We consider a network with three nodes including two source nodes, denoted by S1 and S2, and one relay node R. A half-duplex system is assumed and all nodes are equipped with one antenna. Information is exchanged between S1 and S2 with the help of R, which is completed in two phases. In the first phase, the MA phase, both source nodes send the differentially encoded signals to the relay, and in the second phase, the BC phase, the relay broadcasts the superimposed signals back to both source nodes.
Let z
i
(k) ∈ Ω,i ∈ {1,2} denote the symbol to be transmitted by source node S
i
at discrete symbol time k, where Ω represents a unity power M-PSK constellation set. As single-differential modulation is used, the signal z
i
(k) sent by source S
i
is given as
(1)
In the MA phase, two terminals simultaneously transmit the differentially encoded information to the relay. For simplicity, we assume that the fading coefficients and the carrier offsets keep constant over the frame of length L and change independently from one frame to another[7, 10]. The received signal at the relay at time k is then
(2)
where h
i
,i ∈ {1,2} denotes the complex channel gain with zero mean and unit variance between S
i
and R;, T is the symbol interval and is the Doppler shift introduced between S
i
and R. nr(k) stands for a zero mean complex Gaussian random variable with variance, and P
i
denotes the transmit power at source S
i
.
In the BC phase, the relay R amplifies yr by a factor α and then broadcasts its conjugate, denoted by back to both S1 and S2 with transmit power Pr. The corresponding signal received by S1 at time k, denoted by y1(k), can then be written as
(3)
For the decoding simplicity at S1, we can obtain the conjugate of (3) as
(4)
where and the equivalent noise.
Similarly, the received signal at S2 can be expressed as
(5)
Given that S1 and S2 are mathematically symmetrical, as shown in (3) and (5), for simplicity, we only discuss the signal detection at S1 in the following.
Since the relay has no knowledge of CSI, we cannot obtain the amplification factor α directly. We may rewrite the received signals at the relay in a vector format as
(6)
where Yr = [yr(1),⋯,yr(L)]T,nr = [nr(1),⋯,nr(L)]T,, i ∈ {1,2}. We therefore have
(7)
where and. α can thus be approximated at high signal-to-noise-ratio (SNR) as
(8)
Similar to (6), the received signals at source S1 can also be rewritten in the vector format as
(9)
where and.
It is shown in (3) that the signal received at source S1 is a complex superimposed signal; therefore, the application of conventional single- or double-differential detection on point-to-point communication link to TWR is not straightforward. It is difficult to decode the expected information z2(k) if we cannot subtract the self-information s1(k) from y1(k) when μ is unknown, due to the lack of CSI at S1. Therefore, we propose a three-step approach in the single-differential detection for TWR with carrier offsets: step 1, the self-information of μ s1(k) is subtracted from y1(k), the most important step in the whole detection procedure. Step 2, the carrier frequency offset is estimated and compensated. Step 3, signal z2(k) differentially decoded using the single-symbol single-differential detector.
Step 1: self-information subtraction
Since terminal S1 knows its own transmitted signal, μ needs to be estimated before we can subtract the contribution of μ s1(k) from y1(k). We thereby propose a simple estimation method as follows
(10)
By taking the expectation of, given that s1(k) and s2(k) are independent and have the same distribution, we can approximately obtain
After obtaining the estimation of μ, we can easily subtract the self-information of s1(k) as
(13)
Step 2: carrier offset estimation
A frequency offset estimation method was introduced in[13], which is effective in removing the impact of carrier frequency offsets, independent of data symbols and channel gains. However, training symbols are required to be transmitted at the beginning of each frame to solve the ambiguous estimation problem. In this case, two training symbols are enough to provide a good estimation of the carrier offsets. Then, the signals received at S1 can be rewritten as
(14)
(15)
where P is the number of training symbols. Define the training symbols as s
i
(-P) = 1 and P = 2, we have
(16)
Since ν is also a complex value, the following transformation is made
(17)
where. Then, the estimation of ω can be obtained as.
Step 3: single-symbol single-differential detection
With the estimation of the carrier frequency offset, the frequency offset effect is compensated, and the received data after compensation can be expressed as
(18)
Consider the generalized likelihood ratio test (GLRT) detection of the multiple symbols. Define
(19)
The GLRT algorithm for detection of can be obtained by minimizing the following metric:
(20)
Performing the minimization of the metric over ν results in the following decision algorithm:
(21)
Let be the detection results during the observation length of N for the signal transmitted by S2. Then, by differential decoding, we can recover the z2(k-n) as
(22)