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Orthogonal Frequency Division Multiplexing (OFDM) is readily employed in wireless communication to combat the intersymbol interference (ISI) effect with limited success because as the capacity of MIMO systems increases, other destructive effects affect the propagation channels and/or overall system performance. As such, research interest has increased, on how to improve performance in the mediums where fading and ISI permeate, working on several combinatorial techniques to achieving improved effective throughput. In this study, we propose a combined model of the Space-Time Trellis Code (STTC) and Single-Carrier Frequency Domain Equalization (SC-FDE) to mitigate multiple-fading and interference effects. We present analytical performance results for the combined model over spatially correlated Rayleigh fading channels. We also show that it is beneficial to combine coding with equalization at the system’s receiving-end ensuring overall performance: a better performance over the traditional space-time trellis codes.

Wireless communication is constantly expanding in scope, complexity, and high demand of data usage thereby boosting its research profile. As demand rises, there is a prevailing need to address multipath fading and intersymbol interference (ISI) [

Tarokh et al. [_{t} number of transmit antennas, and in time, over t symbol periods yielding (n_{t} × t) codeword and coding rate that is a fraction of the number of sequence over the symbol period.

The performance of any wireless communication system is a measure of the percentage number of bits in error. The performance index of space-time-code is determined by the BER performance: a measure of the distance attributes of the code [

Considering a space-time coding system with n_{t} transmit antenna and n_{R} receive antenna over spatially correlated Rayleigh fading channel. The received sequence is denoted by

R = H X + N (1)

where X, H, and N denote the transmitted signal, channel matrix, and additive white noise, respectively, and corresponding dimensions n t × T where T is the symbol durations through n_{t} transmit antennas, n t × n t , and n r × 1 . Suppose the channel is known to the receiver, and a codeword c = c 1 1 c 1 2 ⋯ c 1 n c 2 1 c 2 2 ⋯ c 2 n ⋯ c l 1 c l 2 ⋯ c l n was transmitted and the receiver decides erroneously in favor of signal e = e 1 1 e 1 2 ⋯ e 1 n e 2 1 e 2 2 ⋯ e 2 n ⋯ e l 1 e l 2 ⋯ e l n , then a difference matrix can be obtained as [

B ( c , e ) = [ e 1 1 − c 1 1 e 2 1 − c 2 1 ⋯ e l 1 − c l 1 e 1 2 − c 1 2 e 2 2 − c 2 2 … e l 2 − c l 2 ⋮ ⋮ ⋱ ⋮ e 1 n − c 1 2 e 2 n − c 2 n ⋯ e l n − c l n ] (2)

In the Rayleigh fading channel, the Ricean factor equals to zero. Therefore, the average pairwise error probability (PEP, or simply P ( C → E ) ) between two arbitrary codewords C and E over independent and identically distributed Rayleigh fading channels is written as [

P ( C → E ) = 1 π ∫ 0 π / 2 ( det ( I T n r n t + η ϑ Δ ) ) − 1 d β (3)

= 1 π ∫ 0 π / 2 ∏ i = 1 r ( ϑ Δ ) ( 1 + η λ i ( ϑ Δ ) ) − 1 d β (4)

where

ϑ is the spatio-temporal correlation matrix;

r ( ϑ Δ ) is the rank of ϑ Δ ;

η is the effective signal-to-noise-ratio, SNR;

β is intergrated over the maximum at β = π 2 . Equation (4) is obtained from

the Gaussian Q-function.

Ensuring that the code is not rank-deficient, suppose the Rayleigh fading channel is stable per frame, Equation (4) can thus be reduced to [

P ( C → E ) = 1 π ∫ 0 π 2 ∏ i = 1 r ( C R ) ( 1 + η λ i ( C R ) ) − 1 d β (5)

when R = R r ⊗ R t . R r and R t are the receive and transmit correlation matrices, respectively. At high SNR, Equation (5) results in

P ( C → E ) ≅ 1 π ∫ 0 π 2 η − r ( C R ) ∏ i = 1 r ( C R ) λ i − 1 ( C R ) − 1 d β (6)

In space-time coding, for slow fades, the rank-determinant criterion is usually used [

P ( C → E ) ≅ ∏ i = 1 r ( C R ) ( 1 + p 4 λ i ( C R ) ) − n r (7)

However, for high SNR using the rank-determinant criterion, Equation (7) leads to

P ( C → E ) ≤ ( p 4 ) − n r r ( C R ) ∏ i = 1 r ( C R ) λ i − n r ( C R ) (8)

where λ i − n r ( C R ) is the n r th power of λ i ( C R ) .

