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1122
IEEE
Transactions on Consumer Electronics,
Vol.
44,
No.
3,
AUGUST
1998
PILOTTONE SELECTION
FOR
CHANNEL ESTIMATION IN A MOBILE
OFDM
SYSTEM
Rohit Negi and
John
Cioffi
Information Systems Laboratory
Stanford University, Stanford,
CA
94305
Abstract-
Channel estimation for mobile
OFDM
systems
requires transmission
of
pilot tones. This paper addresses
the important issue of selecting these pilot tones,
so
as to
achieve
a
good quality estimate. It is shown that the best set
of
tones to be used are those which are equally spaced.
Fur-
thermore, it is shown using the case of
a
first order Markov
channel, that it
is
more efficient to use
a
few pilot tones
in
all symbols, rather than use all tones
as
pilot tones
in
some
symbols.
KeywordsaFDM,pilot tones,channel estimation,mobile
I. INTRODUCTION
Recently, there has been considerable interest in
us-
ing Orthogonal Frequency Division Multiplexing (OFDM)
transmission for mobile wireless channels ([1],[2]). OFDM
transmission invariably requires an estimation of the chan-
nel frequency response (i.e. the gains
of
the OFDM tones).
Blind channel estimation techniques try to estimate the
channel without any knowledge of the transmitted data.
However, whereas blind estimation methods are attractive
because of the possible savings in training overhead, they
are only effective when a large amount of data can be col-
lected
(so
that stochastic estimation can be made reliably).
This is clearly a disadvantage in the case of mobile (es-
pecially, point-to-point) wireless because the time-varying
channel would preclude accumulation of a large amount of
data. Therefore, it would seem that channel estimation for
this case would need training data.
The conventional method of estimating the channel re-
sponse for a wireline channel is to use straight-forward
frequency domain estimation. However, this is probably
an overkill for the mobile wireless case. The reason is
that, given a specific wireless scenario (e.g. macro-cellular
wireless), the channel length is limited to a few samples.
The OFDM system can be designed with a cyclic prefix
of this length,
so
that the need for a Time-domain equal-
izer
(TEQ) to shorten the channel, is eliminated. Indeed,
most of the proposals for OFDM-based wireless systems
assume that this
is
the case [3],[4]. Given this fact, it
would be appropriate to estimate the channel in the time-
domain because there are fewer parameters in the impulse
response than in the frequency response. Given the limited
amount of training data that can be sent to estimate the
time-varying channel, limiting the number of parameters
to be estimated would increase the accuracy of the estima-
tion. This is the thrust of the estimation techniques used
in
[4],[5],
for example.
In an OFDM system, the training data has to be sent on
The
work was
supported
by
NSF
grant
NCR-9628185
select tones, which are called
pilot tones.
The question
then arises as to which tones should be used as pilot tones,
and the impact of pilot tone selection on the quality of the
estimate. This paper presents some results addressing this
issue.
The organization of the paper is as follows. The prob-
lem is formulated in Section
11.
In Section
I11
we present
the results on pilot tone selection.
It
will be shown that
equally-spaced tones are the best set to use as pilot tones.
Furthermore, we derive the Minimum Mean Square Error
(MMSE) expressions for the channel estimate for the case
of a first
order Markov channel. Simulation results are
presented in Section IV and we conclude in Section
V.
Fig.
1.
Basic
OFDM
system
11.
PROBLEM
FORMULATION
A.
Notations
Standard notations are used in this paper. Bold letters
denote vectors and matrices. Upper-case and lower-case
letters denote frequency domain and time domain variables
respectively. Other notation are as follows.
(.)t
Transpose
(.)*
Hermitian
E{.}
Expectation operator
tr{.} Trace operator
I
.
I
Absolute value
11
.
/I
%norm
I,
(:)
Estimate of
(.)
(.)(Pz)
n
x
n
Identity matrix
Vector with pilot tone subscripts
B.
OFDM
Sgstem
The basic baseband-equivalent OFDM system is shown
in Figure
1.
Each OFDM
symbol
consists of a packet of N
data points, that are carried on
N
frequency
tones
respec-
Manuscript received June
17,
1998 0098 3063/98
$10.00
1998
IEEE
1123
Negi and Ciofi: Pilot Tone Selection
for
Channel Estimation in
a
Mobile OFDM System
tively. The system equations can be written as
yk,m
=
Hk,mXk,m
4-
vk,m
=
0,1,.
