Open access peer-reviewed chapter

Coherent Receiver for Turbo Coded Single-User Massive MIMO-OFDM with Retransmissions

By K. Vasudevan, Shivani Singh and A. Phani Kumar Reddy

Submitted: November 10th 2018Reviewed: March 17th 2019Published: April 20th 2019

DOI: 10.5772/intechopen.85893

Downloaded: 434


Single-user massive multiple-input multiple-output (MIMO) systems have a large number of antennas at the transmitter and receiver. This results in a large overall throughput (bit-rate), of the order of tens of gigabits per second, which is the main objective of the recent fifth-generation (5G) wireless standard. It is feasible to have a large number of antennas in mm-wave frequencies, due to the small size of the antennas. This chapter deals with the coherent detection of orthogonal frequency division multiplexed (OFDM) signals transmitted through frequency-selective Rayleigh fading MIMO wireless channels. Low complexity, discrete-time algorithms are developed for channel estimation, carrier and timing synchronization, and finally turbo decoding of the data at the receiver. Computer simulation results are presented to validate the theory.


  • 5G
  • channel capacity
  • channel estimation
  • single-user massive MIMO
  • OFDM
  • spatial multiplexing
  • retransmissions
  • synchronization
  • turbo codes

1. Introduction

The main objective of the fifth-generation [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] wireless communication standard is to provide peak data rates of 10 gigabit per second (Gbps) for each user, ultra-low latency (the time duration between transmission of information and getting a response) of less than 1 ms, and, last but not the least, very low bit error rates (BER) (<1010). High data rates are essential for streaming ultrahigh definition (4k) video. Low latency is required for future driverless cars and remote surgeries. An important feature of the 5G network is that it involves not only people but also smart devices. For example, it may be possible to control a microwave oven or geyser located in the home, from the office. High data rates are feasible by using a large number of transmitting antennas. For example, if each transmit antenna transmits at a rate of 100 megabits per second (Mbps), then using 100 transmit antennas would result in an overall bit-rate of 10 Gbps. This technique of increasing the overall bit-rate by using a large number of transmit antennas is also known as spatial multiplexing (not to be confused with spatial modulation [16, 17, 18, 19, 20], wherein not all the transmit antennas are simultaneously active). This is illustrated in Figure 1, where the ithtransmit antenna sends Cibits of information and each of the receive antennas gets C/Nbits of information, in each transmission (see Proposition A.1 and A.2 in [21]). It must be noted that a large array of transmit antennas can also be used for beamforming [22, 23] and beam steering (the ability to focus the transmitted signal in a particular direction, without moving the antenna), which is not the topic of this chapter. In fact, the basic idea used in this chapter is captured in the following proposition.

Figure 1.

Illustration of spatial multiplexing for N × N MIMO.

Proposition 1.1 Signals transmitted and received by antennas separated by at least λ/2(λ=c/νwhere cis the velocity of light and νis the carrier frequency) undergo independent fading.

A typical massive MIMO antenna array is shown in Figure 2. The black dots denote the antennas, and the circles denote obstructions used to prevent mutual coupling between the antennas. While spatial multiplexing is a big advantage in massive MIMO, the main problem lies in the high complexity of data detection at the receiver. To understand this issue, consider the signal model:

Figure 2.

A massive MIMO antenna array.


where R˜CN×1is the received vector, H˜CN×Nis the channel matrix, SCN×1is the symbol vector drawn from an M-ary 2D constellation, and W˜CN×1is the additive white Gaussian noise (AWGN) vector. Here Cdenotes the set of complex numbers. Due to Proposition 1.1, the elements of H˜are statistically independent. Moreover, if there is no line-of-sight (LOS) path between the transmitter and receiver, the elements of H˜are zero-mean Gaussian. The elements of W˜are also assumed to be independent. The real and imaginary parts of the elements of H˜and W˜are also assumed to be independent. Now, the problem statement is find Sgiven R˜. There are several methods of solving this problem, assuming that H˜is known.

  1. Perform an exhaustive search over all the MNpossibilities of S. This is known as the maximum likelihood (ML) approach, which has got an exponential complexity.

