MIMO-OFDM channels can be estimated in either the time or the frequency domain. The received time-domain signal can be directly used to estimate the channel impulse response; fre- quency-domain channel estimation is performed using the received signal after processing it with the FFT. Here, we review both the time- and the frequency-domain channel-estimation methods, assuming that each channel is clear of interference from the other transmit antennas, which can be ensured by using the pilot designs described previously. Thus, the antenna indices and are neglected in this section, and these techniques are directly applicable to single- antenna OFDM systems as well.
Figure 5.17 Three different patterns for transmitting training signals in MIMO-OFDM
Figure 5.18 Training symbol structure of preamble-based and pilot-based channel estimation methods
TX
Training Signal
Time
T1 T2
NoSignal TX Training Signal NoSignal
Frequency f1
TX
f2
f3
f4
f1
f2
f3
f4
Data Pilot Null
TX
Training Signal 1
Training Signal 2
Orthogonal
(a) Independent Pattern (b)Scattered pattern (c) Orthogonal pattern
Preamble User Data
OFDM Packet (time domain)
1 OFDM Symbol 3 OFDM Symbols
Frequency
Time
Frequency
Time
2D (Time-Frequency) Interpolation
1D Frequency Interpolation
1D Time Interpolation d
e s a b - t o l i P d
e s a b - e l b m a e r P
TrainingSymbol Data Symbol
i j
5.7.2.1 Time-Domain Channel Estimation
Channel-estimation methods based on the preamble and pilot tones are different due to the dif- ference in the number of known symbols. For preamble-based channel estimation in the time domain with a cyclic prefix, the received OFDM symbol for a training signal can be expressed with a circulant matrix as
(5.67)
wherey and n are the L samples of the received OFDM symbol and AWGN noise, is the lth time sample of the transmitted OFDM symbol, and is the ith time sample of the channel impulse response. Using this matrix description, the estimated channel can be readily obtained using the least-squares (LS) or MMSE method. For example, the LS—that is, zero forcing—estimate of the channel can be computed as
(5.68) sinceX is deterministic and hence known a priori by the receiver. When pilot tones are used for time-domain channel estimation, the received signal can be expressed as
(5.69) whereXP is a diagonal matrix whose diagonal elements are the pilot symbols in the frequency domain, is a DFT matrix generated by selecting rows from DFT matrix F according to the pilot subcarrier indices, and
. (5.69)
Then, the LS pilot-based time-domain estimated channel is
(5.70) y=
(0) ( ) 0 0
0 (0) ( ) 0
(1) ( ) 0 (0)
h h v
h h v
h h v h
⎡
⎣
⎢⎢
⎢⎢
⎤
⎦
⎥⎥
⎥⎥⎥
⎡ −
⎣
⎢⎢
⎢
⎤
⎦
⎥⎥
⎥ +
− − +
x L x
x x L x L x L v
x x x
( 1) (0)
=
(0) ( ) ( 1) ( 1)
(1) (0)
n
(( ) ( 2)
( ) ( 1) ( )
(0) ( )
L x L v
x L x L x L v
h h v
− +
− −
⎡
⎣
⎢⎢
⎢⎢
⎤
⎦
⎥⎥
⎥⎥
⎡
⎣⎣
⎢⎢
⎢
⎤
⎦
⎥⎥
⎥ +
+
n
Xh n
= ,
x l( ) h i( )
hˆ
hˆ = (X X* )−1X y* ,
y=F X Fh* P +n,
F (P×v) (L×v)
[ ] = 1
( 2 ( 1)( 1)/ ) Fi j,
L exp−j πi− j− L
hˆ = (F X X F* *P P )−1F X Fy* *P .
5.7.2.2 Frequency-Domain Channel Estimation
Channel estimation is simpler in the frequency domain than in the time domain. For preamble- based frequency-domain channel estimation, the received symbol of the lth subcarrier in the fre- quency domain is
(5.71) Since is known a priori by the receiver, the channel frequency response of each sub- carrier can easily be estimated. For example, lth frequency-domain estimated channel using LS is (5.72) Similarly, for pilot-based channel estimation, the received symbols for the pilot tones are the same as Equation (5.71). To determine the complex channel gains for the data-bearing sub- carriers, interpolation is required.
