Improved joint time and frequency domain channel estimation

One problem associated with method described above is that the estimation of a significant tap may be affected by the other significant taps. An improved method was then proposed in [4]. The main idea is to conduct a re-estimation for all significant taps. After all the taps has been estimated, we can conduct the re-estimation of one tap by subtracting the channel response contributed from the other taps and using the LS method shown above.

Using the approach, the estimation accuracy can be improved. The procedure is summarized below. Figure 3-10 show the flowchart of the method. Note that the channel taps can be re-estimated more than one time until the satisfactory result is obtained.

Interpolated frequency domain channel estimate

IFFT

Tap searching algorithm

LS algorithm

Low pass filter

Compute LSE Locate tap

Find the tap value we locate

Subtract the located tap from original response in time domain

LSE>R?

STOP No YES

STEP 1: With the channel estimation method described above, we get L significant

taps. We can order

h

_k’s and start from the maximum one.

STEP 2: With the FFT operation, we obtain the frequency response for a subcarrier of

the other taps (the subcarrier index is

i

), denoted as

h

_i

'

.

STEP 3: For a specific pilot position, say p, we can subtract the response contributed

by

h

_p

'

. Let the transmitted pilot signal be

x

pand the received pilot signal be

y

_p. We then have

' '

p p p p

y = y − x h

(3-14)

STEP 4: Using all y

_p

'

s, we can conduct the LS channel estimator to re-estimate the

designated tap.

STEP 5: Check if all taps are re-estimated. If no, then select the next largest tap and go

to STEP 2.

Figure 3-10 Improved joint time and frequency domain channel estimation

The dotted block is the improved LS algorithm which will be used in the next Chapter. It contains the iterative channel tap searching and improved LS operation.

STOP

Channel estimation result in Figure 3-9

Subtract the response in

frequency domain

of the selected tap from

Chapter 4

Proposed Joint Time and Frequency Channel Estimator for DVB-T Systems

As discussed in Chapter 3, the channel estimation with a two-dimensional interpolation requires 7 OFDM symbols. The computational complexity is high and the memory to store the OFDM symbols is large. For one symbol in the first three or the last three OFDM symbols (in a frame), we cannot collect 7 symbols (three before and three after) for channel estimation and the performance in these areas will degrade. Another problem is that the channel cannot have large variation in the selected 7 OFDM symbols. Thus, the methods described in Chapter 3 are only applicable in slow-fading environments. In this chapter, we will propose a new method to overcome this problem for DVB-T systems.

The proposed method uses only one OFDM symbol to conduct the channel estimation.

As known, the pilot density is low in DVB-T systems. Direct estimation will result in poor performance. Our idea is to use decisions as pseudo pilots, which will be referred to as pseudo pilots in the sequel, such that the pilot density can be effectively enhanced. The proposed algorithm can be summarized as follows.

1. Obtain an initial channel estimate with one OFDM symbol.

2. Use the estimate with the LS algorithm to conduct data detection.

3. Use some detected data as pseudo pilots and conduct channel re-estimation.

4. Use the re-estimated channel with the LS algorithm to conduct data re-detection.

In the following subsection, we will describe each step in details.

4.1 Initial channel estimation with one OFDM symbol

As discussed in Chapter 3, we also conduct an initial frequency-domain channel estimate by pilot information and transfer it to time domain. However, we do not conduct interpolation. Since the pilot density is low, the initial estimate is not accurate. As known, the pilot subcarriers are uniformly spread in the frequency domain. If we only estimate the channel response with those in pilot subcarriers, it is equivalent to conduct a sampling on the frequency-domain channel response. As a result, the time-domain channel response will be periodic.

As known, sampling a frequency domain signal makes its time-domain signal periodic.

If the sampling rate is not high enough, aliasing will occur. Figure 4-1 shows the aliasing problem of the channel estimation.

Figure 4-1 Aliasing in initial channel estimation

Let the sampling period be K. It is simple to see that the period of the time- domain channel estimate, denoted with

D , will be N/K. In our case, K=12. Consider the 2K mode,

₁ we have N=2048. Thus, we know that the period of the time-domain channel estimate is about 171 (2048

170.67 171

12 ≅ ≅ ). Let the maximum delay spread of the channel be

D .

