P.Poornima^{1}, G. Laxminarayana^{2} and D. Srinivas Rao^{3}^{ }

^{1}Associate Professor, Dept. of ECE, Sphoorthy Engineering College, Hyderabad, India

^{2}Professor, Dept. of ECE, Anurag College of Engineering, Hyderabad, India

^{3} Professor, Dept. of ECE, JNTUH CEH, Kukatpally, Hyderabad, India

*Abstract** *

*In this paper, we analyzed a numerical evaluation of the performance of MIMO radio systems in the LTE network environment. Downlink physical layer of the OFDM-MIMO based radio interface is considered for system model and a theoretical analysis of the bit error rate of the two space-time codes (SFBC 2×1 and FSTD 4×2 codes are adopted by the LTE norm as a function of the signal to noise ratio. Analytical expressions are given for transmission over a Rayleigh channel without spatial correlation which is then compared with Monte-Carlo simulations. Further evaluated channel capacity and simulation results show throughput almost reaches to the capacity limit*

*Keywords*

* **Channel Capacity, MIMO, OFDM, LTE Downlink, Spate time Block Coding and Throughput.*

**1.Introduction**

LTE is designed to meet carrier needs for high-speed data and media transport as well as high-capacity voice support well into the next decade. The LTE PHY employed some advanced technologies like Orthogonal Frequency Division Multiplexing (OFDM) and Multiple Input Multiple Output (MIMO) to provide downlink speed up to 100 Mbps [1-3]. The combination of OFDM and MIMO with spatial multiplexing (SM) scheme is viewed as a viable means to achieve high user information rates.

In cellular networks, the same radio spectrum is reused in different cells, in order to improve the network capacity [5,6]]. Several authors have considered the effect of increasing the density of base stations in the network, and improvements in the network capacity with increasing base station density have been revealed [7]. For a MIMO LTE system, a significant spectral efficiency improvement of 17% over SIMO LTE for static scenarios is achieved though slightly over 29% performance drop occurs when a mobility of 29 mph is introduced [9,10,11]. Hence it is established that in deploying diversity techniques, spectral efficiency is not the target rather diversity gain is. In [11, 12], a performance of 2x2STBC and SM are evaluated using typical OFDM based system Receiver Transmitter Channel Coding Modulation Mapper S/P Converter IFFT P/S Converter Channel Decoding Modulation Damper P/S Converter IFFT S/P Converter Wireless Channel with AWGN throughput and BER as metrics of measurement. In [13, 14, 15] the performance of MIMO in LTE is viewed from the capacity perspective.

Despite of these advantages due to the combination of OFDM and MIMO in the physical layer, there are many problems that are to be answered to achieve data rate as per the standards. The established distribution of resources allows to utilize system capacity for the evaluation of performance [2, 3]. This paper analyses channel capacity for the MIMO-OFDM system using with spatial multiplexing and

This paper is organized as In section II, channel capacity analysis of LTE system with spatial multiplexing and diversity techniques. Section III, A framework on channel capacity with space time coding scheme with SFBC and FSTD is presented. Throughput analysis for the various M-QM schemes under proposed space time coding is represented in Section IV .In Section V, we present numerical analysis and simulation results depicting the performance of the proposed system. Section VI, we conclude.

**2.Channel Capacity Analysis For Lte Systems**

The channel capacity of SISO systems is described by the information theory based on the mathematical frame work introduced by Shannon [5]. In fact, it is the theory of information that communication systems have evolved to their present form. MIMO systems are no exception to this rule since Telstar [6, 7] has extended the work of Shannon to the case of multiple antennas and on OFDM case. As an application of the theory, we will consider first the channel capacity for the spatial multiplexing MIMO scheme followed by the channel capacity of the different diversity schemes.

**A. Channel Model And Channel Capacity Of Spatial Multiplexing Scheme**

MIMO channel capacity for single carrier system and in Rayleigh channel considered in multi antenna systems. Where let us consider vector transmission model with *N _{t} *t transmits antennas and

In the above model, *ρ* is the average signal to noise ratio and *y* represents the received vector of size *N _{r}* ×1 and

Where Cs is the covariance matrix of transmitted signal vector *s* and *I* is the identity matrix with dimension *N _{r}*.

