Interpolation

= F U ,where U is a M × M unitary matrix, F’ is also an optimal precoding matrix. Because the precoder is calculated independently for each subcarrier, the unitary matrix U for each subcarrier is also arbitrarily determined. However, the choice of unitary matrix U has a substantial influence on the performance of an interpolator.

2.4.2 Interpo

mprove the performance of

{ (1), (F F K+1),... (F N− +K 1)}

1. It is not

As we mentioned at Section 2.3.1, the optimal precoder is not unique. That is, if F’

opt n n

n

n

23

se observation, [5] proposes the following interpolation algorithm.

Based on the . We can see that (2-32) is simply a linear interpolator w matrix

ensures that the orthono the precoder after interpolation. The role of the unitary matrix is to solve the non-uniqueness problem. can be found in number of ways, such as maximizing the capacity

(2-33)

Q

i ix.

c

_m=(

m

−1) /

K

,

1 m K≤ ≤ ith an additional

i. After interpolation, a projection is then required. In equation (2-33), F is the projection of Z into ( ,

U N M with respect to the Frobenius norm, and thus it

_t )

where CQ is a codebook for unitary matrix . Note that,

cause a higher computational complexity for the search of the best and more feedback data (for sending the information about ). In [5], a suggested codebook for which contains 4 codewords is shown below:

Q

i a large size of CQ will

Clustering and interpolation both exploit the correlation between neighboring subcarriers. If the channel has a large coherent bandwidth, the data amount for feedb

− −

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢

ack can be significantly reduced by clustering scheme. The interpolation scheme

24

oposed codeword search method

rch is required to find the optimal codeword in the codebook. That is, if we have a codebook with L code

We use a sub-optimal codeword selection criterion which minimizes the chordal distance (Equation 2-20) between the chosen codeword and the ideal (un-quantized) optim

proposed in [5] can further improve the performance of clustering. However, due to the difficulties we mentioned above, the interpolation scheme requires additional feedback information, and also higher computational complexity. When the coherent bandwidth becomes small, these techniques will apparently suffer performance degradation.

2.5 Pr

For conventional codebook-based precoding, an exhaustive sea

words (size = L), we then need to conduct the same operation for L times to find the optimal codeword. At this section, we propose a codeword search method which can reduce about 80% searching complexity with acceptable performance loss.

al precoder.

The simulation shows that this criterion has perfor

With this distance-based codeword selection criterion, a low complexity codeword searc

The proposed codeword search method is composed of the following two steps:

1. Codebook partition: Given any appropriately designed codebook with L

codewords, we first partition this codebook with a simple distance comparison

mance comparable to MSV-SC.

h method becomes possible.

algorithm. After this partition step, the codebook will have a tree structure. Note that, this is an off-line step.

25

2. Codeword searching: With the partitioned codebook known by both the

transmitter and the receiver, a tree searching algorithm can be performed to find an

In codebook partition step, we first find two codewords which have maximum chordal distance. This can be done with an exhaustive search manner. With the two farth

, then X will be referred to group a.

, then X will be referred to group b.

optimal codeword within this partitioned codebook.

est codewords, the other codewords then can be partitioned into two groups with a simple distance comparison algorithm. Assume X is a codeword,

If ( , )

d

_chordal

A X

<

d

_chordal( , )

B X

If ( , )

d

_chordal

A X

>

d

_chordal( , )

B X

Figure 2-6 Partition the codebook into two groups

Then, we can further find two farthest codeword C and D in group a, and two farthest codeword E and F in group b. Following the distance comparison algorithm, group a can be further partitioned into group c and group d, and group b can be partitioned into group e and group f.

Figure 2-7 Partition the codebook into 4 groups

Finally, the codebook will be partitioned into 2, 4, 8, … ,2^k groups , where the

integer k can be define odewords within each

group must be recorded during the partition process. Note that groups at depth k are all contained in groups at depth k-1, in other words, the partitioned codebook has a nested structure.

arching depth k is 3. Note that this codebook is partitioned unequally (number of codewords in each group at same depth is not equal).

d as the searching depth. The farthest c

Figure 2-8 shows an example of codebook partition with codebook size = 64.