.
.,
N
-
1
(1)
Number of OFDM tones
Cyclic prefix length
Maximum length of channel
Frequency subscript (tone number)
Time subscript (OFDM symbol number)
Received signal
Transmitted data
AWGN noise
OFDM tone gains (channel response)
In addition, we have,
where
Hm
=
[HO,m,
,
HN-l,mIt
hm
=
[ho,,,
.
.
.
,
h~-l,,]~
(channel impue response
Q
is the
unitary
Fast Fourier Transform
(FFT)
ma-
trix. Note that the definition of the
FFT
used is
XI,
=
1
Zne-j2flknlN.
fl
Also, for simplicity, we denote
WN
=
e-j2P/N.
In all cases, the time subscript
m
will be dropped when it
is not important, such as in
h,
H,
etc.
C.
Assumptions
The channel is assumed to be shorter than the cyclic
prefix (i.e. maximum channel length is
L
=
U
+
1
taps)
so
that the need for a channel shortening filter (TEQ) is elim-
inated (hence system equation
(1)).
The noise is assumed
to be additive, white and gaussian (AWGN). Therefore, the
MMSE criterion can be used for maximum-likelihood esti-
mation.
No
apriori knowledge
of
the channel gains is as-
sumed. Therefore the MMSE estimator will coincide with
the Zero-Forcing (ZF) estimator. Whereas useful results
can be derived for the case of arbitrary
N
and
L,
the more
mathematically satisfying results are obtained for the case
where
NIL
is an integer. Therefore, this fact will be as-
sumed throughout the paper.
It
is also assumed that the
training data sent
on
each pilot tone has a constant modu-
lus equal to
a.
This is trivially true in the case of QPSK
broadcast, and is clearly the optimum training data for the
power spectrum density limited case (where each pilot tone
should
carry
maximum
power).
D.
Problem Formulation
The transmitter uses certain tones (called 'pilot tones')
in
a
particular symbol to transmit known data. The chan-
nel impulse response
hm
can be estimated using the MMSE
criterion, given knowledge of the transmitted and received
signals. The questions that arise are
How many pilot tones are needed per symbol for esti-
What sets
of
pilot tones are better than others?
How does a scheme that uses
some
tones
as pilot tones
in
each symbol
compare with
a
scheme that uses
all
tones
as pilot tones in
some symbols?
We will present some answers to these questions in the
subsequent sections.
mation?
111.
PILOT TONE
SELECTION
A.
Number
of
Pilot Tones
required motivates the following lemma.
The question about the minimum number of pilot tones
Lemma
1:
In
the absence of noise, any L of the
N
avail-
able tones can be used for training
to
recover the channel
h
exactly.
Proof:
Let
{kl,
kz,
.
. .
,
k~}
be the set of
L
tones used
for transmitting training data. The channel gains of those
tones can be found exactly as
HI,^
=
Rki.
Collect these
gains in a vector
H(Pz)
=
HI,^,.
. .
,
HI,,)^.
Then, we can
write
Since the matrix
Q(p')
is a (scalar multiple of a) Vander-
monde matrix with all
L parameters
{Wh}
distinct, hence
it is non-singular
[6],
and therefore,
h
can be found exactly
by inverting it. Also, with less than
L pilot tones, we have
an under-determined system of linear equations, and hence
a non-unique solution
h.
B.
Optimum Set
of
Pilot Tones
Since
L
tones is the minimum number required for exact
channel estimation in the absence of noise, it is assumed
that the transmitter uses
L
pilot tones in each symbol.
In the absence of noise, choice of the
L pilot tones is not an
issue. However, in the presence of noise, the choice of the
L
tones becomes critical.
In fact, the following theorem
illustrates the optimum choice for the set of
L tones for
AWGN case.
Theorem
1:
When the noise is AWGN, then the MMSE
estimate of
h
occurs when the set of
L
pilot tones is one
of
the sets
{i,i+
%,..
.,i
+
v},
i
=
0,1,
...,
$
-
1.
Proof:
Consider the channel equations
(2).
As in the
previous lemma, let
{kl
,
k2,
.
.
