J OINT TIME AND FREQUENCY DOMAIN CHANNEL ESTIMATION WITH WLS

Figure 4-9 shows the block diagram of the proposed time and frequency domain channel estimation method with the SIC-WLS method. The operations conducted in Figure 4-9 is described below.

-10 -2.5 0 2.5 10 15 20

SNR(dB) Weight of each region=

0.01 0.05 0.1 0.3 0.5 0.7 0.9

Figure 4-9 The proposed time and frequency domain channel estimation method with the SIC-WLS algorithm

Initial frequency-domain channel estimate without interpolation

IFFT

The SIC-LS method in Figure 4-2

The nonaliasing part of the time-domain channel estimate Data detection

FFT

N

_re

=0

Iteration Block

N

re

= N

re

+1

START

Frequency-domain channel estimate by pilots and additional

pilots without interpolation

IFFT

The SIC-WLS method

Guard band insertion

Data detection Estimation of complete

time-domain channel

FFT

YES No N

_re

<N

_set

?

STOP

STEP 1: Use the pilot subcarriers to obtain the initial frequency-domain channel

estimate (without interpolation).

STEP 2: Conduct the IFFT of the frequency domain channel estimate to the time

domain.

STEP 3: Find the nonaliasing portion of the initial time-domain channel estimate.

STEP 4: Iteratively find the tap with the maximum value and use the SIC-LS method in

Figure 4-2 to estimate channel response of the tap until the least squared error (LSE) is smaller than the LSE threshold (the LSE threshold is larger here).

STEP 5: Conduct the FFT operation for the estimated channel to obtain the

corresponding frequency-domain channel estimate.

STEP 6: Use the estimated frequency-domain channel result and received signals to

estimate transmitted symbols and make data detections. Decisions at some designated subcarriers are used as pseudo pilots.

STEP 7: Insert the channel response in the guard band from the previously estimate

channel response when Nre≠0.

STEP 8: If the number of re-estimation has not reached the preset value, go to STEP 2.

Notice that the SIC-LS method in STEP 4 is replaced by the SIC-WLS method discussed in Section 4.4 during channel re-estimation.

Chapter 5

Proposed Time-Variant Channel Estimation

We have discussed the channel estimation problem in linear time invariant systems in Chapter 3 and Chapter 4. It has been shown that the OFDM technique has good performance for quasi-static frequency-selective fading channels. A channel is quasi-static if it remains time-invariant during an OFDM period. For this scenario, the performance of the channel estimation methods described in Chapter 3 and Chapter 4 work well. However, in high-mobility wireless environments, the channel becomes time variant within one OFDM symbol. The orthogonality of subcarriers in one OFDM symbol is no longer held and this causes the ICI. The frequency domain channel response is no longer a diagonal matrix as that shown in (2.10). From [8], the received signal with ICI for a subcarrier can be expressed as

1

Rewrite (5.1) using a matrix form, we can have

00 01 02 0( 1) where

y

is the frequency domain received signal vector,

x

is the frequency domain transmitted signal vector, M is the frequency domain ICI channel matrix, and

w

is the frequency domain AWGN noise vector. The channel estimation problem becomes more

involved. In this chapter, we will study the channel estimation problem in time-variant channels. We propose to use LS/WLS algorithm to identify the channels. Figure 5-1 shows an example of one time-variant channel tap. As we can see, the channel is variant within one OFDM symbol.

Figure 5-1 One time-varying channel tap

5.1 Linear approximation method of time-variant channel

The linear approximation method uses a straight line to fit the variation of one time-variant channel within one OFDM symbol, as shown in Figure 5-2. A straight line can be specified by two unknowns (the start point and its slope) and the number of unknowns for a complete channel is then limited. The approximation error of the the linear model has been shown to be small for the normalized Doppler up to 20% [6]. The normalized Doppler is defined as the maximum Doppler spread divided by the subcarrier spacing. Denote the maximum Doppler spread as

f . Then, we have

_D

C D

f v f c

= ×

(5.4) Where v is the mobile speed,

f is the carrier frequency, and

_C c is the speed of light. In the next section, we will use the LS method to identify the parameters of the linear model.

