This document gives a quick introduction of time-lective Rayleigh fading process and common channel estimation techniques.
Doppler Frequency and Rayleigh Fading Process Generating Rayleigh Fading Processing Channel Estimation by Low-Pass Filtering Hard/Soft Decision Feedback
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Doppler Frequency and Rayleigh Fading Process
Let ()x t be the complex fading process for the desired ur. Measurements indicate that the complex fading coefficient ()x t is a random quantity that changes slowly over time. So the mathematical nature of ()x t is a narrowband random process which has correlation over time. In the ca of Rayleigh fading, ()x t is a complex Gaussian narrowband process, which can be modeled as the output of a low pass filter excited by temporally white complex Gaussian noi. The low pass filter is often referred as the shaping filter, becau it determines the power spectrum shape and the temporal correlation function of the fading process. In the most widely ud Jakes' model, ()x t is assumed to have the following temporal correlation function,
0{()()}(2)H d E x t x t J F τπτ−=, where 0(*)J is the 0-th Besl function of the first kind and d F
is the physical Doppler frequency. The corresponding power spectrum is
()||d X f f F =≤. In current cellular systems, typical d F ranges from 5Hz to
300Hz, depending on the specific situation. For example, for a carrier frequency c f of 2GHz and a mobile speed v of 30 m/c (68 mile/hour),
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830210200310c d vf F Hz c
××===×, where c is speed of electromagnetic wave in the air. If d F is bigger than 100Hz, it is often referred to as "fast fading". Below we concentrate
on the digital receiver and still u parameters in the above example. Suppo that the symbol duration is limited to s T . If the baud rate 1()baud s R T =
is 40k /per cond, then the fading rate normalized to data rate is 320014010200
d
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d baud F f R ===×. Roughly speaking, th
e channel does not change much over 200 symbols. Since the fading is so slow at the symbol level, we can often neglect the change o
f the fadin
g process over one symbol duration s T and assume that the fading process remains constant over a
symbol, i.e., ()()s n x t x nT x ==, for (1)s s nT t n T ≤<+. According to the correlation in the
continuous ca, the correlation function is 0{}(2)H n n m d E x x J f m π−=, where
d d d s baud
F f F T R =
= is the normalized Doppler frequency. The power spectrum of n x
is ()||d X f f f =≤. At last, we point out that Jakes’ model in fact is mathematically derived rather than synthesized from field measurement. While Jakes’ model does fit some field measurement, other models from measurement
campaign might be more accurate in specific cas. For example, another normalized correlation function appears in a contribution to IEEE 802.16 specification for broadband wireless access, who power spectrum is
24()1 1.720.785,||d X f f f f f =−+≤.
Generating Jakes’ Rayleigh Fading Processing
A common method to generate n x is to sum up veral sinusoids, as first suggested in
Jakes’ book. However, this method in fact generates a deterministic process and not a truly random process. Nevertheless, it is still widely ud due to its simplicity. Here are the C++ implementation and COM implementation of a modified Jakes’ model propod in this paper by P. Dent et al . Another way is to pass white complex Gaussian noi n u
through the following shaping filter, ()||d h f f f =≤,
which is the square root of the power spectrum of n x . This filter is highly nonlinear
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and approximation has to be sought. Since the shaping filter has infinite impul
respon, it is natural to ek an IIR filter for approximation. JTC recommends using a 32 order IIR filter to approximate ()h f with normalized fading rate d f of 12
and
then generating slower fading process by interpolating the output from the IIR filter.
A simpler method is to u an autoregressive (AR) filter to approximate ()h f , who coefficients can be calculated from the correlation function of n x via the Yuler-
Walker equation.
Channel Estimation by Low-Pass Filtering
The model for flat fading channel can be expresd as, n n n n y x d v =+, where n y is the received
signal and n v is the complex additive white Gaussian noi. n d is the n th
transmitted symbol and in this documentation is limited to BPSK signals taking values of 1±. We first assume that the transmitted data is known to the receiver, for
example, n d is either a known pilot symbol or a highly reliable decision. So we can
alternatively write the model as H H n n n n n y d x v d =+, where we want to recover the
narrowband process n x buried in white noi H n n v d .
