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Variable pilot assisted channel estimation in MIMO-OFDM system with STBC and different modulation modes

Abstract

High speed and low latency have become the development trend of mobile communication network. Orthogonal frequency division multiplexing (OFDM) technology has become the key technology of mobile communication network due to its efficient transmission method. Space time block coding (STBC) introduces time and space correlation in the signals sent by different antennas, so that diversity gains and coding gains are obtained for the multiple input multiple output-OFDM (MIMO-OFDM) receiver without sacrificing bandwidth. STBC is a signal coding technology that combines higher transmission rates and it is a combination of space and time transmission signals. STBC uses the spatial diversity of multiple transmitters and multiple receivers in space to improve the capacity and information rate of the OFDM system. It proceeds parallel multiplex transmission in the transmission channel, which effectively improves spectrum utilization. For obtaining diversity gains and spectrum efficiency, this paper introduces a variable pilot assisted channel estimation method with different modulation modes. In this paper, the new type pilot sequences with different modulation orders are proposed for improving spectrum efficiency. The proposed variable pilot assisted channel estimation method in MIMO-OFDM systems is adopted with different low and high order modulations. Simulation results show that the proposed variable pilot assisted channel estimation method has 1–2 dB signal-to-noise ratio (SNR) gains for bit error rate (BER) performance of MIMO-OFDM systems.

Introduction

Orthogonal frequency division multiplexing (OFDM) system is characterized by its robustness to the multipath induced inter-symbol interference (ISI) [1, 2]. Foschini et al. [3, 4] proved that antenna diversity can be exploited to significantly enhance channel capacity by using multiple antennas at both transmitter and receiver. Recently, space time block coding (STBC) and space time block decoding (STBD) have been proposed as an efficient way to achieve this capacity improvement for the multiple input multiple output-OFDM (MIMO-OFDM) system, which is recognized as the promising technology for 5G and 6G mobile communications.

In MIMO-OFDM systems, like STBC-based transmit diversity system, channel state information (CSI) [5] is required for coherent detection and decoding. CSI can be obtained in two categories. The one does not require pilots, and the other is based on pilots which are known beforehand at the receiver. In STBC MIMO-OFDM systems, symbols are transmitted from different antennas simultaneously, and the received signal is the superposition of these signals. The designed pilot sequence is orthogonal among transmitting antennas [6]. Optimal pilot sequences were proposed in [7, 8] for MIMO-OFDM systems. Chern et al. [9] proposed a new pseudo random cyclic postfix-OFDM system, but the disadvantage is that the cyclic postfix sacrifices the frequency bandwidth. Del Peral-Rosado et al. [10] proposed an optimal placement of the pilot subcarriers about the mean square error (MSE) of the least square (LS) channel estimation for single input single output-OFDM systems. Younas et al. [11] proposed the MIMO-OFDM system based on adaptive LS channel estimation method with pilot aided sequences in the correlated time varying channels. It simplifies the LS channel estimation by utilizing the proposed pilot construction, which avoids matrix inversion. Linear minimum mean square error and singular value decomposition channel estimation methods are widely utilized in multi-user MIMO-OFDM systems [12, 13], but they inevitably bring many matrix calculations in frequency domain.

