2D Wiener channel estimation performance analysis with diamond-shaped

2D Wiener channel estimation performance analysis with diamond-shaped
pilot-symbol pattern in MC-CDMA systems
Carlos Ribeiro1,2, Atílio Gameiro1
Instituto de Telecomunicações / Universidade de Aveiro, Campo Universitário, 3810-193 Aveiro, Portugal
Instituto Politécnico de Leiria, Morro do Lena, Alto Vieiro, 2411-901 Leiria, Portugal
Phone: +351-244820300, e-mail: [email protected]
Abstract1— The influence of the pilot density and distribution
on the channel estimation mean squared error of MC-CDMA
systems is addressed. The adopted frame structure uses the
optimum 2-dimensional diamond-shaped pilot grid and the
receiver performs the estimation using a 2-dimensional Wiener
interpolator filter, designed for the “Worst Case” scenario.
Simulation results put in evidence that the system’s performance
is strongly dependent on the pilot density and distribution and
that real-time knowledge of the received signal SNR may be used
to improve the performance.
Multicarrier code division multiple access systems (MCCDMA) performance depends heavily on the ability of the
receiver’s channel estimator to extract accurate channel state
information. Blind estimation techniques that need to gather a
large amount of information to perform the estimation exhibit
a poor performance in mobile systems where the channel
varies rapidly under the influence of Doppler’s effect and
multipath propagation. To achieve better performance pilotaided channel estimation techniques are commonly preferred.
Among these, the 2 dimension (2D) Wiener interpolator
filters [1] stand out as the one that guarantees the channel
estimation minimum mean squared error (MSE).
The pilot-aided channel estimation is the centre of Section
II, where the optimum 2D diamond-shaped pilot pattern is
introduced. In Section III, the 2D Wiener interpolator is
explained and the filters used in the simulations are presented.
The performance analysis of channel estimation scheme is the
aim of Section IV, firstly introducing the main characteristics
of the simulation tool and then presenting the attained results.
Finally, in Section V conclusions are drawn on this channel
estimation scheme and some suggestions are given on how to
improve its performance.
Channel state information is crucial for the MC-CDMA [2]
receiver’s equalizer to implement its algorithm and its
performance is closely dependent on the accuracy of that
information. This information is acquired by the channel
estimator in the receiver. Pilot-aided techniques perform the
channel estimation with the help of deterministic data
Parrt of this work was developed in the WISQUAS project, supported by the
European Union in the framework of CELTIC with contract no. CP2-035.
symbols that are inserted in the transmitted symbols. These
deterministic data symbols are known as pilots.
The MC-CDMA systems use a frame structure where the
pilots are multiplexed in time and frequency, Figure 1, by the
transmitter. The frame is made-up of Ns symbols, and each
symbol contains Nc carriers. Pilots replace data symbols in
some of the carriers as depicted in Figure 1. Pilots are
commonly spread in a regular structure (rectangular in Figure
1), Nf carriers apart in frequency and Nt symbols in time.
Other structures that can be found in the literature [2], [3] are
diagonal, random and diamond shaped.
The pilots are used by the channel estimator to get the Least
Squares estimates (LS), H LS , of the channel in the pilot
positions ( n ', i ') Î P ,
H LS n ' i ' =
Rn ' i ' + N n ' i '
= H n ' i ' + n ' i ' , "(n ' i ') Î P , (1)
Sn 'i '
Sn 'i '
where H ni is the complex value of the channel in position
( n, i ) of the frame, Rni , S ni and N ni are, respectively, the
received symbol, the transmitted symbol and the noise
component in the carrier ( n, i ) .
The initial LS estimates can be interpreted as noisy samples
of the channel and to be able to recover the channel state
information, the pilot distances Nf and Nt should fulfill the 2D
Nyquist Theorem, with cut-off frequencies given by the
minimum coherence time and minimum coherence bandwidth
[2], that are, respectively, functions of the mobile maximum
speed and channel multipath maximum delay,
Nf < Delaymax Df
Nt < f Dopplermax TS
where Delaymax is the maximum delay of the propagation
channel, Df is the carrier separation, f Dopplermax is the
maximum Doppler frequency and TS is the MC-CDMA
symbol duration.
