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ECE 562: Advanced Digital Communications
Lecture 19: Sequential Communication over a Slow Fading
Wireless Channel
Introduction
Consider the simple slow fading wireless channel we arrived at in the previous lecture:
y[m] = hx[m] + w[m], m = 1,... , N. (1)
Here N T is the time scale (the product of N , the number of time samples communicated over, and T , the sampling period) of communication involved. The channel coefficient h is well modeled as independent of m if N T is much smaller than the coherence time Tc of the channel. This is a very common occurrence in many practical wireless communication systems and we begin our study of reliable communication over the wireless channel. We will start with sequential communication, much as we started out with the AWGN channel (cf. Lecture 3). The goal of this lecture is to be able to compare and contrast the simple slow fading channel in Equation (??) with the familiar AWGN channel. In particular, we will calculate the unreliability of communicating a single bit as a function of the SNR. The main conclusion is the observation of how poorly the unreliability decays with increasing SNR. This is especially stark when compared to the performance over the AWGN channel.
Comparison with AWGN Channel Model
The channel model in Equation (??) is quite similar to that of the AWGN channel model. But there are differences as well, with some aspects being more important than others.
- The transmit and receive symbols are complex numbers (pair of voltages) as opposed to the real numbers in the AWGN channel. This is a relatively minor point and poses hardly any trouble to our calculations and analysis (as seen further in this lecture).
- The channel “quality” h can be learnt by the receiver (through the transmission of pilot symbols, cf. Lecture 9), but is not known a priori to the communication engineer. This is a very important difference, as we will see further in this lecture. The important point is that the calculation of the unreliability level is now to be calculated in terms of the knowledge the communication engineer has about h: the statistical characterization of h.
In this lecture we make the following suppositions about the channel quality h:
- the channel quality h is learnt very reliably at the receiver. Since we have several, namely N samples to communicate over, we could spend the first few samples in transmitting knows voltages (pilots) thus allowing the receiver to have a very good estimate of h. We will suppose that the receiver knows h exactly, and not bother to model the fact that there will be some error in the estimate of the channel as opposed to the true value.
- a statistical characterization of h is available to the communication engineer: we study the relation between unreliability level and SNR in the context of a simple statistical model: Rayleigh fading.
Sequential Communication
Suppose we transmit a single information bit every time instant, using the modulation sym- bols ±
E. We decode each bit separately over time as well. Focusing on a specific time m, we can write the received complex symbol as
< [y[m]] = < [h] x[m] + < [w[m]] , (2) = [y[m]] = = [h] x[m] + = [w[m]]. (3)
This follows since x[m] is real (and restricted to be ±
E). Since the receiver known < [h] and = [h], a sufficient statistic of the transmit symbol x[m] is (cf. Lecture 5):
y˜ def = < [h] < [y[m]] + = [h] = [y[m]] (4) = < [h∗y[m]] (5) = |h|^2 x[m] + ˜w. (6)
Here we have denoted h∗^ as the complex conjugate of h: the real part of h and h∗^ are identical, but the imaginary part of h∗^ is the negative of that of h. Further,
w˜ def = < [h] < [w[m]] + = [h] = [w[m]] (7)
is real valued and has Gaussian statistics: zero mean and variance 12 |h|^2 σ^2. Now the ML receiver to detect x[m] from ˜y is very clear: Equation (??) shows that the relation is simply that of an AWGN channel and we arrive at the nearest neighbor rule:
decide x[m] = +
E if ˜y > 0 , (8)
and vice versa if ˜y ≤ 0. The corresponding error probability is
Q
2 |h|^2 SNR
where we have written SNR = E/σ^2 as usual.
Average Unreliability vs SNR
Now we, as communication engineers, are in a position to decide what value of SNR to operate at given the need for a certain reliability level of communication. To do this, it will help to simplify the expression in Equation (??): using the Taylor series expansion
√ 1 + x ≈ 1 +
x 2
, x ≈ 0 , (14)
we can approximate the expression in Equation (??) as
Pe ≈
4 ASNR
, ASNR 1. (15)
Now we see the benefits of increasing SNR: for every doubling of SNR, the unreliability level only halves. This is in stark contrast to the behavior of the AWGN channel where the unreliability level squared for every doubling of SNR (cf. Lecture 3). To get a feel for how bad things are, let us consider a numerical example. If we desire a bit error probability of no more than 10−^10 and have an attenuation of A = 0.01, then we are looking at a required SNR of 25 · 1010! This corresponds to astronomical transmit powers – clearly physically impossible to meet.
Looking Ahead
We have seen that the performance of sequential communication is entirely unacceptable at physically attainable SNR levels. This is serious motivation to look for better strategies. One possibility is, continuing along the line of thought early in this course, to study block communication schemes. Block communication schemes improved the reliability level while maintaining non-zero communication rates. The goal of block communication was primarily to better exploit the statistical nature of the additive noise by mapping information packet directly to the transmit symbol vector (of high dimension). As such, it was a way to deal with additive noise. In the wireless channel, we have an additional source of noise: the channel quality h itself is random and it shows up in a multiplicative fashion. So, it is not entirely clear if block communication which was aimed at ameliorating the effects of additive noise would work too well in dealing with multiplicative noise. We get a better feel for this aspect in the next lecture where we see that block communication cannot improve the performance significantly beyond the expression in Equation (??).
