The channel estimation problem is related to the channel model assumed, itself determined by the physical propagation characteristics, including the number of transmit and receive antennas, transmission bandwidth, carrier frequency, cell configuration and relative speed between eNodeB and UE receivers. In general,
• The carrier frequencies and system bandwidth mainly determine the scattering nature of the channel.
• The cell deployment governs its multipath, delay spread and spatial correlation characteristics.
• The relative speed sets the time-varying properties of the channel.
166 LTE – THE UMTS LONG TERM EVOLUTION The propagation conditions characterize the channel correlation function in a three- dimensional space comprising frequency, time and spatial domains. In the general case, each MIMO (Multiple-Input Multiple-Output) multipath channel component can experience dif- ferent but related spatial scattering conditions leading to a full three-dimensional correlation function across the three domains. Nevertheless, for the sake of simplicity, assuming that the multipath components of each spatial channel experience the same scattering conditions, the spatial correlation is independent from the other two domains and can be handled separately from the frequency and time domain correlations.
This framework might be suboptimal in general, but is nevertheless useful in mitigating the complexity of channel estimation as it reduces the general three-dimensional joint estimation problem into independent estimation problems.
For a comprehensive survey of MIMO channel estimation the interested reader is referred to [12].
In the following two subsections, we define a channel model and its corresponding correlation properties. These are then used as the basis for an overview of channel estimation techniques which exploit channel correlation in the context of the LTE downlink.
8.3.1 Time-Frequency Domain Correlation: The WSSUS Channel Model
The Wide-Sense Stationary Uncorrelated Scattering (WSSUS) channel model is commonly employed for the multipath channels experienced in mobile communications.
Neglecting the spatial dimension for the sake of simplicity, let h(τ;t)denote the time- varying complex baseband impulse response of a multipath channel realization at time instant tand delayτ.
When a narrowband signalx(t)is transmitted, the received narrowband signaly(t)can be written as
y(t)=
h(τ;t)x(t−τ )dτ (8.2)
Considering the channel as a random process in the time directiont, the channel is said to be delay Uncorrelated-Scattered (US) if
E[h(τa;t1)∗h(τb;t2)] =φh(τa;t1, t2)δ(τb−τa) (8.3) with E[ã] being the expectation operator. According to the US assumption, two CIR componentsaandbat relative respective delaysτaandτbare uncorrelated ifτa=τb.
The channel is Wide-Sense Stationary (WSS)-uncorrelated if
φh(τ;t1, t2)=φh(τ;t2−t1) (8.4) which means that the correlation of each delay component of the CIR is only a function of thedifferencein time between each realization.
Hence, the second-order statistics of this model are completely described by its delay cross-power densityφh(τ;t)or by its Fourier transform, the scattering function
Sh(τ;f )=
φh(τ;t)e−j2πf tdt (8.5)
REFERENCE SIGNALS AND CHANNEL ESTIMATION 167 withf being the Doppler frequency. Other related functions of interest include themultipath intensity profile
ψh(τ )=φh(τ;0)=
Sh(τ;f )df thetime-correlation function
φh(τ )=
φh(τ;t)dτ and theDoppler power spectrum
Sh(f )=
Sh(τ;f )dτ
A more general exposition of WSSUS models is given in [13]. Classical results were derived by Clarke [14] and Jakes [15] for the case of a mobile terminal communicating with a stationary base station in a two-dimensional propagation geometry.
These well-known results state that
Sh(f )∝ 1
fd2−f2
(8.6) for|f| ≤fd,fd=(v/c)fcthe maximum Doppler frequency for a mobile with relative speed v, carrier frequencyfcand propagation speedcand
φ¯h(t)∝J0(2πfdt) (8.7)
whereJ0 is the zeroth-order Bessel function. The autocorrelation function is obtained via the inverse Fourier transform of the Power Spectral Density (PSD) (Figure 8.5 shows the PSD of the classical Doppler spectrum described by Clarke and Jakes [14, 15]). The squared magnitude of the autocorrelation function corresponding to Clarke’s spectrum model is plotted in Figure 8.6 where the variableφ along thex-axis is effectively a spatial lag (in metres) normalized by the carrier wavelength. The Clarke and Jakes derivations are based on the assumption that the physical scattering environment is chaotic and therefore the angle of arrival of the electromagnetic wave at the receiver is a uniformly distributed random variable in the angular domain. As a consequence, the time-correlation function is strictly real-valued, the Doppler spectrum is symmetric and interestingly there is adelay-temporal separability property in the general bi-dimensional scattering functionSh(τ, t). In other words,
Sh(τ;f )∝ψh(τ )Sh(f ) (8.8) or equivalently
φh(τ;t)∝ψh(τ )φh(t) (8.9) When the mobile is moving in a fixed and known direction, as for example in rural or suburban areas, the WSSUS mobile channel is in general non-separable, but can be considered to be separable when the direction of motion averages out because each multipath component is the result of omnidirectional scattering from objects surrounding the mobile, as one would expect in urban and indoor propagation scenarios. Separability is a very important assumption for reducing the complexity of channel estimation, allowing the problem to be separated into two one-dimensional operations.
168 LTE – THE UMTS LONG TERM EVOLUTION
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Figure 8.5 Normalized PSD for Clarke’s model.
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Figure 8.6 Autocorrelation function, flat Rayleigh fading, Clarke’s Doppler.
8.3.2 Spatial Domain Correlation: The Kronecker Model
While the previous section addresses frequency and time correlation induced by the channel delays and Doppler spread, the spatial correlation arises from the spatial scattering conditions.
Among the possible spatial correlation models, for performance evaluation of LTE the so- called Kronecker model is generally used (see [10]). Despite its simplicity, this correlation- based analytical model is widely used for the theoretical analysis of MIMO systems and yields experimentally verifiable results when limited to two or three antennas at each end of the radio link.
REFERENCE SIGNALS AND CHANNEL ESTIMATION 169 Let us assume the narrowband MIMO channelNRx×NTxmatrix for a system withNTx
transmitting antennas andNRxreceiving antennas to be
H=
h0,0 ã ã ã h0,NTx−1
... ...
hNRx−1,NTx−1 ã ã ã hNRx−1,NTx−1
(8.10)
The narrowband assumption particularly suits OFDM systems, where each MIMO channel componenthn,mcan be seen as the complex channel coefficient of each spatial link at a given subcarrier index.
The matrixHis rearranged into a vector by means of the operator vec(H)= [vT0,vT1, . . . , vTN
Tx−1]T wherevi is theith column ofHand{ã}Tis the transpose operation. Hence, the correlation matrix can be defined as
CS=E[vec(H)vec(H)H] (8.11) where{ã}H is the Hermitian operation. The matrix in (8.11) is the full correlation matrix of the MIMO channel.
The Kronecker model assumes that the full correlation matrix results from separate spatial correlations at the transmitter and receiver, which is equivalent to writing the full correlation matrix as a Kronecker product (⊗) of the transmitter and receiver correlation matrices:
CS=CTx⊗CRx (8.12)
Typical values assumed for these correlations according to the Kronecker model are discussed in Section 21.3.7.