Fundamentals of Wireless Communication

Chapter 3Point-to-Point Communication: Detection, Diversity and Channel Uncertainty In this chapter we look at various basic issues that arise in communication over fading channels.. We

EE 290S: Fundamentals of Wireless Communication

Course Notes U.C Berkeley Fall 2002

Instructor: David Tse Co-written with: Pramod Viswanath

September 18, 2002

Chapter 3

Point-to-Point Communication:

Detection, Diversity and Channel Uncertainty

In this chapter we look at various basic issues that arise in communication over fading channels We first start with analyzing uncoded transmission in a narrowband fading channel We study both coherent and non-coherent detection We see that in both cases, the error probability performance is very poor compared with that in an non-faded AWGN channel The basic reason is because there is a significant probability that

the channel is in a deep fade This motivates us to look at various diversity techniques

which improve upon this performance The diversity techniques discussed operate over time, frequency or space, but the basic idea is the same By sending signals that carry the same information through different paths, multiple independently faded replicas of data symbols can be obtained at the receiver end and more reliable reception can be achieved Finally, we study the impact of channel uncertainty on the performance of diversity combining schemes We will see that in some cases, having too many diversity paths can have an adverse effect due to channel uncertainty

The emphasis of this chapter is on concrete techniques for communication over fading channels to familiarize ourselves with the basic issues In Chapter 4 we take a more fundamental and systematic look and use information theory to derive what is

the best performance one can achieve.

3.1.1 Noncoherent Detection

To understand the basic issues in communicating over wireless channels, we start with

a very simple detection problem in a fading channel For simplicity, let us assume a flat

fading model where the channel can be represented by a single discrete-time complex

filter tap h[0, m], which we abbreviate as h[m]:

where w[n] ∼ CN (0, N0) We assume Rayleigh fading; i.e h[m] ∼ CN (0, 1), where

we normalize the variance to be 1 For the time being, however, we do not specify

the dependence between the fading coefficients h[m]’s at different times m nor do we assume any prior knowledge the receiver has of h[m]’s (This is sometimes called

non-coherent communication.)

First consider binary antipodal signaling with amplitude a, i.e x[m] = ±1, each with probability 1/2, and {x[m]} is iid This signaling scheme fails completely, even in the absence of noise, since the phase of the received signal y[m] is uniformly distributed between 0 and 2π under both hypotheses, and the received amplitude is similarly

independent of the hypothesis It is easy to see that phase modulation is similarly flawed In fact, signal structures must be used in which either different signals have different magnitudes, or coding between symbols is used

Next consider a form of binary pulse-position modulation where, for each pair of time samples, either transmit

xA :=

µ

x[0]

x[1]

¶

=

µ 1 0

¶

(3.2) or

xB =

µ 0 1

¶

Note that this is a simple form of orthogonal modulation We would like to perform detection based on:

y :=

µ

y[0]

y[1]

¶

This is a simple hypothesis testing problem, and it is straightforward to derive the MAP (maximum a posterior) rule:

Λ(y)

ˆ

H=0

≥

<

ˆ

H=1

0,

where Λ(y) is the log likelihood ratio:

Λ(y) := ln

½

f (y|H0)

f (y|H1)

¾

(3.4)

and f (y|H i ) is the probability density function of y given hypothesis H i It can be

seen that given H0, y[0] ∼ CN (0, a2 + N0) and y[1] ∼ CN (0, N0) and y[0], y[1] are independent Similarly, given H1, y[0] ∼ CN (0, N0) and y[1] ∼ CN (0, a2 + N0) and

y[0], y[1] are independent Hence the likelihood ratio can be computed to be:

