Time-Variant Distortions in OFDM

Analysis of multicarrier transmission in time-varying channels
Suhas N. Diggavi *
Durand, Information Systems Labomtory,
Stanford University, Stanford, CA 94305, USA.
Email:suhasQrascakr.stanfonl.edtl - Tel.: (415) 725-6099 - Fax: (415) 723-8473
Abstract
The transceiver design i s dependent on the apriori information
available about the communication environment. We consider
two cases: channel completely known at the receiver and tmns-
mitter (via feedback); and only channel statistics are known at
the transmitter and the channel is known at the receiver. We
develop an information-theoretic analysis of these scenarios by
examining the mutual information expressions. We analyze a
vector coding scheme suitable for time-varying chunnels and il-
lustrate its asymptotic optimality. We examine the eqected
mutual information for slowly fading channels and investigate
the OFDM tmnsceiver structure. We then derive the average
mutual information for OFDM in time-varying environments.
This allows us to study the efiect of time-variation on OFDM
packet-site design. We illustrate the transmission overhead re-
quirements through a numerical example.
I Introduction
The explosive growth of wireless communications is cre-
ating the demand for high-speed, reliable and spectrally
efficient communicationsover the wireless medium. There
are several challenges in attempts to provide high-quality
service in this dynamic environment. These pertain to
cliaiiiicl time-variation and the limited spectral bandwidth
avai1;it)le for transmission. In this paper we are concerned
ivith the role of timevariation in transceiver design. We
consider two cases: channel completely known at the trans
iriitter and receiver; and channel unknown at transmitter
aut1 known at the receiver.
Tlierc has been a large body of work devoted to data
trarisinission over frequency selective timeinvariant chan-
iiels [l].There has been recent interest in reliable transmis
sioii in time-varying channels (2, 3, 41. We concentrate on
an information-theoretic analysis of transceiver structures.
111this paper we mainly specialize to narrowband modula-
t iori schemes. There has been considerable recent interest
iii the use of multiple transmitter and receiver sensors. We
vi11 describe a general channel model which accounts for
iriiiltiple inputs and outputs.
X brief outline and summary of the paper is as follows.
In Section I1 we establish the discretetime model suitable
'This research was supported in part by the Department of the
Arn~y,Army Research Office, under Grant No. DAAH04-95-1-0249
and an Okawa Foundation fellowship.
for time-varying channels. In Section I11we study the sce-
nario when the channel is time-varying but is completely
known both at the transmitter and at the receiver. Here
we establish the capacity of this channel and propose a
structure based on a vector coding scheme, which is illus-
trated to be asymptotically optimal. However the require-
ment of knowing the channel perfectly at the transmitter
is quite onerous. Therefore in Section IV we study the
case where only the receiver is able to track the channel.
We establish the achievablerate for Orthogonal F'requency
Division Multiplexing (OFDM) in time-varying channels.
Using this result we can examine the influence of channel
time-variation on OFDM transceiver design. In Secti6 V
we provide a numerical example illustrating the packet size
requirements in OFDM transceivers. We conclude with a
discussion of the main ideas in Section VI.
I1 Channel model
The input data {z(k)}is passed through the filter g(2) to
produce the transmitted signal s(t). The received signal
can be written as,
yc(t) = Jhc(t;T ) S ( t -T)dT +n(t) (1)
where h,(t;7)is the impulse response of the time-varying
channel. We collect sufficient statistics through Nyquist
sampling. A careful argument about the sampling rate r e
quired for time-varying channels can be found in [3]. In
this paper we assume that this criterion is met and there
fore we have the following discretetime model:
U - 1
Y(k) =YC(kG)= w;m)z@-m)+n(k) (2)
m=O
where h(k;m) represents the sampled time-varyingchannel
impulse response (which combines the transmit filter g(t)
with the physical channel h,(t;T ) ) . The approximation of
having afinite impulseresponsein (2) can be made asgood
as we need by choosing Y [3]. In this paper we evaluate
performancebased on the discretetime model given in (2).
For the matrix channel when the input and the output are
vectors, we can generalize the above model to,
U - 1
y(k)= H ( k ; m ) x ( k-m) +n(k) (3)
msO
0-7803-3925-8/97$1 0.000 1997 IEEE 1191