For virtual multiple input single output-OFDM systems, Wang et al. [14] proposed an efficient resource allocation method to provide high transmission throughput. The base station allocates subcarriers to transmit node with the highest magnitude of channel frequency response (CFR) on the subcarriers. In time varying channels, Youssefi et al. [15] proposed a new approach to achieve optimal training sequences (OTS) in terms of minimizing the MSE for spectrally efficient MIMO-OFDM systems. The OTS are equal powered and spaced, and orthogonal positioned. However, in practical OFDM based systems with low pass filter, to avoid the transmitted signals being distorted, the subcarriers that fall in the roll-off region of the filter are usually not used, which are often referred to as virtual subcarriers [16, 17]. The existence of virtual subcarriers breaks the equal spaced property of conventional pilot sequences and the system performance can be increased. Kenarsari-Anhari et al. [18] formulated the power allocation as linear programming problem in coded OFDM systems. Wang et al. [19] proposed a low complexity power allocation method in the cooperative OFDM networks which greatly improves the system throughput. The equivalent channel power gain is calculated and the power among each subcarrier is allocated through Lagrange optimization method. In MIMO-OFDM systems, the frequency resources are very valuable and sparse. To improve the reliability of communication and spectrum efficiency, the paper proposes a variable pilot sequence assisted channel estimation method in MIMO-OFDM system, which suppresses the ISI and inter-carrier interference (ICI) in time and spatial distribution effectively. Our main contributions are as follows: (1) We propose a method that can change the pilot insertion interval according to the modulation order of the signal, which can flexibly utilize spectrum resources and flexibly select the pilot amplitude. (2) We combine STBC and variable pilot method, which can be applied to different modulation schemes, such as binary phase shift keying (BPSK), quadrature phase shift keying (QPSK), 8-phase shift keying (8PSK), 16-quadrature amplitude modulation (16QAM), and 64-quadrature amplitude modulation (64QAM). (3) Simulation results are provided to demonstrate the effectiveness and superiorities of the proposed method and it show that the proposed variable pilot assisted channel estimation method has 1–2 dB signal-to-noise ratio (SNR) gains for bit error rate (BER) performance of MIMO-OFDM systems.

The remainder of this paper is organized as follows. The MIMO-OFDM system model is presented in Section 2. Section 3 introduces variable pilot assisted channel estimation. Section 4 illustrates simulation results and comparison analysis. Section 5 concludes the paper.

MIMO-OFDM system model

Figure 1 illustrates the 2 × 2 MIMO-OFDM system model, at the transmitter, the input bits are grouped and mapped according to a pre-specified constellation modulation scheme, which likes BPSK, QPSK, 8PSK, 16QAM, and 64QAM [20] that the modulation order ε is 1, 2, 3, 4, and 6, respectively. The higher order modulation mode is, the more bits an OFDM symbol carries and the more information the MIMO-OFDM system transmits. That is, higher order modulation improves the transmission efficiency of MIMO-OFDM. In this paper, the modulation order of 1, 2, 3, 4, 6 is considered in the MIMO-OFDM system. However, high order modulation causes a problem that phase interference occurs during symbols demodulation among different constellation points, resulting in demodulation errors among some signals. However, the combination of STBC and OFDM technology overcomes the disadvantage effectively.

Fig. 1.
figure 1

2 × 2 MIMO-OFDM system model

System model

Figure 2 illustrates pilot allocation in time, frequency, and space in STBC MIMO-OFDM system. Due to the superimposed noise on the signals and pilots in the time, frequency, space domains, and the mutual interference among the transmitting and receiving antenna pairs, there induces inevitable mistakes among the transmitting and receiving signals. These mistakes are reduced by different modulation methods and pilot intervals. After constellation modulation, the modulated symbols are sent to space-time encoder. After STBC in the transmitter, the comb type pilots are inserted with fixed intervals of subcarriers. As shown in Fig. 1, the inverse fast Fourier transform (IFFT) converts \({\varvec{S}}_{1}\) and \({\varvec{S}}_{2}\) from frequency domain to time domain, denoted as \({\varvec{s}}_{1}\) and \({\varvec{s}}_{2}\), respectively. The time domain signals can be expressed as:

$${\varvec{s}}_{\alpha } (n) = \frac{1}{\sqrt N }\sum\limits_{k = 0}^{N - 1} {{\varvec{S}}_{\alpha } {(}k{\text{)e}}^{{\frac{\text{j}2\uppi kn}{N}}} } ,\quad {0} \le k \le N - 1,$$
(1)
Fig. 2
figure 2

Pilot allocation in time, frequency, and space in STBC and STBD of MIMO-OFDM system

where \(\alpha = 1,2\) is the index of transmitting antennas, \(N\) is length of IFFT, \(k\) is the index of subcarriers.