As is the covariance matrix of transmitted signal vector *s* and *I* is the identity matrix with dimension *N _{r}*.

For fading channel, the channel matrix *H* is a random matrix and hence the associated channel capacity is also a random variable. To deal with the random behavior of the channel of the channel, the average of the above equation over the distribution of *H* with the given name of erotic MIMO channel capacity can be defined as:

T the above derivation of the erotic MIMO channel capacity does not provide information to choose the covariance matrix of *s* (*C _{s}*) to get the maximum mutual information. To be able to compute the maximization, it should be clarified if the transmitter, the receiver, or both have perfect knowledge of the channel state information (CSI).If the channel matrix

With such covariance matrix, the erotic MIMO channel capacity becomes]:

**B.Channel Model And Channel Capacity Of Diversity Schemes**

The erotic channel capacity in (7) is valid for channel matrix where different signals are transmitted independently and hence cannot be applied directly to the space time block coding where the signals are transmitted form of spatial block codes. In fact, an STBC scheme with Nt transmit with *N _{t}* transmit antennas and

Where gig represents a linear combination of the signal constellation components and their from conjugates. The gig is the *i ^{th}* transmit antenna in the

Where, and is the *S* × 1 complex matrix after STBC decoding and *S* represents the number of transmitted symbols. And is the transmitted *S* ×1 complex vector with each entry having energy *E** _{s}*/

which means that the channel matrix is converted into a scalar channel and hence the ergodic capacity of the equivalent STBC channel in (9) is given by:

or equivalently, by inserting (10) into (11):**
**If the probability density function of the instantaneous SNR

**3.Channel Capacity Analysis For Space Time Codes:**** **

**A. Sfbc Channel Capacity Analysis**** **

Since OFDM system converts the multipath channel into *N* frequency flat-fading sub- channels, the average capacity over flat-fading sub-channel that is defined by *C** _{k}* is determined, then, the overall average channel capacity over

Where, *k* represents the subcarrier index and χ ϵ{SFBC, FSTD}. The average capacity achieved by a SFBC-OFDM scheme is calculated using the probability density function of the instantaneous SNR:

Substituting the PDF in the equation of average capacity, we obtain:

The integral in equation (18) can be evaluated using the following:

Using (20), the closed form of (18) becomes

**B. FSTD Channel Capacity Analysis**

**4.Throughput Performance With M-Qam Modulation Schemes**** **

In SISO OFDM systems, depends on the available bandwidth and the parameters associated with the OFDM signal, such as the number of subcarriers and the modulation order (QPSK, 16-QAM, 64-QAM). For a given frequency bandwidth (*B*) the maximal coded data throughput in bits per second can be approximated by the following simple equation:

where N_{FB} is the number of Frequency Blocks in the given frequency bandwidth (*B*), N_{SC} is the number of subcarriers in one Frequency Block, N_{OFDM }is the number of OFDM symbols in one sub frame, *N _{b}* is the number of bits in one subcarrier; ECR is the Effective Code Rate, and

Where, N_{SC} is the number of subcarriers in one Resource Block and is equal to 12, N_{OFDM }represents the number of OFDM symbols in one sub frame and for normal cyclic prefix, it is equal to 14. N_{FB} is the number of resource blocks (RB) that fit into the selected bandwidth (for example in 5 MHz bandwidth we found 25 Resource Blocks). The values of N_{FB} corresponding to the selected bandwidth (*B*) are shown in Table 1.

Finally, *T** _{sub}* is the duration of one sub frame and is equal to 1 ms. To take into account the losses due to the added cyclic prefix and reference symbols, a correction factor

Where *T** _{cp1}* is the duration of the first cyclic prefix of the first OFDM symbol in one slot and is equal to 5.2μs.

**Figure 1.** Structure of the symbols in one slot with Normal Cyclic Prefix.

Where, *T**cp1* is the duration of the first cyclic prefix of the first OFDM symbol in one slot and is equal to 5.2μs. *T**cp2* is the duration of the cyclic prefix of the next six OFDM symbols in one slot as shown in Figure 1 and is equal to 4.7μs. Finally, *N**slot* is the number of slots within the duration of one frame and is equal to 20. The results of calculation of the system capacity based on different bandwidth values are shown in Figure 3.