The maximum se

Figure 2-8 Example of codebook partition

After codebook partition step, the tree structure is determined and stored. In online codeword search, we can then perform a tree search algorithm to locate the

26

27

optim

opt

al codeword is assum

opt

assumed to be within group c. Repe

Assume a codebook can be partitioned equally at each level, then we can easily express the complexity of the proposed codeword searching algorithm as follows:

al codeword within this partitioned codebook. First, we calculate the ideal optimal precoder Fopt by SVD of channel matrix H. Then, we find a codeword Fi

which has minimum chordal distance to Fopt. The first step is to compare ( , )

chordal opt

d A F

and

d

_chordal( ,

B F

_opt), where A and B are two farthest codewords within the codebook (the largest group). If F is nearer to the codeword A, the

be within group a. Then, further compare ( , )

chordal opt

d C F

and

d

_chordal( ,

D F

_opt), where C and D are two farthest codewords within the group a. If F is nearer to the codeword C, the optimal codeword is at this process, and finally we can find the optimal codeword. Using the algorithm, we can significantly reduce the searching complexity.

optim ed to

2 2^k

Searching complexity= k+ L (2-37)

Table 2-1 shows the searching complexity for L = 64 and L = 128.

Table 2-1 Searching complexity for equally partitioned codebook

Note that k = 0 corresponds to exhaustive search. Note that this complexity indicates

the number of chordal distance calculations. Although it’s difficult to partition the codebook equally, simulation shows that the average searching complexity with unequally partitioned codebook will approach to this result listed in table 2-1.

28

One problem with the proposed algorithm is that codeword searching error will occur under some situations.

Figure 2-9 Codeword searching error

As we show in Figure 2-9, the codeword Fi has minimum chordal distance to ideal

Table 2-2 shows the average complexity ratio and average distance error. The comp

optimal precoder Fopt. However, the optimal codeword is determined in the wrong group. This is because Fi and Fopt are too close to the partition edge. In this case, if we assume searching depth is two, the codeword chosen by tree search algorithm will be Fj, which is a sub-optimal codeword. The codeword searching error causes performance loss compared to exhaustive search.

lexity ratio is defined as the complexity of proposed tree search algorithm divided by complexity of exhaustive search. The complexity of exhaustive search is

equal to the size of codebook, L. The distance error can be defined as:

( , ) ( , )

chordal j opt chordal i opt

d F F

−

d F F

(2-38)

where Fi is the exhaustively searched codeword , and Fj is the codeword chosen by tree search algorithm. If there is no codeword searching error, the distance error will be zero.

Table 2-2 Searching com ratio and distance error

From table 2-2, we can realize that increasing the size of codebook, L, will decre

In order to lower the probability of codeword searching errors, we can modify the o

plexity

ase the complexity ratio and distance error. For instance, if L = 64, k = 3, the complexity for tree search algorithm only requires 21.77% of that for exhaustive search. If we increase L to 128, the complexity ratio can be further reduced to 17.25%.

Besides, increasing the searching depth k will decrease the complexity ratio but cause higher distance error. For L = 64, k = 3 or k = 4 will be good choices. Further increase the searching depth will not reduce complexity but will incur serious performance loss (high probability of codeword searching error).

riginal slightly. As the codebook partition method described above, we first find two codewords which have maximum chordal distance. The other codewords then can be partitioned into two groups with a modified distance comparison algorithm.

29

Figure 2-10 Modified codebook partition

Define a new factor called the overlap threshold, denoted as ε. Assume that X is a codeword,

If d_chordal( , )A X −d_chordal( , )B X <

ε

, then X will be referred to both groups, a and b.

If d_chordal( , )A X −d_chordal( , )B X >

ε

, then the algorithm is unchanged:

( , ) ( , )

chordal chordal

d A X

<

d B X , then X will be referred to group a.

( , ) ( , )

chordal chordal

d A X

>

d B X , then X will be referred to group b.

