Chapter 3 Standards of IEEE 802.11n (WWiSE) and IEEE 802.16a
3.2 IEEE 802.11n WWiSE Proposal
3.2.3 The Code Structure of Space-time Block Code
From Table 3.3, the transmission model of space-time block code can be described in following equations. Assume the channel responses and are static in a STBC symbol (two consecutive OFDM symbols).
h0 h1
* *
When the number of data streams is three and four, we use the following space-time block codes, H3 and H4, to encode the data. The decoders for H3 and H4
are derived in the Appendix of [14].
( )
Chapter 4
Channel Estimations for MIMO OFDM Systems
4.1 Preamble Design for MIMO OFDM Systems
In [15], several types of pilot arrangements are proposed for MIMO OFDM systems. In this thesis, three of them will be discussed. In this section, these pilot arrangement methods will be introduced briefly. The pilot arrangements concerned are all-pilot preamble, space-time coded preamble, and scattered preamble, respectively. The same spatial arrangement may combine different time-frequency preamble formats, such as block type (802.11n) and comb type (802.16a with STC).
A simple category of pilot arrangement is shown in Figure 4.1. After a general study, channel estimation in both 802.11n and 802.16a systems will be discussed.
Figure 4.1 Classification of pilot arrangement in MIMO OFDM
4.1.1 Scattered Preamble
This preamble format is proposed in [15]. The scattered pilot preambles organize subcarriers in a single all-pilot-preamble symbol into several groups for different antennas. The transmission signal on each antenna can be expressed in the form of (4.1) and (4.2). The illustration of scattered preamble and data symbols is shown in Figure 4.2.
For pilot symbol assisted modulation (PSAM) OFDM symbol
(4.1)
1
2
( 0 0
(0 0 )
X P d d P d d
X P d d P d d
=
=
)
) ) where P is pilot tone and d is data tone
For block type OFDM preambles (802.11n-like)
(4.2)
1
2
( 0 0 0 0
(0 0 0 0
X P P P P
X P P P P
=
=
An OFDM packet
Data symbols coded by STBC
All-pilot preamble
Pilots for first antenna Pilots for second
antenna
Figure 4.2 Frame structure of a 2x1 MIMO OFDM system with scattered pilot
4.1.2 Space-time Coded Preamble
According to [2], space-time block code (STBC) can be applied to MIMO system so that the diversity of multiple antenna systems can be utilized. If the transmitted symbols are known, one can obtain the channel response from space-time coded OFDM symbols. The transmission scheme of this space-time coded preamble is depicted in Figure 4.3, and it can be seen how channel estimator (for preambles) and combiner (for data symbols) work. Table 4.1 lists transmission sequence of the space-time symbols between two antennas, and Figure 4.4 shows total arrangement of a whole packet in this kind of pilot arrangement.
Figure 4.3 Receiving and decoding structure of a 2x1 space-time coded system
Table 4.1 Training symbol arrangement of space-time coded preamble Antenna 0 Antenna 1
time t P0 P1
time t+T -P1* P0*
An OFDM packet
Data symbols coded by STBC
Space-time coded preable P0
P1
-P1*
P0*
P0-P1*
P1 P0*
Figure 4.4 Frame structure a 2x1 MIMO OFDM system with space time coded pilot
4.1.3 All-pilot Preambles
In a SISO OFDM system with packet transmission, the all-pilot preambles are often used. As introduced in Section 3.2.2, 802.11a system adopts this frame structure to perform channel estimation with its LTF preambles. If 802.11n system is needed to backward-compatible to 802.11a, the 802.11n system must reserve the feature of all-pilot preambles. As explained in section 3.2.2, WWiSE uses cyclic-shift version of original LTF for antennas other than the first one. However, this structure may experience severely co-channel (CCI) effect because pilots from different antennas occupy the same tones at the same time. For scattered preambles and space-time coded preambles mentioned previously, this problem can be avoided by tone-interleaving skills and space time block coding. The issue of CCI cancellation will be discussed later in this chapter.
4.2 Channel Estimation Techniques for MIMO OFDM System
In [16], the authors mainly introduce the methods to detect channel response on pilot tones based on LS and MMSE methods for SISO OFDM systems. For MIMO OFDM systems, the channel estimation problems may be more complicated. Due to the special structure of space-time coding and co-channel interference, some additional processing must be integrated into the MIMO OFDM system to solve these problems. In this section, we will study channel estimation methods for MIMO OFDM systems.