At receiver, the received frequency domain signals can be expressed as:

$${\varvec{Y}}_{\beta } (k) = \sum\limits_{\alpha = 1}^{2} {{\varvec{H}}_{\alpha ,\beta } {(}k{)}{\varvec{S}}_{\alpha } {(}k{)}} + {\varvec{W}}{(}k{)},\quad{ 0} \le k \le N - 1,$$
(2)

where \(\beta = 1,2\) is the index of receiving antennas, \({\varvec{W}}{(}k{)}\) is the additive white Gaussian noise (AWGN) of the wireless channels. After removing the cyclic prefix (CP), the time domain received OFDM signals are transformed into frequency domain by fast Fourier transform (FFT), which can be expressed as:

$${\varvec{Y}}_{\beta } (k) = \frac{1}{\sqrt N }\sum\limits_{n = 0}^{N - 1} {{\varvec{y}}_{\beta } {(}n{\text{)e}}^{{\frac{ - \text{j}2\uppi kn}{N}}} } ,\quad {0} \le n \le N - 1.$$
(3)

In the receiver of MIMO-OFDM system, the frequency domain symbols are decoded in STBD decoder. The signals at multiple receiving antennas are decoded using CSI provided by the channel equalizer. Therefore, channel equalizer is a critical module of MIMO-OFDM systems. At last, the binary information bits are obtained after the BPSK, QPSK, 8PSK, 16QAM, and 64QAM constellation demodulation.

STBC and STBD

According to Alamouti criterion, the space-time coded signals are allocated to two antennas. After STBC [2], the transmitted OFDM codeword is expressed as:

$${\varvec{S}} = \left[ {\begin{array}{*{20}c} {{\varvec{S}}(i)} & { - {\varvec{S}}(i + 1)^{*} } \\ {{\varvec{S}}(i + 1)} & {{\varvec{S}}(i)^{*} } \\ \end{array} } \right],$$
(4)

where i denotes the ith OFDM symbol; \(( \cdot )^{*}\) denotes the conjugation operation. According to Eq. (4), the transmitting OFDM symbol on the first transmitting antenna is expressed as:

$${\varvec{S}}_{1} = [\begin{array}{*{20}c} {{\varvec{S}}(i)} & { - {\varvec{S}}(i + 1)^{*} } \\ \end{array} ].$$
(5)

Correspondingly, the transmitting OFDM symbol on the second transmitting antenna is expressed as:

$${\varvec{S}}_{2} = [\begin{array}{*{20}c} {{\varvec{S}}(i + 1)} & {{\varvec{S}}(i)^{*} } \\ \end{array} ].$$
(6)

Diversity gains can be obtained by using STBC technology at the transmitter. STBC is a key technology to improve the reliability of MIMO-OFDM systems.

According to the estimated CFR, the OFDM symbols after STBD in the receiver can be expressed as:

$$\left[ \begin{gathered} \hat{\varvec{S}}(i) \\ \hat{\varvec{S}}(i + 1) \\ \end{gathered} \right] = \left[ \begin{gathered} \frac{{{\varvec{Y}}_{1} (i)\hat{\varvec{H}}_{1,1}^{*} (i) + {\varvec{Y}}_{1}^{*} (i + 1)\hat{\varvec{H}}_{2,1} (i) + {\varvec{Y}}_{2} (i)\hat{\varvec{H}}_{1,2}^{*} (i) + {\varvec{Y}}_{2}^{*} (i + 1)\hat{\varvec{H}}_{2,2} (i)}}{{\hat{\varvec{H}}_{1,1} (i)\hat{\varvec{H}}_{1,1}^{ * } (i) + \hat{\varvec{H}}_{2,1} (i)\hat{\varvec{H}}_{2,1}^{ * } (i) + \hat{\varvec{H}}_{1,2} (i)\hat{\varvec{H}}_{1,2}^{*} (i) + \hat{\varvec{H}}_{2,2} (i)\hat{\varvec{H}}_{2,2}^{*} (i)}} \\ \frac{{{\varvec{Y}}_{1} (i)\hat{\varvec{H}}_{2,1}^{*} (i) - {\varvec{Y}}_{1}^{*} (i + 1)\hat{\varvec{H}}_{1,1} (i) + {\varvec{Y}}_{2} (i)\hat{\varvec{H}}_{2,2}^{*} (i) - {\varvec{Y}}_{2}^{*} (i + 1)\hat{\varvec{H}}_{1,2} (i)}}{{\hat{\varvec{H}}_{1,1} (i)\hat{\varvec{H}}_{1,1}^{ * } (i) + \hat{\varvec{H}}_{2,1} (i)\hat{\varvec{H}}_{2,1}^{ * } (i) + \hat{\varvec{H}}_{1,2} (i)\hat{\varvec{H}}_{1,2}^{*} (i) + \hat{\varvec{H}}_{2,2} (i)\hat{\varvec{H}}_{2,2}^{*} (i)}} \\ \end{gathered} \right].$$
(7)