Least-squares channel estimation is often not very robust in high-interference or noisy envi- ronments, since these effects are ignored. This situation can be improved by averaging the LS estimates over numerous symbols or by using MMSE estimation. MMSE estimation is usually more reliable, since it forms a more conservative channel estimate based on the strength of the noise and statistics on the channel covariance matrix. The MMSE channel estimate in the fre- quency domain is
(5.73)
whereH and Y here are the point DFT of H and the received signal on each output subcar- rier, and the estimation matrix A is computed as
(5.74)
and is the channel covariance matrix, and it is assumed that the noise/interfer- ence on each subcarrier is uncorrelated and has variance σ2. It can be seen by setting σ2= 0 that if noise is neglected, the MMSE and LS estimators are the same.
One of the drawbacks of conventional Linear MMSE frequency-domain channel estimation is that it requires knowledge of the channel covariance matrix in both the frequency and time domains. Since the receiver usually does not possess this information a priori, it also needs to be estimated, which can be performed based on past channel estimates. However, in mobile applica- tions, the channel characteristics change rapidly, making it difficult to estimate and track the chan- nel covariance matrix. In such cases, partial information about the channel covariance matrix may be the only possibility. For example, if only the maximum delay and the Doppler spread of the channel are known, bounds on the actual channel covariance matrix can be derived. Surprisingly, the LMMSE estimator with only partial information often results in performance that is compara- ble to the conventional LMMSE estimator with full channel covariance information. The perfor- mance of these channel-estimation and tracking schemes for WiMAX are provided in Chapter 11.
Y l( ) =H l X l( ) ( )+N l( ).
X l( )
H lˆ( ) =X l( )−1Y l( ).
Hˆ =AY, L
A=RH(RH+σ2(X X* ) )− −1 1X−1, RH = [E HH*]
5.8 Channel Feedback
As shown in previous sections, closed-loop techniques, such as linear precoding and transmit beamforming, yield better throughput and performance than do open-loop techniques, such as STBC. The key requirement for closed-loop techniques is knowledge of the channel at the trans- mitter, referred to as transmit CSI. Two possible methods exist for obtaining transmit CSI. First, CSI is sent back by the receiver to the transmitter over a feedback channel. Second, in TDD sys- tems, CSI can be acquired at the transmitter by exploiting channel reciprocity, or inferring the downlink channel from the uplink channel, and can be directly measured. Our discussion focuses on the feedback channel, namely on an efficient technique based on quantized feedback [43]. Quantized feedback will be discussed for linear precoding, but it is applicable for other types of closed-loop communication, such as beamforming [44], adaptive modulation [69], or adaptive STBC [42].
The development of quantized precoding is motivated by the need for reducing the channel feedback rate in a MIMO linear precoding system. Ideally, the transmit precoder would be informed by the instantaneous and exact value of the matrix channel between the transmit and receive antenna arrays. But accurate quantization and feedback of this matrix channel can require a large number of bits, especially for a MIMO-OFDM system with numerous antennas, subcarriers, and a rapidly varying channel. Quantized precoding techniques provide a solution for this problem by quantizing the optimal precoder at the receiver. Specifically, the precoder is constrained to be one of distinct matrices, which as a group is called a precoding codebook.
If the precoding codebook of matrices is known to both the receiver and the transmitter, only bits of feedback are required for indicating the index of the appropriate precoder matrix.
The number of required feedback bits for acceptable distortion is usually small, typically 3–8 bits. Figure 5.19 illustrates a quantized precoding system.
Typically, the precoding codebook is designed to minimize the difference between the quan- tized precoder and the optimal one, which is referred to as the distortion. The MMSE is a typical distortion measure; another is the Fubini-Study distance
(5.75)
whereA and B are two different matrices. Other possible distortion measures include chordal distance and the projection 2-norm, but these distortion measures do not easily allow for optimal precoding codebooks to be derived analytically and so are usually computed using numerical methods, such as the Lloyd algorithm [25]. These techniques have been shown to provide near- optimal performance even with only a few bits of channel feedback [43].
The effectiveness and efficiency of quantized precoding has led to its inclusion in the WiMAX standard, which has defined precoding codebooks for various channel configurations.
It also should be noted that in the WiMAX standard, channel sounding, a method for obtaining transmit CSI through reciprocity, has been defined for TDD systems.
N N log2N
d( , ) =A B arccos det| (AB*) |,
5.9 Advanced Techniques for MIMO
In addition to the single-user MIMO systems that use diversity, beamforming, or spatial multi- plexing, these techniques can be combined and also used to service multiple users—mobile sta- tions—simultaneously. In this section, we briefly look at some of these advanced concepts for increasing the capacity, reliability, and flexibility of MIMO systems.