Doubly Differential Communication Assisted with Cooperative Relay Na Yi, Yi Ma, and Rahim Tafazolli Centre for Communication Systems Research University of Surrey, UK, GU2 7XH. Emails:{n.yi, y.ma, r.tafazolli} @surrey.ac.uk Abstract— Doubly differential modem turns out to be a promis- ing technology for coping with unknown frequency offsets with the pay of signal-to-noise ratio (SNR). In this paper, we propose to compensate the SNR loss by employing the detection-forward cooperative relay. The receiver can employ two kind of combiners to attain the achievable spatial diversity-gain. Performance anal- ysis is carefully investigated for the Rayleigh-fading channel. It is shown that the SNR-compensation is satisfied for the large-SNR range. Index Terms— Cooperative relay, doubly differential, detection-forward, carrier frequency offset. I. I NTRODUCTION Carrier frequency offsets (CFOs) caused by Doppler shift and oscillator instability can affect significantly the overall performance of digital communication systems. State-of-the- art approaches for solving this problem can be classified into two main categories, i.e., coherent approaches and differential- coherent approaches. The coherent approaches are usually based on the carefully designed CFO estimators, which have been intensively investigated for the end-to-end communi- cations (e.g., [1]-[2]). Recently, the CFO estimation issue is mainly focused on the network scenario, i.e., multiuser and multi-link cases (e.g., [3]-[4]). On the other hand, the imperfect CFO estimation and compensation can still affect the overall system performance. The differential-coherent ap- proaches mainly refer to the doubly differential communi- cations, which have been originally proposed by Okunev in 1979 [5]. Afterwards, this technique has been applied in many advance wireless systems such as multi-antenna systems (e.g., [6]) and block transmissions (e.g., [7]). The major advantage of the doubly differential communication is its bypass of the CFO and channel estimation. However, the pay for this nice feature is the considerable loss in the received signal-to-noise ratio (SNR) (up to 7.8 dB loss [8]). In the wireless networks, distributed nodes can establish efficient cooperation to exploit the distributed spatial diversity. The node cooperation is usually based on some classical relaying protocols such as selection detection-forward (DF), amplify-forward, etc. [9]-[11]. While the sub-channels are orthogonal with each other, the receiver can employ the The work reported in this paper has formed part of the Core 4 Research Programme of the Virtual Centre of Excellence in Mobile and Personal Communications, Mobile VCE, www.mobilevce.com, whose funding support, including that of EPSRC, is gratefully acknowledged. Fully detailed technical reports on this research are available to Industrial Members of Mobile VCE. maximum-ratio combine (MRC) to enjoy the maximum co- operative diversity-gain. This impressive result motivates us to employ the half-duplex cooperative relay to compensate the SNR loss inherent in the doubly differential communica- tions. In this paper, we consider the relay operating in the selection DF mode, i.e., the relay can forward the received symbols when the received SNR is larger than a threshold. The receiver can employ two kind of combiners to attain the achievable cooperative diversity-gain. Considering the single- relay case, the overall performance is carefully investigated for the slow Rayleigh-fading channel. The selection of the SNR threshold is also carefully studied for the performance optimization. Computer simulations are carried out for various channel setup, which can clearly disclose the effect of the SNR compensation. II. DOUBLY DIFFERENTIAL COMMUNICATION AND SNR COMPENSATION A. Doubly differential Communication Here, we introduce briefly the work philosophy for the double differential communication. Prior to transmission, the sender first format the information-bearing bits into an M × 1 symbol block with N -PSK modulation, i.e., s = [s 0 ,s 1 , ··· ,s M-1 ] T . The symbols are fed into the doubly differential modulator [8], which relates the input s to the output x as x m =x * m-1-τ x m-1 x m-τ s m , (1) where m stands for the symbol index, τ ( 1) for the lag, the superscript * for the conjugate, and the terms x -1-τ , x -τ , x -1 , |s m |, are set to 1 for convenience. Then, the symbols x m are transmitted through the sender-destination (SD) channel h (sd) ( (sd) stands for the SD channel). Denote ν to be the CFO normalized by the block duration. The received symbols at the destination are expressible as y (sd) m = e j2πνm M h (sd) x m + v (sd) m , (2) where v stands for the Gaussian noise with zero mean and variance of N o . At the receiver, the doubly differential de- modulation can be implemented as below z (sd) m =y (sd) m-1-τ  y (sd) m-τ y (sd) m-1  * y (sd) m , (3) ≈   h (sd)    4 s m +¯ v (sd) m , (4) 978-1-4244-1645-5/08/$25.00 ©2008 IEEE 644 Authorized licensed use limited to: University of Surrey. 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