Simplifying Eq. (7), we can obtain:

$$\left[ \begin{gathered} \hat{\varvec{S}}(i) \\ \hat{\varvec{S}}(i + 1) \\ \end{gathered} \right] = \left[ \begin{gathered} \frac{{\hat{\varvec{Y}}_{1} (i)\hat{\varvec{H}}_{1,1}^{*} (i) + \hat{\varvec{Y}}_{1}^{*} (i + 1)\hat{\varvec{H}}_{2,1} (i) + \hat{\varvec{Y}}_{2} (i)\hat{\varvec{H}}_{1,2}^{*} (i) + \hat{\varvec{Y}}_{2}^{*} (i + 1)\hat{\varvec{H}}_{2,2} (i)}}{{|\hat{\varvec{H}}_{1,1} (i){|}^{2} + |\hat{\varvec{H}}_{2,1} (i){|}^{2} + |\hat{\varvec{H}}_{1,2} (i){|}^{2} + |\hat{\varvec{H}}_{2,2} (i){|}^{2} }} \\ \frac{{\hat{\varvec{Y}}_{1} (i)\hat{\varvec{H}}_{2,1}^{*} (i) - \hat{\varvec{Y}}_{1}^{*} (i + 1)\hat{\varvec{H}}_{1,1} (i) + \hat{\varvec{Y}}_{2} (i)\hat{\varvec{H}}_{2,2}^{*} (i) - \hat{\varvec{Y}}_{2}^{*} (i + 1)\hat{\varvec{H}}_{1,2} (i)}}{{|\hat{\varvec{H}}_{1,1} (i){|}^{2} + |\hat{\varvec{H}}_{2,1} (i){|}^{2} + |\hat{\varvec{H}}_{1,2} (i){|}^{2} + |\hat{\varvec{H}}_{2,2} (i){|}^{2} }} \\ \end{gathered} \right],$$
(8)

where \(\hat{\varvec{S}}(i)\) is the ith symbol after STBD; \({\varvec{Y}}_{\beta } (i)(\beta = 1,2)\) is the ith symbol received by the βth receiving antenna; \(\hat{\varvec{H}}_{\alpha ,\beta } (\alpha = 1,2, \, \beta = 1,2)\) is the estimated CFR between αth transmitting antenna and βth receiving antenna.

Variable pilot assisted channel estimation

Various comb-type pilot sequences

The comb pilot interval can be selected according to the frequency selectivity or time selectivity of the fading environment. When the total subcarrier number \(N = 2048\), there are six options for pilot interval n, which are 2, 4, 8, 20, 50, and 100, respectively. Figure 3 shows the optional comb type pilot, where (a–d) represent comb type pilot \({\varvec{x}}_{1}\), \({\varvec{x}}_{2}\), \({\varvec{x}}_{3}\), and \({\varvec{x}}_{4}\), respectively. Assuming the pilot matrix is \({\varvec{a}} = [\sqrt {2.3} ,\, - \sqrt {2.3} ]\), the light blue circle represents the pilot signal with \(a_{1} = \sqrt {2.3}\), and the brown circle represents the pilot signal with \(a_{2} = - a_{1} = - \sqrt {2.3}\). The element value in \({\varvec{a}}\) changes according to the variation of fading channels, the transmitted symmetric pilot matrix \({\varvec{x}}\) is denoted as:

