A space-frequency block codes MIMO single-carrier code-frequency-division multiple access system

Journal Pre-proofs Regular paper A Space-Frequency Block Codes MIMO Single-Carrier Code-Frequency-Division Multiple Access System Abdullah Y. Alamri, Ehab F. Badran PII: DOI: Reference:

S1434-8411(19)31273-7 https://doi.org/10.1016/j.aeue.2019.153053 AEUE 153053

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International Journal of Electronics and Communications

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17 May 2019 24 December 2019

Please cite this article as: A.Y. Alamri, E.F. Badran, A Space-Frequency Block Codes MIMO Single-Carrier Code-Frequency-Division Multiple Access System, International Journal of Electronics and Communications (2019), doi: https://doi.org/10.1016/j.aeue.2019.153053

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A Space-Frequency Block Codes MIMO SingleCarrier Code-Frequency-Division Multiple Access System Abdullah Y. Alamri1 and Ehab F. Badran2

Abstract In this paper, a novel space frequency block codes MIMO single-carrier code-frequency-division multiple access (SFBC MIMO SC-CFDMA) transceiver is proposed. The proposed SFBC SC-CFDMA system allocates a unique spreading code for each user. The proposed SFBC SC-CFDMA system offers robustness against the effect of carrier frequency offsets (CFOs). The proposed SFBC SC-CFDMA system is suggested to improve both of the peak-to-average power ratio (PAPR) performance and the bit error rate (BER) performance. The simulation results show that the proposed SFBC SC-CFDMA system has superior improvements in the BER performance over the traditional SFBC SC-FDMA system. Moreover, the PAPR performance of the proposed SFBC SC-CFDMA system is better than that of the traditional system for the case of a localized mapping scheme. Furthermore, BER analysis for the proposed SFBC SC-CFDMA and traditional SFBC SC-FDMA system over a frequency-selective Rayleigh fading channel is presented for the case of zero-forcing (ZF) equalizer and QPSK modulation. Keywords

Peak-to-Average Power Ratio. Bit Error Rate. Carrier Frequency Offset. Space-Frequency

Block Codes. Single-Carrier Frequency-Division Multiple Access. 1 Introduction In CDMA systems, spread spectrum signals are resistive to noise and other interference. Due to that, each user data is multiplied by a unique code sequence that allows the user to spread the information signal across the assigned frequency band. There are a lot of the unique properties of the spread spectrum technique that cannot be found in other techniques. One unique property is its ability to eliminate multi-path interference. It increases multi user handling capacity. It also has a low power spectral density since signal information is

spread over a large frequency band [1]. A single-carrier frequency-division multiple access system (SC-FDMA), which is a single-carrier communication technique, has similar performance as orthogonal frequency-division multiple access (OFDMA) system. However, OFDMA system suffers from the high peak-to-average power ratio (PAPR). Thus, PAPR reduction techniques, [2-6] and the references therein, have to be utilized in OFDMA system. Therefore, SC-FDMA, due to its lower PAPR, became widely accepted primarily. SC-FDMA is currently common in uplink of Long-Term Evolution (LTE). SC-CFDMA is a single-carrier code-frequency-division multiple access scheme, where the symbols of different users are first spread across the frequency band using unique spreading sequences, and SC-FDMA modulation is applied. Several advantages exist in combining CDMA with SC-FDMA modulation. The multiple access interference (MAI) is reduced due to orthogonal spreading codes. Also, both mappings and overlapping subcarriers coexist by using the orthogonal sequence spread spectrum technique. Meanwhile, SC-CFDMA system is used for the uplink control channel in 3GPP2 Ultra Mobile Broadband (UMB) [7]. The performance of SC-CFDMA and MC-CDMA systems has been compared in [8], and the performance of hybrid subcarrier mapping, localized with interleaving, based for uplink over AWGN channel has been given. It has been shown that SC-CFDMA is better than MC-CDMA in PAPR for uplink, and its BER performance is close to MC-CDMA. SC-CFDMA and other multiple access schemes, namely OFDMA, MCCDMA and SC-FDMA, have been compared in [9]. It was shown that SC-CFDMA is better than SC-FDMA in PAPR, and its BER performance is the same as SC-FDMA. Meanwhile, it has been shown that the performance of SC-CFDMA system is better than both of MC-CDMA and OFDMA systems performance. In space frequency block codes MIMO technique, the modulated signal is encoded into two different signals in the frequency domain, which improves the reliability of communication in fading channels [10]. However, the SFBC SC-FDMA system suffers from PAPR performance degradation [11]. The PAPR performance of the Spatial Multiplexing (SM) SC-FDMA and Space Frequency block Codes (SFBC) SC-FDMA uplink system for different subcarrier mapping and pulse shaping roll-off factors, has been discussed in [12]. Many techniques have been proposed to minimize the PAPR in SFBC MIMO SC-FDMA systems [11, 13, 14]. SFBC MIMO technique has been investigated with orthogonal frequency division multiplexing (OFDM) system, MC-CDMA system and SC-FDMA system. To the best of our knowledge, the SC-CFDMA system with SFBC technique has not been studied yet. As well, the bit error rate (BER) analysis of the SC-FDMA with SFBC technique has not been studied yet. This paper proposes SFBC SC-CFDMA system and presents its BER analysis compared to the conventional SFBC SC-FDMA systems with CFOs and without CFOs. The proposed SFBC SC-CFDMA system allocates a unique spreading code for each user. In reference [15], a blind inter-carrier interference compensation in MIMO SC-interleaved FDMA system using firefly algorithm was proposed to cancel the effect of CFOs. The

problem of channel estimation was investigated in references [16-19]. In this paper, it is considered that the fading channel samples are independent and perfectly known at the receiver side, i.e. perfect channel state information (CSI) is considered. This paper proposes SFBC SC-CFDMA system which utilizes MIMO equalization process with carrier frequency offsets compensation in the frequency domain to reduce the impacts of the multipath channel and CFOs. Simulation results show that, the proposed SFBC SC-CFDMA system offer better robustness against CFOs. In addition, the multiple access interference (MAI) is reduced due to orthogonal spreading codes. Simulation results with and without CFOs show that, the proposed SFBC SC-CFDMA system has superior improvement over SFBC SC-FDMA system in BER performance. Moreover, the proposed SFBC SC-CFDMA system has improvement over SFBC SC-FDMA system in PAPR performance for the case of a localized mapping scheme. The list of abbreviations and the list of symbols, used in this paper, are shown Table 1 and Table 2. Table 1 List of Abbreviations AWGN Additive White Gaussian Noise BER Bit Error Rate RC Raised-Cosine RRC Square-Root-Raised-Cosine CFOs Carrier Frequency Offsets CDMA Code Division Multiple Access SC-FDMA Single-Carrier Frequency-Division Multiple Access SC-CFDMA Single-Carrier Code-Frequency-Division Multiple Access SC-WDMA Single-Carrier Wavelet-Division Multiple Access OFDMA Orthogonal Frequency-Division Multiple Access MC-CDMA Multi-Carrier Code Division Multiple Access