The use of 2D pilot grids has been shown to outperform
conventional 1D pilot patterns [1]. Among all regular
patterns, diamond-shaped pilot pattern was proven to deliver
Fig.1. MC-CDMA frame structure.
Fig. 2. 2D diamond-shaped pilot pattern.
the best performance [3], [4]. Figure 2 shows an example of a
MC-CDMA frame with this pilot pattern.
The diamond-shaped pattern, like any regular pilot pattern,
can be represented using 2 basis vectors V1 and V2 [5],
V 1 = [i1 n1]
V 2 = [i 2 n 2] ,
The optimum pilot pattern for a given pilot density is
achieved [3], [4] when n1 = 0 and i 2 = i1 2 . Such a pattern
is exemplified in Figure 2, where i1 = Nt and n 2 = Nf ,
resulting in,
é Nt ù
é Nt ù
V1 = ê ú
V 2 = ê 2ú .
êë Nf úû
Defining the matrix V,
é Nt Nt ù
2ú ,
V = [V 1: V 2 ] = ê
ëê 0 Nf ûú
the pilot density D is inversely proportional to the pilot
spacing and is defined [5] for any regular 2D pattern as,
D = det (V )
The estimation error is inversely proportional to the size of
the set of LS estimates used [7]. The LS estimates used to
estimate each position should be the ones closer to estimated
position, according to the rules defined in [7].
The calculation of the optimum filter coefficients for each
point to estimate is based on the Wiener-Hopf equations [2],
θn - n '',i -i '' =
wn ',i ',n,ifn '-n '',i '-i '' , " {n '', i ''} Î t ni , (8)
{n ',i '}Ît ni
where the filter’s input-output cross-correlation θn - n '',i - i '' is
= ( NtNf ) .
This estimation filter assures the minimization of the
channel estimation MSE, when the channel samples are
corrupted by noise [2], [3], [6].
The use of this type of static filter assumes that the
propagation channel is wide-sense stationary for the duration
of the MC-CDMA frame. The optimum filter coefficients
vary from estimated point to estimated point (time and
frequency), by what the filter is usually described as shiftvariant.
The filter input is made up of the LS estimates of the
channel in the pilot positions. The filter output Hˆ ni is
channel’s discrete transfer function estimation and it’s
expressed as,
Hˆ ni =
wWienn ',i ',n ,i H LS n ' i ' , "t ni Î R ,
{ } ni
where wWienn ',i ',n ,i represents the 2D shift-variant Wiener filter,
com NTap coefficients per estimated position and t ni is the
set of LS estimates used to get the estimate of the
( n, i )
position of the frame. t ni is a sub-set of set of pilots present
in each frame P . The size of t ni is necessarily coincident
with the length of the filter, t ni = NTap .
defined by,
θn - n '',i -i '' = E H ni H LS
= Rf (n - n '') Rt (i - i '') ,
n '' i ''
considering that the noise and the transmitted symbols are
statistically independent. Rf and Rt are, respectively, the
frequency and time channel correlations.
The filter’s input correlation fn '- n '',i '- i '' is defined as,
fn '- n '',i '-i '' = E H LSn ' i ' H LS
n '' i ''
= q n '- n '',i '-i '' +
d ( n '- n '', i '- i '' )
= Rf (n '- n '') Rt (i '- i '') +
, (10)
d ( n '- n '', i '- i '' )
where g P is the average signal-to-noise ratio per path.
Considering that each element of the MC-CDMA frame is
estimated using every pilot in it, given a frame with Ns
symbols and Nc carriers per symbol, using a 2D shift-variant
Wiener filter with NTap coefficients, the total number of
coefficients needed for the frame will be,
NTotal = NTap .Ns.Nc .
In the analysis performed in this document, channel
estimation is performed using 2 orthogonal 1D Wiener
interpolator filters. First filter is applied in the frequency
domain and the second in time domain, using the estimates
obtained from the first. These filters perform similarly to the
2D Wiener interpolator filter [1], with a considerably lower
complexity. The filters are designed assuming uniform delay
and Doppler power spectrums (worst case scenario:
maximum expected channel delay and mobile terminal (MT)
speed) and the calculation of its coefficients is based on (8),
using all the pilots present in the frame. The average signalto-noise ratio per path is known and used in the filter project.