$${\varvec{x}} = \left[ \begin{gathered} {\varvec{x}}_{1} \\ {\varvec{x}}_{2} \\ {\varvec{x}}_{3} \\ {\varvec{x}}_{4} \\ \end{gathered} \right] = \left[ {\begin{array}{*{20}c} { - a_{1} } & {a_{1} } & {a_{1} } & { - a_{1} } \\ {a_{1} } & { - a_{1} } & { - a_{1} } & {a_{1} } \\ { - a_{1} } & {a_{1} } & { - a_{1} } & {a_{1} } \\ {a_{1} } & { - a_{1} } & {a_{1} } & { - a_{1} } \\ \end{array} } \right],$$
(9)
Fig. 3
figure 3

Variable comb type pilots \({\varvec{x}}_{1}\), \({\varvec{x}}_{2}\), \({\varvec{x}}_{3}\), and \({\varvec{x}}_{4}\)

where the first kind of comb type pilot in \({\varvec{x}}\) is \({\varvec{x}}_{1}\), the second kind of comb type pilot is \({\varvec{x}}_{2}\), the third kind of comb type pilot is \({\varvec{x}}_{3}\), and the fourth kind of comb type pilot is \({\varvec{x}}_{4}\).

Supposing the transmitting OFDM symbols on each data subcarriers are \({\varvec{e}}_{i} (1 \le i \le N_{{\text{d}}} )\) and \(N_{{\text{d}}}\) denotes the number of transmitting OFDM symbols, which means \({\varvec{e}}_{i} = [e_{1} ,\,e_{2} ,\,e_{3} ,\, \cdots ,\,e_{{N_{{\text{d}}} }} ]\). Under comb type \({\varvec{x}}_{1}\) insertion, and pilot interval \(n = 2\), the symmetric matrix of the transmitting OFDM symbols on four consecutive subcarriers \({\varvec{\lambda }}_{m}\) are denoted as:

$${\varvec{\lambda}}_{m} { = }\left[ \begin{gathered} \,\,\,{\varvec{A}} \\ {\varvec{s}}_{4m - 3} \\ \,\,\,{\varvec{B}} \\ {\varvec{s}}_{4m - 2} \\ \,\,\,{\varvec{B}} \\ {\varvec{s}}_{4m - 1} \\ \,\,\,{\varvec{A}} \\ \,\,{\varvec{s}}_{4m} \\ \end{gathered} \right] = \left[ {\begin{array}{*{20}c} { - a_{1} } & { - a_{1} } & { - a_{1} } & { - a_{1} } \\ {e_{16m - 15} } & {e_{16m - 14} } & {e_{16m - 13} } & {e_{16m - 12} } \\ {a_{1} } & {a_{1} } & {a_{1} } & {a_{1} } \\ {e_{16m - 11} } & {e_{16m - 10} } & {e_{16m - 9} } & {e_{16m - 8} } \\ {a_{1} } & {a_{1} } & {a_{1} } & {a_{1} } \\ {e_{16m - 7} } & {e_{16m - 6} } & {e_{16m - 5} } & {e_{16m - 4} } \\ { - a_{1} } & { - a_{1} } & { - a_{1} } & { - a_{1} } \\ {e_{16m - 3} } & {e_{16m - 2} } & {e_{16m - 1} } & {e_{16m} } \\ \end{array} } \right],{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 1 \le m \le 124,$$
(10)

where \(m\) is the number of symmetric matrices in 2 × 2 MIMO-OFDM systems. The transmitting pilots on subcarriers are \({\varvec{A}}\) and \({\varvec{B}}\). The transmitting OFDM symbols on four consecutive subcarriers are \({\varvec{s}}_{4m - 3}\), \({\varvec{s}}_{4m - 2}\), \({\varvec{s}}_{4m - 1}\), and \({\varvec{s}}_{4m}\), respectively. When \(m = 125\), the seventh and eighth line in matrix \(\varvec{\lambda}_{m}\) is all zero.