SM SFBC SWBC MIMO MMSE MAI PAPR LTE UMB ZF 3GPP2

Spatial Multiplexing Space Frequency Block Codes Space Wavelet Block Codes Multiple Input Multiple Output Minimum Mean Square Error Multiple Access Interference Peak-To-Average Power Ratio Long-Term Evolution Ultra-Mobile Broadband Zero-Forcing 3rd Generation Partnership Project 2

Table 2 List of Symbols 𝑲 𝑳 𝑳𝑪 𝑳𝒉 𝑪𝒌 𝑺𝒌 𝑸 𝑪𝒑 𝑴𝒌 𝑴𝑻𝒌 𝜼 𝝁 𝝈𝟐 𝑵, 𝑵𝒄 𝑴 𝒌𝒏 𝓕𝑵𝒄 𝓕𝑯 𝑴 𝒉𝒌𝒋𝒊 𝑯𝒌𝒋𝒊 𝜠𝒌𝒋𝒊

Number of active users Length of spreading code vector Length of Cyclic prefix vector Length of Channel impulse response The 𝑘th user specific spreading matrix The 𝑘th user specific de-spreading matrix Max. number of users that can transmit simultaneously Cyclic prefix matrix The 𝑘th subcarrier mapping matrix The 𝑘th subcarrier de-mapping matrix Complex Gaussian noise vector Mean Noise variance Number of subcarriers each user Total number of subcarriers Column vectors of length 𝑁𝑐 𝑁𝑐 × 𝑁𝑐 FFT matrix 𝑀 × 𝑀 IFFT matrix Channel impulse response Circulant channel matrix Diagonal matrix which describes the CFOs

𝓕𝑀×𝐿ℎ 𝜀𝑗𝑖𝑘 𝜦𝑘2𝑁𝑐 𝜳𝑘 𝚪𝑗𝑖𝑘 𝜰𝑗𝑖𝑘 𝒙𝑘 𝑿𝑘 ̌𝑘 𝒙 𝓨 𝑃𝑒 Es Eb Q(⋅) 𝒪(. ) (. )−1 (. )𝑇 (. )𝐻 (. )∗ 𝑰𝑁 𝟎(𝑀−𝑁)×𝑁

Submatrix CFOs of the kth user normalized Joint MIMO equalization Effective frequency domain channel matrix Frequency domain equivalent channel matrix MIMO interference matrix Input signal vector The FFT of 𝒙𝑘 Transmitted signal vector Time domain received signal Probability of symbol error rate Average symbol energy Average bit energy Q-function 𝒪 −function Inverse of a matrix Transpose of a matrix Complex conjugate transpose of a matrix Complex conjugate of a matrix Identity matrix Zero matrix

This paper is structured as follows; section 2 presents the proposed SFBC SC-CFDMA system model. The BER analysis of the proposed SFBC SC-CFDMA and the conventional systems are presented in section 3. the computational complexity is discussed in section 4. Section 5 and Section 6 present the simulation results and the conclusions of this paper. 2 The Proposed SFBC SC-CFDMA System Model The uplink block-based system model for 2×2 SFBC SC-CFDMA transceiver is shown in Fig. 1. K users and one base station are assumed. A. SFBC SC-CFDMA Transmitter The proposed SFBC MIMO SC-CFDMA transmitter first modulates the kth user data into complex-valued 𝑇

𝑘 data symbols vector 𝒙𝑘 = [𝑥0𝑘 , 𝑥1𝑘 , … , 𝑥𝑁−1 ] with a length of 𝑁. Then, the vector 𝒙𝑘 is spread by 𝑘th user

SFBC Encoder

specific spreading code as follows

Input Data

Modulation

Spreading

NC-FFT

Mapping

M-IFFT

Add CP

Front end

Mapping

M-IFFT

Add CP

Front end

Channel

MIMO Detection

Despreading

SFBC Decoder

Demodulation

Nc-IFFT

Rx User 1 DeMapping

M-FFT

Remove

DeMapping

M-FFT

Remove

CP

CP

Front end

Front end

Output Data

Fig. 1 2×2 proposed SFBC SC-CFDMA system model. 𝒅𝑘 = 𝑪𝑘 𝒙𝑘 ,

(1)

where 𝒅𝑘 is a 𝑁𝑐 × 1 spread data vector. 𝑪𝑘 is the 𝑘th user specific spreading matrix of size 𝑁𝑐 × 𝑁 which is defined as 𝒄𝑘 𝟎 𝟎 𝒄𝑘 𝑪𝑘 = [ ⋮ ⋮ 𝟎 ⋯

⋯ 𝟎 ⋯ 𝟎 ], ⋱ ⋮ 𝟎 𝒄𝑘

(2)

𝑇

where 𝒄𝑘 = [𝒄𝑘,1 𝒄𝑘,2 … 𝒄𝑘,𝐿 ] is the spreading code vector of length 𝐿 for the 𝑘th user. The complex-valued data symbols 𝒅𝑘 is converted to frequency domain by applying 𝑁𝑐 -point FFT, with 𝑁𝑐 = 𝑁𝐿, as follows 𝑿𝑘 = 𝓕𝑁𝑐 𝒅𝑘 ,

(3)

where 𝓕𝑁𝑐 is 𝑁𝑐 × 𝑁𝑐 FFT matrix. The discrete frequency domain vector 𝑿𝑘 = [𝑋1𝑘 , 𝑋2𝑘 , . . . , 𝑋𝑁𝑘𝑐−1 , 𝑋𝑁𝑘𝑐 ] is then encoded with SFBC encoder into two output vectors 𝑿1𝑘 and 𝑿𝑘2 . The SFBC encoder output vectors can be represented as [20] ∗ 𝑇

