Chapter 4 Iterative Receiver
4.2 Tracking Stage:
[ ] { [ ] [ ] }
[ ] { [ ] [ ] }
*
*
Step1: Set preassumed number of paths & observation window Calculate & by
Step2 : Estimate multipath delays and comples gains coarsely:
while
Fig. 4 The MPIC-based decorrelation method in the initialization stage
4.2 Tracking Stage: β -tracker
Through the initialization stage, we are able to obtain information on the number of paths
( )j i,
(
Np)
κ ≤ , the multipath delays τˆl( )j i, , and the multipath complex gains μˆl( )j i, , for
{
1, , ( )j i,}
l∈ … κ . Without loss of generality, we assume that the multipath delays do not vary over the duration of each OFDM frame. In this stage, we also suppose that the channel is quasi-static within one OFDM data symbol, which means the multipath complex gains of channel model are unchanged, so as to estimate the average channel variations of each path.
[ ]
Accordingly, the corresponding channel estimator Hˆ( )j i, k k,
(
for the frequency response
[ ]
( )In this stage, a
β
-tracker followed by DF DFT-based channel estimation is applied to track channel. We start by utilizing the decision feedback (DF) DFT- based channel estimationmethod, using the ML criterion [9] [10][11][12], which is provided in the following:
( ), ( ),
(
( ),, ( ),,)
1 ( ),,( )
( ) 1 ( ),the estimated channel frequency response between the ith transmit antenna and the jth receive antenna at the vth iteration, Xˆ i =diag X
{
ˆ i[Θ1],...,Xˆ i[ΘΘ]}
Θ Q
consists of the decision data symbols in which is a subset of used to track channel variations and Xˆ [( )i k can be ] obtained by applying the previously estimated channel frequency response to the V-BLAST data detection with respect to the received signals. Fd DFT( ),j i, represents the Θ×κ( )j i,
truncated IDFT matrix which can be equivalently expressed as
j i j i H
d IDFT = d DFT
F F , and FDFT(j i,) is the N×κ( )j i, truncated DFT matrix. For simplification, the
subscripts “DFT” and “IDFT” are omitted, then Fd DFT( ),j i, , Fd IDFT( ),j i, and FDFT(j i,)
(
are replaced by
)
, j i
Fd ,
( )
Fd( )j i, H, and F(j ,i) respectively. We assume that the data interference signals from other transmit antennas can be reconstructed and cancelled perfectly from the received signals( )
,( )j
R . Thus, the refined signals corresponding to the j i
( ) ( )
[ ]
( )[ ]
( )[ ]
( )[ ]
( )[ ]
th antenna pair can be represented as Moreover, since the refined signals from different antenna pair do not interfere with each other, we can perform the channel estimation for each transceiver antenna pair and, for simplification, the superscript “
( )
j, i ＂ and “( )
i ＂are dropped hereafter. Therefore, the DF-DFT based channel estimation method can be expressed as follows:( )
1This is, however, not a good solution in fast time-varying channels because decision data symbols easily induce the error propagation effect. Therefore, pilot tones play an important role in fast fading channels. While in the slowly fading channels, decision data symbols are more reliable than pilot tones due to the number of data subcarriers are much more than pilot tones. Hence, we can modify the first iteration by adopting pilot tone as well as decision data symbols simultaneously to perform channel estimation at the first iteration. This, we modify the first iteration with v=1 by the
β
-tracker in which the current channel can be weighted with the channel estimated over data symbols in the previous time slot and the channel estimated over pilot signals in the current time slot due to that the current channel is related to the previous channel, which depends on how fast the channel varies. Therefore, Theβ
-tracker for the first iteration (v=1) can be described as follows:where represents the channel estimated over data symbols in previous time slot, denotes the channel estimated by DF DFT-based channel estimation over pilot subcarriers in which is the diagonal matrix of pilots, is the corresponding received pilot signals, and is the J×κ( )j ,i truncated DFT matrix over pilot subcarriers. Likewise, since the received pilot signals from different antenna pair do not interfere with each other, we can also perform the channel estimation o for each transceiver antenna pair. In order to initialize the channel estimator of (4.8), the CSI estimated in the last iteration of previous time slot has to be taken as the initial value of the CSI for the current time slot.
Then, we can construct an MMSE cost function over all subcarrier indices, and derive the minimum β as follows:
where is the true value of current channel frequency response. Each component of β can be derived separately in the following. From (4.8) and (4.9), we have
( )
Let the estimation of be the summation of the true previous value plus a noise term, and the estimation of
v−1
H
H also be the summation of the true current value plus a noise term P
as follows:
( )
Furthermore, the autocorrelation function of the channel frequency response is as follows:
(4.14)
where denotes the zero-order Bessel function of the first kind, and f is Doppler frequency in hertz, σh2 is the channel power, and is the time duration of one OFDM symbol after adding the guard interval.