  2. Pre-multiply R˜with H˜1. This is known as the zero-forcing approach and has a complexity of the order of 2N3(N3complexity for computing the inverse and another N3for matrix multiplication). This approach usually leads to noise enhancement and a poor symbol error rate (SER) performance.

  3. The third approach, known as sphere decoding, has polynomial complexity (C0×NC1, where C0>1and C1>3) and has been widely studied in the literature [24, 25, 26, 27, 28, 29, 30, 31, 32, 33].

Data detection in single-user massive MIMO systems using retransmissions, having a complexity of Nrt×N3, where Nrtis the number of retransmissions, has been proposed [34], where it was assumed that H˜is known at the receiver. In this work, which is an extension of [34], we present a coherent receiver for massive MIMO systems, where not only H˜but also the carrier frequency offset and timing are estimated. Moreover, the signal model in Eq. (1) is valid for flat fading channels. When the channel is frequency selective (the length of the discrete-time channel impulse response is greater than unity), orthogonal frequency division multiplexing needs to be used, since OFDM converts a frequency-selective channel into a flat fading channel (length of the discrete-time channel impulse response is equal to unity) [35]. To this end, the channel estimation and carrier and timing synchronization algorithms developed in [36] for single-input single-output (SISO) OFDM, [37, 38] for single-input multiple-output (SIMO) OFDM, and [21, 39] for multiple-input multiple-output (MIMO) OFDM are used in this work. In [40], a linear prediction-based detection of serially concatenated QPSK is presented, which does not require any preamble. The prospect of using superimposed training [41] in the context of massive MIMO looks quite intimidating, since the signal at each receive antenna is already a superposition of the signals from a large number of transmit antennas.

This work is organized as follows. Section 2 presents the system model. The discrete-time receiver algorithms are presented in Section 3. The computer simulation results are discussed in Section 4, and the chapter concludes with Section 5.

2. System model

The transmitted frame structure is shown in Figure 3(a). The signal in the blue boxes is sent from transmit antenna nt. The signal in the red boxes is sent from other antennas. Note that in the preamble phase, only one transmit antenna is active at a time, whereas in the data phase, all transmit antennas are active simultaneously. In practice, each transmit antenna could use a different preamble. However, in this work, we assume that all transmit antennas use the same preamble. The signals in Figure 3(a) are defined as follows (similar to [21]):

Figure 3.

(a) Frame structure for k th retransmission. (b) Signal from transmit antenna n t . (c) Receiver for the data phase.


The term iin the above equations denotes the ithsubcarrier, ndenotes the time index, and 1ntNis the index to the transmit antenna. Note that in this work, the same preamble is transmitted one after the other by each of the transmit antennas, as shown in Figure 3(a). In [21], different preambles are transmitted simultaneously from all the transmit antennas. The channel coefficients h˜k,n,nr,ntassociated with the receive antenna nr(1nrN) and transmit antenna nt(1ntN) for the kthretransmission are CN02σf2(CNdenotes a circularly symmetric Gaussian random variable) and satisfy the following relations [21]:


where “*” denotes complex conjugate and δKis the Kronecker delta function. Observe that Eq. (3) implies a uniform power delay profile. Even though an exponential power delay profile is more realistic, we have used a uniform power delay profile, since it is expected to give the worst-case BER performance, as all the multipath components have the same power [21]. The channel is assumed to be quasi-static, that is, h˜k,n,nr,ntis time-invariant over one frame (retransmission). The length of all the N2channel impulse responses is assumed to be Lh, which is proportional to the difference between the longest and shortest multipath [21]. The channel span assumed by the receiver is [21, 36, 39].


The length of the cyclic prefix or suffix is [21, 36, 39].


The length of the preamble is Lp, and the length of the data is Ld. The AWGN noise samples w˜k,n,nrfor the kthretransmission at time nand receive antenna nrare CN02σw2and satisfy


The noise and channel coefficients are assumed to be independent. The frequency offset ω0is uniformly distributed over 0.030.03radians, and the ML frequency offset estimator searches in the range ω0,maxω0,maxradians [42] where


For convenience, and without loss of generality, we assume that ω0is constant over Nrtretransmissions.