As shown in Figure 2-10, X is the nearest codeword to the ideal optimal precoder

F

opt. However, with the original partition method, they will be partitioned into different groups. It will cause codeword searching error when performing tree search algorithm. For the modified partitioned method, we refer X to both groups and thus avoiding the error. Obviously, a higher overlap threshold will result in a higher searching complexity since each group size is enlarged. Table 2-3 shows the average

30

complexity ratio and average distance error with this modified codebook partition algorithm for ε = 0.05

Table 2-3 Searching complexity ratio and distance error with modified codebook partition algorithm

Compared to table 2-2, we can find that the average distance error is smaller for different codebook size and different searching depth. However, the complexity ratio increases significantly, also. Thereby, how to choose an appropriate overlap threshold becomes a critical problem.

31

Chapter 3 Time domain CSI feedback

In this chapter, we propose time domain CSI feedback methods. Under some channel conditions, the proposed time domain CSI feedback method only requires a small amount of data. Even in the application of procoding, our method is comparable to the conventional precoder feedback scheme such as clustering. For time varying channels, we incorporate a differential pulse code modulation (DPCM) scheme in our time domain CSI feedback method such that the required feedback data can be further reduced.

3.1 Least squares method

We begin with an example of a 4 by 2 MIMO channel model shown in Figure 3-1. As we can see, the MIMO channel contains 8 single input single output (SISO) channels. We refer each SISO channel from one transmit antenna to one receive antenna as a Tx-Rx channel pair.

Figure 3-1 4 by 2 MIMO channel model 32

Figure 3-2 shows a typical time domain channel response for one Tx-Rx channel pair.

Figure 3-2 A time domain channel response

A simple way to quantize the time domain channel response is to directly quantize the complex value and delay for each channel tap, individually. That is, quantize a1, b1, a2, b2, …, a6, b6, P1,P2,…, P6 , if there are 6 channel taps. However, it may require large amount of quantization bits. Therefore, we propose to quantize the overall time domain channel response, jointly. For this purpose, we first shorten the channel taps by removing insignificant taps for each Tx-Rx channel pair. Then, we sort the shortened channel based on magnitude, as shown in Figure 3-3.

Figure 3-3 Shortening and sorting the time domain channel response

After shortening and sorting, the magnitude of time domain channel response will have high correlation between each channel tap. Thus, we can apply least squares

33

(LS) to fit these sorted taps with a straight line or a higher-order polynomial curve and thus avoid the quantization of each channel tap. Notice that, the delay information is quantized before shortening. The delay information fed from receiver back to transmitter can be used to recover the original taps before shortening and sorting.

Besides, the sorting operation is based on magnitude, thus the phase information must be quantized with other scheme. Because the phase for each tap is i.i.d. and has uniform distribution, we simply apply an uniform quantizer for the phase information.

Least squares (LS) method is a well-known curve fitting method. Given N observed data, we can find a straight line or a higher-order polynomial curve to fit these data with a minimum squared errors. Figure 3-4 shows an example of linear fitting:

Figure 3-4 First-order polynomial least squares fitting

We express the N observed real-valued data as:

[

1 2

]

T

x x x

N

=

X

" (3-1)

The LS method finds a parameter vector θ to minimize the squared error vector ( )

J θ , which can be written as:

( ) ( ) (^T

J θ

=

X - Kθ X - Kθ) (3-2)

34

where Kθ is the fitting vector, and K is a known observation matrix. For the first-order polynomial fitting, K can be written as:

1 0

For the second-order polynomial fitting, K can be written as:

2

If the gram matrix K K^T is non-singular, then the least squares solution will be:

ˆ_LS =( ^T )⁻1⋅

θ K K K X^T (3-5)

Figure 3-5 shows an example of the first-order polynomial fitting for 6 sorted channel taps.

Figure 3-5 First-order polynomial fitting for sorted channel taps 35

A second-order polynomial fitting for 6 sorted channel taps is shown in Figure 3-6.

Figure 3-6 Second-order polynomial fitting for sorted channel taps

Obviously, the second-order polynomial fitting has higher accuracy (lower squared-errors) compared to the first-order polynomial fitting.