Small scale propagation in MIMO-OFDM system

Under Rayleigh fading channel [21], the coherence time \(t_{{\text{c}}}\) of MIMO-OFDM system is defined as:

$$t_{{\text{c}}} = \frac{9}{{16{\uppi }f_{{\text{d}}}^{{2}} }},$$
(11)

where \(f_{{\text{d}}}\) is the Doppler frequency shift, and \(f_{{\text{d}}}\) is expressed as [22]:

$$f_{{\text{d}}} = \frac{v}{\lambda },$$
(12)

where \(v\) is the speed of mobile station and \(\lambda\) is the wavelength. When the Doppler frequency shift \(f_{{\text{d}}}\) is 20 Hz, 60 Hz, 80 Hz, 120 Hz, 160 Hz, and 180 Hz, respectively, the corresponding coherence time \(t_{{\text{c}}}\) is 4.4762 × 10−4 s, 4.9736 × 10−5 s, 2.7976 × 10−5 s, 1.2434 × 10−5 s, 6.9941 × 10−6 s, and 5.5262 × 10−6 s, respectively. As \(f_{{\text{d}}}\) increases, \(t_{{\text{c}}}\) decreases gradually, and the multipath channel will experience deep fading. Table 1 shows the corresponding relation between \(f_{{\text{d}}}\) and \(t_{{\text{c}}}\) in AWGN channel. Figure 4 illustrates large scale propagation and small scale propagation of MIMO-OFDM systems. When \(f_{{\text{d}}}\) increases, it induces time selective propagation. The time selective propagation includes large-scale propagation and small-scale propagation. To suppress the ISI which is caused by small-scale propagation, the pilots are inserted in comb type manner through the subcarriers in the frequency domain with variable pilots.

Table 1 The corresponding relation between fd and tc in AWGN channels
Fig. 4
figure 4

Large scale propagation and small scale propagation of MIMO-OFDM system

Simulation results and comparison analysis

In MIMO-OFDM system, the transmitted signal and received signal are independent of each other, thus the probability of signal interference caused by AWGN or multipath is greatly reduced. The theoretical analysis would be verified through simulation. The simulation is performed in static AWGN channel and it adopts five kinds of modulation modes. The comb type pilots \({\varvec{x}}_{1}\), \({\varvec{x}}_{2}\), \({\varvec{x}}_{3}\), and \({\varvec{x}}_{4}\) are inserted in the proposed MIMO-OFDM systems, and the parameters of MIMO-OFDM system are shown in Table 2.

Table 2 Parameters of MIMO-OFDM system

The BER curve of the system could be drawn by making the statistics of BER for every OFDM frame under different SNR. In the statistical process of BER, the bit stream of each frame output by the constellation symbol demodulation module is compared with the bit stream contained in each frame transmitted, and the error bits are demodulated and accumulated to obtain the sum of the cumulative number of error bits per frame. And then the number of error bits in all transmitted OFDM data frames is accumulated, and the total number of bits transmitted in all transmitted OFDM data frames is calculated. The whole process that antenna transmit signals and antenna receive signals is cyclic, and the total number of error bits is obtained. With a certain fixed SNR, the calculated BER of the MIMO-OFDM system is denoted as:

$$r_{{{\text{BER}}}} = \frac{{\sum\nolimits_{u = 1}^{L} {N_{u} } }}{{LN_{t} }},$$
(13)

where \(N_{u}\) is the number of error bits in the uth OFDM frame under the condition that the maximum number of cycles is \(L\). Assuming that the MIMO-OFDM system transmitter transmits equal data bits in each OFDM transmitted in each frame is equal, and \(N_{t}\) represents the number of bits transmitted in each OFDM frame.

