Communication over Fading Channel
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component and s(t) is the small-scale-fading component.
Classification of fading channels (qualitative): two types/classes
 Reflection
 Ground reflection (2-ray) model
 Diffraction
 Knife-edge diffraction model
 Multiple knife-edge diffraction model
 Scattering
 Marginal probability density distribution: Rayleigh, Ricean,
Nakagami
 Second-order statistics: Doppler spectrum
 Autoregressive (AR) model
o Per Doppler rate f_d, small-scale fading channels can be classified into
 The symbol duration is smaller than the coherence time
(roughly, 1/Doppler rate), i.e., low Doppler rate.
 Coherence time is a statistical measure of the time duration
over which the channel impulse response is essentially
invariant. It quantifies the similarity of the channel
response at different times. In other words, coherence time
is the time duration over which any two received signals (at
different times) have a strong correlation.
 The symbol duration is larger than the coherence time, i.e.,
high Doppler rate.
 The higher the Doppler rate is, the higher the degree of
diversity is. For sufficiently high degree of diversity (high
f_d), the fast fading channel can support constant bit rate,
i.e., the fast fading channel becomes an ergodic channel.
This is because the randomness of the channel gain is
averaged out due to high degree of diversity (independent
signal paths).
 The signal bandwidth is smaller than the coherence
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Coherence bandwidth is a statistical measure of the range
of frequencies over which the channel can be considered
&quot;flat&quot;, i.e., having approximately equal gain and linear
phase. In other words, coherence bandwidth is the range of
frequencies over which any two frequency components
have a strong correlation.
 Coherence bandwidth B_c is approximately 1/(50*
sigma_t), where sigma_t is the root-mean-square (rms) of
 The signal bandwidth is larger than the coherence
bandwidth.
o Per ergodicity, small-scale fading channels can be classified into
 Ergodic channel
 The channel gain process is ergodic, i.e., the time average
is equal to the ensemble average. In other words, the
randomness of the channel gain can be averaged out
(removed) over time. So long-term constant bit rates can
be supported (like AWGN channels).
 Non-ergodic channel
 The channel gain is a random variable and does not change
with time. The channel gain process is stationary but not
ergodic, i.e., the time average is not equal to the ensemble
average. In other words, the randomness of the channel
gain can not be averaged out (removed) over time. So
long-term constant bit rates can not be supported.
o The large-scale-fading component m(t) in the received signal r(t), is a
random variable. m(t) can be characterized by its mean (deterministic
part) and the deviation from its mean (random part)
o Deterministic model of the mean of m(t):
 Power law decay: the mean received power is inversely
proportional to the n-th power of Transmit-Receive separation
distance, where $n$ takes integer values from 2 to 5.
 Free space model: the path loss follows inverse-square law.
o Stochastic model of the deviation from the mean of m(t):
 Log-normal shadowing: the deviation from the mean is a lognormal random variable.
 The random model is to consider the fact that the surrounding
environmental clutters may be vastly different at two different
locations having the same T-R separation distance.
Performance metrics:
o Average Signal-to-Noise Ratio (SNR): expectation of the instantaneous
SNR, which is a random variable.
 What if I'm not interested in average behavior? I don't care too
much about the average delay because my packets may have high
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probability of experiencing very long delay. So I'm interested in
the CDF of the delay. For example, what is the delay of 99percentile?
 Another example is channel coding. For a convolutional code, I'm
interested in not only the minimum distance of the code but also
the CDF of the distance (note the Hamming distance is not a
random variable).
o Outage probability, denoted by p_{outage}:
 It is the Cumulative Distribution Function (CDF) of the
instantaneous SNR or the instantaneous channel capacity, which
are random variables.
o Average bit error probability: expectation of the Q-function, where the
SNR is a random variable.
Outage capacity, denoted by C(p_{outage}):
o It is the inverse function of CDF of the instantaneous channel capacity.
Ergodic capacity: expectation of the instantaneous channel capacity, log(1+SNR),
assuming that the fading process is white. What if the fading is a Markovian
process?
OFDM system designers care about frequency diversity (coherence frequence),
time diversity (want independent resolvable delay spreads), use RAKE in time
domain.