.
,
k~}
be the set of L pilot
tones. Then,
(4)
holds. Let the AWGN variance per time
sample be
ai.
This is also the noise variance in each tone.
Because of the AWGN assumption, and also because each
1124
EEE Transactions
on
Consumer
Electronics,
Vol.
44,
NO.
3,
AUGUST
1998
pilot tone carries data of
constant modulus
&,
hence the
MMSE (which is also the
ZF)
estimate of
h
is given by
(5)
where
R(p')
=
.
.
and
s(p')
=
[sk,, . . .
,
SkLlt,
and the inverse of Q(p') exists,
as
seen in Lemma
1.
Since the
{Vk}
are i.i.d. random variables, hence
so
are the
{Sk},
with the same variance. Therefore, the Mean Square
Error (MSE) in the channel estimate can be calculated as
}
E{llh
-
-tr{Q(p')-'u21
1
Q(p')-*
SL
E,
Note that Q(p') depends on the choice of the set of pilot
tones. Thus, the design pnroblem is to choose the set that
minimizes the MSE,
E{llh
-
hll'}.
This can be achieved
as follows.
First, note that
/1
z z z\
\z z
z
1)
(a constant)
tr{Q(P')*Qb')}
=
-
L2
N
Let the eigenvalues of (Q(p')*Q(p'))-' be denoted by
{X,,p
=
I,
2,
.
. .
,
L}.
Then, the eigenvalues of Q(p')*Q(p')
are
{
l/Xp,p
=
I,
2,
.
.
.
,
L}.
Since the trace
of
a matrix is the sum of its eigenvalues,
hence, the relaxed optimization problem is,
minimize
eAp
subject to
p=
1
L
x--.
1
L2
Since the above matrices are non-negative definite, hence
all the eigenvalues are non-negative. Thus, the mini-
mization occurs iff all the eigenvalues
{A,}
are identical.
This occurs iff Q(pl)*Q(p')
=
$L.
The sets
of
tones
{i,i+
-
1,
are the
only sets that have
a
Q(p') that satisfy this orthogonality
condition.
The
MMSE
in channel estimate, when such
a
tone set is
used, is
,...,
if
v},
i
=
O,l,
...,
Notice that the optimal tone sets are the sets of the
equally-spaced tones. Hereafter, such a set of tones will be
referred to as the 'optimally-spaced tones set'. Roughly
speaking, when the pilot tones are clustered close to each
other, the Vandermonde matrix is 'more' ill-conditioned,
and hence there is
a
noise enhancement effect in interpo-
lating the
h
from the
R(p').
The proof also indicates the noise-enhancement penalty
one would incur if the optimally-spaced tones are not used.
This may be of interest when evaluating the use of other
tone sets.
C. Comparison
of
Two
Schemes
Our final objective is to use a scheme that uses a few
(L)
pilot tones in each symbol,
for
time-varying channels.
In that case, one would form an instantaneous estimate
of the channel impulse response in each symbol, and use
it to update the average channel estimate, in some fash-
ion. For simplicity, this scheme is labelled as scheme
A.
For the AWGN case, any of the optimally-spaced tones set
provides an equally good estimate of the channel. It will,
therefore be assumed that for the AWGN case, in scheme
A,
one uses
a
fixed
set of optimally-spaced tones.
One could also think of a scheme, where one uses all the
tones in
a
symbol as pilot tones, periodically. i.e. Peri-
odically, one entire symbol would be used just to transmit
training data, and the channel estimate would be obtained
from this. This estimate would then be used for all symbols
until the next pilot symbol is received. Call this scheme
B.
It would appear that scheme
B
has the advantage that the
channel would be more accurately estimated, because more
number of pilot tones are being used in a pilot symbol. The
disadvantage, of course, is that when the channel is time-
varying, the fixed estimate would degrade in quality, as
compared to the new channel response. It seems intuitive
that scheme
A
is probably more advantageous than scheme
B.
As a first step towards proving this intuition, we state
the following lemma.
Lemma
2:
In an AWGN environment, for a
time-
invariant
channel, the efficiency in the use of the pilot
tones is the same for
a.
scheme that transmits a few pilot
tones in each symbol (scheme
A),
and for
a
scheme that
uses all pilot tones in one symbol periodically (scheme
B),
provided that the former scheme uses the optimally-spaced
tones.
ie. the MMSE in the channel estimate is the same
for both schemes, when the 'total number' of pilot tones
transmitted is the
same.