Figure 5. shows BER performance of QPSK modulated 2 × 2 MIMO-OFDM system under AWGN with comb type pilot \({\varvec{x}}_{1}\) insertion. As shown in Fig. 5, the \(n = 2\) BER curve has the worst performance; \(n = 4\) BER curve is the second; \(n = 8\) BER curve is the third; \(n = 20\), \(n = 50\), and \(n = 100\) BER curves nearly all have better BER performance. For example, at the target \(r_{{{\text{BER}}}} = 10^{ - 3}\), \(n = 50\) BER curve outperforms \(n = 2\) BER curve 2.15 dB SNR gains.

Fig. 5
figure 5

BER performance of QPSK modulated 2 × 2 MIMO-OFDM system under AWGN with comb type pilot \({\varvec{x}}_{1}\) insertion

Figure 6. shows BER performance of 8PSK modulated 2 × 2 MIMO-OFDM system under AWGN with comb type pilot \({\varvec{x}}_{3}\) insertion. As shown in Fig. 6, the \(n = 2\) BER curve has the worst performance; \(n = 4\) BER curve is the second; \(n = 8\) BER curve is the third; \(n = 20\) BER curve is the fourth; \(n = 50\) BER curve is the fifth; and \(n = 100\) BER curve has the best performance. At the target \(r_{{{\text{BER}}}} = 10^{ - 3}\), the SNR gap between \(n = 4\) BER curve and \(n = 100\) BER curve is 1.1 dB.

Fig. 6
figure 6

BER performance of 8PSK modulated 2 × 2 MIMO-OFDM system under AWGN with comb type pilot \({\varvec{x}}_{3}\) insertion

Figure 7. shows BER performance of 16QAM modulated 2 × 2 MIMO-OFDM system under AWGN with comb type pilot \({\varvec{x}}_{2}\) insertion. As shown in Fig. 7, \(n = 2\) BER curve has the best performance; \(n = 4\) BER curve is the second; \(n = 8\) BER curve is the third. It can be seen that \(n = 20\), \(n = 50\), and \(n = 100\) nearly all have better BER performance. At the target \(r_{{{\text{BER}}}} = 10^{ - 3}\), \(n = 8\) BER curve outperforms \(n = 4\) BER curve about 0.8 dB SNR gains.

Fig. 7
figure 7

BER performance of 16QAM modulated 2 × 2 MIMO-OFDM system under AWGN with comb type pilot \({\varvec{x}}_{2}\) insertion

Figure 8. shows BER performance of 64QAM modulated 2 × 2 MIMO-OFDM system under AWGN with comb type pilot \({\varvec{x}}_{4}\) insertion. As shown in Fig. 8, \(n = 8\), \(n = 20\), \(n = 50\), and \(n = 100\) near all have better BER performance. At the target \(r_{{{\text{BER}}}} = 10^{ - 4}\), \(n = 20\) BER curve outperforms \(n = 2\) BER curve 2.4 dB SNR gains. In the paper, it is recommended that the optimal pilot interval value of \(P\) in the proposed variable pilot assisted MIMO-OFDM system is \(P \in [8,\,100]\).

Fig. 8
figure 8

BER performance of 64QAM modulated 2 × 2 MIMO-OFDM system under AWGN with comb type pilot \({\varvec{x}}_{4}\) insertion

Conclusion

This paper studies a variable pilot assisted channel estimation method in STBC MIMO-OFDM system with different modulation modes. The variable pilot patterns are proposed to estimate the transmitted OFDM symbols by effectively allocate comb type pilot sequences through the frequency subcarriers. The proposed variable pilot patterns work well for MIMO-OFDM systems. The novelty of the paper is that it saves the spectrum resources by using STBC and variable pilot insertion intervals. Moreover, the performance of MIMO-OFDM systems is improved as can be observed from the simulation results. In the future, we will apply the proposed variable pilot assisted channel estimation method in cognitive MIMO-OFDM systems [23, 24], ultra-wideband MIMO-OFDM systems [25], China multimedia mobile broadcasting MIMO-OFDM systems [26], massive MIMO-OFDM systems [27, 28], and underwater acoustic OFDM systems [29].

Availability of data and materials

Not applicable.