OFDM Symbols

……

Data Data

CP CP

time

Figure 5-2 Linear approximation of a time-variant channel tap

5.2 Time domain LS time-variant channel estimator

In Section 5.1, we approximate a time-variant channel tap by a straight line; we call the channel a linear time variant (LTV) channel. An LTV channel tap can be expressed as

k

( )

k k

h n = h + × n a

(5.5)

where

h n

_k

( )

denote the response of the

k

’th tap of a channel at time instant

n

,

h

_kis

the starting value, and

a

_k is the variation slope. Notice that we let

n

be zero at the start point of an OFDM symbol. Let the complete channel have Nc taps. We can then express the received an OFDM symbol as:

(

_v

) +

y = H + D A x w

(5.6) where

y

is the time domain received signal vector,

x

is the time domain transmit signal vector,

w

is the time domain AWGN noise vector, H is a circulant matrix with

0 1 1 The Estimation of Slope

of the Second Symbol The Estimation of Slope

of the First Symbol

……

OFDM Symbols

Data Data

CP CP

The Estimation of Start Point of the First Symbol

The Estimation of Start Point of the Second Symbol

time

[0,1, ,

N

_c 1]^T

= −

v as its diagonal elements .

Transforming the time domain received signal vector into the frequency domain, we have

y Gy

is the frequency domain received signal vector,

0 1 1

[ , , ,

_c

]

^T

c N

N x x x

₋

= =

x Gx

is the frequency domain transmitted signal vector,

0 1 1

w Gw

is the frequency domain AWGN noise vector,

h

=

^H

0 0

0 0

0 0 0

Combining the result of (5.10) and (5.14) and ignoring the noise term, we can rewrite (5.9) into the matrix form as:

= ^H + ^{H H} ⇒

We can finally rewrite (5.15) as we want to estimate. Now, we can use (5.17) to conduct the LS estimation. Since only the data on pilot subcarriers are available, we then select them from the rows of Y and ⎡⎣

B V

⎤⎦ when using the LS algorithm. Here, we assume that the number of significant taps is much smaller than Nc, and the delay of each significant tap is known. In other words, many elements in

h

_a will be zero. Only do the columns of B , V corresponding to the significant taps need to be considered. Figure 5-3 depicts the entries we need to consider when conducting the LS algorithm for (5.17).

Figure 5-3 Entries need to be considered in (5.17)

After removing irrelevant elements in (5.18), we can have

k k

p

⎡ ⎣

pk pk

⎤ × ⎦

a

= ×

a

y = B V h Q h

(5.19)

where

y

_p

= ⎣ ⎡ Y Y

_m0

,

_m1

,... Y

_mM−1

⎤ ⎦

^T is the frequency domain received signal vector on pilot subcarriers,

B

_pkis a sub-matrix of

B

whose rows are determined by the positions of pilot subcarriers, and columns by the positions of significant taps, and

V

_pkis a sub-matrix of V whose rows and columns are determined as those of

B

_p and

0

,

1

,...

1

,

0

,

1

,...

1

k K K

T a

= ⎣ ⎡ ⎣ ⎡ h h

k k

h

k ₋

⎤ ⎡ ⎦ ⎣ a a

k k

a

k ₋

⎤ ⎦ ⎤ ⎦

h

contains the parameters we want to

estimate and then, we have

0 0 0 1

, 0 1

M

i ≤ ≤

i M

− is pilot locations, M is the number of pilots, ,0

k

_i ≤ ≤

i K

− is channel 1 taps’ positions and K is the number of channel taps.

The LS solution is can then be obtained by

(

^H

)

^{-1 H}

a_k p

h = Q Q Q y

(5.20) Compared to the LS channel estimate for the time-invariant channel, the number of parameters in (5.20) is doubled. This is because for each tap we need to estimate two parameters, the value of the starting value and the variation slope. As a result, the accuracy of the estimation result is affected. To improve the performance, we can use decisions as pseudo pilots as discussed in Section 4.2 and Section 4.3. As the number of pilots increases, the performance of the LS estimate can be improved. As we did in Chapter 4, the channel estimation with decisions can be made iteratively and the estimation performance can be further improved. Figure 5-4 shows the LS estimation for the time-variant channel with decisions.