Note that the total
number
of
pilot tones
is
defined as the product of the number of pilot
tones per symbol and the number of such symbols trans-
mitted.
Proof:
Let the AWGN noise variance per time sam-
ple be
n:.
This is also the noise variance in each tone.
Consider scheme
B, where all tones in a symbol are pilot
tones. From
(2),(3),
and because each pilot tone carries
data that has
a
constant modulus
G,
we can write
1
R
Qlh+-S
4%
Negi
and
Cioffi:
Pilot Tone Selection
for
Channel Estimation in a Mobile
OFDM
System
1125
where
R
=
[&,
. . .
,
RN-~]~
and
S
=
[SO,.
.
.
,
SN-#.
Since the
{vk}
are i.i.d.
random variables, hence
so
are
the
{sk},
with the same variance.
Further, we have
QTQ1
=
IL.
Therefore, the MMSE estimate of
h
is given
by
h
=
QTR
1
=
h+Q;zS
Therefore, the MMSE in the channel estimate can be cal-
culated as
1
E{llh
-
hl12}
=
---tr{$Ta?I~Qi)
E,
Compare this MMSE with the MMSE obtained for
scheme A, where
L optimally-spaced pilot tones are used in
each symbol. Denoting the :Instantaneous channel estimate
in successive symbols by
hl
,
h2,
. . .
,
for
a
time-invariant
with AWGN, the optimal way to combine these estimates
is
-M
1
h
=
-Eh
m
M
m=l
Now, to compare the efficiency of the two schemes in util-
ising the training data, choose
M
=
NIL,
so
that both
schemes use
a
total of
N
tones for training. Then, the
MMSE in channel estimate is
a:
L
EX
- -
-
which is the same as the MMSE obtained for scheme B.
W
Thus, we've shown that for a
time-invariant channel,
distributing the pilot tones into different symbols, performs
as good sending them all in one symbol.
Now, we focus on the more interesting case of a mo-
bile time-varying channel. The objective is to show that
scheme
A
can better track a
time-varying
channel, than
scheme B. The previous lemma indicates that scheme A
should certainly do no worse. By using a
forgetting factor
in combining the estimates from consecutive symbols, it
should be possible to do better than scheme B.
To
see
this,
we
can
compare the steady
state
MMSE
for
the two schemes for a first order Markov channel, which
should adequately represent
a
slowly time-varying channel.
Consider the following first order Markov channel
([7])
hm+l=
ah,
+
wm,
Rw
=
E{wmwL}
(7)
where the vector sequence
w,
is uncorrelated with it-
self and with the noise vector sequence
s,.
Further,
a
<
1,
a
M
1
is assumed (slow time variations).
Scheme A uses the following algorithm to estimate the
channel
For OFDM symbol
m,
r(~l)
m
=
Q(P~)--~R(P~)
m
(8)
h,
=
ph,-I
+
(1
-
/3)@
(9)
where
,B
is a forgetting factor.
Rk')
and
Q(p')
are as de-
fined in Section 111-B.
The algorithm can be initialized
with
ho
=
0.
It should be remarked that the above algorithm is equiv-
alent to the LMS algorithm (and also the
RLS
algorithm
with infinite state variance) for minimizing the tap error.
Scheme B uses
a
similar forgetting factor in updating the
channel estimate, but only when the next pilot symbol is
received.
It
keeps the channel estimate fixed until the next
pilot symbol is received.
One can find the steady state MMSE in the channel esti-
mate (which would include the tracking error of the algo-
rithm), and select
,B
so
as to minimize it. This analysis is
presented in Appendix A.
The results are stated in the following theorem.
Theorem
2:
The MMSE of the
scheme
A
is
channel estimate for
where
17s
.J
173
Popt
=
1
+
-
-
Vs
+
-
2
4
The MMSE of the channel estimate for scheme B is
I
N2
-17s
L2
17s
=
Dopt
=
1
+
9
-
?js
+
-
7id?
