Communication over Fading Channel 

• The received signal r(t) = m(t)*s(t), where m(t) is the large-scale-fading component and s(t) is the small-scale-fading component.
• Classification of fading channels (qualitative): two types/classes
• Large-scale fading (path loss)
• Small-scale fading
• Fading channel modeling (quantitative/mathematical characterization)
• Large-scale fading (deterministic process)
• Reflection
• Ground reflection (2-ray) model
• Diffraction
• Knife-edge diffraction model
• Multiple knife-edge diffraction model
• Scattering
• Radar cross section model
• Small-scale fading (stochastic process)
• Marginal probability density distribution: Rayleigh, Ricean, Nakagami
• Second-order statistics: Doppler spectrum
• Autoregressive (AR) model
• Taxonomy of small-scale fading channels
• Per Doppler rate f_d, small-scale fading channels can be classified into
• Slow fading: 
• 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.
• Fast fading: 
• 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).
• Per delay spread, small-scale fading channels can be classified into
• Flat fading or non-frequency-selective fading:
• The signal bandwidth is smaller than the coherence bandwidth (roughly, 1/maximum delay spread).
• Coherence bandwidth is a statistical measure of the range of frequencies over which the channel can be considered "flat", 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 delay spread.   
• Frequency-selective fading
• The signal bandwidth is larger than the coherence bandwidth.
• 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.
• Large-scale fading channels
• 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)
• 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.
• Stochastic model of the deviation from the mean of m(t):
• Log-normal shadowing: the deviation from the mean is a log-normal 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:
• 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 probability of experiencing very long delay.   So I'm interested in the CDF of the delay.   For example, what is the delay of 99-percentile?   
• 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).
• 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.
• Average bit error probability: expectation of the Q-function, where the SNR is a random variable.
• Outage capacity, denoted by C(p_{outage}):
• 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), use FT/IFT in frequency domain; spread spectrum system designers care about time diversity (want independent resolvable delay spreads), use RAKE in time domain.
• Use matlab to compute the ergodic capacity of Rayleigh fading channel
• 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).
• 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:

• Judge the performance of a system from three perspectives:
• Expection of the performance metric:
• E.g., the average of the received SNR for BPSK modulation/demodulation. 
• E.g., Ergodic capacity.
• 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.
• CDF or outage probability (percentile):
• E.g., the outage probability of the received SNR for BPSK modulation/demodulation.
• E.g., outage capacity.
• Bufferless vs. buffer
• The outage probability (for the bufferless case) is not equal to the probability of busy server (for the buffered case).
• We can use outage probability to get intuition about the buffered case (or large deviation theory) but note this is not rigorous.
• Power control to combat fading

1. (Knopp & 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>\lambda_i, \gamma_i/\gamma_j > \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 > \lambda_i/\lambda_j, for all j \neq i$ is a bit similar to GPS scheduling (without equality).

• 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 delay-sensitive applications such as wireless video communication.

2. (Bettesh & 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.
• 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 & 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 & 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.

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Fast fading (time diversity) is useful. From the physical-layer point of view, the higher the time diversity, the higher the channel coding gain $\gamma_{code}$ if the codeword length is fixed and the number of interleaved bits is fixed. If the number of interleaved bits is infinite (in the limit), a fading channel with any finite nonzero coherence time can be converted into a discrete-time channel with independent channel gains. Hence, in the literature, interleaved Rayleigh fading channel is meant to be i.i.d. Rayleigh fading.

Interleaving could be applied to a single codeword by pseudo-random permutation; interleaving can also be applied to multiple codewords. A simple interleaver is to write multiple codewords row-wise and read the codewords column-wise.