IEEE SIGNAL PROCESSING LETTERS, VOL. 22, NO. 7, JULY 2015 905 The Effects of Multiple Carrier Frequency Offsets on the Performance of Virtual MISO FSK Systems Muddassar Hussain and Syed Ali Hassan, Member, IEEE Abstract—In this letter, a virtual multiple-input single-output (VMISO) network employing non-coherent frequency shift keying (FSK) is considered. In the VMISO network, spatially separated single-antenna nodes transmit the same information cooperatively to a single receiver where each received signal is affected by an independent carrier frequency offset (CFO). The existing works in this area assume a perfect carrier synchronization between the nodes. However, in this letter, the effects of CFOs on the perfor- mance degradation of this network are analyzed. For that purpose, the expression for the probability of symbol error has been derived. The results indicate that the performance is degraded due to CFOs, which is dependent upon the magnitude of CFOs, signal-to-noise ratio (SNR), number of transmitting nodes, and the modulation order of FSK. At high SNR, the CFOs affect the system severely and the performance margin is minimum. Index Terms—Carrier frequency offset (CFO), coopera- tive communication, non-coherent FSK, virtual multiple-input single-output (VMISO). I. INTRODUCTION C OOPERATIVE TRANSMISSION (CT) is an emerging paradigm in wireless communication, which combats the effects of multipath fading by employing spatial diversity. CT has become a promising candidate for wireless sensor networks (WSNs) because of its inherent advantages such as energy-ef- ficiency [1] and range extension [2]. Since the transceivers for WSNs should be cost-effective and energy-efficient, a low-com- plexity modulation scheme is desirable [3]. One such scheme is non-coherent frequency shift keying (FSK), with many advan- tages such as constant envelope, efficient amplification at the transmitter, and a simple receiver design using envelope detec- tion. Because radios in WSNs are low-powered, non-coherent FSK scheme is used along with non-coherent combining tech- niques such as equal gain combining (EGC) to provide array gain and range extension [4]. However, to keep a low receiver complexity, sophisticated circuitry cannot be deployed at the re- ceiver, thereby introducing problems such as CFO that degrade the performance of these systems. The effects of CFO on the performance of FSK systems are previously studied in [5] and [6]. In [5], a single-input single- Manuscript received October 14, 2014; accepted November 19, 2014. Date of publication November 26, 2014; date of current version December 11, 2014. The authors gratefully acknowledge the grant from National ICT R&D Fund, Pakistan, for sponsoring this research work. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Chandra Ramabhadra Murthy. The authors are with the School of Electrical Engineering and Computer Sci- ence (SEECS), National University of Sciences and Technology, Islamabad, Pakistan (e-mail: muddassar.hussain@seecs.edu.pk; ali.hassan@seecs.edu.pk). Digital Object Identifier 10.1109/LSP.2014.2375172 output (SISO) system employing non-coherent FSK is consid- ered. The effects of timing and frequency errors on the error probability are presented for a noise-only channel. The results show a significant degradation on the performance of the system due to CFO and timing error. In [6], the authors have analyzed a two-hop network consisting of a source, a relay node and a destination. The authors derived the expression for error prob- ability for both coherent and non-coherent FSK systems under the impacts of the CFO. In [7], the effect of CFO and channel estimation errors on the performance of multiple-input multiple-output (MIMO) system using orthogonal frequency division multiplexing (OFDM) is analyzed. Theoretical and simulation results of symbol error probability are presented for the MIMO-OFDM-based system, which show a considerable performance degradation with in- creased CFOs. To the best of authors’ knowledge, no work has been done for analyzing the impacts of CFOs on the performance of non-co- herent FSK-based systems when they are used in a multi-hop co- operative network. Most previous works, e.g., [8]–[10] and the references therein, assume perfect carrier synchronization be- tween the nodes. This assumption, however, is not valid for prac- tical systems and the CFO plays its role in performance degrada- tion [11]. Since in a multi-hop cooperative network, many spa- tially distributed transmitters transmit the same message to all nodes of the next hop, hence for each receiving node a VMISO system is created. Therefore, in this letter, we characterize the performance of a one hop VMISO network employing non-co- herent orthogonal FSK. Specifically, multiple nodes transmit the same information to a single receiving node where each re- ceived signal is affected by a CFO. The timing synchroniza- tion is achieved using the techniques discussed in [12] and [13]. When CFOs are introduced in the network, the receiver branches no longer remain orthogonal and a correlation is introduced. The expressions for the correlation between the decision vari- ables and the probability of symbol error are derived mathemat- ically. Both the simulation and theoretical results are provided that show a significant performance degradation which is depen- dent on the magnitude of CFOs. II. SYSTEM MODEL Consider a VMISO communication system with transmit- ting nodes employing -ary frequency shift keying ( -FSK) modulation with non-coherent detection. The signal transmitted by the th node is given as (1) where is the signal energy, is symbol period, is carrier frequency and is the -FSK frequency corresponding to the th transmitted symbol. The CFO and the carrier phase of the th 1070-9908 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.