The numerical results in terms of bits/s/Hz are shown in Figure 2. In 4×2 FSTD MIMO analysis, the results of Monte- Carlo simulation of channel capacity of the two diversity schemes including SISO.

**Figure 2.** Capacity for SISO, SFBC-OFDM and FSTD-OFDM Schemes

The dashed line curve in Figure 2, represents the channel capacity of SISO scheme using our analysis. The diamond shaped points represents the SISO capacity obtained with Monte-Carlo simulation. The theoretical channel capacity of 2 × 1 STBC is shown by the red curve in Figure 2, whereas the black square line represent the capacity obtained by Monte-Carlo simulation. As expected, in the high SNR regions, only a small improvement of the channel capacity of the 2×1 SBC with respect to the SISO case is observed. Specifically, for SNR = 34 dB the improvement of 2×1 STBC is about 0.5 bits/s/Hz with respect to SISO. For low SNR regions, the channel capacity of the SISO case and the 2×1 STBC is almost the same. The theoretical channel capacity of the 2×2 STBC scheme is shown by the purple dashed line in Figure 2. The improvement with respect to the SISO case is more important. For the same SNR = 34 dB, the improvement is about 1.6 bits/s/Hz. It can be observed from Figure 2, that numerical evaluation results obtained from capacity analysis match closely to the capacity results obtained from Monte-Carlo simulations.

The results in Figure 3, represent the limit of channel capacity of a system with continuous valued inputs and continuous valued outputs of complex signals for different channel bandwidth (*B*) as defined in LTE (Table 1). These limits can be considered as an upper bound of real data throughput of a system with discrete valued multilevel/phase inputs. It can also be noticed from the Figure 3 how the Shannon capacity increases as the bandwidth increases.

**Table 1**. Number of Resource Blocks.

**5.Numerical Results**

Data throughput of SFBC-OFDM and FSTD-OFDM MIMO schemes for different M-QAM modulation schemes using the LTE simulator are depicted. The LTE simulator allows the calculation of coded and un-coded data throughput. For coded calculations, the transmitted signal is coded using the Turbo Code as defined in LTE. The overall effective code rates for different modulation schemes are defined in LTE and are given in Table 2. The Monte-Carlo simulation are presented in Figure 4,5 and 6 are compared to coded and un-coded data throughput for the SISO configuration. Figure 4, represents the results for QPSK modulation. The results of 16-QAM are presented in Figure 5 while those relative to 64-QAM modulation in Figure 6.

** Table 2**. ECR and Modulation Order for CQI.

For QPSK modulation, the un-coded data throughput of SISO configuration is shown by the ball blue curve. It can be observed that as the SNR increases the data throughput increases and it reaches its maximum at an SNR of almost 48 dB. The coded data throughput of the same modulation (QPSK) and represented by the blue curve in Figure 4 which reaches its maximum with only 10 dB. As expected from the coding process, the coded data throughput is less than the un-coded data throughput. In fact, the coded and un-coded data throughputs are related by the effective code rate used for channel coding. As shown in Table 2, the ECR for QPSK (CQI = 6) is equal to 0.5879, which represents the ratio of the useful transmitted data rate to the overall coded data after channel coding. An ECR of 0.5879 means that only 0.5879 of the transmitted data are useful data. The remaining data are used only for error protection and detection. This principle is well represented in the results of Figure 4, where the coded data throughput is found to be 4.9 Mbps whereas the un-coded data throughput is 8.3 Mbps, which represents, effectively and as expected, a code rate of 0.5879. The benefits of using the channel coding (Turbo coding in LTE)is clearly shown in Figure 4, where the data throughput of coded scheme is much closer to the channel capacity limit.

