We consider the transmission of MIMO-OFDM symbols synthesized according to the PHY layer specifications of the IEEE 802.16e (Mobile WiMAX) standard. We assume OFDM symbols with *N* subcarriers and a cyclic prefix of *N*
_{
g
} samples are transmitted. In total, each OFDM symbol occupies *N*
_{
t
}=*N*+*N*
_{
g
} samples. OFDM symbols are grouped in frames of *K* consecutive symbols, including a symbol reserved as a preamble. We also assume multiple antennas at transmission and reception, i.e., MIMO-OFDM transmissions.The number of transmit and receive antennas is *M*
_{
T
} and *M*
_{
R
}, respectively. OFDM symbols are spatially multiplexed over the *M*
_{
T
} transmit antennas except the preamble which is transmitted only through the first antenna.

Let \({\mathbf {s}}_{k}^{(m)} \in \mathbb {C}^{N \times 1}\), *k*=1,…,*K*, *m*=1,…,*M*
_{
T
} be the column vector that represents the *N* complex-valued information symbols transmitted in the *k*th OFDM symbol over the *m*th antenna. Such vectors contain data, pilot, and guard symbols. Similarly, \({\mathbf {x}}_{k}^{(m)} \in \mathbb {C}^{N_{t} \times 1}\) contains the *N*
_{
t
} samples corresponding to the *k*th OFDM symbol transmitted over the *m*th antenna.

Elaborating the signal model of a MIMO-OFDM system, the discrete-time representation of the transmitted MIMO-OFDM symbols is:

$$ {\mathbf{x}}_{k} = \left({\mathbf{I}}_{M_{T}} \otimes {\mathbf{G}}_{1}{\mathbf{F}}^{H}\right) {\mathbf{s}}_{k}, \quad k=1,\ldots, K, $$

((1))

where \({\mathbf {s}}_{k} = \left [ {{\mathbf {s}}_{k}^{(1)}}^{T},{{\mathbf {s}}_{k}^{(2)}}^{T},\cdots,{{\mathbf {s}}_{k}^{(M_{T})}}^{T} \right ]^{T}\) is the *N*
*M*
_{
T
}×1 column vector containing the information, pilot, and guard symbols transmitted in the *k*th MIMO-OFDM symbol; **F** is the standard *N*×*N* discrete Fourier transform (DFT) matrix; **G**
_{1} is a *N*
_{
t
}×*N* matrix which appends the *N*
_{
g
} samples of the cyclic prefix; ⊗ denotes the Kronecker product; \({\mathbf {I}}_{M_{T}}\) is the *M*
_{
T
}×*M*
_{
T
} identity matrix; and \({\mathbf {x}}_{k} = \left [ {{\mathbf {x}}_{k}^{(1)}}^{T},{{\mathbf {x}}_{k}^{(2)}}^{T},\cdots,{{\mathbf {x}}_{k}^{(M_{T})}}^{T} \right ]^{T}\) is the *N*
_{
t
}
*M*
_{
T
}×1 column vector with the samples of the *k*th MIMO-OFDM symbol. We assume the samples in \({\mathbf {x}}_{k}^{(m)}\) are serially transmitted over the *m*th antenna at a sampling rate *F*
_{
s
}=1/*T*
_{
s
} where *T*
_{
s
} is the sampling period.

The information symbol vectors are constructed as \({{\mathbf {s}}_{k}^{(m)}={\mathbf {P}}_{k}^{(m)}{\mathbf {p}}_{k}^{(m)}+{\mathbf {D}}_{k}^{(m)}{\mathbf {d}}_{k}^{(m)}}\) where \({\mathbf {p}}_{k}^{(m)}\) is a *P*×1 vector containing the pilot symbols in the *k*th OFDM symbol transmitted over the *m*th transmit antenna, whereas \({\mathbf {p}}_{k}^{(m)}\) is the *N*×*P* matrix that defines the positions of the pilots in such a symbol. Similarly, \({\mathbf {d}}_{k}^{(m)}\) is the *D*×1 vector containing the data symbols, and \({\mathbf {d}}_{k}^{(m)}\) is the *N*×*D* matrix that defines their positions. Data symbols are the output of a quadrature amplitude modulation (QAM) constellation mapper whose inputs are the channel encoded source bits. Note that *P*+*D*<*N*, with *N*−(*P*+*D*) being the number of guard subcarriers. Matrices \({\mathbf {p}}_{k}^{(m)}\) and \({\mathbf {d}}_{k}^{(m)}\) consist of ones and zeros only and are designed so that data and pilots are assigned to different subcarriers. According to the Mobile WiMAX standard, pilot subcarriers allocated in the *m*th antenna are set to 0 in all other antennas.