Now we can briefly describe the time domain feedback scheme with LS fitting method as follows:

1. Quantize the delay and phase information first. Then, shorten and sort the channel taps using their magnitude for each Tx-Rx channel pair.

2. Apply LS method to fit the sorted magnitude response for each Tx-Rx channel pair.

3. Feedback the quantized LS parameters together with the quantized phase and delay information to the transmitter side.

At transmitter, we can reconstruct the time domain channel response using the LS parameters, quantized phases, and delay information. If precoding is conducted at

36

transmitter, the precoding matrix for each active subcarrier can be obtained by performing singular value decomposition (SVD) on the reconstructed frequency domain channel response.

3.2 Discrete cosine transform method

In this section, we propose another time domain CSI feedback scheme using discrete cosine transform (DCT). Assume that x(n) is a real-valued sequence. A one-dimensional DCT can be expressed as follows.

1

It has been shown that many physical signals can be accurately reconstructed using only a few of their DCT coefficients. Therefore, it is useful in data compression.

We can easily extend the one dimensional DCT in (3-6) to two-dimensional DCT as:

1 1

Comparing (3-6) and (3-7), we can see that the two-dimensional DCT is equivalent to two one-dimensional DCTs, performed along one dimension followed by another DCT in the other dimension. Two-dimensional DCT is a very common

37

method for image compression.

As discussed in Section 2.1, shortening and sorting for each Tx-Rx channel pair will generate correlations between the channel taps. Therefore, a joint quantization strategy such as LS method proposed in the previous section can be applied to quantize the channel taps with only a few parameters. Note that, each Tx-Rx channel pair is quantized independently in the LS fitting method. However, correlations also exist between different Tx-Rx channel pairs. For small separation distance between the receive antennas, the channel pairs from one transmit antenna to different receive antennas are often very similar. Therefore, this inspires us to apply a two-dimensional quantization scheme such as DCT to quantize multiple Tx-Rx channel pairs.

After shortening and sorting to each Tx-Rx channel pair, we can collect the sorted taps for all channel pairs and regard them as a two-dimensional response. An example is shown in Figure 3-7.

Figure 3-7 Sorted taps for the entire MIMO channel 38

Now we can perform two-dimensional DCT to the sorted magnitude response in Figure 3-7. The transformation result is shown in Figure 3-8.

Figure 3-8 Two-dimensional DCT

Apparently, there are two significant parameters at positions (1,1) and (1,2). We can extract these two parameters to reconstruct the sorted magnitude response by inverse two-dimensional DCT, as shown in Figure 3-9.

39

Figure 3-9 Reconstructed magnitude response with two DCT parameters

Since the response in Figure 3-9 is reconstructed with two most significant parameters, distortion is unavoidable. Figure 3-10 shows the reconstructed sorted magnitude response with six most significant parameters. Distortion is obviously lowered compared to that in Figure 3-9.

Figure 3-10 Reconstructed magnitude response with six DCT parameters

We briefly describe the proposed time domain feedback scheme with DCT as follows:

1. Quantize the delay information and phase information first. Then, shorten and sort the channel taps based on magnitude for each Tx-Rx channel pair.

2. Apply DCT to the entire sorted magnitude response, and extract the most significant parameters.

3. Feedback the quantized DCT parameters together with the quantized phase and

40

delay information to the transmitter side.

3.3 Differential pulse code modulation

Differential pulse code modulation (DPCM) is a technique which is often used in speech coding or audio coding. It exploits the correlations between input signals and the quantization bits can be reduced significantly compared to conventional pulse code modulation (PCM). Conventional PCM is an instantaneous quantization scheme.

That is, to quantize the signal at different time independently. When the signal to be quantized varies slowly, conventional PCM is not efficient.