Λ(y) = {|y[0]|

2− |y[1]|2} a2

(a2+ N0)N0 . (3.5) The optimal rule is simply to decide H0 if |y[0]|2 > |y[1]|2 and decide H1 otherwise Note that the rule does not make use of the phases of the received signal, since the

random phases of the channel gains h[0], h[1] render them useless for detection

Geo-metrically, we can interpret the detector as projecting the received vector y onto each

of the two possible transmitted vectors xA and xB and comparing the energies of the

projections Thus, this detector is also called an energy or a square-law detector It is somewhat surprising that the optimal detector does not depend on how h[0] and h[1]

are correlated

We can analyze the error probability of this detector By symmetry, we can assume

that H0 is the correct hypothesis Under H0, y[0] and y[1] are independent circular symmetric complex Gaussian random variables with variances a2+ N0 and N0

respec-tively As shown in the exercises , |y[0]|2, |y[1]|2 are therefore exponentially distributed

with mean a2+N0 and N0respectively.1 The probability of error can now be computed

by direct integration:

p e = P©|y[0]|2 > |y[1]|2|H1

ª

=

·

2 + a

2

N0

¸−1

We can define:

SNR := a2

N0

as the average received signal to noise ratio per dimension.2 Then the error probability is:

This is a very discouraging result To get an error probability p e = 10−3 would

require SNR ≈ 1000 (30 dB) Stupendous amounts of power would be required for

more reliable communication

Before we explore further the root cause of the poor performance of this detector,

we note that there are other ways to perform noncoherent modulation and detection

1Recall that a random variable U is exponentially distributed with mean µ if its pdf is f U (u) =

1

µ e −u/µ

2 Whenever we refer to “dimension”, we implicitly mean a complex dimension We will also use the term “degree of freedom” interchangeably with “dimension”.

Here, we did not assume any relationship between consecutive channel gains, but if we assume that they do not change much from symbol to symbol, differential phase shift keying (DPSK) can be used to convey information in the relative phases of consecutive transmitted symbols The performance of DPSK is analyzed in the exercises

3.1.2 Coherent Detection

Why is the performance of the detector so bad? It is instructive to compare its per-formance with detection in AWGN channel without fading:

For antipodal signaling (BPSK) , x[m] = ±a, the error probability is easy to compute:

p e = Q

Ã

a

p

N0/2

!

= Q ³√

2SNR

´

where Q(·) is the complementary cumulative distribution function of a N(0, 1) random variable It is known that Q(x) decays exponentially with x; more specifically,

1

√

2πx

µ

1 − 1

x2

¶

e −x2/2 < Q(x) < e −x2/2 , x > 1. (3.10)

Thus, the detection error probability decays exponentially in SNR in the AWGN chan-nel while it decays only inversely with the SNR in the fading chanchan-nel To get error probability of 10−3, one only needs an SNR of about 7 dB in an AWGN channel Compared to detection in the AWGN channel, the detection problem considered in

the previous section has two differences: the channel gains h[m]’s are random, and the

receiver is not assumed to know them Suppose now that the channel gains are tracked

at the receiver so that they are assumed to be known at the receiver (but they are still random) In practice, this is done either by sending a known sequence (called a pilot

or training sequence) or in a decision directed manner The accuracy of the tracking

depends of course on how fast the channel varies For example, in a narrowband 30kHz system (such as IS-136) with a Doppler spread of 100Hz, the coherence time T c is 300 symbols and in this case there should be plenty of time to estimate the channel with minimal overhead expended in the pilot 3 For our purpose here, let us assume the channel estimates are perfect

Knowing the channel gains, coherent detection of BPSK can now be performed on

a symbol by symbol basis, exactly as in the AWGN case other than a scaling by the

3 The channel estimation problem for a broadband system with many taps in the impulse response

is more difficult; see Section 3.4.2.

channel gain at the receiver If the transmitted symbol is x[0] = ±a, then for a given value of h[0], the error probability of detecting x[0] is:

Q

Ã

a|h[0]|

p

N0/2

!