The performance analysis of the effect of the pilot density
and distribution in the mean squared error of the channel
estimation was carried out using a MC-CDMA system
simulation tool with single antenna receiver and transmitter.
The main characteristics of the simulation tool are presented
in Table I.
The adopted MC-CDMA frame always uses the optimum
2D diamond-shaped pilot pattern, with 128 symbols per
frame. The pilot distances adopted for simulations where
Nt Î {2, 4, 6,K, 20} and Nf Î {1, 2,K, 20} .
Fig. 3. Examples of frequency responses of simulated BRAN E channel.
The simulated wireless channel followed the time related
parameters from ETSI BRAN-E channel model [8], with an
impulse response given by,
h ( i ,t ) =
L -1
åa d (t - t ) ,
l =0
where L is the number of paths, ai l is the complex value of
path l in instant i and t l is the delay of path l. The paths are
assumed to be statistically independent, with normalized
L -1
average power,
= 1 , where s l 2 is the average power
l =0
of path i.
Table I - Simulation tool main characteristics.
Carrier frequency
OFDM symbol duration
Carrier spacing
Cyclic prefix
Number of carriers
Spreading factor
» 26 m s
20% OFDM symbol duration
Figure 3 shows examples of the simulated channel,
respectively (from left to right), for MT speed of 30km/h,
100km/h and 300km/h. Figure 4 shows the channel estimation
MSE performance surfaces as function of the pilot distances
Nt and Nf,. Pilot density ranges 50% (Nf=1, Nt=2) to 0,25%
(Nf=20, Nt=20). Columns (from left to right) represent the
MT speeds of 30km/h, 100km/h and 300km/h, respectively.
Rows (from top to bottom) represent the power efficiency,
Eb N 0 , of 10dB, 20dB and 30dB, respectively.
Each surface has a larger dot that points out the optimum
value in each surface and a set of smaller dots that show the
pairs (Nt,Nf) over which the achieved channel estimation
MSE is not higher than 3dB from the optimum.
Table II summarizes the achieved channel estimation
minimum MSE in the scenarios depicted in Figure 4.
Table III summarizes the optimum pilot densities D and
associated optimum pilot distances pairs (Nt,Nf) in the
scenarios depicted in Figure 4.
Observing the surfaces in Figure 4 it is obvious that the
achieved channel estimation MSE is strongly dependent on
the pilot distribution and density. Looking at each row, it’s
clear that the performance of the channel estimator remains
fairly constant with the MT speed. Both the optimum value of
density and the set of points that will result in a near optimum
performance are nearly the same for the 3 simulated speeds.
In addition, the optimum density is always surrounded by a
set of near optimum points. The achieved MSE varies little
with the pilot distance in time Nt, and a value in the range
[2,…,8] will most likely result in a near optimum
performance of the channel estimator.
The main issue is that the MSE surface does not evolve
smoothly with Nf and looking at surfaces in columns is easy
to see that there‘s no value of Nf that will achieve a near
optimum performance for the various simulated values of
power efficiency.
A frame structure with a static pilot grid will not achieve a
near optimum channel estimation. If better performance is
mandatory then real-time knowledge of the received signal
SNR is needed and pilot distance Nf must dynamically adapt.
The pilot distance Nf should vary inversely with the estimated
received signal SNR, but always fulfilling the Nyquist
Simulations carried out with BRAN-C and BRAN-A
channel models showed a similar behavior pointing out that,
for this channel estimation scheme, the rules adopted for the
choosing of Nt and Nf will also be valid.
Table II - Achieved channel estimation minimum MSE.
MT Speed
Table III - Optimum pilot density D and pilot distances in time and
frequency (Nt,Nf).
MT Speed
Fig. 4. Channel estimation performance surfaces.
Some care should be taken when finding the pilot distances
Nt and Nf using the method proposed in [1] and [2], as it not
always provides the best results.
The simulation results presented in this paper showed that
the performance of the 2D Wiener interpolator used for
channel estimation is strongly dependent on the pilot symbols
density and distribution. Some rules are given that most likely
will result in a near optimum channel estimation for a variety
of MT speeds and wireless channels. This work puts in
evidence the need for a dynamic frame structure and the realtime estimate of system parameters like received signal SNR.
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