Abbreviations

AWGN:

Additive white Gaussian noise

BER:

Bit error rate

BPSK:

Binary phase shift keying

CFR:

Channel frequency response

CP:

Cyclic prefix

CSI:

Channel state information

FFT:

Fast Fourier transform

ICI:

Inter-carrier interference

IFFT:

Inverse fast Fourier transform

ISI:

Inter-symbol interference

LS:

Least square

MIMO:

Multiple input multiple output

MSE:

Mean square error

OFDM:

Orthogonal frequency division multiplexing

OTS:

Optimal training sequences

P/S :

Parallel to serial

QPSK:

Quadrature phase shift keying

SNR:

Signal-to-noise ratio

STBC:

Space time block coding

STBD:

Space time block decoding

S/P:

Serial to parallel

8PSK:

8-phase shift keying

16QAM:

16-quadrature amplitude modulation

64QAM:

64-quadrature amplitude modulation

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Acknowledgements

The authors thank Mingtong Zhang and Ruiguang Tang for their kind help and valuable suggestions in revising this paper.

Funding

This work was supported in part by the Joint Fund of Shandong Provincial Natural Science Foundation (No. ZR2021LZHP003), in part by the Shandong Provincial Natural Science Foundation, China (Nos. ZR2020MF008, ZR2021MF060), in part by the 16th Student Research Training Program (SRTP) at Shandong University, Weihai (No. A21246), in part by the Education and Teaching Reform Research Project of Shandong University, Weihai (No. Y2021054), and in part by the Science and Technology Development Plan Project of Weihai Municipality in 2020.

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Authors and Affiliations

Authors

Contributions

QW, XZ, CW, and HC conceived the algorithm and designed the experiments; HC designed computer programs; QW and CW performed the experiments; HC prepared tables and figures; XZ and HC analyzed the experimental results; XZ and CW drafted the manuscript; QW, XZ, and HC revised the manuscript. All authors read and approved the final manuscript.

Authors’ information

Qun Wu was born in Heze city, Shandong province, China, in 1998. She received the B.E. degree in electronic information engineering from Jiangxi University of Science and Technology, China, in 2020. She is currently pursuing the M.E. degree in information and communication engineering with Shandong University, China. Her current research interests include wireless communication theory and technology, intelligent information processing, and computer vision.

Xiao Zhou received the B.E. degree in automation from Nanjing University of Posts and Telecommunications, China, in 2003; the M.E. degree in information and communication engineering from Inha University, Korea, in 2005; and the Ph.D. degree in information and communication engineering from Tsinghua University, China, in 2013. She is currently an associate professor and a master's supervisor with Shandong University, Weihai, China. Her current research interests include wireless communication technology, image communication, digital image/video processing and analysis, intelligent information processing, computer vision, and artificial intelligence.

Chengyou Wang received the B.E. degree in electronic information science and technology from Yantai University, China, in 2004, and the M.E. and Ph.D. degrees in signal and information processing from Tianjin University, China, in 2007 and 2010, respectively. He is currently an associate professor and a master's supervisor with Shandong University, Weihai, China. His current research interests include digital image/video processing and analysis, intelligent information processing, computer vision, deep learning, artificial intelligence, and wireless communication technology.

Hai Cao received the B.S. degree in computer science and technology from Shandong University, Weihai, China, in 2003, and the M.E. degree in computer applications from Shandong University, China, in 2006. He is currently a lecturer with Shandong University, Weihai, China. His current research interests include software theory and engineering, digital image processing, machine learning, and wireless communications.

Corresponding author

Correspondence to Hai Cao.

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Wu, Q., Zhou, X., Wang, C. et al. Variable pilot assisted channel estimation in MIMO-OFDM system with STBC and different modulation modes. J Wireless Com Network 2022, 45 (2022). https://doi.org/10.1186/s13638-022-02126-2

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Keywords

  • Orthogonal frequency division multiplexing (OFDM)
  • Space time block coding (STBC)
  • Additive white Gaussian noise (AWGN)
  • Multiple input multiple output (MIMO)
  • Bit error rate (BER)