2
Here,
MuSEbest
is defined as the MMSE in the channel
estimate just after
a
pilot symbol (this is the best useful
estimate), while
M&?SE,,,,t
is defined as the MMSE just
before
a
pilot symbol
(this
is
the
worst
useful estimate).
Proof:
See Appendix A
Normalized
MMSE
for scheme
A,
and normalized
MMSEbest, MMSE,,,,t
for scheme B are compared in
Figure 2 for
a
typical case, where
L
=
8,
N
=
128,
SNR
1126
(per tone)
=
10
dB
(see Section
IV
for definitions). In gen-
eral, not unexpectedly, the performance of scheme
A
lies
between the best and worst cases of scheme
B.
However,
for high values of degree of non-stationarity
qs,
which in-
dicates a fast varying channel, scheme
A
performs better
than even the best case
of scheme
B.
For moderate val-
ues of
qs,
scheme
A
performs significantly better than the
worst case of scheme
B. This is important, because the
worst case will be the limiting factor in any transmission
design.
IV.
SIMULATION
RESULTS
We consider
a
typical case, where
L
=
8,
N
=
128
and
QPSK
transmission in AWGN. For example, the
system parameters may be: symbol period=160
psec,
bandwidth=810
kHz,
carrier frequency=l
GHz,
channel
length=9.0
pec. The MMSE was calculated by averaging
over 50 independent runs.
To illustrate the equivalence of schemes
A
and
B
for the
time-invariant case (Lemma 2), we consider the channel
h
=
[0.5,1.2,0.8,0.4
+
0.5j,0.3j,0.3,0.2,0.1]t and
SNR
(per tone)
=
&,llh/12/Noi
=
10
dB.
Figure
3
shows the
Normalized MMSE
(E{
llh-hl12}/E{
l\hl12})
in the channel
estimates for scheme
A
(using various sets of pilot tones)
and scheme
B,
over time. It can be seen that the optimally-
spaced tones are the best sets to use in scheme
A.
Also,
the MMSE decreases equally rapidly for both schemes
A
and
B,
which shows that both schemes are equally efficient
in the use of pilot tones.
Figures
4
and 5 show the Normalized MMSE for first order
Markov channels. The MMSE at the pilot symbol has been
suppressed because this cannot be used anyway. The opti-
mum forgetting factors were used in all cases. The channel
in Figure
5
varies faster than that in Figure
4.
It
is obvi-
ous that the better tracking ability
of
scheme
A
allows for
superior performance as compared to scheme
B,
especially
in the case of the faster varying channel.
Also,
it is again
obvious that the optimally-spaced set
of
tones should be
used in scheme
A.
Figure
6
shows the Normalized MMSE for
a
channel that
varies according to the well-known Jakes' model
[8].
The
doppler frequency was set at
100
Hz
(vehicle speed of
about
70
mph).
The forgetting factors for both schemes
were optimized by trial and error.
It
can be seen that for
this case too, scheme
A
performs significantly better than
the worst case of scheme
B.
EEE Transactions
on
Consumer
Electronics,
Vol.
44,
No.
3,
AUGUST
1998
V.
CONCLUSION
This paper answered some questions pertaining to the
choice
of
pilot tones in a mobile OFDM system. It was
shown that, from an efficiency point of view, it is better
to choose to transmit
a
few pilot tones in each symbol
rather than clump them together in one symbol. This al-
lows for better tracking of the channel variations.
It
was
also shown that for the former case, one should choose the
equally-spaced tones sets as pilot tones, to avoid the noise
enhancement effect in interpolating the channel impulse
response from the frequency response.
APPENDIX
I.
PROOF
OF
THEOREM
2
Define the tap error
E,
=
h,
-
h,
and define its auto-
correlation
R,
=
E{E~E&}.
Then
e,
=
phmp1
+
(1
-
p)rkz)
-
h,
using (9)
=
&-I
+
(1
-
P)(hm
+
1
---Q(pz)-lS(p'))
m
-
h,
using (5),(8)
VE
=
/3hm-i
-
/3(ah,-l
+
~~-1)
+
=
PE,-l
-
/3wm-1+
1
(1
-
P)-Q(tT)-lSg')
+
P(l
-
u)h,-l
4%
Therefore, due to the uncorrelatedness properties of
U,,
and neglecting the
(1
-
U)
term in the equation, we can
write
NO,"
LE,
R,
=
P2R,-1
+
P2RU
+
(1
-
P)2-I~
(1
-
/3)2 NU:
R,
+
____-
IL
P2
R,
=
1
-/32
1
-
(52
LE,
1/31
<
1
is required for convergence.