The Wide-Sense Stationary Uncorrelated Scattering
(WSSUS) model is quite commonly used in describ-
ing wireless communication channels [5]. In this
model, the channel impulse response is modeled as
a Gaussian stochastic process with the property that
IE[H(k;n)HH(k;m)] = E[H(k;n)HH(k;m)]6[n- m]. We
invoke this model in Section IV to gain insight into trans-
mission over fading channels.
I11 Completely known channel
In this section we assume that the channel is completely
known at the receiver and the transmitter (through feed-
back). In subsection A we propose a vector coding struc-
ture suitable for timevarying channels. In order to study
optimality of this scheme we examine the achievable rates
of timevarying channels in Section B. Using this result it
is easy to see that a vector coding structure is asymptoti-
cally optimal.
A Vector Coding Structure
In this section we propose a simple extension of the vector
coding (VC) structure proposed in [6]to the time-varying
case. We write the scalar block channel representation
developed in Section I1 as,
YN = HNXN+Z N . (4)
Here the N x ( N + v ) matrix HN does depend on the
timeblock chosen and for simplicity of notation we have
suppressed this dependence. We denote the SVD of HN
by HN = uN[AN~o]vH,, where UN and VN are unitary
matrices of appropriate dimensions. If we assume that
the noise Z N is white then we can create parallel inde-
pendent channels by using the right singular vectors VN
for transmission and the left singular vectors UN for the
receiver. Concatenating powerful ISI-free coding schemes
ivitli the channel decomposition we can come quite close
to the rates specified in Proposition III.1. The transmit,
rcwipe vectors and the codes need to vary with time ac-
cordiiig to the channel variations. The complexity of this
scliemc is O(Ar3AI3)per symbol of block length N for M
trwsinit/receive sensors. The question we ask is whether
this structure retains the optimality properties of VC in
timeinvariant channels. To this end we need to study
the capacity of the discrete-time time-varying channel de-
scribed iii (3).
B Achievable rates
Tl~riiriaiii difference between a timeinvariant channel and
this c u e is that we have equivalently a non-stationary noise
process (as will be seen soon). Therefore, the AEP used
iii the random coding arguments of [7] would not be appli-
cable. The proof is based on a random coding argument
based on an AEP for Gaussian non-ergodic processes de-
veloped by Cover and Pombra (81.
Proposition 111.1 Let CN be a number used for a block
of length N which is defined as,
where yjN)are the eigenvalues ofA N 1 U ~ K ~ ” ’ u ~ A ~ ’and
(.)+ = maz(-,O). Also A satisfies CEl(A- ylN))+ =
NP. There exists a sequence of (2N(CN-t),N)codes with
P,“’ +o as N -+ 00, for E > 0. Conversely, for e > o any
sequence of (2N(CN+C),N ) codes has P,” bounded away
from zem for all N .
Proof:
as,
We rewrite (4) using the SVD of the channel HN
Y N = U N [ A N I O ] V E X N+Z N (6)
Now, we ca.n write VEXN= [X$,OTIT without any loss
of generality as the last D dimensions do not contribute
to the mutual information. Also, as UN is a unitary ma-
trix, we c& multiply both sides of (6) without any loss
in information. Therefore, using these co-ents we can
rewrite (6) as Y N= A N X N+Z N , where Y N= UEYN
and 2~= UEZN.Note that the power constraint is now
tary transformation. By construction AN is a diagonal
matrix with positive entries and hence we can invert this
matrix and obtain Y N= X N +Z N , where Y N= A;’~N
and Z N = AN’ZN. Therefore, we have reduced the time-
varying IS1 channel into a time-varying additive Gaussian
noise channel. We note that the covariance matrix of Z N
is given by K(N)= A & l U $ K ~ ~ ’ U ~ A ~ l .Therefore, using
the Theorem 1in [SIwe have the coding theorem and its
converse.
stated as trace(KXN)(NI 5 N P as we have just done a uni-
Z N
The flat-fading AWGN scalar channel given by,
y(k) = h(k;O)z(k)+n(k) (7)
In this we can clearly see that K&Y is a diagonal matrix
with entries {&=}. Therefore, as we waterfill over its
eigenvalues we can see that we can interpret this loosely
as “waterfillingin time” as we modulate the power of the
codewordsaccordingto the strength of the channel at that
time. This has been observed in the context of finite state
time-varying channels in [2].
For the matrix channel we can model the block channel
for a block of N received vectors as y N = HNXN+Z N .
Notice that the form of this model remains identical to (4)
and therefore, it is easy to extend the argument of Thee
rem 111.1 to this case. To summarize, we need to design
non-stationary codebooks matched to the channel time-
variations and noise statistics. Although we did not need
any asymptotic statement on CN to impart a significance
to this quantity, we do not know conditions under which
it would converge.
1192