Figure 3-11 Quantization to the prediction error

The main idea of DPCM is to quantize the prediction error of signal, rather than the instantaneous signal itself. For example, assume we have exact value for S(n-2) and S(n-1), then S(n) can be predicted with a linear predictor. Let the predicted signal be . If the signal varies slowly with time, the prediction error is often very small and can be quantized efficiently. Figure 3-12 shows the block diagram of a simple DPCM scheme:

( )

S n



S n

( )− 

S n

( )

41

Figure 3-12 Open-loop DPCM

It is an open-loop DPCM scheme, where P denotes predictor, Q denotes the quantizer, and k is the discrete time index. However, this open-loop scheme will cause an accumulation of reconstruction errors. At transmitter, the prediction error e(k) can be expressed as:

( ) ( ) _T( )

e k

=

S k

− 

S k

(3-8)

where S(k) is the input signal, and

S k

_T( ) is the prediction signal at transmitter.

( ) ( 1)

S k

T = ⋅

h S k

− (3-9) From (3-8) and (3-9), we can write the input signal S(k) as:

)

( ) ( 1) (

S k = ⋅h S k− +e k (3-10) Iterating (3-10) for k =1,2,…K, we have

(2) (1) (2) (0) (1) (2)

Equation (3-11) expresses the input signal S(K) in terms of the initial value S(0) and

prediction error e(k)’s. Now we turn to the receiver side. At receiver, the

42

reconstructed signal ˆ( )

S k

can be written as:

43

(3-12)

where ( is the prediction signal at receiver and is the quantized ˆ( ) _R( ) _q( ) where q(k) denotes the quantization error. Using (3-13) and (3-14), we can rewrite

) (3-15)

Iterating (3-15), for k =1,2,…K, we have

[ (1) (1)] [ (2) (2)]

Equation (3-16) express the reconstructed signal in terms of the initial value , prediction error e(k)’s, and quantization error q(k)’s. Assume that

. Comparing (3-11) and (3-16), we can write the reconstruction error (3-12) as:

( ) ˆ( )

Therefore, for an open-loop DPCM scheme, the reconstruction erro

roblem, a closed-loop DPCM shown in Figure 3-13 is often applied:

r will accumulated, as shown in equation (3-17).

In order to solve this p

Figure 3-13 Closed-loop DPCM

For closed-loop DPCM, nsmitter and that at receiver are m

the prediction signal at tra ade identical. That is,

S k

_R( )=

S k

_T( )=

S k

( ) on the previous reconstructed signal. We can express the reconstruction error

( ) ( )

45

From (3-18), we can find that the reconstruction error at time = k is identical to that at time = k. Therefore, for a closed-loop DPCM, the reconstruction

will not happen.

ple to demonstrate the effectiveness of the method. We use the spatial channel model (SCM) [12],

to the transmitter. Besides, the delay for each tap is assumed to be time-invariant. That is, on

slowly with time. Figure 3-14 and 3-15 show a variation of the two LS parameters A and B for linear fitting (Y=A+BX).

error accumulation

So far, we have described how to apply DPCM to quantize a slowly-varying signal. In many scenarios, the variation of channel taps is slow. We can then apply the DPCM method to further reduce the feedback data. Here, we use an exam

provided by 3GPP, as our time-varying channel model. The SCM channel model gives 6 non-zero taps for each Tx-Rx channel pair and their values change with time.

For our application, we let the speed for mobile station be 20 km/hr.

We assume that the channel is quasi-stationary, which means that the channel is time-invariant in one OFDM-symbol. For our system, one frame consists of 10 OFDM symbols. For each frame, only the CSI for the first OFDM symbol is fed back

ly the magnitude and phase information of the first OFDM symbol are fed back to the transmitter for each frame.

Now we can combine the DPCM scheme with the time domain CSI feedback schemes described in Section 3.1 and 3.2. The parameter for LS or DCT method now can be considered as a signals varied

Figure 3-14 Variation of parameter A

Figure 3-15 Variation of parameter B

Figure 3-16 shows the phase variation for one tap.

46

Figure 3-16 Variation of phase for one tap

Then, we quantize the parameters and phases by DPCM scheme with a linear

Then, we quantize the parameters and phases by DPCM scheme with a linear

在文檔中 應用於MIMO-OFDM系統之前置編碼搜尋與時域通道資訊回傳 (頁 32-0)