Performance Evaluation of Spectral Efficiency Hybrid Precoding and Combining Algorithm for Millimeter Wave -MIMO Systems

Abstract

Multiple input multiple output (MIMO) system with Millimeter Wave spectrum is currently used in most wireless applications and all cellular system to provides high data rates. with using large antenna array which is possible by decrease the wavelength to achieve high beamforming gain and improve the spectral efficiency. in this paper, used low complexity with hybrid precoding at the transmitting side and combining at the receiver side with limited feedback system, by using the concept of orthogonal matching pursuit (OMP) in single and multi-user cases, and compared the results with analog only beasmstring. The results of simulation showed that when used Minimum Mean Square Error (MMSE) precoders performed better than other hybrid precoding approaches, in addition the MMSE hybrid precoding /combining technique offers higher spectral efficiency compared with analog only beamstring.

1 Introduction

One of the most important technologies in communication field is Millimeter wave, which in range 30-300 GHZ because it has short wavelength which allows a huge number of antennas to be installed and supported high speed links [1].

The fully digital precoding in mmwave system with large antennas arrays required high cost and power consumption [2], also the pure analog solutions have severe performance limitations. So, the best solution is hybrid analog / digital beamforming to overcome the limitations of pure digital or analog beamforming for single user and multi user scenario. In addition, the advantage of the hybrid is the digital precoding in transmitter and combiner at receiver to reduce the precision of residual multi stream in the analog [3].

Previous studies give a basic overview on hybrid precoding in [3] proposed beamforming solution for MIMO system based on kalman precoder by two steps firstly, calculate the RF precoding/combining matrix secondly, design the digital baseband precoder at the BS. While author in [4] presented hybrid precoding structure based on switching networks and author in [5] proposed a hybrid precoding and combining with MMSE and the rate fairness among users, [6] provide the algorithm solution based on orthogonal matching pursuit (OMP), while the author in [7] pointed to the low complexity for only multi user hybrid precoding structure using MMSE and author in [8] present the method to increase the spectral efficiency and analyse the numerical results for performance of spatially spare mmwave system.

In this paper, low complexity used with the hybrid precoding at transmitter and combining at the receiver terminal based on limited feedback since no practical availability of the channel knowledge. Based on the concept of orthogonal matching pursuit (OMP) and compared is with only analog beamstring and single user, the performance analysed in single path channel and Multi paths channel to improve the performance of system.

In addition, studied the MMSE precoder and compared with analog only beamforming and ZF.

This process can be applied in outdoor environment with Non line of sight (NLOS) scattering medium in order to send multiple data streams through the channel.

The organization of the paper comes as follows: Sect. 2 introduces the System Model. Section 3 explains the channel model. Section 4 describes simulation results. Section 5 affords a conclusion. Finally, Sect. 6 presented a appendix.

2 System Model

Considered the single-user mmwave MIMO system as describe in Figs. 1 and 2 in which the base station is connected with ${N}_{t}$ transmission antenna and ${R}_{F}$ chain communicates N_s data stream to N_r receiving antennas. in condition as $Ns$ ≤ $NRF$ ≤ $Nt$. in transmitter side, firstly data streams ${N}_{s}$ are transmitted to ${N}_{RF}{R}_{F}$ chains by an ${N}_{RF }\times {N}_{S}$ digital precoding matrix ${F}_{BB}$, then an ${N}_{t}$× ${N}_{RF}$ analog precoding matrix ${F}_{RF}$ is usual transmit these data streams to ${N}_{t}$ transmitting antennas [12].

Fig. 1

Block diagram of the system model [10]

Fig. 2

Transmitter array structure with fully-connected structure [9]

In a fully connected hybrid beamforming structure, each RF chain is connected with all antennas, and the transmitted signal on each of the $NRF$ digital transceivers go through $NtRF$ paths (mixer, power amplifier, phase shifter, etc.) and summed up before being connected with each antenna as shown in Fig. 2. The fully-connected structure can make full use of the degrees of freedom (DOF) of precoding provided by the RF chains.

Accordingly, can be expressed the Transmitted signal in [9] as:

$$x={F}_{RF}{F}_{BB}S$$

(1)

where; $S$ is symbol vector $({N}_{s}\times 1 )$

${F}_{RF}={N}_{TX}\times {N}_{RF}$ is analog precoding matrix.

${F}_{BB}={N}_{RF}\times {N}_{S}$ is digital precoding matrix.