During the preamble phase, the signal at receive antenna nr, for the kthretransmission, can be written as (for 0nLp+Lcp+Lh2)


where “” denotes linear convolution, s˜5,nis depicted in Figure 3(a), h˜k,n,nr,ntdenotes the channel impulse response between transmit antenna ntand receive antenna nrfor the kthretransmission, and


The subscript “p” in Eqs. (8) and (9) denotes the preamble. Note that any random carrier phase can be absorbed in the channel impulse response. We have


where “Lp” denotes an Lp-point circular convolution, “Lp” denotes the Lp-point discrete Fourier transform (DFT) or the fast Fourier transform (FFT), and


Due to the presence of the cyclic suffix, we have


Assuming perfect carrier and timing synchronization (ω0is perfectly canceled and the frame boundaries are perfectly known) at the receiver, the signal at the output of the Lp-point FFT for the ith(0iLp1) subcarrier and receive antenna nr, due to the preamble sent from transmit antenna ntduring the kthretransmission, is


where H˜k,i,nr,ntand W˜k,i,nr,nt,pdenote the Lp-point FFT of h˜k,n,nr,ntand w˜k,n,nr,nt,p, respectively. The average SNR per bit corresponding to Eq. (13) is


where [36]


where Ais a constant to be determined and it is assumed that each sample of each receive antenna gets 2/NNrtbits of information during the preamble phase (see Proposition A.2 in [21]).

During the data phase, the signal for the kthretransmission at receive antenna nrcan be written as (for 0nLd+Lcp+Lh2)


where s˜6,n,ntis depicted in Figure 3(a) and


The subscript “d” in Eqs. (16) and (17) denotes data. Assuming perfect carrier and timing synchronization at the receiver, the signal at the output of the Ld-point FFT for the ith(0iLd1) subcarrier and receive antenna nr, during the kthretransmission, is


where H˜k,i,nr,ntand W˜k,i,nr,ddenote the Ld-point FFT of h˜k,n,nr,ntand w˜k,n,nr,d, respectively. The average SNR per bit corresponding to Eq. (18) is


where [36].


and it is assumed that each receive antenna gets 1/2Nrtbits of information in each transmission [34]. We impose the constraint that


Let us now compare the average power of the preamble with that of the data, at the transmitter. The average power of the preamble in the time domain is


where Ais defined in Eqs. (15) and (21). Similarly, the average power of the data in the time domain is


Therefore, the radio frequency (RF) amplifiers at the transmitter must have a dynamic range of at least (note that the RF amplifiers have to also deal with the peak-to-average power ratio (PAPR) problem [43, 44, 45, 46, 47, 48, 49])


Let us now consider the case where the preamble power is equal to the data power at each transmit antenna. From Eqs. (22) and (23), we have [36]


Substituting for Afrom Eq. (25), we obtain the average SNR per bit of the preamble phase and the data phase as


In other words, the average SNR per bit of the preamble phase would be less than that of the data phase by 6 dB. In what follows, we assume that Ais given by (21).

3. Receiver algorithms

The receiver algorithms have been adapted from [21, 36, 37, 39] and will be briefly described in the following subsections.

3.1. Start of frame and frequency offset estimation

The first task of the receiver is to detect the presence of a valid signal, that is, the start of frame (SoF). The SoF detection and coarse frequency offset estimation are performed for each receive antenna 1nrN, transmit antenna 1ntN, and retransmission 1kNrtas given by the following rule (similar to Eq. (17) in [21]: choose that value of m̂knrntand ν̂knrntwhich maximizes


where r˜k,m,nr,nt,pis given in Eq. (8) and


for 0lB1, where land B1[21] are positive integers and ω0,maxis given in Eq. (7). Observe that m̂knrntsatisfies Eqs. (18) and (19) in [21]. The average value of the frequency offset estimate is given by


3.2. Channel estimation

We assume that the SoF has been estimated using Eq. (27) with outcome m0given by (assuming the condition (19) in [21] is satisfied for all k, nr, and nt)


and the frequency offset has been perfectly canceled [36, 38]. Observe that any value of k, nr, and ntcan be used in the computation of Eq. (30). We have taken k=nr=nt=1. Define [21, 36, 39].