Maximization of the rank of the error matrix C R gives the diversity gain.

Coding gain is obtained by maximizing the ∏ i = 1 r ( C R ) λ i − n r ( C R ) quantity.

Consider a single carrier block transmission after serial binary bits being mapped into parallel bits, and the cyclic prefix (CP) inserted into the blocks of bits, as shown in

Assuming s(t) was encoded into n_{t} streams of data x(t) expressed as

X ( t ) = [ X 0 X 1 ⋯ X t ⋯ ] = [ x 0 1 x 1 1 ⋯ x t 1 ⋯ x 0 2 x 1 2 ⋯ x t 2 ⋯ ⋮ ⋮ ⋱ ⋮ ⋯ x 0 N T x 1 N T ⋯ x t N T ⋯ ] (9)

and when appended with CP of length L becomes

[ x t − 1 1 x t 1 x 0 1 x 1 1 ⋯ x t − 1 2 x t 2 x 0 2 x 1 2 … ] (10)

At the receiver, the received signal is given by

r t j = ∑ i = 1 n h i j ( t ) x t i ( t ) + η t j (11)

where j ( = 1 , 2 , ⋯ ) is the number of receive antennas, η t j is the effective SNR additive white Gaussian noise at time t of antenna j; x t i ( t ) is the transmitted signal from i number of transmit antenna; h i j ( t ) is the complex channel coefficient. By considering the channel to be slowly fading, then Equation (11) can be written as

R = H C P [ X i ] + η (12)

where H C P is a block-wise circulant square matrix of size T × T in the form:

H C P = [ H ( 0 ) H ( L − 1 ) H ( L − 2 ) ⋯ H ( 1 ) H ( 1 ) H ( 0 ) H ( L − 1 ) ⋯ H ( 2 ) ⋮ ⋮ ⋮ ⋱ ⋮ H ( L − 1 ) H ( L − 2 ) ⋯ … H ( 0 ) ] (13)

Obtaining the singular value decomposition, H C P becomes F H Λ C P F , where Λ C P is a diagonal matrix whose elements are obtained by a block-wise FFT of [ H ( 0 ) H ( 1 ) ⋯ H ( L − 1 ) ] , F is the Discrete Fourier Transform (DFT) matrix.

Λ C P = ∑ l = 0 L − 1 H ( l ) e − j 2π T k l (14)

The eigenvectors of H C P are independent of channel matrices H(l) [

Y ( f ) = F [ R ( f ) ] = Λ C P [ X ( f ) + [ η ( f ) ] (15)

Performing equalization using the minimum mean square, which minimizes the mean square error between the estimated and received symbols, and assuming an MMSE equalizer coefficient, we write Equation (15) as

y = ρ W H C P X + η w (16)

where ρ is the signal-to-noise ratio, W is the equalizer coefficient, and η w , noise as a result of equalization. Performing inverse FFT on Equation (16) results

F − 1 ( y ) = F − 1 ρ W H C P F − 1 x + F − 1 η w (17)

which is consistent with [

The Bit Error Rate (BER) evaluation of the combined model over spatially correlated Rayleigh fading channels was done.

MIMO channel type | 3GPP ITU Pedestrian A |
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Fading distribution | Rayleigh |

FFT size | 512 |

Channel bandwidth | 5 MHz |

Cyclic prefix length | 40 |

Modulation scheme | QPSK |

Antenna configuration | 2 × 2 |

Channel coding | None |

Channel estimation and equalization | Minimum Mean Square Error (MMSE) |

It was observed from the results shown in

^{−4}, the PEP obtained for STTC-MMSE had a gain of 6 dB compared to the STTC.

The operation of having to decode the transmitted symbol in the time domain could explain the gain. The probability that the decoder would select an erroneous signal was low due to the fact that equalization and the FFT/IFFT operations were carried out before the STTC decoding. More studies are continuing on large antenna configuration.

This paper has examined the performance of wireless communication when space-time trellis code is combined with Equalization in single-carrier transmission. In combining the two techniques, it was observed that BER increases marginally for STTC-MMSE compared to the basic STTC. Pairwise Error Probability shows an improved performance of STTC-MMSE over the traditional STTC. The error analysis obtained shows the viability of implementing a combination of diversity with frequency equalization in wireless communication.

Adebanjo, I.A., Olasoji, Y.O. and Kolawole, M.O. (2017) Single Carrier Frequency Domain Equalization with Space-Time Trellis Codes. Communications and Network, 9, 164-171. https://doi.org/10.4236/cn.2017.93011