We next assume that the previous OFDM waveforms are transmitted over a time-varying MIMO channel. Such a channel is represented by:

$$ {\fontsize{8.8pt}{9.6pt}\selectfont{\begin{aligned} {}h^{i,j}(t,\tau) \,=\, \sum_{l=1}^{L} h^{i,j}_{l}(t)\delta\left(\!\tau - \tau^{i,j}_{l}\!\right)\!, i=1, \ldots, M_{R} \ \text{and}\ j = 1, \ldots, M_{T}, \end{aligned}}} $$

((2))

where *h*
^{i,j}(*t*,*τ*) is the channel impulse response between the *j*th transmit antenna and *i*th receive antenna, while \(h^{i,j}_{l}(t)\) and \(\tau ^{i,j}_{l}\) are the *l*th path gain and delay of *h*
^{i,j}(*t*,*τ*), respectively. Elaborating on the discrete-time signal model of a MIMO-OFDM system, the MIMO channel during the transmission of the *k*th OFDM symbol is represented by the block matrix:

$$ {\mathbf{H}}_{k} = \left[ \begin{array}{cccc} {\mathbf{H}}_{k}^{1,1} & {\mathbf{H}}_{k}^{1,2} & \cdots & {\mathbf{H}}_{k}^{1,M_{T}} \\ {\mathbf{H}}_{k}^{2,1} & {\mathbf{H}}_{k}^{2,2} & \cdots & {\mathbf{H}}_{k}^{2,M_{T}} \\ \cdots & \cdots & \cdots & \cdots \\ {\mathbf{H}}_{k}^{M_{R},1} & {\mathbf{H}}_{k}^{M_{R},2} & \cdots & {\mathbf{H}}_{k}^{M_{R},M_{T}} \\ \end{array} \right], $$

((3))

where \({\mathbf {H}}_{k}^{i,j} \in \mathbb {C}^{N_{t} \times N_{t}}\) are matrices representing the channel impulse response between the (*i*,*j*) antenna pair whose entries are:

$$ {}{\mathbf{H}}^{i,j}_{k}(r,s) = h^{i,j}(((k-1)N_{t} + N_{g} + r - 1)T_{s}, \text{mod}(r-s,N)T_{s}). $$

((4))

Note that \({\mathbf {H}}_{k}^{i,m} {\mathbf {x}}_{k}^{(m)}\) represents the time-convolution between the signal transmitted over the *m*th antenna and *h*
^{i,m}(*t*,*τ*).

Having in mind the previous channel representation, the discrete-time received signal is given by:

$$ {\mathbf{r}}_{k} = \left({\mathbf{I}}_{M_{R}} \otimes {\mathbf{F}}{\mathbf{G}}_{2}\right) {\mathbf{H}}_{k} {\mathbf{x}}_{k} + {\mathbf{w}}_{k} = {\mathbf{G}}_{k}{\mathbf{s}}_{k}+{\mathbf{w}}_{k}, $$

((5))

where **G**
_{2} is a *N*×*N*
_{
t
} matrix which represents the removal of the cyclic prefix, **G**
_{
k
} is a block matrix containing the channel frequency response between each antenna pair, and \({\mathbf {w}}_{k} \in {\mathbf {C}}^{N_{t} \times 1}\) is a vector of independent complex-valued Gaussian-distributed random variables with variance \({\sigma ^{2}_{w}}\). When the channel is time invariant, all submatrices in **G**
_{
k
} are diagonal. However, when the channel is time-variant, entries different than zero appear outside their main diagonals, hence introducing ICI between the transmitted subcarriers. In such case, Equation 5 is rewritten as:

$$ {\mathbf{r}}_{k} = \bar{{\mathbf{G}}}_{k}{\mathbf{s}}_{k}+{\mathbf{z}}_{k}+{\mathbf{w}}_{k}, $$

((6))

where \(\bar {{\mathbf {G}}}_{k}\) is a block matrix with the main diagonals of the submatrices of **G**
_{
k
} and \({{\mathbf {z}}_{k} = ({\mathbf {G}}_{k}-\bar {{\mathbf {G}}}_{k}){\mathbf {s}}_{k}}\) represents the ICI in the received signal. Note that in multi-antenna systems, ICI occurs not only among subcarriers but also among different transmit antennas.

When transmitting MIMO-OFDM symbols over time-varying channels, the amount of ICI relates to the normalized Doppler spread given by *D*
_{
n
}=*f*
_{
d
}
*T*, where *f*
_{
d
} is the maximum Doppler frequency and *T*=*N*
*T*
_{
s
} is the duration of an OFDM symbol excluding the cyclic prefix. According to our previous proposal [18], it is possible to adjust the parameter *T* by time interpolation by a factor *I*, yielding an OFDM symbol duration *T*
^{I}=*I*
*T*
_{
s
}
*N*. Therefore, the normalized Doppler spread impacting the time-interpolated OFDM signal is:

$$ {D^{I}_{n}}=f_{d} T^{I} = f_{d} I T_{s} N = \frac{T_{s} N I f_{c} v}{c} = \frac{T_{s} N f_{c}}{c} v^{I}, $$

((7))

with *f*
_{
c
} the carrier frequency, *c* the speed of light, and *v*
^{I}=*I*
*v* the emulated speed as a result of an actual measurement speed *v* and a time-interpolation factor *I*. Consequently, enlarging the symbol length *T*
^{I} allows for the emulation of a velocity *v*
^{I}, which is *I* times higher than the actual speed of the receiver, namely *v*.

Figure 1 shows the graphical representation of a measurement setup designed according to the previous premises. Interpolating the MIMO-OFDM waveforms prior to its transmission allows for emulating high-velocity conditions while conducting low-velocity experiments. Due to the interpolation step, signals over the air will suffer from severe ICI degradation, although the maximum Doppler frequency is low.

It should be noticed that interpolation does not allow for a perfect recreation of high-mobility channels because the signals over-the-air have a reduced bandwidth and are less sensitive to the channel frequency selectivity.

Nevertheless, note that in this work, we are mostly interested in conducting experiments to test the performance of ICI cancellation methods in WiMAX receivers over real-world channels rather than channel equalization methods which can be tested in static experiments. Time interpolation does not reproduce the exact conditions of high mobility scenarios but provide a cost-efficient approximation to them.