∗

∗

𝑿1𝑘 = 𝑫 𝑿𝑘 − 𝑬 𝑿𝑘 = [𝑋1𝑘 , −𝑋2𝑘 , … , 𝑋𝑁𝑘𝑐−1 , −𝑋𝑁𝑘𝑐 ] , ∗

∗ 𝑇

∗

𝑿𝑘2 = 𝑭 𝑿𝑘 + 𝑮 𝑢𝑿𝑘 = [𝑋2𝑘 , 𝑋1𝑘 , … , 𝑋𝑁𝑘𝑐 , 𝑋𝑁𝑘𝑐−1 ] ,

(4𝑎) (4𝑏)

with 𝑫 = [𝒌1𝑇 ; 𝟎1×𝑁𝑐 ; 𝒌𝑇3 ; 𝟎1×𝑁𝑐 ; … ; 𝒌𝑇𝑁𝑐−1 ; 𝟎1×𝑁𝑐 ],

(5𝑎)

𝑬 = [𝟎1×𝑁𝑐 ; 𝒌𝑇2 ; 𝟎1×𝑁𝑐 ; 𝒌𝑇4 ; … ; 𝟎1×𝑁𝑐 ; 𝒌𝑇𝑁𝑐 ] = 𝑰𝑁𝑐 − 𝑫,

(5𝑏)

𝑭 = [𝒌𝑇2 ; 𝟎1×𝑁𝑐 ; 𝒌𝑇4 ; 𝟎1×𝑁𝑐 ; … ; 𝒌𝑇𝑁𝑐 ; 𝟎1×𝑁𝑐 ],

(6𝑎)

𝑮 = [𝟎1×𝑁𝑐 ; 𝒌1𝑇 ; 𝟎1×𝑁𝑐 ; 𝒌𝑇3 ; … ; 𝟎1×𝑁𝑐 ; 𝒌𝑇𝑁𝑐−1 ],

(6𝑏)

𝑁

𝑐 where {𝒌𝑛 } 𝑛=1 are column vectors of length 𝑁𝑐 ,with all-zero entries except at the 𝑛𝑡ℎ entry equal one.

Each output vector of the 𝑁𝑐 -FFT process is then mapped into 𝑀 orthogonal frequencies, 𝑀 = 𝑄𝑁𝑐 , 𝑄 is the bandwidth expansion factor. There are totally 𝑀 = 𝑄𝑁𝑐 subcarriers and a subset of the 𝑁𝑐 subcarriers is assigned for each user. Q is the maximum number of users that can transmit, simultaneously. The 𝑀- IFFT process is next applied to convert the discrete frequency domain output of mapping process to time domain. ̌𝑘𝑖 , which Afterwards, the cyclic prefix of length 𝐿𝑐 is added to the IFFT output signal. Therefore, the signal 𝒙 is the transmitted signal for the kth user in ith transmit antenna branch, can be expressed as 𝑘 ̌𝑘𝑖 = 𝑪𝑝 𝓕𝐻 𝒙 𝑀 𝑴𝑘 𝑿𝑖 ,

𝑖 = 1,2,

(7)

̌𝑘𝑖 is a vector of size 1 × (𝑀 + 𝐿𝑐 ). 𝓕𝐻 where 𝒙 𝑀 is an IFFT matrix of size 𝑀 × 𝑀. 𝑴𝑘 is a 𝑀 × 𝑁𝑐 representing the subcarrier mapping of matrix for the kth user. 𝑪𝒑 = [𝑪0 𝑰𝑀 ]𝑇 is the cyclic prefix adding matrix of size

(𝑀 + 𝐿𝑐 ) × 𝑀, where 𝑪0 = [𝟎𝐿𝑐×(𝑀−𝐿𝑐) 𝑰𝐿𝑐 ]𝑇 . The subcarrier mapping matrix 𝑴𝑘 for both the interleaved mapping (IM) and the localized mapping (LM) can be expressed as 𝑴𝑘 = {

[𝟎(𝑘−1)×𝑁𝑐 ; 𝒌1𝑇 ; 𝟎(𝑄−1)×𝑁𝑐 ; 𝒌𝑇2 ; … ; 𝒌𝑇𝑁𝑐 ; 𝟎(𝑄−𝑘)×𝑁𝑐 ] [𝟎(𝑘−1)𝑁𝑐×𝑁𝑐 ; 𝑰𝑁𝑐 ; 𝟎(𝑀−𝑘𝑁𝑐 )×𝑁𝑐 ]

IM,

(8𝑎)

LM.

(8𝑏)

B. SFBC SC-CFDMA Receiver After removing the cyclic prefix, the received signals vector to include the impact of the channel and carrier frequency offsets, can be expressed as 𝐾

̌𝑘𝑖 + 𝜼𝑗 , 𝓨𝑗 = ∑ 𝜠𝑗𝑖𝑘 𝑯𝑗𝑖𝑘 𝒙

(9)

𝑗 = 1,2,

𝑘=1

where 𝓨𝑘𝑗 is a 2𝑀 × 1 the received signal vector at the jth receive antenna. 𝜼𝑗 is a 2𝑀 × 1 the complex Gaussian noise vector at jth receive antenna. 𝑯𝑗𝑖𝑘 is an 𝑀 × 𝑀 circulant matrix defining the multipath channel between the ith transmit antenna and the jth receive antenna. The circulant matrix can be expressed as 𝑯𝑗𝑖𝑘 = 𝑘 𝑘 𝑇 𝓕𝐻 𝑀 𝑑𝑖𝑎𝑔(𝓕𝑀×𝐿ℎ 𝒉𝑗𝑖 )𝓕𝑀 , where 𝒉𝑗𝑖 = [ℎ𝑗𝑖 (0) ℎ𝑗𝑖 (1) ⋯ ℎ𝑗𝑖 (𝐿ℎ − 1)] is the channel impulse response of

length 𝐿ℎ and 𝓕𝑀×𝐿ℎ is the submatrix that contains the first 𝐿ℎ columns of 𝓕𝑀 . 𝜠𝑗𝑖𝑘 ∈ ℂ𝑀×𝑀 is a diagonal 𝑘

matrix which describes the CFOs of the kth user with entries, [𝜠𝑗𝑖𝑘 ]𝑚,𝑚 = 𝑒 𝑗2𝜋𝑚 𝜀𝑗𝑖 /𝑀 , 𝑚 = 0, … . , 𝑀 − 1. 𝜀𝑗𝑖𝑘 is the CFOs of the kth user normalized to the subcarriers spacing. Then, the M-point FFT, the subcarriers demapping and the joint MIMO equalization operations are performed successively on the received signal {𝓨𝑘𝑗 }