₂

Magnitude

D

1

D

2

t

……

D

₁

is the end of first period

D

2

is the maximum delay of channel tap

N/12

Thus, we can see that aliasing occurs when

D

₂ ≥

D

₁.

4.2 Tap search and estimation for initial channel estimate

As shown, there may be aliasing in the initial channel estimate. The channel response in the aliasing area cannot be recovered. Thus, we conduct tap searching only in the non-aliased area, and combine the LS methods described in Figure 3-10 and 3-8 to conduct the initial estimate. Given an initial frequency-domain channel estimate, Figure 4-2 shows the LS channel estimation method we use. Note that the method is essentially a successive interference cancellation (SIC) approach. For easy reference, we call the LS channel estimation method in Figure 4-2 as a SIC-LS method.

Figure 4-2 The SIC-LS channel estimation method

An initial frequency-domain channel

estimation without interpolation

Subtract the located tap from original response in

of the located tap from

The located tap The nonaliasing part of the time-domain channel estimate SIC_LS

method

The LSE threshold here should be higher since the response in the initial channel estimate is not complete (some taps may be in the aliasing area). Figure 4-3 shows the complete block diagram of the proposed initial channel estimation method and the procedure is summarized in the following.

Figure 4-3 Proposed initial channel estimation method

STEP 1: Use pilot subcarriers to obtain an initial frequency-domain channel estimate.

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

channel estimate.

STEP 3: Find the non-aliasing area of the initial time-domain channel estimate and use

the SIC-LS method in Figure 4-2 to locate and identify the channel taps in the area.

Using the channel estimation method outlined above, we can obtain an initial time-domain channel estimate. Note that the channel response may be incomplete. However, since the aliasing area is not large and the power of the channel taps in the area may be small,

Initial frequency-domain channel estimate without interpolation

IFFT

The SIC-LS method in Fig. 4-2

Locate tap &

Find the tap value

The nonaliasing part of the time-domain channel estimate

STOP

we can use the estimate to recovery data with an acceptable error probability. Let

ˆ x

i

(

y

_i

= h x

_{i i}

+ w

_i, _ˆ^{i ˆ}_i

i

y x

h = ) be the estimated symbol for a subcarrier in subcarrier index

i

.We

have

y

_i

= h x

_{i i}

+ w

_i, and then

ˆ

h

i is the estimated channel response at that subcarrier,

x

i is the transmitted frequency domain signal, and

ˆ

x

iis the corresponding estimated signal.

To recover the original transmit symbol, we can make decision based on

ˆ

x

i. Let the detected data be

ˆ

x

de. Figure 4-4 shows the procedure of data detection.

Figure 4-4 The block diagram of data detection

4.3 Proposed channel estimation with pilots and decisions

We now can use the detected data:

ˆ

x

de’s as pseudo pilots and conduct the channel estimation of the whole channel. Note that as long as aliasing does not occur in the time-domain channel estimate, we can recovery the whole channel response. As a result, no

ˆ

de

x

Frequency Domain Time Domain Channel estimation result in Fig. 4-3

IFFT

1/8 of the FFT size (256 for the 2K mode), a pilot density of 1/6 will avoid the aliasing at the time-domain channel estimate. The performance comparison for different pilot densities will be shown in simulations. If we use all decisions (in all subcarriers) as pseudo pilots, we may have the best performance. However, the computational complexity is also highest. To compare the channel estimation result in Chapter 3, we may let the pilot density be 1/3.

With the pseudo pilots, we can use the channel estimation method described above again.

Note that if the pilot density is 1/3, the period in the time-domain channel estimate

is ₁ 683

3

D

=

Nc

≅ which is actually much longer than the CP size. Thus, we can assume that

all the channel taps will be in the non-aliasing region. Note that one can conduct the channel re-estimation more than once.