Figure 5-4 The LS time-variant channel estimation with decisions

0

Pilot & pseudo pilot tone

Rewriting Equation (5.19), we can obtain positions of pilots and decisions. And, the LS solution now becomes

( )

^'

k

'

H -1 H

a

' '

p

h = Q Q Q' y

(5.22) Since not all decisions are correct, the performance of the LS channel estimator may not always satisfactory. So, we can use the WLS algorithm as discussed in Section 4.4 to solve the problem.

5.3 Time domain time-variant channel estimation with WLS

We have discussed the WLS algorithm in Section 4.4. The weighting matrix is denoted by a diagonal matrix

D

=

diag d d

( , , ,_{0 1}

d

_J−1),

j

=

j j

_o, , ,₁

j

_J−1 are the positions of pilots and pseudo pilots. Figure 5-5 shows how the weights are added in (5.20).

Figure 5-5 Weighting method in WLS algorithm

0 0 0 0

Pilot & pseudo pilot tone

Thus, (5.21) now becomes

' ' ' _k, _k,

p

⎡ ⎣

p k p k

⎤ × ⎦

a wls

= '' ×

a wls

Dy = D B V h Q h

(5.23) where

Q ''

= ⎣

D B

⎡ _{p k'}

V

_{p k'} ⎤⎦ and the solution of the WLS channel estimate is then

( )

, '

k

H -1 H

a wls

'' '' ''

p

h = Q Q Q Dy

(5.24)

Here, we use a simple weight scheme: the weight of a pilot is set to be 1 and the weights for decisions are smaller than one and all the same. The decision weight for different modulation schemes and different ICI levels may be different and it will be determined in the simulation chapter. The comparison of the LS and WLS channel estimator will also be shown in the simulation chapter.

5.4 Time-variant channel estimation by time domain WLS channel estimator

Figure 5-6 shows the block diagram of the complete scheme for the time-variant channel estimation. As discussed in Section 4.1, we only identify the channel taps in the non-aliasing region. The detail operation is summarized in the following procedure.

Figure 5-6 Time Domain time-variant WLS channel estimation

STEP 1: We assume that the channel taps’ positions in the nonaliasing area can be

identify.

STEP 2: Use the LS algorithm discussed in Section 5.2 to estimate the parameters of the

time-variant channel.

STEP 3: Construct the ICI matrix M from Equation (5.7).

STEP 4: Use the zero forcing equalizer to obtain estimate transmit symbols

(

y

=

Mx + w , x M y ) and make decisions as discussed in Section 4.1.3.

ˆ = ⁻¹ Using decisions as pseudo pilots; here, we let the pilot density be 1/3 as that we did in Section 4.2.

STEP 5: Assume that al1 the channel taps’ positions are known (discussed in Section

4.3).

The positions of the channel taps in the the nonaliasing part

LS alogorithm to estimate these taps’ start points and slopes

by pilots

Data detection

Data detection

Weighted LS to estimate these Taps’

start points and slopes by pilots and pseudo pilots

Construct the frequency domain ICI matrix

The positions of all channel taps Construct the frequency domain

ICI matrix

STEP 6: Use WLS algorithm discussed in Section 5.3 to re-estimate all the parameters

of the channel taps.

STEP 7: Construct the ICI matrix M from (5.7).

STEP 8: Use the zero forcing equalizer to re-estimate transmit symbols ( x M y ).

ˆ = ⁻¹

STEP 9: Go to STEP 4 if the number of re-estimation, N

re is less than a preset number

Nset.

Chapter 6

Simulation Results

In this Chapter, we conduct simulation to evaluate the performance of the proposed joint time and frequency domain channel estimator in both the time-invariant and time-variant channels. In our channel model, there are 6 channel taps (paths) and the average power of these 6 taps will be determined according to the signal-to-noise ratio (SNR) used. The bit-error-rate (BER) is used as the performance index. We consider two OFDM systems; one is the standard DVB-T system and the other is an OFDM system we define. For the former system, we use two types of channels with different delay spreads.

The maximum delay (MD) for the first one is smaller than 64Ts and that for the second one is between 171Ts and 256Ts. The channel estimate for the first-type channel in the DVB-T system will not be aliased while that for the second-type channel will be aliased. We refer the first-type channel as Channel A and the second-type of channel as Channel B. The MD of the channel used for our defined system is between 43Ts and 64Ts and the corresponding channel estimate will be aliased. We refer this type of channel as Channel C. The FFT size is set as 2048 in the DVB-T system and the FFT size is set as 512 in our defined system.