= Q³p

2|h[0]|2SNR

´

(3.11)

We average over the random gain h[0] to find the overall error probability For Rayleigh fading when h[0] ∼ CN (0, 1), we find:

p e = E

h

Q³p

2|h[0]|2SNR

´i

= 1 2

Ã

1 −

r SNR

1 + SNR

!

. (3.12)

(See the exercises.) Figure 3.1 compares the error probabilities of coherent BPSK and noncoherent orthogonal signally over the Rayleigh fading channel, as well as BPSK over the AWGN channel We see that while the error probability for BPSK over AWGN channel decays very fast with the SNR, the error probabilities for the Rayleigh fading channel are much worse, whether the detection is coherent or noncoherent In fact, for high SNR, the error probability for coherent BPSK is:

p e ≈ 1

which also decays inversely proportional to the SNR, as in the noncoherent orthogonal signaling scheme (c.f (3.7)) There is a 6 dB difference between the two schemes Thus, we see the main reason why detection in fading channel has poor performance

is not because of the lack of knowledge of the channel at the receiver It is due to the fact that there is a significant probability that the channel is very poor More specifically,

by inspecting (3.11), we see that errors occur with significant probability when the

channel gain |h[0]|2 is of the order or less than 1/SNR At high SNR,

and so this latter event occurs with probability approximately 1/SNR When the channel gain is much larger than 1/SNR, the conditional error probability decays very rapidly, exponentially in |h[0]|2SNR Thus, at high SNR, the typical way for errors

to occur is when the channel gain is small, of the order or less than 1/SNR, rather

than when the additive noise is large, which occurs much more rarely because of the exponential tail of the Gaussian In contrast, in the AWGN channel the typical and indeed the only possible way for errors to occur is for the additive noise to be large Thus, the error probability performance over the AWGN channel is much better The approximate analysis above seems pretty hand-waving, but can in fact be made

precise (See the exercises ) Even though the error probability p e can be directly computed in this case, the approximate analysis provides much more insight as to how

−10 −5 0 5 10 15 20 25 30 35 40

10−15

10−10

10−5

10

SNR (dB)

Non−coherent orthogonal signaling Coherent BPSK

Figure 3.1: Performance of coherent BPSK vs noncoherent orthogonal signaling over Rayleigh fading channel vs BPSK over AWGN channel

typical errors occur Understanding typical error events in a communication system often suggest how to improve it Moreover, the approximate analysis gives some hint

as to how robust the conclusion is to the Rayleigh fading model we assumed In fact, the only aspect of the Rayleigh fading model that is important to the conclusion is the

fact that P {|h[0]|2 < ²} is proportional to ² for ² small This holds whenever the pdf

of |h[0]|2 is positive and continuous at 0

3.1.3 Diversity

We see from the above coherent detection example that the root cause of the poor performance is that reliable communication depends on the strength of a single signal path, and with significant probability that path will be in a deep fade A natural solution to improve the performance is to ensure that the information symbols pass through multiple signal paths, each of which fades independently, such that reliable communication is possible as long as some of the paths are strong This technique is

called diversity, and it can dramatically improve the performance over fading channels.

There are many ways to obtain diversity Diversity over time can be obtained via

coding and interleaving: information is coded and the coded symbols are dispersed over

time in different coherence periods so that different parts of the codewords experience independent fades Analogously, one can also exploit diversity over frequency if the channel is frequency-selective In a system with multiple transmit or receive antennas spaced far enough apart, diversity can be obtained over space as well In a cellular network, macrodiversity can be exploited by the fact that the signal from a mobile can be received at two base-stations Since diversity is such an important resource, a wireless system typically uses several means of diversity We will see that although the basic principle of achieving diversity is the same in the different modes, specific issues arise that are peculiar to particular modes of diversity

We will survey several diversity techniques in the next few sections The simplest

diversity schemes are based on repetition coding: the same information symbol is

trans-mitted over several signal paths While repetition coding achieves a diversity gain, it is usually quite wasteful of degrees of freedom of the system More sophisticated schemes

can increase the data rate and achieve a coding gain beyond the diversity gain.