Figure 1: Channel Estimation via Low-pass Filtering
The engineering approach is low-pass filtering of the modulation-free signal H n n y d . Low-pass filtering itlf is a well studied problem, both mathematically and
practically. If the temporal correlation of the fading process is known, Wiener filter is the optimal filter. Particularly, if the fading process is generated by the IIR filter or AR filter, a Kalman filter can carry out the estimation recursively, which is computationally efficient. Note that temporal correlation is a function of the
normalized Doppler frequency and is unknown unless the vehicle speed is known. If the vehicle speed is unknown, adaptive algorithms can be ud to extract the narrowband process n x from the background noi. A popular adaptive filter is the
linear predictor, where LMS or RLS is ud to minimize the prediction error. There are also methods to estimate the coefficients of the AR filter when Kalman filtering is carried out. Approaches above are pursuing the optimality of the channel estimator by adapting the low-pass filter to the bandwidth/shape of the shaping filter. As a practical matter, the low-pass filter can be designed as a fixed filter to handle the maximum Doppler frequency possible, for example, a brick-wall low-pass filt
er with bandwidth max f . This worst-ca design has been shown numerically to be robust in
various situations and the performance degradation from "optimal" filters is minimal. The reason is that typically the maximum normalized Doppler frequency is fairly
small (for example, 1
in the first paragraph) and the fixed low-pass filter can
200
average over sufficient number of symbols to obtain accurate channel estimate for detection. An "optimal" filter could have averaged over even more symbols to further improve the channel estimate, but bit-error-rate is dominated by the additive noi人非圣贤孰能无过什么意思
v
n and will not be significantly lowered by the better channel estimate. On the other hand, if the bandwidth of the fixed low-pass filter is smaller than the true normalized Doppler frequency, the channel estimate can not keep up with channel variation and the BER degradation is more vere. S
ince channel estimation only incurs moderate complexity, most cellular systems employ coherent demodulation to take advantage of its 2-3dB gain over non-coherent demodulation.A uful BER expression can be found in the Appendix C of "Digital Communications" by John G. Proakis, which takes into account the impact of channel estimation error when maximal ratio combining is ud. This article by me further analyzes the impact of channel estimation error in the prence of strong co-channel interference.
Hard/Soft Decision Feedback
Up to now, we have assumed that the symbol
d is somehow known and w
器械体操
e u n d
n
to remove the modulation from
y. While we can inrt many pilot symbols to make
n
this assumption true, the overhead consumes a lot of energy and bandwidth. A more attractive approach is to u decisions as addition pilot symbols as follows. Suppo the transmission is in blocks. After the initial channel estimate is carried out, the data block is detected. While we might make wrong symbol decisions, we still remove the modulation with the decisions and re-estimate the channel using the low-pass filter. Hopefully the correct decisions can outweigh wrong decisions and output of the low-pass filter is a better channel estimate than the initial channel estimate. This better channel estimate in turn can lead to better decisions. This procedure can be repeated veral times. As found out by many practitioners, this iterative hard decision feedback method does provide decent BER even if the number of known pilot symbols is quite small. In fact, if the shaping filter is an IIR/AR filter and the channel estimator is a Kalman filter, it can be formally proved that each iteration can increa the likelihood of the demodulated symbol quence, which justifies the validity of this intuitive idea of hard decision feedback.
Figure 2: Iterative Receiver with Soft Decision Feedback
In a fast fading environment, further improvement can be obtained by using "soft" decisions. That is,
if the decision is unreliable, it is discounted in re-estimating the channel. For BPSK signals, the implementation is to multiply n y with {}H n E d , the expected value of the (conjugate of the) symbol. The rationale is that the clor the signal is to the decision boundary, the more unreliable the decision is and the smaller the amplitude of expected value of the symbol. So by using soft decision (the expected value) rather than hard decision, an unreliable decision has less impact in re-estimating the channel. Similar to the hard decision feedback ca, if the shaping filter is an IIR/AR filter and the channel estimator is a Kalman filter, it can be formally proved that each iteration can increa the a posteriori probability of the channel estimate, which justifies the validity of this idea of soft decision feedback. Interested readers are referred to Chapter 4 of my thesis for the proof. Through extensive simulations we find that when the fading is slow (say, 150
d f <), hard decision feedback has similar BER performanc
e as soft decision feedback, whereas i
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g is fast, soft decision feedback is better than hard decision feedback. Our conjecture is that when fading is slow, the channel estimator can average over many symbols, so the effect of a few wrong d
ecisions can be averaged out. However, if the fading is fast, the channel estimator can only average over limited number of symbols and the effect of wrong decisions is muc
h harder to be averaged out.
Conquently soft decision feedback is more favorable in a fast fading environment.
A rigorous proof of why soft decision back is better still remains elusive, becau it needs to deal with the convergence property of the expectation-maximization algorithm.
from the Previous Stage
Soft Limiter
