MIMO channel model

Continuous time channel impulse response

Let us consider a wireless communication system with N_t transmit and N_r receive an-tennas. This is referred to as a N_t × Nr MIMO system. In the following, the index

t = 1, . . . , N_t refers to transmit antennas and r = 1, . . . , N_r to receive antennas. Let us denote the continuous-time impulse response between transmit antenna t and receive antenna r by

h^(c)_tr,k(τ_c) = h^(tx)_t (τ_c)∗ h^(ch)_tr,k(τ_c)∗ h^(rx)r (τ_c), (2.1) where ∗ stands for the convolution operator, and h^(tx)t (τ_c) and h^(rx)_r (τ_c) are the transmit and receive pulse shaping filters, respectively. We denote by τc the continuous time-delay.

In the above equation, h^(ch)_tr,k(τ_c) is the impulse response of the propagation channel cor-responding to MIMO branch tr, sampled at time kT_s+g and delay lag τ_c. The channel is assumed to be constant within the time duration of one OFDM block, which is equal to T_s+g. This assumption is called block fading model. The index k refers to the OFDM block number. OFDM systems are usually designed using rectangular pulse shape. How-ever, other pulse shapes may be useful, for instance, to lower side lobes in frequency domain [254, 269]. This is beneficial in reducing interference as OFDM with a cyclic pre-fix can prevent intercarrier interference but does not combat inter-channel interference if rectangular pulses are used.

Discrete time channel impulse response

Taking samples at t_s = 1/(N Bsc), where Bsc is the subcarrier bandwidth and N the total number of subcarriers, the discrete-time channel impulse response (CIR) for MIMO branch tr is obtained as

h_tr(k, l) = h^(c)_tr,k(τ_c) 

τc=lts

, l = 0, . . . , P − 1, k ∈ N, (2.2) where P denotes the total length of the OFDM block in samples, including the cyclic prefix.

Indices k and l correspond to OFDM block and tap numbers, respectively. For a given block index k, the quantity h_tr(k, l) is referred to as the l^th channel tap and is linked to the physical propagation environment. Under the idealistic assumption of sample-spaced CIRs, h_tr(k, l) corresponds to a single propagation path. In practice, this assumption does not always hold, and consequently h_tr(k, l) contains energy from multiple physical paths.

This introduces correlation among the channel taps and may affect channel estimation algorithms if they assume uncorrelated channels [113, 16.4.2].

Due to the block fading assumption, the channel taps are constant within the time duration of one OFDM block, which is equal to T_s+g= P t_s. Furthermore, we assume that the CIR has L_h non-zero taps and is no longer than the cyclic prefix of length LCP. At block time instance k, one may express the CIR corresponding to MIMO branch tr as the following vector of size L_h× 1:

h_tr(k) = [h_tr(k, 0), h_tr(k, 1), . . . , h_tr(k, L_h− 1)]^T . (2.3) The channel coefficients in time domain h_tr(k, l), for l = 0, . . . , L_h− 1, are assumed to be zero-mean complex circular Gaussian random variables, which leads to Rayleigh fading.

Channel taps are considered to be correlated in time, and may be dependent or indepen-dent from each other. The average power and delay profiles, 

E

|htr(k, l)|² Lh−1 l=0 and {τtr(k, l)}^L_l=0^h−1, respectively, are determined by the propagation environment. Note that the multipath nature of the channel leads to frequency selectivity, while the mobility-induced Doppler spectrum translates into time-selectivity and correlation over time [220, Ch.14]. Thus, one is dealing with time-frequency dispersive/selective channels, which need to be estimated and tracked over time for successful data transmission. The Ricean

and Nakagami-m distributions are among other well-known and frequently used statisti-cal models for fading [220, Ch.14]. The Rice model includes a line-of-sight (LOS), unlike Rayleigh fading. Both models can be considered as special cases of the Nakagami fading model. Simulation studies in this work have been conducted with independent Rayleigh fading, with correlation over time. The Doppler spectrum is assumed to follow Jakes’

model [134].

Practical implementations in both wireless and wireline transmissions experience quite long CIRs. In this case, channel shortening is an interesting option to improve effective data rates as a smaller guard interval (i.e., cyclic prefix) may be used. The idea is to linearly equalize the CIR to a much shorter target impulse response which length may be fixed a priori [30, 204].

Discrete time MIMO channel vector

We may stack the MIMO channel coefficients at time instance k into a column vector of size N_tN_rL_h× 1 as follows:

h(k) =

h^T₁₁(k), . . . , h^T_Nt1(k), . . . , h^T_1r(k), . . . , h^T_Ntr(k), . . . , h^T_1Nr(k), . . . , h^T_NtNr(k)T

. (2.4) Channels corresponding to different transmit and receive antenna pairs in MIMO systems usually exhibit similar delay profiles. In this thesis, CIR vectors in each branch are assumed to be independent and identically distributed. In practice this means that the scattering environment is rich and the antennas are placed further apart than the coherent distance.

High correlation in MIMO channel branches is detrimental in terms of reduced capacity and higher bit error rates, and leads to lower system performance.

Correlation among MIMO branches

MIMO channel models, especially the ones validated by measurement campaigns [143, 282], are of great importance in algorithm design. In the Kronecker model [143], the channel correlation matrix is given by the Kronecker product between the transmitter and receiver correlation matrices. The main assumption is that the correlation properties at the two link ends are separable. The Kronecker model has become popular, despite its limitations. Indeed, it was shown recently to under-estimate the MIMO channel capacity [282]. The Weichselberger model (W-model) [282] provides more advanced modeling and differs from the Kronecker model in the sense that correlation features at the link ends are not necessarily decoupled. In realistic scenarios, both uncorrelated and correlated channels are encountered. Recently, the third generation partnership project (3GPP) proposed the spatial channel model (SCM) and extended SCM (SCME) [285]. The IEEE task group in charge of 802.11n addresses channel modeling issues as well [131].

MIMO systems allow improving the spectral efficiency (bits/s/Hz) tremendously as they promise a linear improvement in the capacity proportional to the number of antennas.

Indeed, channel capacity was shown to be proportional to min{Nt, N_r} [207, 261]. Spatial multiplexing allows creating multiple data streams between transmitter and receiver arrays without additional bandwidth or increased total power. Channel diversity of order up to N_t× Nr may be obtained via transmit diversity techniques such as, e.g., space-time coding [126]. This results in improved quality of the radio link. Finally, increased array gain over SISO systems allows improving the coverage and SNR. Both multi-antenna systems and multicarrier modulation are needed for filling the technology gaps for future broadband wireless networks.