Freescale Semiconductor Application Note
AN3059Rev. 0, 1/2006
2016年属猴是什么命This application note gives an overview of the channel estimation strategies ud in orthogonal frequency division multiplexing (OFDM) systems. Section 1 describes the protocols associated with OFDM systems and the problems pod by such systems. Section 2 through Section 5 describe the various types of channel estimation methods for u in such systems. The implementation complexity and system performance of the methods are studied and compared in
Section 6, measuring performance in terms of symbol error rate (SER).
Channel Estimation in OFDM Systems
by Yushi Shen and Ed Martinez
CONTENTS
.23Block-Type Pilot 44Co
mb-Type Pilot 75Other Pilot-Aided 107Conclusions. (148)
References (15)
OFDM Background
1OFDM Background
OFDM is becoming widely applied in wireless communications systems due to its high rate transmission capability with high bandwidth efficiency and its robustness with regard to multi-path fading and delay [1]. It has been ud in digital audio broadcasting (DAB) systems, digital video broadcasting (DVB) systems, digital subscriber line (DSL) standards, and wireless LAN standards such as the American IEEE® Std. 802.11™ (WiFi) and its European equivalent HIPRLAN/2. It has also been propod for wireless broadband access standards such as IEEE Std. 802.16™ (WiMAX) and as the core technique for the fourth-generation (4G) wireless mobile communications. The u of differential pha-shift keying (DPSK) in OFDM systems avoids need to track a time varying channel; however, it limits the number of bits per symbol and results in a 3 dB loss in signal-to-noi
ratio (SNR). Coherent modulation allows arbitrary signal constellations, but efficient channel estimation strategies are required for coherent detection and decoding.
There are two main problems in designing channel estimators for wireless OFDM systems. The first problem is the arrangement of pilot information, where pilot means the reference signal ud by both transmitters and receivers. The cond problem is the design of an estimator with both low complexity and good channel tracking ability. The two problems are interconnected. In general, the fading channel of OFDM systems can be viewed as a two-dimensional (2D) signal (time and frequency). The optimal channel estimator in terms of mean-square error is bad on 2D Wiener filter interpolation. Unfortunately, such a 2D estimator structure is too complex for practical implementation. The combination of high data rates and low bit error rates in OFDM systems necessitates the u of estimators that have both low complexity and high accuracy, where the two constraints work against each other and a good trade-off is needed. The one-dimensional (1D) channel estimations are usually adopted in OFDM systems to accomplish the trade-off between complexity and accuracy [1–7]. The two basic 1D channel estimations are block-type pilot channel estimation and comb-type pilot channel estimation, in which the pilots are inrted in the frequency direction and in the time direction, respectively. The estimations for the block-type pilot arrangement
can be bad on least square (LS), minimum mean-square error (MMSE), and modified MMSE. The estimations for the comb-type pilot arrangement includes the LS estimator with 1D interpolation, the maximum likelihood (ML) estimator, and the parametric channel modeling-bad (PCMB) estimator. Other channel estimation strategies were also studied [8–12], such as the estimators bad on simplified 2D interpolations, the estimators bad on iterative filtering and decoding, estimators for the OFDM systems with multiple transmit-and-receive antennas, and so on.
2Baband Model
The basic idea underlying OFDM systems is the division of the available frequency spectrum into veral subcarriers. To obtain a high spectral efficiency, the frequency respons of the subcarriers are overlapping and orthogonal, hence the name OFDM. This orthogonality can be completely maintained with a small price in a loss in SNR, even though the signal pass through a time dispersive fading channel, by introducing a cyclic prefix (CP).
A block diagram of a baband OFDM system is shown in Figure 1.
Baband Model
The binary information is first grouped, coded, and mapped according to the modulation in a “signal mapper.” After the guard band is inrted, an N-point inver discrete-time Fourier transform (IDFT N ) block transforms the data quence into time domain (note that N is typically 256 or larger). Following the IDFT block, a cyclic
extension of time length T G , chon to be larger than the expected delay spread, is inrted to avoid intersymbol and intercarrier interferences. The D/A converter contains low-pass filters with bandwidth 1/T S , where T S is the sampling interval. The channel is modeled as an impul respon g(t) followed by the complex additive white Gaussian noi (AWGN) n(t), where αm is a complex values and 0 ≤ τm T S ≤ T G .
Equation 1
At the receiver, after passing through the analog-to-digital converter (ADC) and removing the CP, the DFT N is ud to transform the data back to frequency domain. Lastly, the binary information data is obtained back after the demodulation and channel decoding.
Let and denote the input data of IDFT block at the transmitter and
the output data of DFT block at the receiver, respectively. Let and denote the sampled channel impul respon and AWGN, respectively. Define the input matrix and the DFT-matrix,
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Equation 2
where . Also define , and .
Figure 1. A Digital Implementation of a Baband OFDM System.
s(t)nt)
r(t)
Input Data
Output
Data
Channel Coding and Modulation (Signal Mapper)
Guard Band Inrtion
S/P
P/S
I D F T
CP Inrtion
D/A
Fading
Channel g(t)
A/D
CP Deletion
Guard Band Deletion
Channel Decoding and Demodulation
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P/S
D F T
S/P
Y
y
. . .. . .
. . .