IV Fading channel with no feed-
back
In Section 111we considered a scenario where the chan-
nel was completely known at the transmitter and receiver.
This would require enormous feedback rates. Also the need
to adapt continuously and its associated complexity make
the vector coding structure less attractive. In this section
we assume no feedback of the channel impulse response
from receiver to the transmitter. Hence the transmitter
only knows the average statistics of the channel. In sub-
section A we examine the achievable rate for multiple in-
put and output sensors when the channel is slowly time-
varying. In Section B we derive the expected mutual infor-
mation of OFDM transceivers for fast fading channels. We
specialize the results of Section B to the WSSUS model in
Section C. It is important to note that we assume Gaus
sian input codebooks and hence maximizing this mutual
information rate does not necessarily lead to channel ca-
pacity.
A Slowly Fading Channels
Reliable transmission rates for fading channels has been
studied extensively in literature, see [4] and references
therein. The most common assumption made in studying
these schemes is that of slow timevariation, i.e. Band-
width >> Doppler spread. The rate of reliable information
for the scalar channel has been derived in [4] in terms of
the expected mutual information. We develop an extension
of this result to the matrix channel. We assume that the
channel is slowly time-varying so that it is time-invariant
over a block of N samples. In the frequency domain the
received signal can be written as,
.Y(n)= H(n)X(n)+Z(n) (8)
where Y ( . ) , H ( . ) , X ( . )and Z(-) are the DFT of
y(.),h(.),x(.)and z(.).The mutual information with the
given Channel State Information (CSI).is,
-I(Y, H;X ) = E- log(lH(n)S(n)HH(n)/a2+11) (9)
hr N
Heiice using the slow time-variation assumption, and let-
tiug N + CO as in [4] we obtain the information rate R
as,
1 1 N-l
n=O
R 5 E[/ l o s ( l H ( f ) S ( f ) H H ( f ) / c r 2+Il)dfI (10)
wlierc S(f)is the input power spectral density. With suf-
ficieiit interleaving we could have independent fading be-
tiveen successive information blocks, so that (10) yields
the achievable rate for slowly time-varying channels. Max-
imizing the functional (10) with respect to S(f)is a hard
problem in general. Schemes using a flat input spectrum
with diversity transmission have been proposed [9]. In the
rest of the paper we will mainly discuss scalar channels
and defer the matrix case extensions to [lo].
B Impact of fast time-variation
In order to reduce the receiver complexity, the OFDRl
transmitter sends information along the Fourier basis. In
time-invariant channels, the Fourier basis allows US to form
parallel ISI-free channels. However, in time-varying chan-
nels, the Fourier basis is not in general an eigenbasis.
This loss of orthogonality causes Inter-Carrier Interfer-
ence (ICI). In this section we derive the information rates
achievable in the presence of ICI. We can write the output
tl :"
U RECEIVER
Figure 1: An OFDM based transmission scheme
of the FFT at the receiver for a time block [-(U -1),N -11
a,
Y(m)=G(m,m)X(m)+ G(m,n)X(n)+Z(m)
nZm
(11)
for m =0,...,N - 1. Here Y(m)corresponds to the mth
frequency bin, {X(m)}is the frequency domain input sym-
bol and Z(m)is the additive noise. We have also defined
G(m,n)as the (m,n)thelement of G = QHQH/N. Here
Q is the DFT matrix and H is the equivalent channel ma-
trix including the effects of the cyclic prefix. We can easily
evaluate G(m,n) as,
.. N-1 U-1
r=O k 0
...
Proposition IV.l The information rate for the OFDM
channel is given by:
1193