The received Signal $y$ at the user is given by in [11]

$$y=\sqrt{\rho } H{ F}_{RF}{ F}_{BB} s+n$$

(2)

where; the average received power is ρ,

H is the channel matrix ϵ ${C}^{{N}_{R}\times {N}_{T}}$ and $n$ is the Additive White Gaussian Noise (AWGN) vector with zero mean and variance ${\delta }^{2}$ equal to 1. Therefore, can be expressed the spectral efficiency R as follows in [9]

$$R={{\text{log}}}_{2} (\left|{I}_{NS}+\frac{\rho }{{6}^{2}{N}_{s}}H {F}_{RF}{F}_{BB}{F}_{BB}^{H}{F}_{RF}^{H} {H}^{H} \right|)$$

(3)

3 Channel Model

Due to the high free—space path loss of the mmwave signal, limited spatial scattering usually occurred. based on saleh -vanzuela channel model, the channel matrix H can be expressed in [13] as follow:

$${\text{H}}=\upgamma {\sum }_{i,l}{a}_{il}{\Lambda }_{r}\left({f}_{il}^{r} ,{\theta }_{il}^{r}\right){\Lambda }_{t}\left({f}_{il}^{t} ,{\theta }_{il}^{t}\right){a}_{r}\left({f}_{il}^{r},{\theta }_{il}^{r}\right){a}_{t}{({f}_{il}^{t} ,{\theta }_{il}^{t})}^{H}$$

(4)

$$\upgamma =\sqrt{{N}_{t}{N}_{r} /{N}_{cl}{N}_{ray}}$$

(5)

where; $\upgamma$ is the normalization factor, $Ncl$ is the scattering cluster and $Nray$ is propagation paths in each cluster, ${a}_{il}$ discusses the complex gain of the $lth$ ray in the $ith$ scattering cluster, ${f}_{il}^{r}$ (${f}_{il}^{t}$) t stand for azimuth angles of arrival (AoAs) and ${\theta }_{il}^{r}$ (${\theta }_{il}^{t}$) elevation angles of departure (AoDs), $\Lambda$_t $\left({f}_{il}^{t} ,{\theta }_{il}^{t}\right)$ and $\Lambda$_r $\left({{\text{f}}}_{{\text{il}}}^{{\text{r}}} ,{\uptheta }_{{\text{il}}}^{{\text{r}}}\right)$ expresses the transmit antenna element gain at a specific AoD and $\Lambda$_r $\left({f}_{il}^{r} ,{\theta }_{il}^{r}\right)$ receive antenna element gain at a specific AoA.${a}_{t}\left({f}_{il}^{t} ,{\theta }_{il}^{t}\right)$ is the normalized vectors of transmit array response at an azimuth angle and ${a}_{r}\left({f}_{il}^{r},{\theta }_{il}^{r}\right)$ is the receive array response at an elevation angle, by using N- elements uniform linear array (ULA). then the array response vector can be expressed in [14] as (6), where $k=2 \pi / \lambda$ and λ is the wavelength of the signal, d the spacing between inter-elements.

$${a}_{ULA}\left(f, \theta \right)= \frac{1}{\sqrt{N}}{[1 , {e}^{JKd {\text{sin}}\left(f\right)},\dots \dots ,{e}^{JKd(N-1){\text{sin}}\left(f\right)}]}^{T}$$

(6)

3.1 The Formulation of a Problem for mm-wave MIMO Hybrid Precoding and Combining for Single Case

Taking in consideration a mm-wave MIMO system Model for hybrid precoding and before combining (for single user)

$${y}_{{N}_{MS} \times 1}=HX+n =H{F}_{BB}{F}_{RF}s+n =HFs+n$$

(7)

RF and BB combiners has been applied, the output signal in [6] become:

$${y}^{\sim }={W}_{BB}^{H}{W}_{RF}^{H}y= {W}_{BB}^{H}{W}_{RF}^{H}Hx+ {W}_{BB}^{H}{W}_{RF}^{H}n$$

(8)

$${y}^{\sim }= {W}_{BB}^{H}{W}_{RF}^{H}H{F}_{BB}{F}_{RF}s+ {W}_{BB}^{H}{W}_{RF}^{H}n=WHFs+Wn$$

(9)

where: combiner W:

$$W={W}_{BB}^{H}{W}_{RF}^{H}, {W}_{BB}^{H}={N}_{S} \times {N}_{RF}, {W}_{RF}^{H}={N}_{RF}\times {N}_{MS}$$

(10)

The main objective for using the optimal hybrid precoding is increase the spectral efficiency of mmWave MIMO systems, and can be formulating the problem as:

$$\left({F}_{BB}^{OUT},{F}_{RF}^{OUT}\right)={\text{argmax}}R$$

(11)

$$\left|{[{F}_{RF}]}_{i,j}\right|=\frac{1}{\sqrt{{N}_{t}}}$$

(12)

$${\Vert {F}_{BB}{F}_{RF}\Vert }_{F}^{2}={N}_{S}$$

(13)

This study depends on orthogonal matching pursuit (OMP) algorithm, where, the OMP is an algorithm to recover of a high-dimensional sparse signal based on a small number of noisy linear measurements. In this algorithm, the AOA/AOD space is divided into grids. Then create a dictionary corresponding to array response vectors of all the possible angles of arrivals/departures. The dictionary size depends on individual resolution where all the received array response vectors corresponding to a specific resolution may be placed [6].