Figure 6-1 shows one example of the channel. Figure 6-2 shows the variation of the 6 taps in the fading environment.

0 50 100 150 200 250 300

Figure 6-1 An example of 6-tap channel

0 2

Figure 6-2 Variation of channel taps in fading environment

6.1 Results of channel estimation in Chapter 3

Table 6-1 shows the parameters of the 6 channel taps used in simulations. The system we consider here is the 2K-mode DVB-T system and the size of the CP is 256 (1/8 of 2048).

Modulation schemes such as QPSK, 16QAM and 64QAM are used. The delays of the channel taps are randomly changed for every 7 OFDM symbols. For a block of consecutive 7 OFDM symbols, they remain the same.

Tap Number Average Power (Lin) Average Power (dB)

1 0.9951 -0.0215

2 0.5727 0.2.421

3 0.4992 -3.0172

4 0.4273 -3.6927

5 0.3558 -4.4880

6 0.2845 -5.4595

Table 6-1 Parameters of multipath fading channel

6.1.1 Results of different interpolation methods

Figure 6-3 shows the performance comparison for different interpolation methods with Channel A. The pilot density here is 1/3 and only the channel response in the frequency domain is estimation. The modulation used is QPSK. Figure 6-4 shows the performance comparison for Channel B. Notice that the positions of the channel taps in a block of 7 OFDM symbols are the same such that the two-dimensional interpolation method can be effectively applied. In the figures, “1D linear” indicates the one-dimensional linear interpolation method, “1D cubic” the one-dimensional cubic interpolation method, “2D linear & linear” the two-dimensional linear interpolation method (linear interpolation both

interpolation method with the linearly interpolation in the temporal domain and cubic interpolation in the frequency domain, “2D cubic & cubic” the two-dimensional cubic interpolation method (cubic interpolation both in the temporal and frequency domains).

We can see from Figure 6-3 and Figure 6-4 that two-dimensional interpolation methods are better than one-dimensional interpolation methods. And the cubic interpolation is better than linear interpolation. Also, the performance of “2D linear & cubic” is similar to “2D cubic & cubic”. This is because the channel taps’ positions in a block of 7 OFDM symbols are the same and the variation of channel among the symbols is small. And since the complexity of operation of linear interpolation is lower, we can use “2D linear & cubic” as the interpolation scheme in the joint time and frequency domain channel estimation methods described in Chapter 3.

5 10 15 20 25 30

10^-4 10^-3 10^-2 10^-1 10⁰

SNR(dB)

BER

1D linear 1D cubic

2D linear & linear 2D linear & cubic 2D cubic & cubic Perfect CH

Figure 6-3 Comparison of different interpolation methods (Chanel A)

5 10 15 20 25 30 10^-4

10^-3 10^-2 10^-1 10⁰

SNR(dB)

BER

1D linear 1D cubic

2D linear & linear 2D linear & cubic 2D cubic & cubic Perfect CH

Figure 6-4 Comparison of different interpolation methods (Channel B)

6.1.2 Results of joint time and frequency domain channel estimation

Figure 6-5 and Figure 6-6 show the performance comparison for the algorithm depicted in Figure 3-9 and Figure 3-10. The modulation scheme is also QPSK. For benchmarking, the performance of the ideal channel is also shown in the Figures.

We can see from Figure 6-5 and Figure 6-6 that the performances of the joint time/frequency domain channel estimation method is good under QPSK, and the performance is improved along with the number of iteration. We also can see that the performance with 6 and 10 iteration is almost the same. This shows that 6 iterations will be sufficient. Figure 6-7 and 6-8 show the performance comparison for QPSK, 16QAM, and 64QAM. These figures show that the performance of the channel estimation is also good with 16QAM and 64QAM.

