5.1. The BER of PSPF
The BER performance of the proposed PSPF scheme with discriminator detection is evaluated using the characteristic function (CF) approach; see . In the analysis, the variances of the fading processes g
(t), and g
(t) in (1) to (6) are denoted as , and , respectively, with and . On the other hand, the variances of the noise processes nR, 1(t), nB, 1(t), nR, 2(t), nA,2(t), nA,3(t), and nB,3(t) in these equations are , and , respectively, with and , where N0 is the noise power spectral density (PSD), B12 the bandwidth of the receive LPFs in Phases I and II, and B3 the bandwidth of the receive LPF in Phase III. In this investigation, B12 is always set to the 99% bandwidth of x
(t) and x
(t), while B3 is either the same as B12, or set to the 99% bandwidth of the relay signal x
(t). Given the nature of the symbol-by-symbol detectors described in the previous section, we take the liberty to drop the symbol index k in d
in the performance analysis.
First, it is observed that the terms D2 in (21) is a quadratic forms of complex Gaussian variables when conditioned on ; refer to the Appendix for the statistical relationships between different parameters in the general channel model
where g(t) and n(t) are, respectively, and is the signal phase, and a(t) and ψ(t) are respectively the amplitude and phase of y(t). Without loss of generality, we assume d
= +1 and hence . By substituting into (A5) and (A8), and with F in (A10) set to the matrix in (21), we can find the two poles of the CF of D2 as following:
where αA,2, βA,2, ρA,2are determined from (A10) under the conditions , and the bandwidth of the receive filter in Phases I and II.
How about the term D3 in (21)? This term can be rewritten as , or as
which is once again a quadratic form of complex Gaussian variables. This quadratic form, however, depends on a number of parameters. First is the data phase derivation . Second, it depends on the forwarded phase derivative , which in turns depends on both and ; refer to (16). Of course, depends on , while depends on , refer to (14) and (15). Note that D2 and D3 are statistically independent. Conditioned on , and , we can determine from (A10) the poles of the CF of D3 as
where are determined from (A10) under the conditions , , and ; B3 the bandwidth of the receive filter in Phase III.
Recall that we assume d
= +1 and hence . In this case, the detector makes a wrong decision when D < 0. Since the characteristic function of D is
the probability that D < 0 is the sum of residues of -ϕ
(s)/s at the right plane poles p2 and Q2, yielding
Finally, since and are random variables given and , respectively, the unconditional error probability can be expressed in semi-analytical form as
where the marginal probability density functions (PDF) and can be determined from (A5) to (A6) in the Appendix.
5.2. BER of 3-DPF and A4-DPF Signals
The two multi-level DPF signals broadcasted by the relay in (11) and (12) are constructed from decisions made by the relay about Users A and B's data. Although different from (10), the exact BER analysis of these signals can still be determined via the characteristic function approach. This stems from the fact that the decision variable D of these DPF schemes are again quadratic forms of complex Gaussian variables when conditioned on the data phase derivatives and , as well as their decoded versions and at the relay. Specifically, the poles of the CF of D2 are identical to those in the PSPF case, and can be found in (28). As for the poles of the CF of D3, we should first replace the term in the Appendix by and then modify the F matrix in (A10) to
The resultant poles are found to be
Where are determined from (A10) under the conditions , , and the bandwidth of the receive filter in Phase III. As in the case of PSPF, the conditional BER is expressed in the form
The only difference between (35) and (31) is that the former is conditioned on the hard decisions and made at the relay, while the latter is based on the soft decisions and . If we let P
be the probabilities that the relay makes a wrong decision about A and B's data, respectively, then the unconditional BER is
= 1 for 3-DPF and N
= 2 for A4-DPF, and the inner summation is over the two different permutations of w
in (13). It should be pointed out the error probabilities P
can be determined by integrating the marginal pdf in (A6) from -∞ to 0 when the data bit is a + 1, or from 0 to +∞ when the data bit is a -1. The end result is of the form [15, 21]
| and |ρ
| are |ρ| in (A5) obtained under the conditions and , respectively.