2

𝑗=1

as 𝑴𝑇 𝓕 𝓨 𝒀𝑘 [ 1𝑘 ] = 𝜦𝑘2𝑁𝑐 [ 𝑇𝑘 𝑀 1 ]. 𝑴𝑘 𝓕𝑀 𝓨2 𝒀2

(10)

The joint MIMO equalization process with carrier frequency offsets compensation is applied in the frequency domain, to reduce the impacts of the multipath channel and CFOs. In this paper, both zero forcing (ZF) and minimum mean square error (MMSE) equalizers are considered, the equalizer-weight matrix 𝚲𝑘2𝑁𝑐 can be expressed as

𝜦𝑘2𝑁𝑐 =

𝜦𝑘 [ 11 𝜦𝑘21

𝑘 𝜦12 ] 𝜦𝑘22

𝐻

−1

( 𝜳𝑘 𝜳𝑘 ) ={

𝐻

𝜳𝑘 ,

−1 1 𝐻 𝐻 ( 𝜳𝑘 𝜳𝑘 + ( ) 𝑰2𝑁𝐶 ) 𝜳𝑘 , 𝑆𝑁𝑅

𝑍𝐹,

(11𝑎)

𝑀𝑀𝑆𝐸,

(11𝑏)

where 𝜳𝑘 is the effective frequency-domain channel matrix of size 2𝑁𝑐 × 2𝑁𝑐 . The structure of 𝜳𝑘 can be presented as 𝝍𝑘 𝜳𝑘 = [ 11 𝝍𝑘21

𝑘 𝑘 𝑘 𝝍12 𝜰11 𝜞11 ] = [ 𝑘 𝑘 𝝍22 𝜰21 𝜞𝑘21

𝑘 𝑘 𝜰12 𝜞12 ], 𝑘 𝜰22 𝜞𝑘22

(12)

where 𝜞𝑗𝑖𝑘 is the frequency domain equivalent channel matrix between the 𝑖 th transmit antenna and the 𝑗th receive antenna for the 𝑘th user, which can be generated as 𝜞𝑗𝑖𝑘 = 𝑑𝑖𝑎𝑔( 𝓕𝑀×𝐿ℎ 𝒉𝑗𝑖𝑘 )𝑴𝑘 of size 𝑀 × 𝑁𝑐 . Also, the MIMO interference matrix 𝜰𝑗𝑖𝑘 of size 𝑁𝑐 × 𝑀 can be generated as 𝜰𝑗𝑖𝑘 = 𝑴𝑇𝑘 𝓕𝑀 𝑬𝑗𝑖𝑘 𝓕𝐻 𝑀. After the joint MIMO equalization process, the SFBC decoding operations are performed and it can be expressed as ̃1𝑘 = 𝑫 𝒀1𝑘 − 𝑬 𝒀1𝑘 ∗ = [𝑌1𝑘 , −𝑌2𝑘 ∗ , . . . , 𝑌𝑁𝑘 −1 , −𝑌𝑁𝑘 ∗ ], 𝒀 𝑐 𝑐

(13𝑎)

̃ 𝑘2 = 𝑭 𝒀𝑘2 ∗ + 𝑮 𝒀𝑘2 = [𝑌2𝑘 ∗ , 𝑌1𝑘 , … , 𝑌𝑁𝑘 ∗ , 𝑌𝑁𝑘 −1 ], 𝒀 𝑐 𝑐 ̃𝑘 = 𝒀

(13𝑏)

̃1𝑘 + 𝒀 ̃ 𝑘2 ) (𝒀 . 2

(13𝑐)

̃ 𝑘 as Then, 𝑁𝐶 -IFFT and de-spreading processes are applied on the resulting signal 𝒀 ̃𝑘 𝒚𝑘 = 𝑺 𝑘 𝓕 𝐻 𝑁𝐶 𝒀 .

(14) 1

where 𝒚𝑘 is an 𝑁 × 1 the received information symbols vector of the kth user and 𝑺𝑘 = 𝐿 (𝑪𝑘 )𝑇 .Lastly, 𝒚𝑘 is demodulated to get the kth user information symbol stream. To estimate the PAPR characteristics, it is important to calculate the Complementary Cumulative Distribution Function (CCDF). The CCDF of the PAPR is the probability that the PAPR of the transmitted signal as in (7) exceeds a given threshold. The PAPR can be expressed as follows

2

𝑃𝐴𝑃𝑅(𝑑𝐵) = 10 log10 (

̌𝑘𝑖 | ) max (|𝒙

1 𝑀−1 𝑘 2 ∑ ̌ 𝑀 𝑚=0|𝒙𝑖 |

).

(15)

3 Bit Error Rate (BER) Analysis The BER analysis both of the proposed SFBC SC-CFDMA and conventional SFBC SC-FDMA systems over Rayleigh fading channels is presented. ZF channel equalization is considered. Consider received signals as in (9), by applying both of 𝑀-point FFT and subcarrier de-mapping to the received signals, the result can be written as 𝒀𝑘 𝝍𝑘 [ 1𝑘 ] = [ 11 𝒀2 𝝍𝑘21

𝑘 𝑴𝑇𝑘 𝓕𝑀 𝒙1𝑘 𝑴𝑇𝑘 𝓕𝑀 𝜼1 𝝍12 ] [ ] + [ ]. 𝑴𝑇𝑘 𝓕𝑀 𝜼2 𝝍𝑘22 𝑴𝑇𝑘 𝓕𝑀 𝒙𝑘2

(16)