4.2.1 Channel Estimation for Scattered Preambles
Scattered preamble described in (4.2) is explored further here. To explain the estimation process, an example is given. The number of transmitter antennas is two, and the total amount of subcarriers in an OFDM symbol is 64. In this case, the frequency domain expression of two OFDM preambles can be described by (4.3).
1 and P P2
m
(4.3)
1 1 1 1
1 0 2 4 62
2 2 2 2
2 1 3 5 63
th
( 0 0 0 0)
(0 0 0 0 )
is the pilot symbol at tone from the antenna
m k
P P P P P
P P P P P
P k
=
=
In preamble symbol which belongs to a transmission packet, the channel effect and additive white noise can be modeled as.
(4.4)
Therefore, the received symbol vector R is
1 2 1 2 1
1 0 2 1 1 2 2 3 1 62 2 63
[ (0) (1) (2) (3) (62) (63) 2]
R= H P H P H P H P H P H P (4.5)
For estimation, the tones may be used for LS
estimation. The general form of estimation is like (4.6). The result can be also applied to channel response for the second antenna, and the extension to more than two antennas is straightforward.
H1 R(0) R(2) R(62)
However, only half the tones are obtained when the estimation in (4.6) is applied. The response on the tones occupied by training symbol from another antenna must be derived with interpolation techniques. In following discussion, some popular interpolation techniques are considered.
4.2.1.1 Piecewise Linear Interpolation
Linear interpolation is quite simple and intuitive among all interpolation skills.
The interpolation skills are based on linearity assumption of unknown subcarrier responses between known two known subcarrier intervals. If known subcarrier data is inserted for each M subcarrier, the segment length is M, and then subcarrier response interpolation in the mth segment can be obtained by
ˆ( ) M l- ˆ( ) l ˆ(( 1) ) ,0
H mM l H mM H m M l M
M M
+ = + + < < (4.7)
4.2.1.2 SPLINE Interpolation
Figure 4.5 Illustration of channel segmentation and the required known parameters for cubic spline interpolation
For generalization, specific-order spline functions can be derived such as the widely used cubic spline for channel interpolation. It is based on the third-order curve-fitting polynomial . Sufficient equations are required to solve this problem because there are total 4N’ unknown variables. All the divisional polynomial coefficients are solved based on continuous assumption at the segment boundary, with the first and second derivative continuity of the pilot channel values on segment boundaries. Therefore, (2N’)+(N’-1)+(N’-1) equations can be set
3 2
i i i i
Y =Ax +Bx +Cx +D
up from the constraints, with two more from the assumption of zero first order derivative value of the very first and last carrier channel value.
4.2.1.3 Transform-Domain Interpolation
DFT-based channel estimators have been proposed in [17,18]. These estimators are based on the techniques performed in transform domain to accomplish the estimation. Fast DFT algorithms can be utilized to reduce the transform complexity.
In the following, we will describe this method in detail. The DFT-based channel estimator has a principal restriction on placement of pilot subcarriers. That is, pilot subcarriers must be equi-spaced along frequency direction. A typical pilot pattern is shown in Figure 4.6. As the figure describes, DFT estimator can be applied only when Df is a constant.In such case, pilot tones can be viewed as a downsampled version of frequency response on all tones.
Figure 4.6 Regular pilot placement
When scattered pilots are used in MIMO OFDM, interpolations are needed to obtain channel response of interleaving tones for different antennas. In this case, transform domain methods are good choices because of the equi-spaced pilot tones in preambles.
Time
Df
Df
pilot
data
F r e q u e n c y
We can find different channels between different antenna pairs with interpolation skills.
In this subsection, we will discuss non-sample spaced channel effect in general wireless channels. As mentioned in Chapter 2, radio channel impulse responses can be modeled as several delay paths with random distributed gain (usually Rayleigh distribution). However, delay intervals are always assumed to be sample spaced. In real transmission environment, this assumption is not true for most cases. In the following parts of this subsection, this effect will be explored while transform-domain interpolation methods are used in MIMO OFDM channel estimation.