R,
is the steady state
matrix.
Define the
degree
of
non-stationarity
[7],
qs
=
w.
This is an indicator of the rate of channel variation. Then,
the steady state MSE in the channel estimate can be found
as
MSE(/3)
is
a
convex function of
/3,
and
so
the optimal
value
of
/3
that minimizes the MSE
is
Popt
=
1
+
-5
-
qs
+
and then
v-7
2
MMSE
MSE(P,,t)
The performance
of
this scheme is to be compared with the
performance of scheme
B.
For the latter scheme, it is clear
that the channel estimate will be best for the symbol that
has
the
pilot
tones, and will steadily deteriorate (because
of the time-varying channel) until the next pilot symbol.
The estimates obtained by this scheme periodically, can be
combined using a forgetting factor.
The analysis for this
case is similar to the analysis presented above, and
so
we
simply state the result
1127
Negi and Cioffi: Pilot Tone Selection
for
Channel Estimation in a Mobile
OFDM
System
0,
-5-
7js
.J
jopt
=
1
+
-
-
7js
+
-
2
-
Oplimumsel
(0,16,32.48.84,80,96.112)
I
Tone
set
j0.15.34,45,69.82.95,118)
I
__
Tone
881
(0,10,40,45,70,75,105,l24)
I
\
.\
!
++
Scheme
B
-
all
tones
evev
161h
symbol
MMSE
=
M%E(&,~)
5
0-
5-
1
where MUSE is the MMSE in the channel estimate
for
the
pilot
symbols
only.
Of interest to
us
is the MMSE in
the channel estimate just after a pilot symbol and in the
estimate just before a pilot symbol. Due to the uncorre-
latedness assumption on
wm,
this can be easily found as
. .
......,
. .
......
I
,
.
......
'
'
,,.,.,
Sdeme
A
-
8
pilot
tones
eveiy
symbol
Scheme
B
(MI
care)
-
128
pllott~e~eveiy
lmh
symbd
Scheme
B
(wont
case)
-
128
p'lot
tones
every
16th
symbol
t
+
-
+
__
++
+
+
+
+
+
+
+
+
N
L
MMSEwOrst
=
MUSE
+
(-
-
l)tr{R,)
leading to the result in Theorem
2.
REFERENCES
[l] W.Y.
Zou,
and Y. Wu,"COFDM:An overview",IEEE Trans.
Broadcasting, vol. 41, pp. 1-8, Mar. 1995.
[2] L.C. Cimini, Jr.,"Analysis and simulation of a digital mobile
channel using orthogonal frequency-division multiplexing"
JEEE
Trans.
Commun., vol.
33,
pp. 665-675, July 1995.
[3]
0.
Edfors,M. Sandel1,J. van de Beek,
S.K.
Wilson,
P.
Ola Bor-
jesson,"OFDM channel estimation by singular value decomposi-
tion", Proc. IEEE VTC'96,vol.
2,
pp. 923-927, Atlanta, GA, Apr.
1996.
[4]
V.
Mignone, and A. Morello,"Novel demodulation scheme for
fixed and mobile receivers",IEEE Trans. Commun., vol. 44, pp.
1144-1151, Sep. 1996.
[5]
J.
Rinne, and M. Renfors,"Pilot spacing in orthogonal frequency
division multiplexing systems on practical channels",IEEE
Trans. Consumer Electron., vol. 42, pp. 959-962, Nov. 1996.
[6] R.A. Horn, and C.R. Johnson,Matriz analysis, New
York:Cambridge University Press, 1990.
[7]
S.S.
Haykin,Adaptive filter theory, Englewood Cliffs,NJ:Prentice-
Hall, 3rd ed., 1996.
[8]
W.C. Jakes,Microwave mobile communications, New York:John
Wiley, 1974.
20
40
80 80
100
120
Number
01
symbols
transmined
-351
0
Fig.
3.
Normalized MMSE in channel estimate of time-invariant
channel, for scheme A (various pilot tones sets) and scheme
B
:
N
=
128, L
=
8,
SNR
=
30
dB
-
optimun
set
(o.t6,32.48.64.80,98,t
12)
. .
