Part IV MULTIPLE ACCESS AND ADVANCED TRANSCEIVER SCHEMES 363
12.1 Demodulator Structure and Error Probability in Additive
12.1.3 Methods for the Computation of Error Probability
In this subsection, we discuss how to compute the performance that can be achieved with optimum receivers. Before going into details, let us define some important expressions: thebit error rate is, as its name says, a rate. Thus, it describes the number of bit errors per unit time, and has dimension s−1. Thebit error ratiois the number of errors divided by the number of transmitted bits; it thus is dimensionless. For the case when infinitely many bits are transmitted, this becomes thebit error probability. In the literature, there is a tendency to mix up these three expressions; specifically, the expression BER is often used when the authors mean bit error probability. This misnomer has become so common that in the following we will use it as well.
Though we have treated a multitude of modulation formats in the previous chapter, they can all be classified easily with the framework of signal space diagrams:
1. Binary Phase Shift Keying (BPSK) signals are antipodal signals.
2. Binary Frequency Shift Keying (BFSK), and Binary Pulse Position Modulation (BPPM), are orthogonal signals.
3. Quadrature-Phase Shift Keying (QPSK),π/4-DQPSK (Differential Quadrature-Phase Shift Key- ing), and Offset Quadrature-Phase Shift Keying (OQPSK) are bi-orthogonal signals.
Error Probability for Coherent Receivers – General Case
As mentioned above, coherent receivers compensate for phase rotation of the channel by means ofcarrier recovery. Furthermore, the channel gainα is assumed to be known, and absorbed into the received signal, so that in the absence of noise,r=s holds. The probability that symbol sj is mistaken for symbol sk that has Euclidean distancedj k fromsj (pairwise error probability) is given as
Prpair(sj,sk)=Q
⎛
⎝ dj k2 2N0
⎞
⎠=Q E
N0(1−Re{ρj k})
(12.24) where theQ-function is defined as
Q(x)= 1
√2π ∞
x
exp(−t2/2) dt (12.25)
This is related to the complementary error function:
Q(x)=1 2erfc
x
√2
(12.26) and
erfc(x)=2Q(√
2x) (12.27)
Equation (12.24) can be found by computing the probability that the noise is large enough to make the received signal geometrically closer to the point sk in the signal space diagram, even though the signalsj was transmitted.
Error Probability for Coherent Receivers – Binary Orthogonal Signals
As we saw in Chapter 11, a number of important modulation formats can be viewed as binary orthogonal signals – most prominently, binary frequency shift keying (FSK) and binary Pulse Posi- tion Modulation (PPM). Figure 12.3 shows the signal space diagram for this case. The figure also shows the decision boundary: if a received signal point falls into the shaded region, then it is decided that a+1 was transmitted, otherwise a−1 was transmitted.
Defining the Signal-to-Noise Ratio (SNR) for one symbols asγs=ES/N0, we get Prpair(sj,sk)=Q
γS(1−Re{ρj k})
(12.28)
=Q(√
γS) (12.29)
Note that since we are considering binary signalingγS=γB.
Demodulation 227
Decision boundary
Select sj Select sk
sk
sj
Figure 12.3 Decision boundary for the selection between skand sj.
Error Probability for Coherent Receivers – Antipodal Signaling For antipodal signals, the pairwise error probability is:
Prpair(sj,sk)=Q
γS(1−Re{ρj k})
(12.30)
=Q(
2γS) (12.31)
For binary signals with equal-probability transmit symbols, pairwise error probability is equal to symbol error probability, which in turn is equal to bit error probability. This is the case, e.g., for BPSK, as well as for MSK with ideal coherent detection (see Chapter 11), and the BER is given by Eq. (12.30). Note that MSK can be detected like FSK, but then does not exploit the continuity of the phase. In this case, it becomes an orthogonal modulation format, and the BER is given by Eq. (12.29). This means deterioration of the effective SNR by 3 dB.
