Space-Time Trellis Coding with Equalization

As we are entering the 5G era, high demand is made of wireless communication. Consistent effort has been ongoing in multiple-input multiple-output (MIMO) systems, which provide correlation on temporal and spatial domain, to meet the high throughput demand. To handle the characteristic nature of wireless channel effectively and improve the system performance, this paper considers the combination of diversity and equalization. Space-Time trellis code is combined with single-carrier modulation using two-choice equalization techniques, namely: minimum mean squared error (MMSE) equalizer and orthogonal triangular (QR) detection. MMSE gives an optimal balance between noise enhancement and net inter-symbol interference (ISI) in the transmitted signal. Use of these equalizers provides the platform of investigating the bit error rate (BER) and the pairwise error probability (PEP) at the receiver, as well as the effect of cyclic prefix reduction on the receivers. It was found that the MMSE receiver outperforms the QR receiver in terms of BER, while in terms of PEP; the QR receiver outperforms the MMSE receiver. When a cyclic prefix reduction test was carried out on both receivers, it yields a significant reduction in BER of both receivers but has no significant effect on the overall performance.


I. INTRODUCTION
The choice of receiver architecture determines the outcome of signals received in multiple input multiple output (MIMO) system.MIMO uses multiple antennae at the base station and serves multiple terminals over the same resourced time-frequency.In wireless communication, signals are sent through space that constantly exhibit fading characteristics, causing ISI (inter-symbol interference) [1], thereby resulting in receiver architectures incorporating equalization techniques, which when properly configured help in reducing bit error rates, as well as obtaining a higher throughput and spectral efficiency.Equalization techniques grow in complexity as the presence of ISI increases [2,3].Equalization is encapsulated in the modulation techniques employed in MIMO system.The multicarrier modulation technique (e.g.orthogonal frequency-division multiplexing, OFDM) has readily found a place in its use in MIMO systems.OFDM based on MIMO (MIMO-OFDM), operating on the principle of orthogonality, is used to mitigate the channel's frequency selectivity.Transmission of signals follows in parallel scheme in sub-channels at Published on September 27, 2019.I. A. Adebanjo is with the Federal University of Technology, Akure, Nigeria (e-mail: iaadebanjo@futa.edu.ng).
M.O.Kolawole is with Jolade Strategic Environmental and engineering Consults, Melbourne, Australia (e-mail: m.kolawole@jolade.com.au)different frequencies.Several works have been carried out on the drawbacks of OFDM, -peak-to-average power ratio (PAPR), receiver's time and frequency offset, leading to increased cost, higher power consumption, and high error rate [4,6].These drawbacks are easily addressed by precoding, clipping and filtering techniques [5].Aside the facts that these drawbacks are addressed, single modulation has found to be appealing in the uplink transmission in MIMO systems (because of its single carrier modulation); it has lesser envelope variation [7].Single-carrier frequency domain equalization (SC-FDE) has comparable performance to OFDM, as well as similarity in complexity of operation: the transmit structure of SC-FDE makes it viable for use in uplink transmission in mobile communication [8], but the receiver structure is quite similar to OFDM since the core of the receiver is greatly dependent on the equalization technique.Several research studies have been carried out on equalization techniques in single and multi-carrier modes of transmission, but central to both modes is the effectiveness of the equalizer algorithm used.As the wireless channel constantly pose danger due to its inherent fading attribute, its strength varies with time, frequency and space, and as high demand is placed on efficient data delivery; several schemes of signal processing modulation are combined to provide an appreciable route of quality service delivery.
In this paper, space-time coding in trellis modulation is combined with single-carrier modulation.For the choice of equalizers, the MMSE equalizer and QR detection were used.MMSE forms the basic building blocks of most known linear equalizers and gives an optimal balance between noise enhancement and net ISI in the transmitted signal [9,10].QR detection is an equalization technique used in Vertical Bell Layered Space-Time (V-BLAST) architecture that factorizes the channel matrix into unitary and upper triangular matrices, making MIMO system to become a causal system [11].Comparison of the bit error rate (BER) performance was done for uplink transmission for both 3G and 4G architectures using the QR detection and MMSE in single-carrier modulation in space-coding.

II. SYSTEM CONCEPTUAL MODEL
Figure 1 shows the System model having NT-input NRoutput MIMO channel model given as y=Hx+η (1) where y is the received vector signal, H is the scattering complex Rayleigh flat fading MIMO channel matrix, x is transmitted signal vector, and η is the additive noise.Cyclic-Prefix (CP) is appended in single carrier transmission to the transmitted data to combat inter-symbol interference (ISI) and the serial conversion is sent through the wireless channel.Whilst the appended CP can also make the received symbol periodic it must not be too large, otherwise transmission efficiency is reduced [9].The circulant channel is decomposed into singular values [12], where  ∈    ×  and  ∈    ×  are unitary matrices of the left and right singular vectors of H, S is a diagonal matrix having non-negative singular values of H; the diagonal matrix transmits the transformed transmitted singular vector V.The decomposed matrix U reverses the transformation at the receive side.
The received vector after CP removal is where j is the number of receive antennas,    is the additive white Gaussian noise,    () is the transmitted signal from i number of transmit antenna,   () is the complex channel coefficient, with 1≤  ≤   and 1≤  ≤   .Equation (3) was received in circular convolution, otherwise written as, Applying FFT operation on the received vector, we have