The complex Gaussian noise vectors are assumed to be 𝜼𝑖 ∼ ℂℕ (0; 𝜎𝜂2 𝑰), 𝑖 = 1,2. By applying ZF channel equalization with carrier frequency offsets compensation as in (11a) to the signal in (16) assuming perfect channel state information, the equalizer output can be written as 𝑘 ̂ 1𝑘 𝒀̈1𝑘 𝑿 𝜦11 [ 𝑘] = [ 𝑘] + [ 𝑘 ̂2 𝜦21 𝒀̈2 𝑿

𝑘 𝑴𝑇𝑘 𝓕𝑀 𝜼1 𝜦12 ][ ], 𝜦𝑘22 𝑴𝑇𝑘 𝓕𝑀 𝜼2

(17)

The operations of SFBC decoding, 𝑀-point IFFT and de-spreading are applied successively, to get the kth user received signal 𝒚𝑘 as in (14) which can be written as 𝒚𝑘 = 𝒙 𝑘 + 𝒏 𝑘 ,

(18)

𝑘 𝑘 ̃𝑘 ̃ , 𝒏𝑘 = 𝑺𝑘 𝓕𝐻 𝑁𝐶 𝑵 = 𝑺 𝒏

(19)

where

̃𝑘 = 𝑵

̃ 1𝑘 + 𝑵 ̃ 𝑘2 ) (𝑵 , 2

̃ 1𝑘 = 𝑫 𝑵 ̂ 1𝑘 − 𝑬 𝑵 ̂ 1𝑘 ∗ , 𝑵 [

𝑘 ̂ 1𝑘 𝑵 𝜦11 ] = [ ̂ 𝑘2 𝜦𝑘21 𝑵

̃ 𝑘2 = 𝑭 𝑵 ̂ 𝑘2 ∗ + 𝑮 𝑵 ̂ 𝑘2 , 𝑵 𝑘 𝑴𝑇𝑘 𝓕𝑀 𝜼1 𝜦12 ] [ ]. 𝜦𝑘22 𝑴𝑇𝑘 𝓕𝑀 𝜼2

(20) (21) (22)

In the following part the effect of the proposed SFBC SC-CFDMA receiver operations on the noise variance is discussed. The FFT, IFFT and subcarrier de-mapping operations do not affect the noise variance. This can be easily verified as follows, for a noise vector 𝜼 ∼ ℂℕ (0; 𝜎𝜂2 𝐈). Variance of 𝓕𝜼 which is the second central moment is given by 𝐸(𝓕𝜼𝜼𝐻 𝓕𝐻 ) = 𝓕𝐸(𝜼𝜼𝐻 )𝓕𝐻 = 𝓕𝐸(𝜼𝜼𝑯 )𝓕𝐻 = 𝓕𝜎𝜂2 𝐈𝓕𝐻 = 𝜎𝜂2 𝓕𝓕𝐻 = 𝜎𝜂2 𝐈.

(23)

where 𝐸[∙] is the expectation operator. While, averaging of two uncorrelated noise vectors 𝝂 ∼ ℂℕ(0; 𝜎𝜈2 𝑰 ) and 𝜼 ∼ ℂℕ (0; 𝜎𝜂2 𝑰) reduces the output variance to the half of the average of the two variances as follows (𝝂 + 𝜼) (𝝂 + 𝜼)𝐻 𝝂𝝂𝐻 + 𝜼𝜼𝐻 1 𝐸( ) = 𝐸( ) = (𝜎𝜈2 𝑰 + 𝜎𝜂2 𝑰), 2 2 4 4

𝐸(𝝂𝜼𝑯 ) = 𝟎.

(24)

Similarly, the variance of the summation of two uncorrelated noise vectors 𝝂 ∼ ℂℕ(0; 𝜎𝜈2 𝑰 ) and 𝜼 ∼ ℂℕ (0; 𝜎𝜂2 𝑰) is the summation of their variances as follows 𝐸((𝝂 + 𝜼)(𝝂 + 𝜼)𝑯 ) = 𝐸(𝝂𝝂𝑯 + 𝜼𝜼𝑯 ) = 𝜎𝜈2 𝑰 + 𝜎𝜂2 𝑰 = (𝜎𝜈2 + 𝜎𝜂2 )𝑰.

𝐸(𝝂𝜼𝑯 ) = 𝟎.

(25)

For the case of multiplying a diagonal matrix 𝚲 of size 𝑁𝑐 × 𝑁𝑐 by a noise vector 𝜼 ∼ ℂℕ (0; 𝜎𝜂2 𝑰). The resultant variance is given by 𝐸((𝜦𝜼)(𝜦𝜼)𝑯 ) = 𝐸(𝜦𝜼𝜼𝑯 𝜦𝑯 ) = 𝜦(𝜎𝜂2 𝑰)𝜦𝑯 = 𝜎𝜂2 𝜦𝜦𝑯 .

(26)

Thus, the average value of the variance in this case will be Nc

2 σ𝛬𝜂

𝜎𝜂2 = ∑|𝛬(𝑛, 𝑛)|2 = 𝛽𝜎𝜂2 . Nc

(27)

n=1

𝐻

As the covariance matrix of 𝒏𝑘 is given by 𝐸 (𝒏𝑘 𝒏𝑘 ) as follows,

𝑘

𝐸 (𝒏 𝒏

𝑘𝐻

𝑘

̃ ) = 𝐸(𝑺 𝒏

𝑘 (𝑺𝑘

̃ 𝒏

𝑘 )𝐻 )

𝑘

𝑘

̃ 𝒏 ̃ = 𝑺 𝐸 (𝒏

𝑘𝐻

)𝑺

𝑘𝐻

𝜎𝑛2̃ = 𝑰 𝐿 𝑁

(28)

Provided the above effects on the noise variance, it is clear that the complex Gaussian noise vectors 𝜼𝑖 ∼ ℂℕ (0; 𝜎𝜂2 𝑰), 𝑖 = 1,2 variances are affected by multiplication by diagonal matrices and additions ̂ 1𝑘 during the equalization process as in (22). Thus, by using the relations (25) and (27), the variances of 𝑵 ̂ 𝑘2 in equation (22) can be written as and 𝑵 𝜎𝑵̂2𝑘 = (β11 + 𝛽12 )𝜎𝜂2 ,

(29)

𝜎𝑵̂2𝑘 = (𝛽21 + 𝛽22 )𝜎𝜂2 ,

(30)

1

2

where Nc

1 2 𝛽𝑖𝑗 = ∑|𝛬𝑖𝑗 (𝑛, 𝑛)| , Nc n=1

i = 1,2,

j = 1,2.