Proof:
where (a)is obtained by the independence of the channel
and the transmitted symbols, (b)is due to the definition of
conditional mutual information, and (c) uses the fact that
In the above proposition we have assumed that we use a
Gaussian input codebook and that the codebooks for each
subcarrier is independent. The easiest coding theorem
proof for the above mutual information rate can be done
by having independent fading on successive transmission
blocks. Though this could be justified by an ideal inter-
leaving assumption, a more general proof can be based on
ergodicity assumptions on the channel impulse response.
Note that we have not made any assumptions on the inde
pendence between G(m,m) and {G(m,n))in this propo-
sition. Also we need only the instantaneous SNR at the
receiver to achieve this rate. This proposition can be e a s
ily extended to the multiple input/output sensors case and
this is deferred to [lo].
conditioned on G (11)is a Gaussian channel.
C The WSSUS Channel
We specialize the results of Section B to the WSSUS model
described in Section 11. This allows us to gain insight into
the impact of fast timevariation on OFDM transceivers.
We notice from (12) and the WSSUS model that {G(m,n))
are jointly Gaussian. We can write the information rate
described in Proposition IV.l as,
R = EG[log(n2+P c IG(m,n>l2)] (14)
n
n f m
1T.p denote E, IG(m,n)I2 = gzg, and
E,,+,,,IC(?n,n)12 = glrfgm. We notice that
g,,, = [G(m,O),... ,G(m,N - 1)IT is Gaussian, and
so is gm which is of dimension N - 1 (constructed by
deleting the element G(m,m)from gm). Hence using the
fact that g, and g, are Gaussian vectors, we can easily
evaluate (14).Using (12) and the WSSUS channel we can
writ e;
. N - 1 N - 1
where rh(rl-r2) = EE[h(rl;l)h*(rz;l)].Let RI = E[smsE]
and R2= E[gmgH,]and for simplicity assume that RI and
Rz have no repeated eigenvalues. Then we can write (14)
as,
N-1
R = - 661)ezp(a2/PX61)))Ej(-a2/PX61))(16)
q=o
N-I
g-1
where E&) = J’, et/tdt is the exponential integral func-
tion [4]. In order to obtain (16) we needed the probability
distribution of gig, and gig,. Rewriting, gig, =
ghHR1+,with + -N(0,I)we can show that the proba-
bility density of gzg, is f(u)= E,=,,6, ezp(-u/At)).
Here (661’) are the residues of the characteristic function
of ggg, at {Xf’}. This is written for the case where the
eigenvalues {A?)} of RIare distinct. Similarly we obtain
the expressions for the probability distribution of ggg,
involving {ai2)}an$ {A?)}. The expression in (16) can be
easily modified for the general repeated eigenvalue case,
though the expression is more complicated and does not
provide much further insight.
Fortransmit diversity the expressionin (16) can be eas-
ily modified. We examine the case when the number of
transmit antennas D is very large and they are far enough
apart to produce independent channels. In this m e when
D +00 we have,
N-1 (1)
Here we have assumed that independent Gaussian code-
books of power P/Dare used at each of the transmit an-
tennas and OFDM subcarriers. A very interesting phe-
nomenon occurs here due to the averaging effects of trans-
mit diversity. If there were no ICI, such averaging would
always increase the information rate by Jensen’s inequal-
ity. However, as both the IC1 and the signal are averaged
it is not necessary that the rate increases. This property
is illustrated in the numerical example given in Section V.
V Numerical Results
In this section we numericallyevaluate the expressions de-
rived in Section IVC. This allows us to plot the informa-
tion rate as a function of Doppler shift and block size. Us-
ing these plots we illustrate the trade-off between receiver
complexity and overhead.
We use a WSSUS channel with three taps, each of
which has energy of 1. We assume a signal bandwidth
of 30kHz. The signal-to-noise ratio (P/a2)is fixed at
1194

20dB and the transmit OFDM spectrum is flat. The
channel time-statistics are assumed to be represented by
Th(k) = Jo(w&T,) where Ja(.)is the Bessel function ofthe
first kind, order 0 and wd = 27rv/A is the Doppler spread,
In Figure 2 we plot the information rate per transmitted
sample, RN/(N +U),as a function of the block length N
and various Doppler spreads. For very low velocity the
time-invariant assumption is quite valid and there is lit-
tle loss due to ICI. However, for larger time-variation the
loss due to IC1 is quite significant. This indicates that
block sizes need to be quite small in fast timevarying
channels and therefore the overhead for OFDM could be
quite large. For reference the information rate for an
AWGN channel with the same channel gain and SNR is
8.23 bits/transmitted sample, and the scalar slowly fad-
ing channel (Section IVA) has an information rate of 7.42
bits/transmitted sample. In Figure 3 we have plotted the
information rates for infinite transmit diversity. Compar-
P aIym (N)->
Figure 3: Information rates with large diversity for various
block sizes and Doppler shifts.
ing this to Figure 2 we see that at high velocities there
is not much gain due to diversity. This shows that the
averaging effect of diversity on the IC1 offsets the gains
of the averaging effect on the signal. For improved per-
formance we would need a multi-tap frequency domain
equalizer, which increases the receiver complexity. This
demonstrates a trade-off between transmission overhead
and receiver complexity.
Figure 2: Information rates for various block sizes and
Doppler shifts.
VI Discussion
In this paper we have developed an information theoretic
analysis of multicarrier transceiver design in wireless com-
munications. We began with a completely known channel
and illustrated the optimality of a vector coding scheme. In
the more realistic case where there is no feedback we devel-
oped the information rate for slow time-variation. We de-
rived the information rate for OFDM in fast time-varying
channels and illustrated the deleterious effects of ICI. The
severe IC1 problem in fast time-varying channels requires
the combination of some equalization with OFDM. This
limits the receiver simplicity advantages of OFDM. The
issue of good transceiver design in time-varying channels
is a difficult one. This paper was meant to develop some
insight into this by studying the achievable information
rates.
References
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1195

Time-Variant Distortions in OFDM

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