Union Bound and Bi-Orthogonal Signaling
ForM-ary modulation methods, exact computation of the BER is much more difficult; therefore, the BER is often upper-bounded by theunion boundmethod. The principle of this bounding technique is outlined in Figure 12.4. The region of the signal space diagram that results in an erroneous decision consists of partial regions, each of which represents a pairwise error – i.e., confusing the correct symbol with another symbol. The symbol error probability is then written as the sum of these pairwise probabilities. Since the pairwise- error regions overlap, this represents an upper bound for true symbol error probabilities. The approximation improves as the SNR increases, because the SER is then mainly determined by regions close to the decision boundaries whereas overlap regions have little impact.
Decision boundary
Selection of wrong symbol
Selection of wrong symbol
Selection of wrong symbol Selection
of wrong symbol
Select
Decision boundary
Decision boundary
Decision boundary
sj sj
Select sj
Select sj Select sj
sj sj
sj
Figure 12.4 Union bound for symbol error probabilities.
Great care must be taken when using equations for the BER of higher order modulation formats from the literature. There are several possible pitfalls:
• Does the equation give the symbol error probability or the bit error probability? Take, e.g., the 4-Quadrature Amplitude Modulation (QAM) of Figure 12.4, and assume furthermore that the signal constellations are Gray-coded, so that the four points represent the bit combinations 00, 01, 11, and 10 (when read clockwise). There is a high probability of errors occurring between two neighboring signal constellations. One symbol error (in one transmitted symbol) then corresponds to one bit error (one error intwo transmitted bits). Thus, bit error probability is only about half symbol error probability.3
• Does computation of the SNR use bit energy or symbol energy? A fair comparison between different modulation formats should be based onEB/N0.
• Some authors define the distance between the origin and (equal-energy) signal constellation points not as √
E, but rather as√
2E. This is related to a different normalization of the expansion
3This demonstrates the drawbacks of mixing up the expressions “bit error probability” and “ bit error rate”. In our example, the BER is the same as the symbol errror rate, while the bit error probability is only half the symbol error probability.
Demodulation 229
functions, which is compensated by different values fornn. The final results do not change, but this makes the combination of intermediate results from different sources much more difficult.
As an example, we compute in the following the BER of 4-QAM. As it is a four-state modulation format (2 bit/symbol), we can invest twice the bit energy for each symbol. The signal points thus have squared Euclidean distance d2=ES=2EB from the origin; the points in the signal constellation diagram are thus at±√
EB(±1±j ), and the distance between two neighboring points isdj k2 =4EB.
We can now consider two types of union bounds:
1. For a “full” union bound, we compute the pairwise error probability with all possible signal constellation points, and add them up. This is shown in Figure 12.4.
2. We compute pairwise error probability using nothing more than neighboring points. In this case, we omit the last decision region in Figure 12.4 from our computations. As we can see, the union of the first two regions (for pairwise error with nearest neighbors) already covers the whole
“erroneous decision region.” In the following, we will use this type of union bound.
Example 12.1 Compute the BER and SER of QPSK.
From Eq. (12.24) it follows that the pairwise error probability is Q 2EB
N0
=Q(
2γB) (12.32)
According to Figure 12.4, the symbol error probability as computed from the union bound is twice as large:
SER≈2Q(
2γB) (12.33)
Now, as discussed above, the BER is half the SER:
BER=Q(
2γB) (12.34)
This is the same BER as for BPSK.
For QPSK, it is also possible to compute the symbol error probability exactly: the probability of a correct decision (in Figure 12.4) is to stay in the right half plane (probability for that event is 1−Q(√2γB)) times the probability of staying in the upper half-plane (the probability for this is independent of the probability of being in the right half plane, and also is 1−Q(√2γB)).
The overall symbol error probability in thus 1−(1−Q(√2γB)2, i.e., SER=2Q(
2γB)
1−1 2Q(
2γB)
(12.35) which shows the magnitude of the error made by the union bound. The relative error is 0.5Q(√2γB), which tends to 0 asγBtends to infinity.
The exact computation of the BER or SER of higher order modulation formats can be com- plicated. However, the union bound offers a simple and (especially for high SNRs) very accu- rate approximation.