The steady-state, preamble part of the received signal for the kthretransmission and receive antenna nrcan be written as [21, 36, 39]




Observe that s˜5is independent of m1and due to the relations in Eqs. (10), (15), and (21), we have


where ILhris an Lhr×Lhridentity matrix. The estimate of the channel is [21, 36, 39]


To see the effect of noise on the channel estimate in Eq. (35), consider


It can be shown that [21, 39]


3.3. Noise variance estimation

The noise variance per dimension is estimated as


3.4. Post-FFT operations

In this section, we assume that the residual frequency offset given by


is such that


so that the effect of inter carrier interference (ICI) is negligible. Let


where m1is defined in Eq. (31). Note that m2is the starting point of the data phase. Define the FFT input in the time domain for the kthretransmission and receive antenna nras


where we have followed the notation in Eq. (16). The Ld-point FFT of Eq. (42) is


where R˜k,i,nr,dis given by Eq. (18). Construct a matrix:


for 0iLd1. Note that from Eq. (18)




which is similar to Eq. (1) in [34]. Let


where Ĥk,iis constructed from the Ld-point FFT of ĥk,nr,ntin Eq. (35) and Ŷk,iis similar to Y˜kin Eq. (6) of [34]. The analysis when


is given in [34]. Let


Note that Y˜iis an N×1matrix, whose ntthelement Y˜i,ntis a noisy version of S3,i,ntin Eq. (18). The matrix


constructed from the elements of Y˜iin Eq. (49) is fed to the turbo decoder. The forward (α) backward (β) recursions for decoder 1 of the turbo code is given by Eqs. (28) and (31) in [34]. The term γ1,i,m,nin Eq. (30) of [34] should be replaced by


where Y˜i,ntis an element of Y˜ntin Eq. (50) and ntis an odd integer. The term σU2in Eq. (51) is given by Eq. (22) in [34] which is repeated here for convenience:




where σ̂w2is given by Eq. (38) and Ĥk,i,nr,ntis obtained by taking the Ld-point FFT of (35). The term Fi,ntin Eq. (51) is given by




The extrinsic information from decoder 1 to decoder 2 is computed using Eqs. (32) and (33) of [34], with γ1,i,n,ρ+nreplaced by γ1,i,n,ρ+n,nt. The equations for decoder 2 are similar, except that γ2,i,m,nin Eq. (34) of [34] should be replaced by


where again ntis an odd integer.

3.5. Throughput and spectral efficiency

Recall from Figure 3(a) that during the preamble phase, only one transmit antenna is active at a time, whereas during the data phase, all the transmit antennas are simultaneously active. Thus the throughput can be defined as [36, 37].


The numerator of Eq. (57) denotes the total number of data bits transmitted, and the denominator represents the total number of QPSK symbol durations over Nrtretransmissions. The symbol rate during the preamble phase and data phase is the same. In the data phase, we are transmitting coded QPSK, that is, in each data bit duration, two coded QPSK symbols are sent simultaneously from two transmit antennas (see Figure 3(b)). Thus, during the data phase, each transmit antenna sends half a bit of information in each transmission. Therefore, the spectral efficiency is


The throughput for various simulation parameters is given in Table 1. Observe that when Lp=Ld/2, LcpLd, and N1, T1/Nrt. In this work, we have used a rate-1/2turbo code, that is, each data bit generates two coded QPSK symbols. The throughput can be doubled by using a rate-1 turbo code, obtained by puncturing.

Simulation parametersThroughput T
Lp = 512
Ld = 1024N = 432.38%
Lcp = 18
Nrt = 2
N = 838.77%
Lp = 4096
Ld = 8192N = 433.21%
Lcp = 18
Nrt = 2
N = 839.84%

Table 1.

Throughput for various simulation parameters.

4. Simulation results

The simulation parameters are given in Table 2. A “run” in Table 2 is defined as transmitting and receiving the frame in Figure 3(a) over Nrtretransmissions. The generating matrix of each of the constituent encoders of the turbo code is given by Eq. (49) in [21]. A question might arise: how does N=4,8correspond to a massive MIMO system, whereas in [34] Nwas as large as 512? The answer is in [34], an ideal massive MIMO was considered, wherein the channel, timing, and carrier frequency offset were assumed to be known, whereas in this work, the channel, timing, and carrier frequency offset are estimated. The estimation complexity and memory requirement increase as N2, for an N×NMIMO system. For example, the memory requirement of Eq. (27) when the number of frequency bins B1=1024[21], preamble length Lp=4096, cyclic prefix length Lcp=18, channel length Lh=10, N=8transmit and receive antennas, and Nrt=4retransmissions is

Lp512, 4096
Ld1024, 8192
N4, 8
Nrt1, 2, 4
B1[21]64, 1024
Runs104, 103

Table 2.