Use matlab to compute the ergodic capacity of Rayleigh fading channel
o E[log(1+g)]=\int_0^infty log(1+g)*exp(-g) dg= -exp(1)*real(-expint(1)) =
0.5963 nats, for E[g]=1, P/N=1. See Page 567 in I. S. Gradshteyn, I. M.
Ryzhik, Alan Jeffrey (Editor), Daniel Zwillinger, Table of Integrals,
Series, and Products,'' Academic Press; ISBN: 0122947576; 6th edition
(July 31, 2000).
o For the same average received SNR, the ergodic capacity of Rayleigh
fading channel is 0.5963/log_e(1+1)=0.5963/0.6931=0.8603 of the
AWGN channel capacity.
Key techniques:
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Judge the performance of a system from three perspectives:
o Expection of the performance metric:
 E.g., the average of the received SNR for BPSK
modulation/demodulation.
 E.g., Ergodic capacity.
o Variance (how random the performance is):
 E.g., the variance of the received SNR for BPSK
modulation/demodulation.
 The degree of randomness can also be seen from the CDF and
PDF.
o CDF or outage probability (percentile):
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E.g., the outage probability of the received SNR for BPSK
modulation/demodulation.
 E.g., outage capacity.
Bufferless vs. buffer
o The outage probability (for the bufferless case) is not equal to the
probability of busy server (for the buffered case).
o We can use outage probability to get intuition about the buffered case (or
large deviation theory) but note this is not rigorous.
1. (Knopp &amp; Humblet, ICC'95) Joint design of power control and
scheduling for multiple users sharing one fading channel.
 Objective: maximize the total throughput of all the users, subject to
average power constraints for each user. Each user can have
different average power constraint. The maximization is over the
transmission power and the transmission order of the users.
 The (multiuser) throughput-optimal power control
$\mu_i(\gamma)$ for each user is
\mu_i(\gamma) = 1/\lambda_i - 1/\gamma_i, if
\gamma_i&gt;\lambda_i, \gamma_i/\gamma_j &gt; \lambda_i/\lambda_j,
for all j \neq i.
\mu_i(\gamma) = 0 otherwise
where $\gamma_i$ is the current channel power gain for user $i$,
$\lambda_i$ is a constant yielded from the power constraint of user
$i$.
The condition $\gamma_i/\gamma_j &gt; \lambda_i/\lambda_j, for all j \neq i$ is a bit similar to GPS scheduling (without equality).
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Use information theory and communication theory
It is called the KH scheme.
Limitation: under this strategy, a user in a fade of an arbitrarily
long period will not be allowed to transmit during this period,
resulting in an arbitrarily long delay; therefore, this strategy
provides no delay guarantees and thus is not suitable for delaysensitive applications such as wireless video communication.
2. (Bettesh &amp; Shamai, VTC'01 Spring) Joint power and rate control for
single user communicating over a block fading channel.
 Objective: minimize a cost function, which is an non-decreasing
function of both the average transmission power and the packet
average transmission delay. The maximization is over the
transmission power and the transmission rate of the users.
 Formulated as an infinite-horizon dynamic program with an
average cost criterion.
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cost = queue_length + weight * transmission_power
 It converts multi-objective to single objective so that we
can use dynamic programming.
 E[cost] = E[queue_length] + weight *
E[transmission_power].
 For each value of the weight, the dynamic program results
in a Pareto-optimal cost {E[queue_length],
E[transmission_power]}.
 Let E[transmission_power] = average power constraint,
then the corresponding E[queue_length] is the minimal
average delay under the specified average power constraint.
 Four policies:
 Variable rate and constant power
 Constant rate and variable power
 Variable rate and variable power
 Variable rate and variable power with transmitter's
knowledge about the channel state information. The
resulting power control is the same as water-filling in time.
 Combine queueuing theory with information theory.
 Limitation:
 The complexity is exponential in the delay/buffer constraint
and the number of users (for the multiuser case);
 The transmission decision can not be made until the whole
channel gain sequence is known; even for truncated
decision, it still has to wait for 5*L slots to make decision,
where L is the memory length of the channel.
 For multiuser scheduling, linear programming has a
complexity of O(K) while dynamic programming has a
complexity of O(N^K), where $K$ is number of users and
$N$ is the number of states of queue length.