Error Probability for Differential Detection
Carrier recovery can be a challenging problem (see Meyr et al. [1997] and Proakis [2005]), which makes coherent detection difficult for many situations. Differential detection is an attractive
alternative to coherent detection, because it renders irrelevant the absolute phase of the detected signal. The receiver just compares the phases (and possibly amplitudes) of two subsequent symbols to recover the information in the symbol; this phase difference is independent of the absolute phase.
If the phase rotation introduced by the channel is slowly time varying (and thus effectively the same for two subsequent symbols), it enters just the absolute phase, and thus need not be taken into account in the detection process.
For differential detection of Phase Shift Keying (PSK), the transmitter needs to provide differ- ential encoding. For binary symmetric PSK, the transmit phase i of theith bit is
i = i−1+
⎧⎪
⎨
⎪⎩ +π
2 if bi = +1
−π
2 if bi = −1 (12.36)
Comparison of the difference between phases on two subsequent sampling instances determines whether the transmitted bitbi was+1 or−1.4
For Continuous Phase Frequency Shift Keying (CPFSK), such differential encoding can be avoided. Remember that in the case of MSK (without differential encoding), the phase rotation over a 1-bit duration is ±π/2. It is thus possible to determine the phases at two subsequent sampling points, take the difference, and conclude which bit has been transmitted. This can also be interpreted by the fact that the phase of the transmit signal is an integral over the uncoded bit sequence. Computing the phase difference is a first approximation to taking the derivative (it is exact if the phase change is linear), and thus reverses the integration.
For binary orthogonal signals, the BER for differential detection is [Proakis 2005]
BER=1
2exp(−γb) (12.37)
For 4-PSK with Gray-coding, it is
BER=QM(a, b)−1
2I0(ab)exp
−1
2(a2+b2)
(12.38) where
a= 2γB
1− 1
√2
, b= 2γB
1+ 1
√2
(12.39) andQM(a, b)is Marcum’s Q-function:
QM(a, b)= ∞
b
xexp
−a2+x2 2
I0(ax) dx (12.40)
whose series representation is given in Eq. (5.31).
Error Probability for Noncoherent Detection
When the carrier phase is completely unknown, and differential detection is not an option, then non- coherent detection can be used. For equal-energy signals, the detector tries to maximize the metric:
|rLPs∗LP,m| (12.41)
so that the optimum receiver has a structure according to Figure 12.5.
4Theoretically, differential detection could also be performed on a non-encoded signal. However, one bit error would then lead to a whole chain of errors.
Demodulation 231
sLP,1(t)
sLP,m(t)
sLP,M(t)
∫T0....dt
∫T
0....dt
∫T0....dt | . |
| . |
| . |
r(t) Select
largest
Figure 12.5 Optimum receiver structure for noncoherent detection.
An actual realization, wherer(t )is a bandpass signal, will in each branch of Figure 12.5 split the signal into two subbranches, in which it obtains and processes the I- and Q-branches of the signal separately; the outputs of the “absolute value” operation of the I- and Q-branches are then added up before the “select largest” operation.
In this case, the BER can be computed from Eq. (12.38), but with a different definition of the parametersaandb:
a= γB
2 1−
1− |ρ|2 , b=
γB
2 1+
1− |ρ|2
(12.42) The optimum performance is achieved in this case if ρ=0 – i.e., the signals are orthogonal.
For the case when|ρ| =1, which occurs for PSK signals, including BPSK and 4-QAM, the BER becomes 0.5.
Example 12.2 Compute the BER of binary FSK in an AWGN channel with γB=5 dB, and compare it with Differential Binary Phase Shift Keying (DBPSK) and BPSK.
For binary FSK in AWGN, the BER is given by Eq. (12.29) withγS =γB. Thus, withγB=5 dB, we have:
BERBFSK=Q(√ γB)
=0.038 (12.43)
For BPSK, the BER is instead given by Eq. (12.37), and thus:
BERBPSK=Q(
2γB)
=0.006 (12.44)
Finally, for differential BPSK, the BER is given by Eq. (12.37), which results in:
BERDBPSK=(1/2)e−γB
=0.021 (12.45)