5 10 15 20 25 30 10^-4

10^-3 10^-2 10^-1 10⁰

SNR(dB)

BER

N_re=0 N_re=1 N_re=6 N_re=10 Perfect CH

Figure 6-5 Performance of the joint time/frequency channel estimate (QPSK, Channel A)

5 10 15 20 25 30

10^-4 10^-3 10^-2 10^-1 10⁰

SNR(dB)

BER

N_re=0 N_re=1 N_re=6 N_re=10 Perfect CH

Figure 6-6 Performance of the joint time/frequency channel estimate (QPSK, Channel B)

5 10 15 20 25 30

Figure 6-7 Performance of the joint time/frequency channel estimate for QPSK, 16QAM, 64QAM (Channel A)

Figure 6-8 Performance of the joint time/frequency channel estimate for QPSK, 16QAM, 64QAM (Channel B)

6.2 Results of channel estimation in Chapter 4

The channel model is the same as the one shown in Table 6-1. The only difference is that the tap positions are changed for each OFDM symbol, and the delays of the channel taps in the aliasing area are set random (one or two taps). The simulation setup is also the same as that in Section 6.1. The channel estimation method is that depicted in Figure 4-5.

6.2.1 Comparison for the choice of pseudo pilots

We have discussed the choice of pseudo pilots in Section 4.3. Figure 6-9 shows the BER performance for the different choices for pseudo pilots. As defined, pseudo pilots are obtained from decisions and they can be erroneous. Figure 6-10 shows the SER of the pseudo pilots used. Figure 6-11 and 6-12 are similar to those in Figure 6-9 and 6-10 except for that Channel B is used. Here, guard band insertion is not conducted.

We can see from Figure 6-9 and Figure 6-11 that the best choice for the addition pilots is to use all detected data as the pseudo pilots. If one out three detected data is used as pseudo pilots, its performance is worse than the previous case. However, in order to compare with the performance of the conventional joint time/frequency domain channel estimator in Chapter 3, we will use the scheme with one out of three detected data as the pseudo pilots. In the simulations below, we will use the setting for simulations.

5 10 15 20 25 30 10^-4

10^-3 10^-2 10^-1 10⁰

SNR(dB)

BER

Sample every 1 subcarrier Sample every 3 subcarriers Sample every 4 subcarriers Sample every 6 subcarriers Perfect CH

Figure 6-9 BER comparison of different pseudo pilot selection schemes (without guard band insertion, Channel A)

5 10 15 20 25 30

10^-3 10^-2 10^-1 10⁰

SNR(dB)

Symbol Error Rate

Sample every 1 subcarrier Sample every 3 subcarriers Sample every 4 subcarriers Sample every 6 subcarriers

Figure 6-10 SER of pseudo pilots for different pseudo pilot selection schemes (without guard band insertion, Channel A)

5 10 15 20 25 30 10^-4

10^-3 10^-2 10^-1 10⁰

SNR(dB)

BER

Sample every 1 subcarrier Sample every 3 subcarriers Sample every 4 subcarriers Sample every 6 subcarriers Perfect CH

Figure 6-11 BER comparison of different pseudo pilot selection schemes (without guard band insertion, Channel B)

5 10 15 20 25 30

10^-2 10^-1 10⁰

SNR(dB)

Symbol Error Rate

Sample every 1 subcarrier Sample every 3 subcarriers Sample every 4 subcarriers Sample every 6 subcarriers

Figure 6-12 SER of pseudo pilots for different pseudo pilot selection schemes (without guard band insertion, Channel B)

guard band insertion. We can see that under the same number of iterations, the performance with the guard band insertion is much better than that without. Notice that we conduct the guard band insertion only when the channel response is completely estimated.

5 10 15 20 25 30

N_re=0 & no insertion N_re=0 & with insertion N_re=1 & no insertion N_re=1 & with insertion N_re=6 & no insertion N_re=6 & with insertion N_re=10 & no insertion N_re=10 & with insertion Perfect CH

Figure 6-13 Performance comparison for the channel estimator with/without guard band insertion (Channel A)

6.2.2 The Results of iterative channel estimation with pilots and pseudo pilots

Figure 6-14 and Figure 6-15 show the performance of the proposed estimator for different number of iterations. The modulation scheme is QPSK. Here, guard band insertion is conducted and the data in the original pilots are used in the SIC-LS algorithm. In other words, the pseudo pilots are only used to help locate channel taps. As we can see, the performance is satisfactory when the number of iteration is 6. Also, its performance is almost as good as that when the number of iteration is 10.