(31)

̂ 1𝑘 and 𝑵 ̂ 𝑘2 are then averaged during the SFBC decoding as in (20), thus, using equation (24) the variance of 𝑵 ̃ 𝑘 can be written as 𝑵

𝜎𝑵̂2𝑘

=

𝜎𝑵̂2𝑘 + 𝜎𝑵̂2𝑘 1

2

4

=

𝜎𝜂2 ∑2i,j=1 βij 4

.

(32)

Finally, the IFFT is applied followed by the de-spreading process as is (19). Thus, final resultant noise variance is given by applying (28). Therefore, the noise variance of 𝒏𝑘 in (18) can be written as σ2𝒏

=

σ2η ∑2i,j=1 βij 4L

.

(33)

For conventional SFBC SC-FDMA systems, noise variances are affected by multiplying diagonal matrices 𝚲𝑖𝑗 , 𝑖, 𝑗 = 1,2, of size 𝑁 × 𝑁, additions during the equalization process and the averaging during the SFBC ⃛ 𝑘 can decoding. Thus, by rewriting the relations (31) and (33) the noise variance of the conventional system 𝒏 be expressed as 𝜎𝑛⃛2

=

𝜎𝜂2 ∑2i,j=1 𝛽̃𝑖𝑗 4

,

(34)

where N

𝛽̃𝑖𝑗 =

1 2 ∑|𝛬𝑖𝑗 (𝑛, 𝑛)| , N

i = 1,2,

j = 1,2.

(35)

n=1

̃𝑘 in (18) and a noise variance as in (33). The symbol error rate For a QPSK modulated data symbol vector 𝒙 for proposed system can be written as Eb 𝑃𝑒 = Q (√ 2 ) 𝜎𝑛

(36)

Similarly, by using a noise variance as in (34) the symbol error rate for the conventional system can be written as Eb ̃ Pe = Q (√ 2 ) σn⃛

(37)

where Es is the average symbol energy for QPSK case. 𝐸𝑏 =

Ẽ𝑘 𝒙

2

is the average bit energy and 𝜎𝑛2 is the noise 1

∞

v2

energy (variance). Q(⋅) is the Q-function which is defined as Q(x) = 2π ∫x e− 2 dv.

4 Computational Complexity The computational complexity of the proposed SFBC SC-CFDMA transceiver depends on the 𝑁𝑐 -FFT and 𝑀-IFFT operations at the transmitter side. While at the receiver side, it depends on the 𝑀-point FFT, FDE process, and 𝑁𝑐 -point IFFT operations. Keeping in mind that, the spreading and de-spreading processes do not contribute to the computational complexity. The 𝑁𝑐 −FFT operation has a complexity of 𝒪(𝑁𝑐 𝑙𝑜𝑔2 𝑁𝑐 ), with 𝑁𝑐 = 𝑁𝐿. The 𝑀-point IFFT operation has a complexity of 𝒪(𝑀 𝑙𝑜𝑔2 𝑀). The computational complexity considered is the number complex multiplications. For the FDE, it has a complexity of 𝒪(20𝑁𝑐 ). Generally, the equalization operation which contains matrix multiplication and matrix inversion for matrices of size 2𝑁𝑐 × 2𝑁𝑐 have a complexity of 𝒪((2𝑁𝑐 )3 ). However, due to the structure of 𝚿𝑘 as in (12) which consists of four diagonal matrices, the computation complexity of multiplying it by its conjugate transpose will be 𝒪(4𝑁𝑐 ) instead of 𝒪((2𝑁𝑐 )3 ). Also, the matrix inversion of 𝑯

a matrix 𝚿𝑘 𝚿𝑘 as in (11) which consists of four diagonal matrices as 𝚿𝑘 has a complexity of 𝒪(8𝑁𝑐 ) instead of 𝒪((2𝑁𝑐 )3 ) as the inversion process can be calculated as the inversion of 2𝑁𝑐 matrices of size 2 × 2, which has a complexity of 𝒪(𝑁𝑐 × 23 ). Thus, as the equalization process has one matrix inversion and three matrix multiplications then its overall complexity will be of 𝒪(20𝑁𝑐 ), as mentioned above. The proposed SFBC SC-CFDMA transmitter has a complexity of 𝒪((𝑙𝑜𝑔2 𝑁𝑐 + 2𝑄 𝑙𝑜𝑔2 𝑄𝑁𝑐 )𝑁𝑐 ) as there are one 𝑁𝑐 -point FFT operation in each branch (see Fig. 1) and one 𝑀-point IFFT, and 𝑄 = 𝑀/𝑁𝐿. While, the proposed SFBC SC-CFDMA receiver has a complexity of 𝒪((𝑙𝑜𝑔2 𝑁𝑐 + 20 + 2𝑄 𝑙𝑜𝑔2 𝑄𝑁𝑐 )𝑁𝑐 ) as there are one 𝑀-point FFT for each branch, one 𝑁𝑐 -point IFFT and FDE for both of the two branches, together. Table 3. Complexity of the proposed SFBC SC-CFDMA and the SFBC SC-FDMA transceivers. Technique

Complexity

proposed SFBC SC-CFDMA

𝒪((𝑙𝑜𝑔2 𝑁𝑐 + 2𝑄 𝑙𝑜𝑔2 𝑄𝑁𝑐 )𝑁𝑐 )

Transmitter SFBC SC FDMA proposed SFBC SC-CFDMA

𝒪((𝑙𝑜𝑔2 𝑁 + 2𝑄 𝑙𝑜𝑔2 𝑄𝑁)𝑁) 𝒪((𝑙𝑜𝑔2 𝑁𝑐 + 20 + 2𝑄 𝑙𝑜𝑔2 𝑄𝑁𝑐 )𝑁𝑐 )

Receiver SFBC SC-FDMA

𝒪((𝑙𝑜𝑔2 𝑁 + 20 + 2𝑄 𝑙𝑜𝑔2 𝑄𝑁)𝑁)