3. (Bettesh &amp; Shamai, PIMRC'98) Joint design of power control and
scheduling for multiple users sharing one fading channel.
 Objective: strike a balance between the throughput and delay
constraints (achieving low delay)
 Method: combine TDMA with multiuser diversity (Knopp &amp;
Humblet scheme) by introducing an inspection scheme, which
checks whether the delay requirement of each user is satisfied.
 It is not formulated as a constrained optimization problem. The
algorithm is simply to combine TDMA with the KH scheme.
 Limitation: no explicit delay guarantee and no optimization.
E[cm / c1 ]  E[c1 ]  ( E[ cm ]) 2  E[cm ] , so we cannot get the relation between E[cm / c1 ]
and E[cm ] / E[c1 ] .
Typically, a bit error model characterizes the performance of modulator/demodulator. A
packet error model characterizes the performance of codec/ARQ system (where modem
is hidden in the system).
Using a bit/packet error model is to simplify the simulation and/or simplify the queueing
analysis.
The assumption for the two state Markov channel (TSMC) model is general.
It does not matter what kind of modulation, detection, channel coding are
used. It's a measurement-based approach. Just record the error process
for a period of time. If the process is stationary, ergodic and
Markovian, then use two-state Markov model (TSMC) to match the real trace
under the criterion of minimum mean square error (MMSE). Zorzi proposed an
algorithm to estimate the parameters (transition probabilities p &amp; q) of the
two-state Markov model. The assumption is that we need the past history of
the error process to do estimation. Actually, I also have similar
assumption. In my estimation, when you dial the number, the mobile
terminal starts to measure the channel gain. Just 2 seconds is enough to
get a good estimate. So when you finish your dialing, your measurement is
also done.
Each user will do this estimation since p &amp; q are different for each user.
Once you get the TSMC, you can do queueing analysis (numerically). I
haven't seen any QoS provisioning work using TSMC. My central question is
how to explicitly guarantee a given QoS triplet {rate, delay, loss
probability}. Have you seen any work on explicit QoS provisioning
A packet level error model is time varying. From s to s+t, where s is the epoch when the
packet head arrives, and t is packet size. We are interested in instantaneous packet error
probability, not average packet error probability. Packet size may be fixed during a
connection life time.
1. Ergodic capacity for communications without delay constraints: like Law of large
numbers (LLN), ergodic capacity notion can be applied to the iid case and non-iid
case (as long as it satisfies LLN). We assume both the transmitter and receiver
know the channel state perfectly.
a. The iid case: Cerg  EG  max I ( X ;Y | G) , where G is the channel gain,
 f X |G ( x|g )
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I ( X ;Y | G ) is the mutual information conditioned on G, and f X |G ( x | g ) is
the pdf of X conditioned on G. The expectation is over G. The empirical
mean of max I ( X ; Y | G ) converges in probability to the expectation.
f X |G ( x| g )
n
p
Cerg .
That is,  max I ( X i ;Yi | Gi ) / n 
i 1
f X i |Gi ( xi | g i )
b. The non-iid case:
f
maxn
X n |G n
n
( x |g )
p
I ( X n ; Y n | G n ) / n 
C erg , where G n is the
channel gain vector of n slots, I ( X n ;Y n | G n ) is the mutual information
conditioned on G n , and f X n |G n ( x n | g n ) is the pdf of X n conditioned on
G n . As long as the channel gain process satisfies WLLN, the max of the
mutual information converges in probability to a deterministic value,
which we call ergodic capacity. This is because the max of the mutual
information is a deterministic function of the channel gain. So if the
channel gain process satisfies WLLN, the max of the mutual information
also satisfies WLLN.
2. Expected capacity for delay-constrained communications
a. Expectation of the capacity
3. Outage capacity for delay-constrained communications
a. Outage probability is CDF of the capacity
4. Effective capacity for delay-constrained communications, taking into account the
queueing behavior.
I.i.d. Rayleigh fading channel: symbol error probability
1
Pe 
4(1  SNR)
For AWGN channel,
Pe  e  SNR / 2
For frequency selective channel with m independent paths,
1

Pe 
? m
 e  SNR / 2
SNR m
(1 
)
2m