TMesel(O,15,34,45,89,~,95,tt8)
--
Tone
set
~0,10,40,45.70,75.105,124/
++
Scheme
B
-all
tones
every
161h
symbol
J
XI
40
EO
80
1W
120 140
-20'
"be,
01
symbol8
transmined
Fig.
4.
Normalized MMSE in channel estimate of first-order Markov
channel
for
schemes A and B
:
a
=
0.999,
N
=
128,L
=
8,
SNR
=
30
dB
(0s
=
0.015,
Popt
=
0.88)
Rohit
Negi
received the Bachelor of Technology degree in Electrical
Engineering
from
the Indian Institute of Technology, Bombay, India,
in 1995, and the M.S. degree in Electrical Engineering from Stan-
ford University, in 1996. He is currently working towards a Ph.D.
in Electrical Engineering, in the Information Systems Laboratory at
Stanford University. His research interests include wireless commu-
nications; in particular, code design for wireless channels, algorithms
for channel estimation, and OFDM systems.
John
M.
Cioffi
is an Associate Professor of Electrical Engineering
at Stanford University. Before joining the Stanford faculty in 1986,
he
was
on the research staff at IBM in San Jose, California, in the
Signal Processing/Coding Group. Between 1978 and 1984, he was
on the technical staff, Advanced Data Communications Department,
at Bell Laboratories in Holmdel, New Jersey.
His
current research
interests are Communications Signal Processing and Coding, with
particular emphasis on their application to data transmission and
storage. He has published over 50 journal papers, over 90 conference
papers, and 15 patents (all licensed).
Dr.
Cioffi is co-founder and chief technical officer of Amati Com-
munications, Inc., San Jose, California.
Dr.
Cioffi won the Best Paper Award from the IEEE Communica-
tions Society in 1990. He has been an NSF Presidential Investigator
1128
0-
IEEE Transactions
on
Consumer Electronics,
Vol.
44,
No.
3,
AUGUST
1998
-
Optimum
set
jo,r8,32.48,84.80.%,112~
...
Tone
ssl
jO.i5,34,45,69,82,95,i18)
..
Tone
Set
(0.1 0,40,45,70.75,105,124]
++
Scheme
0
-all
tones
every
16th
symbol
t
-
Optimum
sa1
(0,16,32.46,64,80,96.112)
...
Tone
set
(0.15,34,45,69.82,95,118~
Tons
sd
(0.1 0,40,45,70,75,105.124]
Scheme
8
-all
tones
every
l8vl
symbol
__
++
+t
-16
0
20
40
MI
80
iw
120
Number
of
symbols
transmined
0
Fig. 5. Normalized MMSE in channel estimate of first-order Markov
channel for schemes A and
B
:
a
=
0.99.
N
=
128. L
=
8.
SNR
=
30
dB
(qs
=
0.15,PoPt
=
0.68)
-25'
Number
01
symbols
transmined
Fig. 6. Normalized MMSE in channel estimate of time-varying chan-
nel (Jakes' model) for schemes A and
B
:
N
=
128,
L
=
8,
SNR
=
30
dB,
doppler=100
Hz
(Popt
=
0.75)
since 1987, and won the Faculty Development Award in 1986-1988
from IBM Research. He was Associate Editor of IEEE Transactions
on ASSP (Adaptive Filtering) 4/64/87; Chief Editor of IEEE JSAC
(Signal Processing and Coding for Recording) 3/92; and Associate
Editor of IEEE JSAC (High-speed Digital Subscriber Lines) 8/91.
He was an Elected Member
of
ASSP Society
DSP
Committee 1987-
1993, and is an IEEE Fellow. He won an Outstanding Achievement
award from the American National Standards Institute for "contri-
butions to ADSL", 10/95, and is the System Requirements Editor
for ANSI for what is known as Very High-speed Digital Subscriber
Lines.
Cioffi
received his Ph.D. in Electrical Engineering from Stanford
University in 1983 under a Doctoral Support Award from Bell Tele-
phone Laboratories, his M.S. in Electrical Engineering from Stanford
University in 1979, and his
B.S.
in Electrical Engineering from the
University of Illinois in 1978.