5 10 15 20 25 30 10^-4

10^-3 10^-2 10^-1

SNR(dB)

BER

N_re=0 N_re=1 N_re=6 N_re=10 Perfect CH

Figure 6-14 Performance comparison for the channel estimator with different numbers of iteration (SIC-LS uses pilots, Channel A)

5 10 15 20 25 30

10^-4 10^-3 10^-2 10^-1

SNR(dB)

BER

N_re=0 N_re=1 N_re=6 N_re=10 Perfect CH

Figure 6-15 Performance comparison for the channel estimator with different numbers of iteration (SIC-LS uses pilots, Channel B)

method with QPSK, 16QAM, 64QAM modulation. The number of iteration is set as 6. As we can see, for each case the performance is almost as good as the perfect channel.

5 10 15 20 25 30

Figure 6-16 Performance comparison for the channel estimator with QPSK, 16QAM, and 64QAM (SIC-LS uses pilots, Channel A)

5 10 15 20 25 30

Figure 6-17 Performance comparison for the channel estimator with QPSK, 16QAM, and 64QAM (SIC-LS uses pilots, Channel B)

Figure 6-18 and Figure 6-19 show the performance of the proposed estimator when

both pilots and pseudo pilots are used in the SIC-LS algorithm. The performances with one, 6 and 10 iterations are almost the same and it is also the same with that of the perfect channel. We also see that the performance is similar to that in Figure 6-14 and Figure 6-15.

5 10 15 20 25 30

Figure 6-18 Performance comparison for the channel estimator with different numbers of iteration (SIC-LS uses original and pseudo pilots, Channel A)

5 10 15 20 25 30

Figure 6-19 Performance comparison for the channel estimator with different numbers of

Figure 6-20 and Figure 6-21 show the performance of the proposed estimator when all the detected data are taken as pseudo pilots. Compared to Figure 6-18 and Figure 6-19 (only one out of three is used as a pseudo pilot), the performance here is better. In Figure 6-20, the performance is still satisfactory even when there is no iteration. In Figure 6-21, only one iteration is sufficient to obtain good performance.

5 10 15 20 25 30

10^-4 10^-3 10^-2 10^-1

SNR(dB)

BER

N_re=0 N_re=1 N_re=3 N_re=6 N_re=10 Perfect CH

Figure 6-20 Performance comparison for the channel estimator with different numbers of iteration (SIC-LS uses original pilots and all decisions, Channel A)

5 10 15 20 25 30 10^-4

10^-3 10^-2 10^-1

SNR(dB)

BER

N_re=0 N_re=1 N_re=3 N_re=6 N_re=10 Perfect CH

Figure 6-21 Performance comparison for the channel estimator with different numbers of iteration (SIC-LS uses original pilots and all decisions, Channel B)

6.3 Results of WLS algorithm

In this section, we report simulation results for the system we have defined. As mentioned, the number of subcarriers and the FFT size is 512 and the pilot density is 1/12. The CP size here is 64 (1/8 of 512) and the guard band size is also 64. The number of pilots here is 39.

Modulation schemes QPSK, 16QAM and 64QAM are used in our simulations. The delays of the channel taps are also set random for different OFDM symbols.

Figure 6-22 shows the performance of the proposed channel estimation (without weighting). The modulation scheme is QPSK. Here one out of three decisions is used as a pseudo pilot, and the SIC-LS method uses pilots only. As we can see, the performance is far from satisfactory even the number of iteration is large. This is because the number of pilots is not large enough such that the SIC-LS method cannot function properly.

5 10 15 20 25 30 10^-4

10^-3 10^-2 10^-1 10⁰

SNR(dB)

BER

N_re=0 N_re=1 N_re=6 N_re=10 Perfect CH

Figure 6-22 Comparison of the proposed channel estimator (no weighting, SIC-LS uses pilots, Channel C)

Figure 6-23 shows the performance of the proposed channel estimation (without

weighting) for the QPSK, 16QAM, 64QAM modulation. Different from that in Figure 6-22, the SIC-LS method now uses the original and pseudo pilots. It is apparent that the

performance has been improved. However, for 16QAM and 64QAM, the performance in high SNR areas seems less satisfactory. Unlike the DVB-T system, the number of pseudo pilots is not large. As a result, erroneous decisions will affect the estimation performance

performance has been improved. However, for 16QAM and 64QAM, the performance in high SNR areas seems less satisfactory. Unlike the DVB-T system, the number of pseudo pilots is not large. As a result, erroneous decisions will affect the estimation performance

在文檔中 應用於時變與非時變OFDM系統之新通道估測法 (頁 57-0)