A comparison of complex multiplications complexity for the proposed SFBC SC-CFDMA transceiver and the conventional SFBC SC-FDMA transceiver is shown in Table 3. As a result, the complexities of both proposed and conventional transmitters systems are close, with no significant difference. While the complexity of the proposed receiver is partially increased as compared to the conventional receiver. However, the partial increase in the proposed receiver complexity in the base station can be tolerated considering the benefits of the proposed system. 5 Simulation and Results The performance of the proposed SFBC SC-CFDMA system is evaluated with and without CFOs over Rayleigh fading channels adopting vehicular A channel model [21]. Simulations and performance evaluation of the proposed SFBC SC-CFDMA system compared to SFBC SC-FDMA system has been investigated using MATLAB simulation program. 𝑘 The carrier frequency offsets were 𝜀11 = 𝑘 𝑘 𝑘 0.1, 𝜀12 = 0.11, 𝜀21 = 0.12, and 𝜀22 = 0.09

he IFFT size and input block size as 𝑀 = 512 and 𝑁 = 32

symbols, and number of users = 2, 4 , are chosen, for both the proposed SFBC SC-CFDMA and the conventional

. The simulation parameters executed in this simulation are listed in Table 4.

Table 4 The Simulation System Parameters Simulation Method

Monte Carlo

Modulation

QPSK – 16 QAM

Cyclic Prefix

20 Samples

Output Block Size (M)

512 Symbols

Input Block Size (N)

32 Symbols

Walsh-Hadamard code length

4

MIMO Technique

SFBC

Equalization Type

MMSE and ZF

Subcarriers Mapping Type

Localized & Interleaved

Channel Model

Vehicular A Outdoor Channels 

Slow Fading



Mobile Speed =120 Km/h.



Doppler Spread =223 Hz



Carrier Frequency =2 GHz.



Channel Length 𝐿ℎ = 7

A. PAPR performance The PAPR performance of the proposed SFBC SC-CFDMA system and the conventional system has been compared for QPSK modulation and pulse-shaping filters and evaluated with four times oversampling. The raised cosine (RC) and the squared root raised-cosine (RRC) pulse shaping have been used. The roll-off factor is 0.22. PAPR performance of the proposed SFBC SC-CFDMA system and the conventional system is illustrated in Fig. 2 and Fig. 3. In Fig. 2, the localized mapping scheme is considered. From Fig. 2, the simulation results show that the PAPR performance of the proposed SFBC SC-CFDMA system is better than the conventional system. When no pulse shaping was applied, the proposed SFBC SC-CFDMA system has a lower PAPR than that of the conventional system by about 3 dB. It is also clear that the proposed SFBC SC-CFDMA system achieves the best PAPR performance with pulse shaping. Moreover, it is also noted that the PAPR performance of the both systems with RC and RRC pulse shaping is identical. In Fig.3, the interleaved mapping scheme is chosen. Simulation results show that the PAPR performance of the proposed SFBC SC-CFDMA system leads to the same as that of the SFBC SC-FDMA system for the case of no pulse shaping and RC pulse shaping at a CCDF = 10−6 . While RRC was applied, the PAPR performance of the proposed system is better than the conventional system by about 0.7 dB at a CCDF = 10−6 . The PAPR performance of the proposed SFBC SC-CFDMA system for localized mapping (LM) provides minimum PAPR compared with the conventional system because the PAPR of the Walsh-Hadamard code has upper bounded by 2𝐿 (PAPR ≤ 2𝐿) [22, 23]. This does not apply if the interleaved mapping (IM) is used because the PAPR performance of the SFBC SC-FDMA conventional system is lower than the PAPR upper bound of the Walsh-Hadamard code. The PAPR performance for the transmitted signal after pulse shaping increases because the pulse-shaping filter has higher side lobes when the roll-off factor is close to zero. Fig. 2 and Fig. 3 show the effect of the pulse-shaping filter on the PAPR performance in the proposed and conventional systems. From these figures, it can be seen that the PAPR increases significantly for interleaved mapping (IM), whereas the PAPR of the localized mapping (LM) system is nearly independent of the pulse-shaping filter. It concludes that the interleaved mapping (IM) will have a trade-off between excess bandwidth and PAPR graph since excess bandwidth increases as the roll-off factor becomes larger [24]. The PAPR performance comparison of the proposed SFBC SC-CFDMA system and conventional SFBC SCFDMA without pulse shaping filter is illustrated in Fig.4 and Fig.5. The localized subcarrier mapping and 16QAM modulation schemes are considered and 𝑁 = 256 and 𝑀 = 1024 are selected for comparison. Without using any reduction techniques, the proposed SFBC SC-CFDMA system has 2 dB reduction over the conventional system at a PAPR = 10−4 . The selective Mapping (SLM) and companding techniques are used to improve PAPR performance for proposed SFBC SC-CFDMA system and conventional system.

Fig. 2 CCDF comparison of PAPR for proposed SFBC SC-CFDMA and conventional systems for localized mapping scheme.

Fig. 3 CCDF comparison of PAPR for proposed SFBC SC-CFDMA and conventional systems for interleaved mapping scheme.

In Fig.4, the selective Mapping (SLM) technique is considered and it is simulated using different number of phase factors U. The U8 and U16 are the number of phase sequences. From Fig.4, it is concluded that the PAPR performance of the proposed SFBC SC-CFDMA system with the SLM technique is the lowest compared to that of the conventional system with SLM and other techniques.

The A-law, µ-law and linear companding transform (LCT) [2] companding techniques are considered in Fig.5. From Fig.5, it is concluded that the PAPR performance of the proposed system with the A-law companding technique is the lowest compared to that of the other techniques. As well, using the µ-law technique, the PAPR performance of the proposed system is better than the conventional system. While using the LCT technique, the PAPR performance of the proposed system is the same as that of the conventional system.

Fig. 4 CCDF comparison of PAPR for proposed SFBC SC-CFDMA and conventional systems with SLM technique for localized mapping scheme and 16QAM.

Fig. 5 CCDF comparison of PAPR for proposed SFBC SC-CFDMA and conventional systems with A-law, µ-law and LCT companding techniques for localized mapping scheme and 16QAM.