Time diversity is achieved by averaging over the fading of the channel over time Typically, the channel coherence time is of the order of 10’s to 100’s of symbols and therefore the channel is highly correlated across consecutive symbols To ensure that the coded symbols are transmitted through independent or nearly independent fading

gains, interleaving of codewords is required For simplicity, let us consider a flat fading

channel We transmit a codeword x of length L and the received signal is given by:

y[`] = h[`]x[`] + w[`], ` = 1, L. (3.15)

Assuming ideal interleaving so that consecutive symbols x[`]’s are spaced sufficiently far apart, we can assume that the h[`]’s are independent.4 The parameter L is commonly called the number of diversity branches.

3.2.1 Repetition Coding

The simplest code is a repetition code, in which x[`] = x[1] for ` = 1, L In vector

form, the overall channel becomes:

where y = [y[1], , y[L]] t , h = [h[1], , h[L]] t and w = [w[1], , w[L]] t

Consider now coherent detection of x[1]; i.e the channel gains are known to the

receiver This is a standard vector Gaussian detection problem, and it is well known that

is a sufficient statistic for the detection problem Thus, this is equivalent to a scalar detection problem with noise (h∗ /khk)w ∼ CN (0, N0) The receiver structure is a

matched filter and is also called a maximal ratio combiner: it weighs the received

signal in each branch in proportion to the signal strength and also align the phases of

the signals in the summing (this is also called coherent combining.)

Consider BPSK modulation, with x[1] = ±a The error probability, conditional on

h, can be derived exactly as in (3.11):

Q³p

2khk2SNR

´

(3.18)

where as before SNR = a2/N0 is the signal-to-noise ratio per degree of freedom We

average over khk2 to find the overall error probability Under Rayleigh fading,

khk2 =

L

X

`=1

is a sum of the squares of 2L independent Gaussian random variables Its distribution

is known as Chi-square with 2L degrees of freedom, and is given by:

f (x) = 1

(L − 1)! x

4This is a slight abuse of notation as we originally denote h[n] as the fading gain at time n Here

we are re-indexing the symbols, assuming that interleaving is already done.

The average error probability can be explicitly computed to be:

p e =

· 1

2(1 − µ)

¸L L−1X

k=0

µ

L − 1 + k k

¶ µ

1 + µ

2

¶k

(3.21)

where

µ :=

r SNR

The error probability as a function of the SNR for different L is plotted in Figure 3.2.

We can see that increasing L can dramatically decrease the error probability For high SNR, (1 + µ)/2 ≈ 1 and (1 − µ)/2 ≈ 1/(4SNR) Furthermore,

L−1

X

k=0

µ

L − 1 + k k

¶

=

µ

2L − 1

L

¶

.

Hence,

p e ≈

µ

2L − 1

L

¶ 1

at high SNR In particular, the error probability decreases as the L th power of SNR, corresponding to a slope of −L in the error probability curve (in dB/dB scale) This

can be explained as follows

As in our analysis in Section 3.1.2, the typical error event at high SNR is when the overall channel gain is small, and this happens with probability

P©khk2 < 1/SNRª

For small x, the pdf of khk2 is approximately

f (x) ≈ 1

(L − 1)! x

and so

P©khk2 < 1/SNRª≈

Z 1 SNR 0

1

(L − 1)! x

L−1 dx = 1

L!

1 SNRL (3.25)

This analysis is too crude to get the correct constant before the 1/SNR L term in eqn

(3.23) but does get the correct exponent L Basically, error occurs when PL `=1 |h[`]|2

is of the order or smaller than 1/SNR, and this happens when all the gains |h[`]|2’s are

small, of the order of 1/SNR Since the probability that each |h[`]|2 is less than 1/SNR

is approximately 1/SNR and the gains are independent, the probability of the overall gain being small is of the order 1/SNR L Typically, L is called the diversity gain of the

system