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g t ()αm δt τm T S –()
m 1
=M
∑
=
X X k []T
=Y Y k []T
=
, N 1–,=()g g n []T =n n n []T
=
, N 1–,=()X diag X ()=⎥
⎥⎥
⎥⎥⎦
⎤
⎢⎢⎢⎢⎢⎣⎡=−−−−)1)(1(0)1()1(000N N N
N N N N
N W W W W F L O L W N i ,k
1N ⁄
()
j –2Πik N ⁄()
=H DFT N g ()Fg ==N Fn =
Block-Type Pilot Channel Estimation
Under the assumption that the interferences are completely eliminated [1–3], you can derive:
Equation 3
This equation demonstrates that an OFDM system is equivalent to a transmission of data over a t of parallel channels.
As a result, the fading channel of the OFDM system can be viewed as a 2D lattice in a time-frequency plane, which is sampled at pilot positions and the channel characteristics between pilots are estimated by interpolation. The art in designing channel estimators is to solve this problem with a good trade-off between complexity and performance.The two basic 1D channel estimations in OFDM systems are illustrated in Figure 2. The first one, block-type pilot channel estimation, is developed under the assumption of slow fading channel, and it is performed by inrting pilot tones into all subcarriers of OFDM symbols within a specific period. The cond one, comb-type pilot channel estimation, is introduced to satisfy the need for equalizing when the channel changes even from one OFDM block to the subquent one. It is thus performed by inrting pilot tones into certain subcarriers of each OFDM symbol, where the interpolation is needed to estimate the conditions of data subcarriers. The strategies of the two basic types are analyzed in the next ctions.
3Block-Type Pilot Channel Estimation
In block-type pilot-bad channel estimation, as shown in Figure 2, OFDM channel estimation symbols are transmitted periodically, and all subcarriers are ud as pilots. The task here is to estimate the channel conditions (specified by or ) given the pilot signals (specified by matrix or vector ) and received signals (specified by ), with or without using certain knowledge of the channel s
tatistics. The receiver us the estimated channel conditions to decode the received data inside the block until the next pilot symbol arrives. The estimation can be bad on least square (LS), minimum mean-square error (MMSE), and modified MMSE.
3.1 LS Estimator
The LS estimator minimizes the parameter , where means the conjugate transpo operation. It is shown that the LS estimator of is given by [2].
Equation 4
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Figure 2. Two Basic Types of Pilot Arrangement for OFDM Channel Estimations
Y DFT N IDFT N X ()g ⊗n +()XFg N +XH N
+===F r e q u e n c y
F r e q u e n c y
Time
Time
S
pilot data
Comb-type pilot estimation
Block-type pilot channel estimation
Block
H g X X Y Y XH –()H
Y XH –()•()H
H H ˆLS X 1–Y X k Y k ⁄()[]
T ==k 0, ..., N 1–=()
Block-Type Pilot Channel Estimation
Without using any knowledge of the statistics of the channels, the LS estimators are calculated with very low complexity, but they suffer from a high mean-square error.
3.2 MMSE Estimator
The MMSE estimator employs the cond-order statistics of the channel conditions to minimize the mean-square error.
Denote by , , and the autocovariance matrix of , , and , respectively, and by the cross
covariance matrix between and . Also denote by the noi variance . Assume the channel vector and the noi are uncorrelated, it is derived that
Equation 5 Equation 6 Equation 7
Assume (thus ) and are known at the receiver in advance, the MMSE estimator of is given by
[2–5]. Note that if is not Gaussian, is not necessarily a minimum mean-square error estimator, but it is still the best linear estimator in the mean-square error n. At last, it is calculated that太阳从西边出来
Equation 8
The MMSE estimator yields much better performance than LS estimators, especially under the low SNR scenarios. A major drawback of the MMSE estimator is its high computational complexity, especially if matrix inversions are needed each time the data in changes.
3.3 Modified MMSE Estimator
Modified MMSE estimators are studied widely to reduce complexity [2–4]. Among them, an optimal low-rank MMSE (OLR-MMSE) estimator is propod in this paper, which combines the following three simplification techniques:
1.The first simplification of MMSE estimator is to replace the term in Equation 8 with its
expectation . Assuming the same signal constellation on all tones and equal probability
on all constellation points, we have
Equation 9
Defining the average SNR as , and the term .
The term , where the signal constellation. For example, for a 16-QAM transmission, .
R gg R HH R YY g H Y R gY g Y σN 2
E N 2{}g N R HH E HH H {}E Fg ()Fg ()H {}FR gg F
H
===R gY E gY H
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{}E g XFg N +()H
{}R gg F H X
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===R YY E YY H
{}XFR gg F H X H
σN 2
I N
+==R gg R HH σN 2
g g
ˆMMSE R gY R YY
1–Y HH =g g ˆMMSE H ˆMMSE Fg ˆMMSE F F H X H ()1–R gg 1–σN 2XF +[]1–Y
==FR gg F H X H XF ()1–σN 2R gg +[]F 1–H ˆ
LS
=R HH R HH σN 2XX H ()1–+
[]1–H ˆ
LS
=X XX H ()1
–E XX H
()1
–{}E XX H ()1
–{}E 1X k ⁄2
{} I
=E X k 2
{}σN 2
⁄=βE X k 2
{}E 1X ⁄k 2
{}⁄=σN 2XX H ()1
–βSNR ⁄()I ββ179⁄=