T
Hence, from (4.12) ~(4.14) , (4.10) is equivalent to
⎡ ⎤ ⎡ ⎤
Similarly, we can obtain
(4.16)
Therefore, apply (4.15)~(4.17) into (4.9) then we can get the following equation:
(4.18)
We can observe from the above equation that as the mobile speed is slow,
and then
It is noted that, after the first iteration, we execute the channel tracking process of (4.7) for the second and subsequent iterations until a stopping criterion holds. The stopping criterion is to check whether the iteration number reaches the maximum value of . The channel
tracking process for the current time slot will be stopped when the above condition holds.
β
Fig.4 outlines the -tracker followed by DF DFT-based channel estimation method to estimate the average channel variations in the tracking stage. We perform the channel estimator described above for each transceiver antenna pair respectively, and then after each iteration we have Hˆ( )j i, = ˆ( )j i,
[ ]
k∈Ω= 0,{
…,N−1}
i= …1, ,NT detect data by performing V-BLAST with MMSE-SIC algorithm on each subcarrier as mentioned in chapter 3.( ) ( )
[ ] { }
β
-tracker followed by DF DFT-based channel estimation method in the tracking stage Fig. 54.3 ICI Estimation
In the presence of Doppler caused by high mobility, due to the ICI term of (2.8), using the estimate of average channel for data detection results in poor performance. This motivates the need to perform ICI estimation.
In this stage, we adopt a linear model with a constant slope over the time duration of one OFDM symbol to approximate time variations of each path. When the linear model is applied, we will derive the frequency domain relationship, similar to (2.8). We first consider the SISO case with one transmit and one receive antenna by omitting the superscript “
( )
j i,)
The received signal of SISO case is as follows:
X k H k k X m H k m Z k
the channel frequency response from the mth subcarrier on the kth one which can be expressed as:
Furthermore, we assume
β
-tracker and canbe expressed as
[ ]
1 ,0 2where μl,0 is the average channel variation of the lth path. Through an IDFT, we have
[ ]
2linearization, knowledge of the channel at one time instant in the symbol is necessary. As mentioned in [7], let
Consider linearization around 1
l 2
h ⎡⎢N − ⎤⎥. Then,
[ ]
⎣ ⎦ h nl
[ ]
can be approximated as follows:
,1 1 ,0 0 1
l l 2 l
h n =μ ⎛⎜⎝n−N + +⎞⎟⎠ μ ≤ ≤ −n N
[ ]
(4.25) Inserting (4.25) into (4.21), we will have
2
Subtract the non- ICI term from the received signals and we will have k
[ ] [ ] [ ]
1[
[ ] [ ] [ ]
For simplification, let
[ ]
and we can modify the cost function as[ ] [ ] [ ]
Direct calculation yields that (4.31) is equivalent to[ ] [ ]
'Now we define Equation (4.35) can be written in matrix form as
(4.36)
Hence, the slope of each path can be estimated by
(4.37) and with the knowledge of the slope of each path, we can reconstruct the channel frequency
response H k m,
]
as shown in (4.26).Now, with the ICI estimation for SISO case, we can apply it to MIMO case by modifying the received signal Y k
[ ]
in (4.20) and perform the ICI estimation for each antenna pair. In MIMO environment, received signal at each receive antenna suffers from inter-antenna interference including intercarrier interference (ICI) as follows:(4.38)
To let the received signal in MIMO case corresponding to the i th antenna pair, named it as refined signal, be the same as Y k
[ ]
in SISO case, we need to subtract the inter-antenna interference including ICI from the received signal R( )j[ ]
k as follows:( )j i
[ ]
antenna pair can be treated as Y k[ ]
in SISO case. However, in the beginning of this stage,( )j i,
[ ]
R k
( )j n,
[
,]
H k m is not available, so we cannot obtain . Therefore, with the simulation in the following, we use another refined signal R( )j i,
[ ]
k(
which is obtained by subtracting inter-antenna interference excluding ICI term from the received signal R j)
[ ]
k as follows to estimate the slope of each path:In the following simulation, we assume that average channel variations μl are known and inter-antenna interference excluding ICI can be perfectly subtracted from received signal
[ ] [ ]
( )j ( )j i,
R k . We adopt the refined signal R k , and perform ICI estimation to estimate the
slope of each path for each antenna pair, and then we have Fd*T=0.02 for 6 path Fd*T=0.02 for 2 path
Fig. 6 NMSE of the channel estimation with the assumption that the average channel variations μl( ),0j i, 0,for l= …,L( )j i, are known versus Eb/No
Fig. 6 shows the normalized mean square error (NMSE) of channel estimation with the assumption that the average channel variations μl( ),0j i, 0,for l= … L, ( )j i,
equals to 0.02 and 0.1, respectively. As can be observed in Fig. 6, the MSE of channel variations are all quite small even in low region. Hence, to perform ICI estimation for MIMO case, we just need to modify (4.20) as follows:
[ ]
( )j i,[ ]
( )j[ ]
( )j n,[ ]
, ( )n[ ]
In the above derivation, we have assumed that the transmitted signals for each antenna are known. However, in reality, the transmitted signals are obtained by applying the V-BLAST detection as shown in chapter 3 to received signals.