According to [19], the continuous channel impulse response can be expressed in (4.8), where v is the total number of channel delay taps, and εl is delay time for each tap. In (4.9), the frequency domain response is obtained by DFT, where τl is the delay interval normalized to sampling period Tc. And Tc equals to T/N.
discrete l l
l l
c
H k e e
k N
T sampling period
ε π τ
To obtain the equivalent discrete-time impulse response, IDFT is performed on .
discrete( )
H k
( ( 1) )
If the delay intervals are all sample-spaced, i.e.{τl}are all integers, can be simplified to
discrete( )
Table 4.2 Channel parameters for indoor wireless channel (Model 1) Tap No. Delay (ns) Delay
(samples)
Power (dB) Amplitude Distribution
1 0 0 0 Rayleigh
2 36 0.72 -5 Rayleigh
3 84 1.68 -13 Rayleigh
4 127 2.54 -19 Rayleigh
Table 4.3 Channel parameters for indoor wireless channel (Model 2) Tap No. Delay (ns) Delay
(samples)
Power (dB) Amplitude Distribution
1 0 0 0 Rayleigh
2 176 3.52 -8 Rayleigh
3 274 5.48 -15 Rayleigh
4 560 11.2 -18 Rayleigh
According to (4.8)-(4.10), the continuous channel response and the discrete channel response can be both illustrated in Figure 4.7 and Figure 4.8. The channel model is cited from [20]. Model 1 is in the environment of typical office, and Model 2 is in an airport hall. In Figure 4.7 and Figure 4.8, one can see that the delay spread of Model 1 is smaller than Model 2, which means that the delay taps are more concentrated in Model 1 than in Model 2. By observing the impulse response plot in Figure 4.7, we can see the aliasing effect in ‘high time’ part due to non-sample spaced channel. However, this effect seems not very obviously in Model 2. To explain this effect, we may examine (4.8)-(4.10) to find the answer. If we check sinusoid ratio part in (4.10), it would become (4.12).
) / ) ( sin(
) sin(
) 1
(n N n N
w
l l
× −
= π τ
πτ (4.12)
Equation (4.12) can be viewed as the gain of interference produced by other taps in continuous impulse response. It is plotted in Figure 4.9. Here, τl= 4.48 and N=64 are chosen as an example. It can be seen that Model 2 has larger attenuations on the multipath taps other than the main path. Additionally, the multipath taps are located at loose positions, so that the introduced aliasing effect is smaller than in Model 1.
Therefore the aliasing effect is not so significant in Model 2.
0 5 10 15 20 25 30 35 40 45 50 55 60 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Tap Delay (samples)
Maginitude
Discrete Channel Impulse Response
Original Continuous Channel Impulse Response
Figure 4.7 Continuous and discrete impulse responses of indoor model 1
0 5 10 15 20 25 30 35 40 45 50 55 60
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Tap Delay (samples)
Maginitude
Discrete Channel Impulse Response
Original Continuous Channel Impulse Response
Figure 4.8 Continuous and discrete impulse responses of indoor model 2
0 10 20 30 40 50 60 70 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7
n (samples)
Maginitude
Figure4.9 Magnitude plot of tap weight of non-sample spaced channel
Unfortunately, DFT-based channel [17] estimator is sensitive to non-sample spaced channel effect. Therefore the DCT-based estimator [19] is applied to this kind of MIMO OFDM system to improve the performance of transform domain channel estimator. In remaining part of this subsection, we will explain how non-sample spaced effect impacts the performance of typical DFT-based estimator. After that, we will show how DCT-based channel estimator compensates this drawback.
The DFT-based estimator starts with Least Square estimation of channel frequency response at pilot subcarriers. The LS estimation is described by (4.13). We assume the maximum delay spread isτmaxTc and ∆ is the minimum integer that is larger thanτmax. Then we shift the channel impulse response by −∆/2 so that the
power of channel impulse response is centered around n = 0. The shift process can be carried out by phase rotation in the frequency domain, as shown by equation (4.14).
( ) ( )
ˆ ( ) ( ) ( ) ( )
( ) ( )
where ˆ ( ) isthe estimated response at subcarrier, ( ) is the received signal,
( ) is the pilot,
where is the number of pilot tones
j k By the concept of interpolation, the estimated channel impulse response is obtained by zero padding. To reduce the aliasing effect, zeros must be padded to the region with less power. Sine we have centered the power around n = 0, zeros are padded in the middle of
{ }
hˆp(n) Mn=−01.After that we perform N-point DFT on , which results in interpolation in frequency domain, as shown below.