Since the guard band does not have any data, theoretically the channel response cannot be obtained. However, if we can conduct the time-domain channel estimate properly, we can obtain the frequency-domain channel response in the guard band. To obtain better result, when conducting channel re-estimation, we can insert the guard band response of the previous estimated channel. To have better performance, we do the guard band insertion only when the frequency domain channel estimation is accurate enough which means that all the channel taps of the time-domain channel estimate are in the nonaliasing area. Let the preset number of the channel re-estimation be Nset and the actual number of re-estimation be Nre. We let Nre=0 for the initial channel estimation. In this iteration, the guard band insertion will not be performed. When Nre≠0, all the channel taps are in the nonaliasing region and we can activate the guard band insertion scheme. The effect of the guard band insertion will also be discussed in the simulation chapter.

Figure 4-5 shows the block diagram of the proposed joint time and frequency domain channel estimation scheme.

Figure 4-5 Proposed joint time and frequency domain channel estimation method The operations can be summarized below:

STEP 0: Use the algorithm in Figure 4-4 to obtain initial data decisions.

STEP 1: Use pilot and pseudo pilot subcarriers (without interpolation) to obtain a

frequency-domain channel estimate.

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

domain.

STEP 3: Find the nonaliasing region of the time-domain channel estimation.

STEP 4: Use the proposed SIC-LS method shown in Figure 4-2 to locate and identify

significant taps.

STEP 5: If N

re<Nset, go to STEP 6. Otherwise, the iteration stops.

STEP 6: Conduct the FFT operation for the identified time-domain channel to obtain its

(a)

(c)

(d) (e)

Frequency-domain channel estimate by pilots and pseudo

pilots without interpolation

Initial data decisions from the result in Figure 4-4

N

re

=0

frequency domain response.

STEP 7: Estimate the transmitted QAM symbols at designated pilot subcarriers and

make decisions.

STEP 8: Insert channel response in the guard band using the channel response estimated.

STEP 9: Let N

re= Nre +1 and go to STEP 1 with a updated new channel estimate.

Notice that there are two differences between the SIC-LS block in Figure 4-5 and Figure 4-2. The block of “LSE>R?” in Figure 4-2 is changed to “Re-estimate all taps?”

since all the channel taps have been initially detected here. As a result, we only have to check if all the taps have been re-estimated. Also, the block “Compute LSE” in Figure 4-2 is not necessary. Besides, the pseudo pilots can also be used in the SIC-LS method. Figure 4-6 shows a graphic example showing the channel response at each of the steps outlined above.

Figure 4-6 (a) is the sketch of the channel estimate by the pilots and pseudo pilots. Figure 4-6 (b) is a sketch of the time-domain channel estimate. Figure 4-6 (c) is the channel taps after refined by the proposed LS algorithm. Figure 4-6 (d) is the frequency-domain channel estimate of Figure 4-6 (c). Figure 4-6 (e) is the sketch of the guard-band insertion (every 3 subcarriers).

(d)

f

Improved LS algorithm

N/3

t

(b)

(c)

N/3 t

f

Data Decision:

D Pilot

Pilot

Pilot Space=12

…………

Channel estimate by original and pseudo pilots

(a)

X ˆ

FFT Operation IFFT Operation

f

Guard band

(e)

D

…

Insertion guard band

that is from (c)

When the number of pilots is not large enough, the proposed channel estimation scheme is still an effective way to obtain accurate channel estimate. We will have more discussions and simulation results in Section 4.4.

4.4 SIC-LS method with weighting

The method of channel estimation of joint pilot and decision data can be used not only in the DVB-T system but also in other OFDM systems. As long as the pilot assignment is periodic, the proposed method can be used. However, if the number of pilot subcarriers is not sufficient, the performance of the proposed method will be affected. For example, if the total number of subcarriers is 512, the CP is 64 (1/8 of 512) and the pilot density is still 1/12.

The total number of the pilots used in this system is 39 which is much less that 142 used in the DVB-T system. As a result, the performance will be degraded in this system. In order to solve the problem of low pilot density, we use the pseudo pilots to conduct the SIC-LS method in Figure 4-5. However, the performance may be still not satisfactory. The problem comes from the decisions, i.e., erroneous decisions. Using erroneous decisions as pilots will degrade the performance of the SIC-LS channel estimation method. One way to alleviate the problem is to use the SIC-LS method with weighting (SIC-WLS), giving more weights to the decisions with higher correct probability. We first consider the LS problem without SIC. From (3.14), we have the received pilot signals as

' , '

p D p

+

y = X G'h w

(4.1) where '

p

=

j

_i,0≤ ≤ − is pilot location and

j J

1 Jis the number of pilots

The weighted LS channel estimation can be formulated as

2

h

is the solution of the weighted LS channel estimate, and

D

=

diag d d

( , , ,

d

) is a diagonal weighing matrix. From (4.1),

we have

Dy = DX

_{p' D p, '}

G'h Dw +

(4.3) Thus, from (3.12), we can get the weighted LS solution as

( )

', , ' , ' , ' '

ˆ

^{H H H -1 H H H}

p wls

=

D p D p D p p

h G X D DX G G X D Dy

(4.4) As mentioned, the weighted LS algorithm put more weights on the decisions having high probability of correctness. How to actually determine D will be the main concern. We proposed two methods to do the job as discussed in the following paragraphies.

The first method uses the estimated symbols to decide the values in

D . We use the

BPSK scheme as our illustration example. Let

ˆ

x

i ₍

y

_i

= h x

_{i i}

+ w

_i,

ˆ ˆ

^{i i}

i

y x

h =

_{) be the}

estimated symbol for a subcarrier in subcarrier index

i

. Since we know the decision of

ˆ x

i

is either 1 or –1. Thus, we can use the distance between the estimated transmitted signals

ˆx

and 1 or -1 to as a measure of the correct probability of the decision and then the corresponding weight. Figure 4-7 shows the proposed first method for the determination of the weight of a decision.

Figure 4-7 The proposed first weighting method

As shown in the figure, we partition the estimated signal value into various design regions and assign a weight for signal falling into the region. We assume that the distance

0 0.35 1.75

-0.35 -1.75

^-1

1

Weight of each region=

0.5 0.7 0.5 0.5 0.7 0.5

between the estimated transmitted signals

ˆx

and 1/ -1 is only affected by noise. So, when the distance between the estimated transmitted signal

ˆ

x

i and 1/ -1 is large, we can assume this estimated transmitted signal

ˆ

x

i, the symbol transmitted in subcarrier index

i

, is affected by noise more seriously. The probability of making wrong decision for the symbol is high so we can put less weight on the decision. On the contrary, when the distance between the estimated transmitted signal

ˆ

x

i and 1/-1 is small, we can assume

ˆ x

i _is

affected by noise lightly. Then we can give more weight for the decision. Notice that the weight of a pilot subcarrier is always 1 (the maximum value of the weights). Figure 4-7 shows one example of the weighting policy.

When the noise level is high, the received signal can cross the decision boundary. As a result, the decision is wrong/right even the distance is small/large. So the performance of this method may not be satisfactory even the averaged SNR is high. It is simple to see that for a subcarrier, the received signal-to-noise (SNR) ratio is the main factor determining the decision error rate. If the SNR is high/low, the error rate will be low/high. We then propose another method for the weight assignment. We assume that the SNR of each subcarrier is known or can be estimated. Similar to the previous approach, we separate the SNR value into various regions and assign a weight for signal falling into a specific region. Notice that we can let the SNR be represented in a dB scale. Also, the weight for the pilot subcarrier is always 1 (the maximum value of the weight). Figure 4-8 shows one example of the weighting policy of the proposed second weighting method. Notice that the weight mapping may be different for different modulation schemes.

Figure 4-8 The proposed second weighting method

We can apply the WLS method in the proposed SIC-LS method discussed in Figure 4-5.

Simulations show that the performance of the proposed SIC-WLS channel estimation method is satisfactory under the BPSK and QPSK modulation schemes. But, the performance is somewhat degraded under the 16QAM modulation scheme. This is because the instantaneous SNR in 64QAM is too low such that the error rate of the decision data is high. We can solve the problem by using all decisions as the pseudo pilots to do the channel estimation and the performance can be greatly enhanced. This will be discussed in our simulation chapter.

4.5 Joint 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

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

在文檔中 應用於時變與非時變OFDM系統之新通道估測法 (頁 42-0)