Simulation parameters.


double precision values. In fact Eq. (27) is implemented using multidimensional arrays in Scilab, instead of using for loops. Note that from Eq. (8), the length of the received signal during the preamble phase is Lp+Lcp+Lh1. If for loops are used, the memory requirement would be


double precision values, which is much less than Eq. (59); however the simulations would run much slower. Does this mean that we cannot go higher than an 8×8MIMO system? The answer is no. The solution lies in using multiple carrier frequencies as illustrated in Figure 4. Observe that with 8×8MIMO and Mcarrier frequencies, we get an overall 8M×8MMIMO system. The bit error rate results for a 4×4MIMO system are shown in Figure 5. The bit error rate results for an 8×8MIMO system are shown in Figure 6. The following observations can be made from Figure 5:

Figure 4.

Massive MIMO using multiple carrier frequencies.

Figure 5.

Bit error rate results for a 4 × 4 MIMO system. (a) L d = 1024 and L p = 512 . (b) L d = 8192 and L p = 4096 .

Figure 6.

Bit error rate results for an 8 × 8 MIMO system. (a) L d = 1024 and L p = 512 . (b) L d = 8192 and L p = 4096 .

  1. There is only 0.75 dB difference in performance between the ideal (id) and estimated (est) receiver, for Ld=1024, Nrt=2, and bit error rate equal to 104. On the other hand, there is hardly any performance difference between the ideal and estimated receiver for Ld=8192. This is because the noise variance (σw2) decreases with increasing Lp,Ld, for a given average SNR per bit, as shown by Eq. (21).

  2. There is only 0.5 dB improvement in performance for Ld=8192over Ld=1024, at a BER of 104.

  3. There is a significant improvement in performance between Nrt=2and Nrt=4, for both Ld=1024and Ld=8192. On the other hand, there is no significant difference in the BER for Nrt=2and Nrt=4in [34]. The reason is because in this work, the channel H˜k,i,nr,ntin Eq. (18) is highly correlated over the subcarrier index i, since it is obtained by taking an Ld-point FFT of an Lh-tap channel (see Eq. (37) of [36]). On the other hand, the channel H˜k,i,jin [34] is independent over all the indices k, i, and j, where kis the retransmission index, idenotes the receive antenna index, and jdenotes the transmit antenna index. See also the discussion leading to Eq. (82) in [21].

  4. There is not much BER performance difference between the 4×4and 8×8MIMO systems. A 4×4MIMO is computationally less complex than 8×8; however the 4×4requires twice the number of carrier frequencies to achieve the same spectral efficiency as 8×8, for the same number of retransmissions.

  5. The BER performance of the 8×8MIMO system with Nrt=4and Ld=8192could not be simulated due to the large amount of memory involved (see Eq. (59)) and Scilab limitations.

5. Conclusions

This work describes the discrete-time algorithms for the implementation of a massive MIMO system. Due to the implementation complexity considerations, more than one carrier frequency is required to obtain a truly single-user massive MIMO system. Each carrier frequency needs to be associated with an 8×8or 4×4MIMO subsystem. The average SNR per bit has been used as a performance measure, which has not been done earlier in the literature. Perhaps the channel can also be estimated using Eq. (27), instead of using Eq. (35). This needs investigation.


The authors would like to thank the high-performance computing (HPC) facility at IIT Kanpur for running the computer simulations.

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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K. Vasudevan, Shivani Singh and A. Phani Kumar Reddy (April 20th 2019). Coherent Receiver for Turbo Coded Single-User Massive MIMO-OFDM with Retransmissions, Multiplexing, Somayeh Mohammady, IntechOpen, DOI: 10.5772/intechopen.85893. Available from:

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