**Figure 4:** Data Throughput for QPSK Modulation

The coded and un-coded data throughput of the 2×1 SFBC are represented by the green and purple curves respectively. In case of un-coded scheme, the improvement with respect to the SISO case is more significant and substantial as shown in Figure 4. However, for coded case, there is no big difference between SISO and 2×1 scheme and the improvement is less significant. One last interesting observation from the results in Figure 4 is the fact that at high SNRs the SISO data throughput is greater than the data throughput for 2×1 and 4×2 diversity schemes. This result is explained by the fact that more overhead data is transmitted in case of diversity schemes as shown in Table 3. Specifically, for SISO scheme the symbol rate is 4000 per sub frame (1 ms) whereas with 2×1 SFBC case, the symbol rate is 3800 per sub frame. The symbol rate of 4×2 FSTD is the smallest one and is found to be 3600 per sub frame and that is why the data throughput of 4 × 2 FSTD is the smallest one at the high SNR region as shown in Figure 4. This observation is valid for both coded and un-coded data throughput. The red and citrine curves represent the coded and un-coded data throughput respectively for the 4×2 FSTD diversity scheme. Same observations as for 2×1 SFBC scheme can be made here. The theoretical channel capacity limits for SISO, 2×1 SFBC and 4×2 FSTD in a Rayleigh channel are also shown in Figure 4 for illustration and to show the effect of channel coding. It is clearly perceptible that the data throughput comes closer to the capacity limit with using channel coding.

**Table 3.4** SISO Generated Data

For 16-QAM modulation (Figure 5), which corresponds to a CQI of 9, the code rate is 0.6016 (Table 2) and hence the un-coded data throughput is higher than the coded data throughput. For SISO scheme, the maximum un-coded and coded data throughput, represented by the ball blue and blue curves in Figure 5, are found to be 15.8 Mbps and 9.5 Mbps respectively, which corresponds as expected to a 0.6 (9.5/15.8) code rate. The maximum data throughput is achieved is achieved with an SNR of 60 scheme whereas only 20 dB is needed to achieve the maximum of data throughput for the coding scheme. It can be observed from Figure 5, that the best. It can be observed from Figure 5, that the best performance in terms of data throughput is achieved with the 4 × 2 FSTD scheme where the maximum data throughput is reached at SNR = 10 dB. To attain the maximum in 2 × 1 SFBC scheme an SNR of 16 dB is required. For SISO scheme an SNR of 20 dB is needed to achieve the maximum data throughput. These results demonstrate the benefits of the diversity schemes. The code rate for CQI 15 (64-QAM Modulation) is 0.9258 which is close to 1. The maximum un-coded and coded data throughputs are found to be 23.5 Mbps and 22 Mbps respectively, which correspond to a code rate of 0.9258.

**Figure 5:** Data Throughput for 16QAM Modulation

The effectiveness of the FSTD scheme is clearly observed in Figure 6 where the maximum data throughput is achieved with 23 dB compared to 32 dB with 2×1 SFBC and 41 dB with SISO. The theoretical channel capacity limit is also shown in the Figure. To compare the coded data throughput with the theoretical limit of channel capacity, the coded data throughput of all modulation schemes and diversity schemes are made in Figure 7, where the coded data throughput for QPSK, 16-QAM, and 64-QAM and for SFBC and FSTD MIMO schemes are plotted.

**Figure 6:** Data Throughput for 64QAM Modulation

**Figure 7:** Coded Data Throughput for M-QAM Modulation

**6.Conclusion**

We presented an analysis and evaluation of the performance of MIMO-OFDM schemes in 4G LTE systems. More specifically, we analyzed performance for two OSTBC diversity MIMO schemes; namely the 2 × 1 SFBC-OFDM and the 4 × 2 FSTD-OFDM in the 3GPP 5 MHz Long Term Evolution (LTE) system over a Rayleigh flat fading channel. To show the BER performance improvement of the MIMO-OFDM schemes, the performance is compared to the SISO configuration. To ascertain the accuracy of the analysis, we also performed Monte- Carlo simulation using an LTE Link Level Simulator. The numerical results and Monte-Carlo simulation results. To complete the analysis, a study of the channel capacity for 2×1 SFBC-OFDM and 4×2 FSTD-OFDM MIMO schemes in Rayleigh flat fading channel has been performed. We use the probability density function of the instantaneous signal to noise ratio to derive analytical expressions for the capacity in terms of exponential integral and Poisson distribution. Effective data throughput as measured through simulation compare well with the system channel capacity limit.

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