)
Finally, the estimated channel frequency response is obtained by removing phase rotation effect from .That is (4.18).
)
The whole estimation process of DFT-based estimator is shown in Figure 4.10. Figure 4.10 DFT-based channel estimator
Basic idea of the DCT-based channel estimator [19] is to make the input data symmetric so that the high frequency component is reduced, and then apply DFT-based interpolation algorithms. As a result, IDCT or DCT based channel estimation will be obtained. Although mirror-duplicating can be done in time domain, it can also be done by defining the extended pilot channel frequency response as [19]
⎪⎪
This design is compatible with the conventional inverse discrete cosine transform. For both two approaches, high frequency components are less significant than DFT-based approach, because the processed data are symmetric. Therefore, interpolation by using
would be better than the original DFT-based estimation.
) ˆ (k H2M
2
With , we can perform its DFT-based interpolation to get the estimated channel frequency response. To achieve IDCT/DCT-based algorithm, each step of DFT-based estimator will be translated to DCT-related operation. First, we perform IDFT on the extended to get the time-domain signal
2 1 2
This shows that the time-domain signal can be obtained by performing IDCT on followed by a constant multiplication. Next, continuing the interpolation by
zero-padding , we can get the corresponding time-domain signal to the target upsampled channel response, which is what we want to solve in the end.
)
Finally, the estimated channel frequency response is obtained by performing DFT on .
) ˆ (
2 n
h N
1
This equation is equivalent to a DCT operation combined with one constant multiplication. It is obvious that in the interpolation process, can be obtained by
IDCT and is the DCT transform of followed by one multiplication.
Therefore, the whole IDFT-based interpolation can be replaced by DCT-based operations. This IDCT/DCT-based channel estimator [19] is shown in Figure 4.11.
We also show the equivalent channel estimations by IDCT-based interpolator and IDFT-based interpolator in Figure 4.12, where
)
Figure 4.11 IDCT/DCT-based channel estimator
)
Figure 4.12 Equivalent channel estimators by IDCT/DCT-based interpolator and IDFT-based interpolator
When we discuss 802.11n-like MIMO OFDM system, the scattered pilot makes the channel estimation easier compared to cyclic-shift preamble (no co-channel interference problem). Once the preamble signal is received, the channel response on pilot tones can be obtained by LS method. After that, we can further obtain responses on all subcarriers with interpolation skills. Transform domain interpolation is a good solution for such equi-spaced pilots. However, conventional DFT-based estimator suffers from non-sample spaced effect, as mentioned in previous part of this subsection. A comparison among different interpolation skills is presented in Chapter 5.
4.2.2 Channel Estimation for Space-time Coded Preamble
As introduced in 4.1.2, channel response can be detected if the pilots from transmitter are known. In Figure 4.3, it shows the special frame structure of space-time coding OFDM mentioned before. The assumption [2] in this system is that the channel responses between two consecutive OFDM symbols are the same. [2] also
assumes that the frequency response is flat in our concerned signal bandwidth, and this is true while OFDM system separates the large transmission band into N narrow subcarriers. Finally, the transmission signal on the kth subcarrier can be expressed as
1 2
According to [21], the received signals at consecutive time slots are and , which are formulated as
y1 y2
The channel response on this subcarrier can be estimated with the following two equations (4.24a) and (4.24b).
* (4.25a) and (4.25b) can be obtained from (4.23).
ˆh ˆh h1 h2
If the two previous equations (4.25a) and (4.25b) are observed, it can be obtained the new estimation error from white noise is turned to
' 1* 1 2 2
' 2* 1 1 2
2 2 2
1 2
( ) ( ) ( ) ( )
( ) ( ) ( )
P k w k P k w k
w k P k P k
⋅ − ⋅
= + (4.26b)
4.2.2.1 Decision-direct Channel Tracking for Space-time Coded Preamble
While the preamble is sent to receiver, the channel is estimated with known preamble. Once the channel is estimated, the residual data symbols are extracted with the estimated channel. A common decision-direct tracking skill can be applied to the space-time coded MIMO OFDM system. The idea is to use the demodulated data subcarriers after demodulation as new pilots, and then estimate the current channel response again with demodulated data. Then channel response is updated with the new estimation.