A. Equalization-MMSE Receiver
Using the rule of orthogonality, since the number of transmit antenna equals the number of receive antenna, the MMSE equalizer coefficient (in matrix form) is given as [13]  = (  H +  −1 ) −1   (6) where   is the Hermitian matrix of the channel matrix, H is the channel matrix, I is the identity matrix and  is the transmission signal-to-noise ratio.
Then channel output is given by where   is the equalized channel matrix.Therefore, equation ( 7) becomes, ŷ = (  H +  −1 ) −1      +  (8) From equation ( 2), we have According to Hermitian theory, the circulant matrix H of the form (ℎ  ) = (ℎ −+1 ) has eigenvalues that are grouping of the coefficients of the channel, having zero mean [13].The SINR is usually the yardstick of evaluating the MMSE output.Following [14], the SINR denoted as  is Since the channel matrix is circulant, the SINR can be expressed in terms of the eigenvalues   as where L is the block length, and k = 1, …, v+1, v is the channel memory length.
The eigenvalues of H are given by [15] where   =  2  ⁄ , ℎ  is the channel coefficients.The eigenvalues of H contain the DFT of the first row and also the inverse DFT of the eigenvalues is the first row of the circulant H matrix [16].
If H is square and non-singular, then  + =  −1 , where  + is the Moore-Penrose pseudo inverse of the channel matrix, then, there will be an inverse of the channel matrix and since H is circulant, therefore the inverse is circulant, making H a simple matrix [15,17].The output of the MMSE detector is the input of the Viterbi decoder which decodes STTC.

B. Equalization-QR Detection
The channel matrix H given in Equation ( 2) is decomposed into where Q denotes unitary matrix that is satisfied with    =   = , where I is an identity matrix and R is the upper triangular matrix.The Schur algorithm [18] can be applied since H is assumed to be full-ranked because nT = nR; therefore, H can be expressed as Following the expression given in Equation ( 3), and by modification with the transform of Q, the received signal becomes, ŷ =      +    (15) According to [19], the Grammian matrix of H is a complex Hermitian Toeplitz matrix  =  * , where = { , } .  = 0,  , =    , and   is obtained by the cyclic convolution of the first column of H with itself.Obtaining deductions from Equations ( 3) and ( 12),

III. SIMULATION PARAMETERS
The simulation parameters employed in the conceptual frame of the research is provided in Table 1.

IV. RESULTS AND DISCUSSION
As observed in Figure 2, the BER performance obtained by the different equalizer output depends relatively on the SNR (signal-to-noise ratio) of interest.At SNR of 24 dB, the QR equalizer gives an approximate increase of 10 −6 over MMSE equalizer at 10 −4 .Also, Figure 3 gives the pairwise error probability (PEP) performance curve of MMSE and QR equalizer.Minimum PEP results in spacetime code when the Euclidean distance is maximized.The diversity order played a key role in obtaining the PEP.For the space-time trellis coding (STTC), PEP gap was maintained evenly.At 10 −4 the PEP obtained for STTC has a gain of 4 dB over STTC-MMSE and 2 dB over STTC-QR.Yet, at SNR of 24 dB, STTC-MMSE gives an approximate PEP of 10 −6 while STTC and STTC-QR gives approximate values of 10 −4 and 10 −5 , respectively.
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.
Figure 4 gives a graphical comparison of the Bit Error Rate performance of the 3G and 4G architecture.It can be observed that space-time trellis coding came be adapted to the 4G architecture.In the 3G architecture, the maximum BER is at SNR of 24dB, while the 4G, the maximum BER is at SNR of 28dB.For the 4G architecture, obtaining an approximate BER of 10 −6 at such a high SNR than 3G architecture is possible when coding, equalization is combined for a 4G network.The 3GPP LTE (long time evolution) system has two levels of CP length: 4.7μs and 16.6μs.An attempt is made to study the effect of CP reduction in the access network performance.A comparison was made in the receiver choice of a STTC-SCFDE system.The CP length was reduced from 8μs to 0.4μs and the results are observed in Figures 5 and 6.There was a significant reduction in BER of both MMSE and QR receivers but has no significant effect on the overall performance.It appears that for both equalizers, cyclic prefix of 20 samples (i.e. 4 μs) got an average response.Aside this, the reduction of the cyclic prefix from 40 samples to 2 samples gives no significant change.As observed, a conclusion can be inferred that space-time coding can effectively handle the effect of ISI in wireless communication, and consequently, a reduction in cyclic prefix does not affect the overall performance of the system.This paper has used space-time coding in trellis modulation combined with single-carrier modulation and two-choice equalizers-QR and MMSE-to obtain a higher throughput and spectral efficiency.MMSE gives an optimal balance between noise enhancement and net ISI in the transmitted signal.There are advantages and disadvantages in the use of these equalizers: MMSE receiver outperforms the QR receiver in terms of bit error rate (BER); while in terms of pairwise error probability, the QR receiver outperforms the MMSE receiver.When a cyclic prefix reduction test was conducted, there was a significant reduction in BER of both MMSE and QR receivers but has no significant effect on the overall performance.Results obtained in employing a 4G architecture depicts the viability of combining coding and equalization methods in combating inherent fading present in the wireless channel.

Fig. 2 .
Fig. 2. Bit Error Rate performance comparison of MMSE and QR Equalizer.

TABLE I :
SIMULATION PARAMETERS Minimum Mean Square Error (MMSE) and Orthogonal Triangular (QR) Detection