for QPSK. That is
Es Ps = 2 Pe − Pe = 3Q 5N 0
2
9 Es − Q 4 5N 0
c ( m) = I q I m + q For 4-PAM with levels of {-3,-1,1,3} we have m=0 10 c ( m) = 0 otherwise Such that C(fT)=10 and the PSD of the 4 level PAM signal becomes Sv ( f ) = 10 Sg ( f ) T
As square QAM constellations can be regarded as two orthogonal independent PAM modulations that are transmitted simultaneously the PSD of QAM is merely twice the PSD of the individual PAM modulations. recall For v(t) is given as
Quadrature amplitude modulation.doc
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orthogonal PAM modulations is straight forward to apply. For moderate SNR’s it is quite tight and therefore used in practice. The union bound for QAM is Es Ps = 2 Pe = 3Q 5N 0
Quadrature Amplitude Modulation (QAM)
While 8 PSK is commonly used, higher order M-ary is not common as the neighbor points get very close which increases the probability of symbol and bit error. To maintain a reasonable BER, the SNR becomes very large which is impractical for most communication links. QAM is a 2D signal space modulation scheme that uses the same basis functions as before:
(
)
E s = 10 E Consequently 3 2Es Pe = Q 2 10 N 0

Let Ps be the probability of symbol error of the 16 QAM modulation. This is easily determined from the union bound based on considering QAM as two independent and
Comparing with BPSK, the Eb/No requirement for a given BER is more than 10log10(5/2)=4dB for the 16 QAM modulation. This is the penalty that is paid for having a more bandwidth efficient modulation. ie 4 bits/sym vs 1bit/sym. Section 6.4 in Haykin outlines the general development of the square QAM constellation performance which follows a very similar development to the 16 QAM analysis given above.
v(t ) =
n = −∞
∑I
∞
n
g (t − nT )
the PSD is Sv ( f ) = 1 S g ( f )C ( fT ) T
where Sg(f) is the ESD of g(t) and C ( fT ) =
m = −∞
∑ c ( m )e
∞
− j 2πfmT
where c(m) is the autocorrelation of the data sequence given by
2
Likewise for the other interior point u=-1 2E Pe|−1 = 2Q N 0 For the end points u=-3 and u=3, the probability of error is half this value as there are only 1 neighboring state for the end points. Hence 2E Pe|−3 = Pe|3 = Q N 0 The total probability of symbol error for the 1D four state PAM system is Pe = Pe|−3 P−3 + Pe|−1 P−1 + Pe|1 P1 + Pe|3 P3 where P-3 is the probability that the input is –3 etc. It is reasonable to assume that
Quadrature amplitude modulation.doc
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Note 16PSK requires higher Eb/No than 16QAM which is why it is not used.
Power Spectral Density of QAM constellations
φ2(t)
φ1(t)
decision boundaries
Quadrature amplitude modulation.doc
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The signals can be written as
S i = si ,1φ1 (t ) + si , 2φ 2 (t ) where s i ,1 ∈ {− 3,1,1,3} E s i , 2 ∈ {− 3,1,1,3} E Note E is not the symbol energy but rather E½ is the shortest distance from a constellation point to a decision boundary. As before, the number of symbols contained per QAM symbol is log2(M) where M is the number of constellation points or signals in the alphabet. The easiest way of obtaining the approximate BER is to view QAM as two orthogonal independent PAM schemes with M levels each. Note that this analysis is only valid for square QAM modulation schemes such as the 16 QAM example above. Consider the 4 level PAM constellation below