To have an alternate physical link (PL), one can purposely vary the power of the bit stream so that the transmitted sequence is equivalent to one formed by multiplexing two
data sources with different powers. If the locations of these two parts in the multiplexed data stream are known, the DN then perform separate comparison and counting based on (2.16.a) and (2.16.b). Although such a two-level amplitude modulation makes pos-sible solving the esrk, erkd ambiguity, allocating unequal powers to different parts of the transmitted data stream is often undesirable. This dilemma can be avoided by creating a virtual link (VL) without modifying the existing link.
A VL can be created by rotating the received I-Q vector counter-clockwise by an angle θ between 0oand 90o. This is equivalent to introducing an artificial phase offset to the received samples which are then used as outputs from another link. Since the noise is circular symmetric, the rotation results in an equivalent signal power degradation cos2θ without altering the noise statistic. Such a virtual SNR loss cannot be accomplished by simply multiplying the BPSK matched filter output by a positive constant less than one.
An alternate method is to add an extra zero-mean white Gaussian noise component to the received in-phase samples. Both schemes give a VL with a smaller γ. The second scheme–the addition of a perturbation term–incurs no hardware increase but requires the estimation of noise power σd2, which is needed in subsequent ML detection anyway.
As the phase-rotation scheme leads to an SNR degradation of magnitude cos2θ, the second scheme has to generate i.i.d. zero-mean Gaussian random samples with variance σv2 = σ2d(1/ cos2θ− 1) to achieve the same SNR loss. Although both approaches achieve the same effect for BPSK signals, the phase-rotating approach cannot produce a VL for noncoherent systems while the method of inserting extra noise suits both coherent and noncoherent applications. Hence, except for the coherent system discussed in this section, we will adopt the noise-injection approach in the following sections.
We use the superscript (v) to indicate that a parameter is associated with a VL, i.e., the kth RD link’s synchronous output samples and their rotated (VL) versions are denoted by yrkd[n], yr(v)
kd[n] and the corresponding ERs by erkdand e(v)r
kd. Since for a BPSK
system operating in a flat Rayleigh fading environment, we have [37]
The two ERs are then related by (1− 2erkd)2
1− (1 − 2erkd)2 = 1 cos2θ
(1− 2e(v)rkd)2
1− (1 − 2e(v)rkd)2. (3.3) Following a procedure similar to that for solving (2.17), we can easily show that the nonlinear system which consists of (2.15), (3.3) and the new cascaded link’s ER equation
Q(v)k = esrk+ e(v)r Based on this solution, we can obtain a complete blind algorithm to estimate the ERs of all component links by using the estimates for Qk and Q(v)k which are computed via (2.18) using another, say lth (l ̸= k) relay link; ER side information is no longer needed. In short, to estimate the triplet (esd, esrk, erkd) associated with an SD and an SRD links without the help of CSI, one needs another independent relay. The auxiliary relay requirement can be waived if one creates a virtual SD link to obtain additional combinational diversities. In general, the rotation angle for producing a virtual SD link can be different from that for a virtual RD link. However, we lose no generality by
assuming both rotation angles are the same, say θ. Denote bybp(vs)r,bps(vr)and bp(vs)(vr)the estimates for the SMPs, Pr(by(v)sd =byrd), Pr(bysd =byrd(v)), and Pr(bysd(v) =byrd(v)), respectively, and by Q = esrd, Q(v) = es(vr)d, the ERs for the SRD and the SR-plus-virtual relay links.
We obtain four nonlinear relations for a single-relay CCN:
bpsr = esdQ + (1− esd)(1− Q) (3.7.a) bp(vs)r = e(v)sd Q + (1− e(v)sd)(1− Q) (3.7.b) bps(vr) = esdQ(v)+ (1− esd)(1− Q(v)) (3.7.c) bp(vs)(vr) = e(v)sd Q(v)+ (1− e(v)sd)(1− Q(v)) (3.7.d)
With the additional PL-VL relation (1− 2esd)2
1− (1 − 2esd)2 = 1 cos2θ
(1− 2e(v)sd )2
1− (1 − 2e(v)sd)2 (3.8) the nonlinear system (3.7.a)–(3.8) yields the closed-form estimators
besd =1 Estimators,besrandberd, can be derived from solving the nonlinear system which includes (2.15), (3.3) and an equation similar to (3.4). An analytic solution of this nonlinear system is obtained by substituting (3.9.b) into (3.6) and then (3.5.a). As has been men-tioned in Section I, we refer to ER estimation algorithms using the approach described in this section as virtual link aided (VLA) estimators. The corresponding estimation procedure is included in Table 3.1.
Note that the SMP formulae (2.14) and (3.7.a)-(3.7.d) are not valid for the SMP between a PL and its virtual version since their outputs are correlated. Actually, this
Table 3.1: A blind ER estimation algorithm for BPSK modulation.
Input: Received samples, y, noise variance, σ2d, and scaling factor values, a(v)sd, a(v)rd.
1: Create virtual SD and RD links by injecting complex Gaussian noise samples with scaling factors, a(v)sd and a(v)r
id. 2: Compute SMPs for all physical, virtual SD-SRD link pairs.
3: Compute bQ, bQ(v), andbesd through (3.9.b) and (3.9.a) with a(v)sd = a(v)rd = cos12θ.
4: Obtainbesr and berd via (3.5.a)-(3.6).
Output: besd, besr and berd.
SMP is the sum of two conditional SMPs defined by (4.28) and (4.29) which are derived in Appendix D. Obviously, a system involves these two nonlinear expressions does not easily render a closed-form solution. On the other hand, a VL can provide a new SMP relation similar to (2.14) with each different PL or its virtual version and a single-relay CCN can offer two uncorrelated VLs to render a basic system that consists of three independent SMP equations, we thus conclude that, by using both virtual RD and SD links, one can estimate all ERs of a single-relay CCN without side information.