Table 5 shows the PAPR performance comparison of the proposed SFBC SC-CFDMA system, conventional SFBC SC-FDMA with and without companding and SLM techniques, OSLM, SSLM, TDLSC schemes [25], Lin’s SFBC scheme [14], and Meng’s SFBC scheme [26]. From Table 5, it is clear that the proposed SFBC SC-CFDMA with companding technique achieves the best PAPR performance. Table 5 CCDF values at a PAPR = 10−4 for the proposed SFBC SC-CFDMA and conventional systems with companding and SLM techniques compared with others techniques without pulse shaping filter. 16QAM - Localized mapping (dB) 8.9

Proposed SFBC SC-FDMA U8

7.8

U16

7.5

Proposed with companding

LCT

4.7

scheme

µ-law

3.2

A-law

2.6

Proposed with SLM scheme

Conventional SFBC SC-FDMA

11

Conventional with SLM

U8

8.6

scheme

U16

8.2

Conventional with

LCT

4.7

companding scheme

µ-law

4.2

A-law

3.4

Meng’s SFBC scheme

9.5

Lin’s SFBC scheme

9.2

SSLM scheme

8.4

TDLSC scheme

8.2

OLSM scheme

7.9

B. BER performance The BER performance of the proposed SFBC SC-CFDMA system and the conventional system has been compared over Rayleigh fading channels adopting vehicular A channel model with and without CFOs. Figures 6 and 7 show the analytical BER performance compared with the simulation BER performance for proposed SFBC SC-CFDMA and conventional SFBC SC-FDMA systems over Rayleigh fading channels with and without CFOs for the case of QPSK modulation and ZF equalizer. The comparison shows that the simulation BER results approach to the analytical BER results for both the proposed SFBC SC-CFDMA and

the conventional systems. Moreover, this comparison shows that the proposed SFBC SC-CFDMA system provides a significant BER performance improvement over the conventional system for both the interleaved and the localized mapping schemes.

Fig.6 BER Performance comparison between analytical results and simulated results for proposed SFBC SC-CFDMA and conventional systems with localized mapping scheme with K=2 users.

Fig.7 BER Performance comparison between analytical results and simulated results for proposed SFBC SC-CFDMA and conventional systems with interleaved mapping scheme with K=2 users.

Fig. 8 and Fig. 9 compare the BER performance of proposed SFBC SC-CFDMA and conventional systems over Rayleigh fading channels, with CFOs and without CFOs. The QPSK modulation and MMSE equalization technique are considered. In Fig.8, localized mapping scheme is considered. From Fig.8, it could be seen that the proposed SFBC SC-CFDMA system achieves better BER performance than traditional system, without CFOs, by about 15 dB at a BER = 10−4. Where, the BER performance of SFBC SC-FDMA system with CFOs is degraded severely. Thus, a BER of 10−4 cannot be reached. In Fig.9, interleaved mapping scheme is considered. From Fig.9, it can be observed that the proposed SFBC SC-CFDMA system achieves better BER performance than traditional system, without CFOs, 8 dB at a BER = 10−4. Where, the BER performance of SFBC SC-FDMA system with CFOs is degraded severely.

Thus, a BER of 10−5 cannot be reached for K=2 and a BER of 10−4 cannot be reached for 𝐾 = 4. However, the proposed SFBC SC-CFDMA system offers better robustness against the effects of CFOs, and consequently enhances the BER performance in MIMO fading channels. As well, in Fig.8 and Fig.9, BER performance of the conventional system without CFOs for each value of 𝐾 = 2, 4 is the same. This is due to the fact that the different orthogonal subcarrier sets for the different users make it possible to avoid the MAI. While the BER performance of the conventional system with CFOs for 𝐾 = 4 is degraded severely compared to that of for 𝐾 = 2. This is due to the result that the CFOs increases MAI between different users. Another noticeable fact is that the BER performance of the conventional system with interleaved mapping (IM) is more sensitive to CFOs than with localized mapping (LM). As Fig.8 and Fig.9 show, for 𝐾 = 2, 4, the BER performance of the proposed SFBC SC-CFDMA system without and with CFOs is the same. This is due to the fact that the proposed system allocates each user a unique spreading code and the different orthogonal subcarrier sets which cancel the effects the MAI.

Fig.8 BER Performance comparison between proposed SFBC SC-CFDMA and conventional systems for localized mapping scheme with K=2 ,4 users.

Fig.9 BER Performance comparison between proposed SFBC SC-CFDMA and conventional systems for interleaved mapping scheme with K=2, 4 users.

Fig.10 BER Performance comparison between the proposed SFBC SC-CFDMA and SWBC SC-WDMA systems for interleaved mapping scheme with K=2 users.

Fig.10 compares the BER performance of the proposed SFBC SC-CFDMA and SWBC SC-WDMA [20] systems over Rayleigh fading channels, with CFOs and without CFOs. The interleaved mapping scheme and MMSE equalization technique are considered. From Fig.10, it is clear that the proposed SFBC SC-CFDMA system achieves the best BER performance compared with SWBC SC-WDMA system. The main reason of the performance improvement in our proposed SFBC SC-CFDMA system is the use of spreading code. As mentioned before, a unique spreading code is allocated for each user.

It worth pointing out that, increasing the number of subcarriers very slightly changes the PAPR and BER performance results, therefore it is omitted.

6 Conclusion In this paper, SFBC SC-CFDMA transceiver has been proposed. BER performance comparisons of the proposed SFBC SC-CFDMA and traditional system, with and without CFOs, over fading MIMO channels, were presented. Simulation results show that the proposed SFBC SC-CFDMA system has a superior improvement of the BER performance compared to the traditional system. Also, the proposed SFBC SCCFDMA system offers better robustness against the effects of carrier frequency offsets (CFOs) for the case of MMSE equalization. The simulation results also show that the proposed SFBC SC-CFDMA system with different techniques has an important improvement of the PAPR performance compared to the traditional SFBC SC-FDMA system. BER analysis of the proposed SFBC SC-CFDMA and the conventional systems was presented. Analytical BER performance and simulated BER performance were very close for both of the proposed and the conventional systems.

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A Space-Frequency Block Codes MIMO SingleCarrier Code-Frequency-Division Multiple Access System Abdullah, Alamri M.Sc. Ph.D. candidate at the Department of Electrical Engineering, Alexandria University, Alexandria 21544, Egypt. [email protected]

Ehab, Badran Ph.D. Professor of Communications and Signal Processing at the Department of Electronics and Communications Engineering, College of Engineering and Technology, Arab Academy for Science, Technology and Maritime Transport, Alexandria 21937, Egypt. [email protected] - Corresponding author

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐ The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: