Frequency Domain Channel Estimation Algorithms

The basic idea of channel estimation (CE) is to use the known value of inserted scattered pilots to estimate the channel responses. Three estimation algorithms will be discussed below.

a) 1-D Channel Estimation

The one dimensional channel estimation algorithm is the simplest of the three algorithms. Since a scattered pilot is inserted every twelve sub-carriers in a symbol, the easiest way is using the two neighboring scattered pilots to interpolate the channel response of the eleven sub-carriers between them. Unfortunately, the variation in frequency domain is very serious and eleven sub-carriers space is too large to lead to an inaccuracy approximation. As Fig. 2.8 shows, the scattered pilots have three sub-carriers space shift for each successive symbol. Due to this characteristic, the space of the two neighboring scattered pilots will reduce to two by collecting the scattered pilots of four continuous symbols. This method supposes the time domain variation during four symbols is very small or even none. As Fig. 4.6 illustrates, the n^th symbol collects the scattered pilots from the (n-3)^th to n^th symbols and uses frequency domain interpolation operation to approximate the channel response in the middle of the two scattered pilots as Fig. 4.7 shows. For the purpose to reduce the hardware complexity, the linear interpolation is adopted as frequency domain interpolation.

Fig. 4.6 Stored scattered pilots of 1-D channel estimation

Fig. 4.7 Frequency domain linear interpolation of 1-D channel estimation The 1-D channel estimation is described mathematically in Eqn. (4.3).

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where the CR(n,m) is the m^th channel response in symbol n, SC(n,m) belongs to scattered pilots and SCexp(m) is the m^th expect value.

This method comes with a risk. If the channel response in time domain changes serious or not as small as expected, the scattered pilots those don’t belong to current

symbol are not adequate to represent the channel response of current symbol. As a result, the wrong channel response is possibly leads to a worse performance.

Therefore, the 1-D channel estimation algorithm only suits for time-invariant channel.

b) 2-D Channel Estimation

The 2-D channel estimation algorithm can take the time domain variation into consideration. In order to conquer the variation in time domain, the 2-D channel estimation uses the scattered pilots before and after the current symbol to interpolate a virtual scattered pilot of current symbol instead of collecting scattered pilots from the other three previous symbols. As Fig. 4.8, Fig. 4.9 and Fig. 4.10 shows, the 2-D channel estimation first collect the scattered pilots from seven continuous symbols and doing time domain linear interpolation to approximate the virtual scattered pilots.

Then use the virtual scattered pilots and scattered pilots itself to do frequency domain interpolation. While the channel response changes linearly or not very serious during continuous five symbols, the 2-D channel estimation algorithm will get the best performance. But the 2-D channel estimation needs to store the sub-carriers at least three symbols before to get enough scattered pilots to do time domain interpolation.

This means a large storage element is required, which is at least 6817×3 sub-carriers for 8K transmission mode.

錯誤! 物件無法用編輯功能變數代碼來建立。

Fig. 4.8 Stored scattered pilots of 2-D channel estimation

Fig. 4.9 Time domain linear interpolation of 2-D channel estimation

Fig. 4.10 Frequency domain linear interpolation of 2-D channel estimation The 2-D channel estimation is described mathematically in Eqn. (4.4).

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c) 2-D Predictive Channel Estimation

For the purpose to solve the requirement of huge storage memories and keep the characteristic to conquer the time domain variation at the same time, a predictive 2-D channel estimation algorithm is offered by using time domain external interpolation instead of internal interpolation [26]. The 2-D predictive channel estimation uses two scattered pilots before current and doing external interpolation to predict the channel response of current symbol as shown in Fig. 4.11, Fig. 4.12 and Fig. 4.13. As a result, the storage element for three symbols’ sub-carriers is saved and an extra storage element for scattered pilots is required. Comparing to the sub-carriers storage element, the storage element for additional scattered pilots is much smaller. But the predictive external time domain interpolation won’t work as good as the internal time domain

interpolation because of the uncertain of prediction. Due to this reason, the 2-D predictive channel estimation is not as good as 2-D channel estimation but better than 1-D channel estimation in time-variant channel.

Fig. 4.11 Stored scattered pilots of predictive 2-D channel estimation

SC(n-7,m+3)

SC(n-3,m+3)

SC(n-6,m+6)t

SC(n-2,m+6)

SC(n-5,m+9)t

SC(n-1,m+9) t CR(n,m+3)

CR(n,m+6) CR(n,m+9)

Fig. 4.12 Time domain linear interpolation of predictive 2-D channel estimation

Fig. 4.13 Freq. domain linear interpolation of predictive 2-D channel estimation The Predictive 2-D channel estimation is described mathematically in Eqn. (4.5).

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3.4.2 Performance Simulation and Comparisons

The performance simulation results are shown in Fig. 4.14. The simulation environment is 2K transmission mode symbols with 1/4 guard interval, zero CFO, and surviving from Rayleigh channel with (a) 0Hz Doppler spread (b) 70Hz Doppler spread. Under the static channel, the predictive 2-D channel estimation algorithm is the worst because of the noise leads to a wrong predictive channel response. For dynamic channel, the predictive 2-D channel estimation algorithm is better then the 1-D channel estimation algorithm because of the channel varies. Overall, the 2-D channel estimation algorithm has the best performance.

19 21 23 25 27 29 Fig. 4.14 BER under (a) static and (b) dynamic Channel

The hardware of the three algorithms is listed in Table 4-4.

Table 4-4 Required storage elements of channel estimation algorithms

1-D 2-D Predictive 2-D

Channel Estimation Channel Estimation Channel Estimation Scattered Pilot

Storage Element

1+142×3 (427) 1+284×3 (745) 1+568×3 (1705)

1+142×3 (472) 1+284×3 (745) 1+568×3 (1705)

1+142×7 (996) 1+284×7 (1738) 1+568×7 (3978) Data Pilot

Storage Element 0

1705×3(5115) 3408×3(10224) 6816×3(20448)

0

Overall 427/745/1725 5587/10969/22153 996/1738/3978

The 1-D algorithm theoretically needs to store four symbols’ scattered pilots. The scattered pilot number is 142/284/568 for 2K/4K/8K defined in [1]. The number for Mode 0 scattered pilots is one more than others. The (n-3)^th symbol’s scattered pilots are only used to compensation for the n^th symbol and can be overwrote by the n^th symbol’s scattered pilots after used. Therefore, the maximum scattered pilot storage is reduced from four to three symbol’s scattered pilots. The 2-D algorithm needs to store additional three symbols’ pilots, which includes the scattered pilots. Therefore, the 2-D algorithm needs to store scattered pilots from the (n-1)^th symbol to the (n-3)^th symbol and all pilots from the (n)^th symbol to the (n+2)^th symbol. For predictive 2-D algorithm, it only needs to store scattered pilots form the (n-1)^th symbol to the (n-7)^th symbol and no data pilot is required to be stored.

Overall, the 2-D algorithm has the best performance with an unreasonable large storage elements requirement. The predictive 2-D algorithm has a better performance than 1-D algorithm under dynamic channel and a little worse performance under static channel. The storage elements required for predictive 2-D algorithm is less than that of the storage elements of mode/GI and boundary detection process. Thus, the predictive 2-D channel estimation algorithm is adopted as the channel estimation algorithm in this thesis.