In a mm-wave system, the multipath dominant components are very few, the problem is solved on path-by-path basis which is an efficient way of doing estimation for better accuracy.

4 4. Simulation Results

In this section, the results have been simulated by MATLAB to describe and analysis the mmwave MIMO system of MMSE hybrid precoding/combining, considering the MIMO system model in the last section, the BS is employing with ${N}_{t}$=16,64 transmit antennas and the ${N}_{r}$=16 receive antennas. The channels are single-path, and Multi-path, the azimuth AoAs/AoDs are assumed to be uniformly distributed in [0; 2π] and the elevation AoAs/AoDs are uniformly distributed in [−π/2, π/2], all parameter are shown in Table 1.

Table 1 Simulation parameter

In this section, studied the performance of MMSE hybrid analog/digital precodering algorithm and compared the results with Zero forcing precoder and analog only beamforming, at different paths.

From Fig. 3 the MMSE hybrid precoding scheme presented results nearly to zero forcing and achieves better spectral efficiency than analog only beamforming. Where at SNR = 10 (dB) the spectral efficiency for analog only beamforming equal to 5.6929 and spectral efficiency for Zero forcing is 10.07 for MMSE equal to 10.098.

Fig. 3

Spectral efficiency vs SNR for single path (L = 1)

While Fig. 4 shows that the results of MMSE hybrid precoding in multi paths case for L = 5, 10, 20, 30. It shows that the spectral efficiency for L = 5 and L = 10 is 7.729, and 6.7595 respectively, while at L = 20 and L = 30, the spectral efficiency is 5.9363, and 5. 5797respectively. It means that when using multi paths the rate of spectral efficiency is decreasing.

Fig. 4

Presented the spectral efficiency for MMSE hybrid precoding VS SNR with different number of paths L = 5, 10, 20, 30

In Fig. 4, Tables 2 and 3 the spectral efficiency decreasing when using multi path with 28% in L = 5, 17% in L = 10, 6% in L = 20. According to the previous results, in order to reduce the effect of paths number, the number of antennas shall be increased as shown in Fig. 6.

Table 2 Spectral efficiency values VS SNR for L = 5,10,20,30

Table 3 Percentage decreased in Spectral Efficiency for L = 30 over L = 5,10, 20

Figure 5 and Table 4 show that the spectral efficiency of MMSE at ${N}_{t}$ =50 is 60.1223 but when using ZF at the same ${N}_{t}$ the spectral efficiency is 41.55, but with analog only beamforming is 24.08. while at high ${N}_{t}$=400 the spectral efficiency is reach to 146.27 in MMSE case, and arrive to 71.711 at ZF case,and become 50.335 at analog ony beamforming.

Fig. 5

Spectral efficiency varying number of transmit antenna in multi-path scenario (L = 10)

Table 4 Spectral efficiency values VS different number of transmit antenna for MMSE, ZF, Analog only beamforming mmwave MIMO

According to the previous results, the MMSE precoding perform better spectral efficiency than the analog beamform and very close from ZF. However with large number of antennas, the MMSE give high significant than other hybrid precoding approaches. So, can be study the MMSE hybrid precoding/combining at the single path and multi paths and compared the results with single user and only analog precoding.

4.1 Single Path Channel L = 1

In this section, the results have been simulated using MATLAB to describe and analysis the efficient of wireless MIMO system over mmWave frequency. Also, comparison between the simulation results for the implemented MIMO Wireless system using hybrid precoding /combining with single user and analog only beamforming at the single path. In addition, the effect of RF chain was studied on the system efficiency with (16 × 16 MIMO System), (64 × 16 MIMO System).

All the above-mentioned trials were applied in order to obtain the most improvement performance for the system. Appendix (A) explain the Matlab code for existing algorithms.

Figure 6 and Table 5 show (16 × 16) MIMO System and single path channel and different RF. It presented the effect of RF quantization on spectral efficiency for single user, hybrid precoding and analog only beamsteering.