Figure 4.13 Illustration of received STBC symbol
In Figure 4.13, a received STBC OFDM symbol is shown to illustrate the decision-direct tracking skill. For preamble and , the channel is estimated by (4.24a) and (4.24b). That is
y1,1 y1,2
*
The data of STBC symbols are extracted from the channel estimates as
* are the decoded data. With these known data, the current channel response can be again estimated and updated by (4.29a) and (4.29b).
*
After that, it can be used to update the previous estimation by the following smoothing process. That is,
(4.30)
, 1,
ˆn i (1 )ˆn i n i h = −α h− +α where α is the forgetting factor
Performance improvement of the improved skills will be shown in Chapter 5.
4.2.3 Channel Estimation with All-pilot Preamble
While using all-pilot preamble in MIMO OFDM system, the main difficulty is the co-channel interference. The received preamble in frequency domain expression is the summation of signals from multiple antennas. For this reason, the interference must be cancelled before channel is estimated. In this subsection, the solution proposed by [22] will be studied first. After that, the complexity issue will be explored. Finally, a modification is proposed for this algorithm.
For convenience of the ensuing discussion, let’s define a MIMO channel response as
(4.31) is time-domain tap index,
is the subcarrier number, is the OFDM symbol index, is the transmission antenna index,
is the number of subcarriers, is the numbe
r of the most significant taps in impulse response, ( , ) is the frequency response of the i MIMO channel, ( , ) is the impulse response of the i MIMO channel
where is the received signal of nR received antenna, is the known preamble signal,
is the additive noise,
is the number of transmission antennas
nR
For a general case (NT transmit antennas), to estimate channel response, one can define the following square error cost function (4.33)
{ }
It can be easily shown that the optimum least-square-error (LSE) solution can be solved from the following normal equation.
(4.34)
Hence (4.34) can be transformed into
(4.37)
Furthermore, one can define the vectors and matrix,
(4.38)
As a result, (4.37) can be rewritten as (4.42).
(4.42) [ ] [ ] [ ]
Q n h n = p n One can solve the equation (4.42) by
(4.43) [ ] 1[ ] [ ]
h n =Q n p n−
The overall process of this LSE MIMO channel estimation technique can be illustrated by Figure 4.14.
Calculate
Figure 4.14 Function structure of LS channel estimation for MIMO OFDM systems using all-pilot preambles
4.2.3.1 Complexity Analysis of LS Channel Estimator for MIMO OFDM System Using All-pilot Preambles
From 4.2.3, it can be found that the complexity of LS estimator is high due to plenty of matrix operations, such as matrix multiplications and inversions. According to [23], the complexity of matrix inversion is proportional to Ο(N03) for general
cases, where is the dimension of the matrix. The detailed numbers of operations with Gaussian elimination are listed in Table 4.4.
N0
Table 4.4 Computational Complexity of matrix inversion with Gaussian elimination (N0 ≡NT×K0)
Operation Division Multiplication Addition Number its inversion is always hard to obtain due to high complexity. A simplified scheme is proposed in [22,24], the main idea is to estimate only K ′0 delay taps in time domain,
[ ] [ ] [ ]
Q n h n = p n (4.48)
[ ] 1[ ] [ ]
h n =Q n p n− (4.49)
4.2.3.2 Proposed Method for the Decision of Significant Number of Channel Taps
In 4.2.3.1, it is shown that the complexity of LS estimator can be reduced if small is chosen. In Chapter 5, the simulation results show that a proper brings good MSE performance, and the simulations exploit the fact that a proper value of is the number of those most significant taps in time-domain. In [22], the
detection of is not studied for this LS estimator. For this reason, we propose a simple method to improve this drawback of the methods.
K0 K0
K0
K0
According to the simulations in Chapter 5, the mean square error is large when is also large. While decreases and is closed to significant length of the channel, MSE is also reduced. An intuitive idea to decide is to use a large initial
According to the simulations in Chapter 5, the mean square error is large when is also large. While decreases and is closed to significant length of the channel, MSE is also reduced. An intuitive idea to decide is to use a large initial