1. Channel Estimation
2012/08/13 蒲俊瑋

2. Outline Introduction Channel Estimation Techniques in OFDM Systems
LS Channel Estimation
Linear Interpolation Channel Estimation
MMSE Channel Estimation
MLS Channel Estimation
Pilot Arrangement in OFDM Systems and Decision-Feedback Channel Estimation
Channel Equalization in Timing Varying Channel

3. Introduction

4. Small Scale Fading
Multi-path channel

5. Channel Impulse Response
A mobile radio channel may be modeled as a linear filter with a time varying impulse response, where the time variation is due to receiver motion in space.
The filtering nature of the channel is caused by the summation of amplitudes and delays of the multiple arriving waves at any instant of time.
LTE, 1 subcarrier = 15K  Doppler effect is not important

6. Channel Impulse Response

7. The Multi-Path Channel Effect
EX:
Data = [ ]
Channel = [1 1]
y = =
path1
path2

8. The Matrix Form of Channel
The wireless stationary channel impulse response is given by , where L is the total number of resolvable paths.
We assume that each tap of the channel impulse responses ,
, are independently distributed complex Gaussian random variables with zero-mean and variance
Each path is Rayleigh fading. Rician channel has the property that the ratio of the main path and other paths is large.

9. The Matrix Form of Channel
The matrix G is constructed as follows:
EX:
Data = [ ]
Channel = [1 1]

10. The Matrix Form of Channel
The matrix is constructed as follows: CP

11. The Matrix Form of Channel
Furthermore, a circular convolution matrix can be obtained:

12. The Transmitted OFDM Signal
After the inverse discrete Fourier transform (IDFT) operation, the ith transmitted OFDM symbol in time domain can be expressed by:
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix.

13. The Received CP-OFDM Symbol
Assuming the synchronization is perfect and CP is adopted, the received ith OFDM symbol can be expressed as:
where denotes the circular convolution and denotes the additive white Gaussian noise (AWGN) vector in the time domain with zero mean and variance We note that there is no ICI and ISI in each OFDM symbol.

14. The Received CP-OFDM Symbol
After DFT operation, the ith received OFDM symbol in the frequency domain can be expressed as:
where is the AWGN in the frequency domain and H is a N × N diagonal matrix denoting the channel response in frequency domain.

15. The Received CP-OFDM Symbol
Any circular matrix can be diagonalized by DFT matrix:
The received signal can be expressed as:

16. Channel Estimation Techniques in OFDM Systems

17. Introduction
In general, channel estimation can be cataloged into three kinds of estimation schemes:
1. Blind 2. Superimposed 3. Pilot-based
The first two structures can obtain some bandwidth merit, but the computational complexity is usually not acceptable in practical realization.

18. Introduction
The pilot-based estimation can be cataloged into two kinds of approaches:
1. The parameters are deterministic but unknown constant, such as maximum likelihood (ML) estimator and least square (LS) estimator.
2. The parameters are random variables, such as minimum mean square error (MMSE) estimator and maximum a posteriori (MAP) estimator.

19. System architecture

20. System architecture 1 Input to Time Domain 2 Guard Interval 3 Channel4
Guard Removal 5
Output to Frequency Domain 6
Output
Channel
ICI
AWGN 7
Channel Estimation
Estimated Channel

21. System Model
Generally, the pilot symbols are multiplexed into an OFDM symbol in frequency domain:
In addition, the power allocation of data and pilot symbols are given by:
ρ: Total Power β: Power Allocation Factor
N: Number of Subcarriers

22. System Model
If system is perfectly synchronized, and the CP is added and removed appropriately, there is no ISI and inter-carrier interference (ICI). As a result, the ith received OFDM symbol after DFT can be expressed as:
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix

23. The LS Channel Estimator
Define , where denotes the received pilot signal.
The channel estimator based on the LS method is given by:
where denotes a diagonal matrix whose diagonal elements are given by

24. Linear and Second Order Interpolation
Linear Interpolation
Second Order Interpolation [1-2]

25. Linear Interpolation Estimated Channel (Channel + Noise)’s Upper Bound
Real Channel
Pilot Subcarrier
(Channel + Noise)’s Lower Bound
Data Subcarrier
Frequency Domain

26. The MMSE Channel Estimator
The MMSE channel estimator is given by [3, 4]
where represents the cross-correlation between all the subcarriers and the pilot subcarriers, and represents the autocorrelation matrix between the pilot subcarriers.