Fig. 6

(a, b) Spectral efficiency VS SNR for 16 × 16 with different RF chain

Table 5 Spectral efficiency for 16 × 16 at different RF chain

In hybrid precoding case, can be noticing that the spectral efficiency at SNR = 10 dB reaches to 7.167 when using 4 RF, but by using 6 RF at the same SNR the spectral efficiency equal to 5.6828. However, in analog only beamstring case, the spectral efficiency arrives to 2.934 and 1.9989 at 4RF, 6RF at the same SNR respectively. It means the performance of analog only beamsteering and hybrid precoding are decreasing with increase RF quantization bits.

Figure 7 and Table 6 present (64 × 16) MIMO system and single path with different RF. It can be explained that the spectral efficiency values when using different number antennas of BS and MS and also varying of RF. For hybrid precoding case with 16 × 16, at SNR = 10 dB the spectral efficiency equal to 7.167 and 5.6828 with 4 RF and 6 RF respectively at the same SNR. But, for hybrid precoding case with 64 × 16, at SNR = 10 dB the spectral efficiency equal to 10.0357 and 9.037 with 4 RF, and 6 RF respectively. That means when using large number of BS antennas, the spectral efficiency is improved.

Fig. 7

(a, b) Spectral efficiency VS SNR for 64 × 16 with different RF chain

Table 6 Spectral efficiency for 64 × 16 at different RF chain

4.2 Multi Path Channel L = 10

In this section, it can be studied the MMSE hybrid precoding /combinig at Multi path channel. Figure 8 shows the (16 × 16) MIMO system with multi path channel L = 10 with different RF. Figure 8 and Table 7 show that the spectral efficiency value at the SNR = 10 dB equal to 3.9669 and 2.84 with 4 RF and 6 RF respectively. It means that when using multi path the rate of spectral efficiency is decreasing, and using large number of RF quantization bits with large number of BS antennas to avoid the decreasing in performance.

Fig. 8

(a, b) Spectral efficiency VS SNR for 16 × 16 with different RF chain

Table 7 Spectral efficiency for 16 × 16, L = 10 with different RF chain

Figure 9 presents (64 × 16) MIMO System and Multi path L = 10 with different RF. The spectral efficiency for analog only beamsteering have been presented in Fig. 9 and Table 8 at SNR = 10 dB is equal to 2.952, and 2.1531 with 4RF and 6RF respectively. While, spectral efficiency for hybrid precoding at SNR = 10 with 4RF dB is equal to 6.4419 and with 6RF it is equal to 5.3236, so that difference between hybrid precoding and the analog only beamsteering decreases at a large number of BS antennas.

Fig. 9

(a, b) Spectral efficiency VS SNR for 64 × 16 with different RF chain

Table 8 Spectral efficiency for 64 × 16, L = 10 with different RF chain

5 Conclusion

This paper gives a comparison analysis and improvement for hybrid precoding at transmitter and combining in receiver mm-wave MIMO system with analog beam steering and digital precoding for single/ multi-use rand single/ multi paths channel based on OMP algorithm. In addition, studied the MMSE precoder and compared with analog only beamforming and ZF.

The MATLAB simulation results gives the MMSE hybrid precoding scheme presented results nearly to zero forcing and achieves better spectral efficiency than analog only beamforming. However, it with multi paths channel reduce the spectral efficiency values by 27,8%, but at used large numbers of antennas the spectral efficiency is improved by Approximately 59%. in addition, when study the relation between the RF chains and behavior of hybrid precoding it found that the spectral efficiency is decreasing with increase RF quantization bits by 20% for (16 × 16) and by 10% for (64 × 16), so, when using increase number of RF chains must be increase the number of antennas to avoid the decreasing of overall performance of systems.

Appendix

Appendix (A) explain the Matlab code for existing algorithms.

Input: system parameter (${N}_{tx }, {N}_{tr},Num of paths,NumRF$,$NumCluster$, $ITER , SNR \left(dB\right)$) Output: Spectral efficiency for (Single user – hybrid precoding – analog only beamsteering) Step 1: Generate user channels Step 2: generate MIMO channel MIMO_Channel = sqrt((Nt*Nr)/(NumRay*NumCluster))*ArrayResponse_RX*diag(Alpha)*ArrayResponse_TX'; Step 3: applied hybrid precoding, Beam steering, single user W_MMSE = ((1/sqrt(SNR(s)))*(F_BB'*F_RF'*H'*H*F_RF*F_BB + (Ns/SNR(s))*eye(Ns))\(F_BB'*F_RF'*H'))'; Step 4: calculate the spectral efficiency for every case Step 5: plot the figures.