1 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.

2 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

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2.1 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

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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.

2.2 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.

3 Variable pilot assisted channel estimation

3.1 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 \({\varvec{x}}_{4}\)

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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.

3.2 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
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Fig. 4
figure 4

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

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4 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
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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

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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

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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

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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

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5 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].