27. The Low-Rank MMSE Channel Estimator
A low-rank MMSE channel estimator is given by [1]:
where is a diagonal matrix with entries
Note that can be viewed as the attenuation of the lth tap of the channel impulse response:
and c can be expressed as [3]:

28. The Realization of MMSE Channel Estimator
In practice, the channel power of the lth transform coefficient can be obtained from the results of the LS channel estimation.
First, the estimate of the channel impulse response can be acquired by taking the IDFT of the channel frequency response obtained from the LS channel estimate:
And then the
is obtained.

29. The Modified LS (MLS) Channel Estimator
The modified LS (MLS) channel estimator is given by
where is a diagonal matrix. The entries of are:
The MLS channel estimator can be considered as a low-pass filter, which is also termed as DFT-based scheme.

30. Pilot Arrangement in OFDM Systems [2] and Decision-Feedback Channel Estimation

31. Introduction
The channel estimation can be performed by either inserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting pilot tones into each OFDM symbol (comb type).
Comb Type Block Type
Freq.
Time

32. LTE Reference Symbol Arrangement
LTE pilot symbol arrangement 1 2

33. Band Edge f Virtual subcarriers Virtual subcarriers DC Active Band … …
Pilot
Data

34. Decision-Feedback Channel Estimation
When the channel is slow fading, the channel estimation inside the block can be updated using the decision feedback equalizer at each sub-carrier.
Decision Directed Channel Estimation (DDCE) is one of the earliest methods studied for OFDM, mainly because of its popularity in legacy systems. In the earlier studies, DDCE was applied mostly in training based systems.

35. Decision-Feedback Channel Estimation
The main idea behind DDCE is to use the channel estimation of a previous OFDM symbol for the data detection of the current estimation, and thereafter using the newly detected data for the estimation of the current channel.
For fast fading, the comb-type estimation performs much better.

36. OFDM Systems in Time-Variant Multipath Channels

37. Introduction
Orthogonal frequency-division multiplexing (OFDM) is generally known as an effective technique for high bit rate applications such as DAB, DVB and WiMAX, since it can prevent intersymbol interference (ISI) by inserting a guard interval and can mitigate frequency selectivity of a multipath channel using a simple one-tap equalizer.

38. Introduction
In an OFDM system, although the degree of channel variation over the sampling period becomes smaller as data rates increase, the time variation of a fading channel over an OFDM block period causes a loss of subchannel orthogonality, resulting in an error floor that increases with the Doppler frequency.
The performance degradation due to the interchannel interference (ICI) becomes significant as the carrier frequency, block size, and vehicle velocity increase.

39. Block Diagram for An OFDM System
The time-domain transmitted signal is given by
The time-domain received signal is then given by

40. ICI Analysis The frequency-domain received signal is then given by
where denotes the frequency-domain noise and
represents the FFT of timing-variant channel, i.e.,

41. ICI Analysis
In the general case where the multipath channel cannot be regarded as time-invariant during a block period, the received signal can be expressed in vector form as
where , , and
with

42. Channel Equalization
After performing channel equalization, the equalized signal can be expressed as
where
The inverse operation increases the system computation complexity.

43. Conclusions
Assuming that the channel is stationary over the period of an OFDM symbol, the conventional frequency-domain equalizer with one-tap in an OFDM system compensates the frequency-selectivity of a multipath fading channel.
The one-tap frequency-domain equalizer cannot eliminate ICI for the case of a time-varying channel.
In time-varying channel, the computation complexity of the frequency-domain equalizer is increased.

44. References
[1] Sinem Coleri, Mustafa Ergen, Anuj Puri, and Ahmad Bahai, “Channel Estimation Techniques Based on Pilot Arrangement in OFDM Systems,” IEEE transactions on Broadcasting, Vol. 48, No. 3, September 2002.
[2] G.-S. Liu and C.-H. Wei, “A new variable fractional sample delay filter with nonlinear interpolation,” IEEE Trans. Circuits and Systems-11: Anulog andDigiral Signal Processing, vol. 39, no. 2, Feb
[3] O. Edfors, M. Sandell, J.-J. van de Beek, S. K. Wilson, and P. O. Borjesson, “OFDM Channel Estimation by Singular Value Decomposition,” IEEE Transactions on Communications, vol. 46, no. 7, pp , Jul
[4] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory, New Jersey: Prentice Hall, 1993, pp

45. Appendix A. LMMSE estimator

46. Appendix A. LMMSE estimator