MIMO Communication Systems

Lecture 3
Digital Modulation, Detection and
Performance Analysis
Prof. Chun-Hung Liu
Dept. of Electrical and Computer Engineering
National Chiao Tung University
Spring 2017

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 Outline
Digital Modulation and Detection (Part of Chapter 5 in
Goldsmiths Book)

Signal Space Analysis

Amplitude and Phase Modulation
Frequency Modulation
Pulse Shaping

Performance Analysis of Digital Modulation over Wireless

Channels (Part of Chapter 6 in Goldsmiths Book)

AWGN Channels
Error Probability
Fading

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 Introduction
Digital modulation and detection consist of transferring information in
the form of bits over a communications channel. The bits are binary
digits taking on the values of either 1 or 0.
Digital modulation consists of mapping the information bits into an
analog signal for transmission over the channel.
Detection consists of determining the original bit sequence based on the
signal received over the channel.
The main considerations in choosing a particular digital modulation
technique are
high data rate
high spectral efficiency (minimum bandwidth occupancy)
high power efficiency (minimum required transmit power)
robustness to channel impairments (minimum probability of bit
error)
low power/cost implementation
There are two main categories of digital modulation: amplitude/phase
modulation and frequency modulation.
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 Signal Space Analysis
Digital modulation encodes a bit stream of finite length into one of
several possible transmitted signals.
The receiver minimizes the probability of detection error by decoding
the received signal as the signal in the set of possible transmitted
signals that is closest to the one received.
Determining the distance between the transmitted and received signals
requires a metric for the distance between signals.
By representing signals as a vector in a vector space, we can have the
metric for the distance between signals.

Signal and System Model

Consider the communication system model shown in Figure 5.1. Every

T seconds, the system sends K = log2 M bits of information through
the channel for a data rate of R = K/T bits per second (bps).

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 Signal Space Analysis

There are M = 2K possible sequences of K bits, and we say that each bit
sequence of length K comprises a message mi = {b1 , . . . , bK } 2 M where
M = {m1 , . . . , mM } is the set of all such messages.
The messages
PM have probability pi of being selected for transmission,
where i=1 pi = 1 .
Suppose message mi is to be transmitted over the channel during the time
interval [0, T).
Each message mi 2 M is mapped to a unique analog signal si 2 S
= {s1 (t), . . . , sM (t)} where si (t) is defined on the time interval
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 Signal Space Analysis
[0, T) and has energy
Z T
Es i = s2i (t)dt, , i = 1, . . . , M.
0

When messages are sent sequentially, the transmitted signal becomes a

sequence of the corresponding
P analog signals over each time interval
[kT, (k + 1)T ) : s(t) = k si (t kT ), where si (t) is the analog signal
corresponding to the message mi designated for the transmission interval

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 Signal Space Analysis
Given the received signal r(t) = s(t) + n(t) , the receiver must determine
the best estimate of which si (t) 2 S .
The goal of the receiver design in estimating the transmitted message is to
minimize the probability of message error:
XM
Pe = 6= mi ; mi sent)p(mi sent)
p(m
i=1

where m 1, . . . , b
= {b K } 2 M is best estimate of the transmitted bit
sequence.

Geometric Representation of Signals: The basic premise behind a

geometrical representation of signals is the notion of a basis set.
Any set of M real energy signals S = (s1 (t), . . . , sM (t)) defined on [0, T)
can be represented as a linear combination ofN M , real orthonormal
basis functions { 1 (t), . . . , N (t)}. These basis functions span the set S .
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 Signal Space Analysis
Each si (t) 2 S in terms of its basis function representation is written as
N
X Z T
si (t) = sij j (t), 0 t < T, where sij = si (t) j (t)dt
j=1 0

is a real coefficient representing the projection of si (t) onto the basis

function j (t) and Z T (
1, i = j
i (t) j (t)dt = . (5.5)
0 0, i 6= j

If the signals {si (t)} are linearly independent then N = M , otherwise N<M.
The minimum number N of basis functions needed to represent any signal
si (t) of duration T and bandwidth B is roughly 2BT. (why?)

The signal si (t) thus occupies a signal space of dimension 2BT.

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 Signal Space Analysis
For linear passband modulation techniques, the basis set consists of the
sine and cosine functions:
r r
2 2
1 = cos(2f c t) and 2 = sin(2fc t)
T T
In fact, with these basis functions we only get an approximation to (5.5),
since Z Z T
T
2 2 sin(4fc T )
1 (t)dt = 0.5[1 + cos(4fc t)]dt = 1 +
0 T 0 4fc T
The second term can be neglected since usually fc T 1.
Z T Z T
2 cos(4fc T )
1 (t) 2 (t)dt = 0.5 sin(4fc t)dt = 0,
0 T 0 4fc T

where the approximation is taken as an equality for fc T 1.

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 Signal Space Analysis
So si (t) can be represented by
r r
2 2
si (t) = si1 cos(2fc t) + si2 sin(2fc t)
T T

The basis set may also include a baseband pulse-shaping filter g(t) to
improve the spectral characteristics of the transmitted signal:
si (t) = si1 g(t) cos(2fc t) + si2 g(t) sin(2fc t).
In this case the pulse shape g(t) must maintain the orthonormal
properties (5.5) of basis functions, i.e. we must have
Z T Z T
g 2 (t) cos2 (2fc t)dt = 1 and g 2 (t) cos(2fc t) sin(2fc t)dt = 0,
0 0

The simplest pulse shape that satisfies

p the above two identities is the
rectangular pulse shape g(t) = 2/T , 0 t < T .

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 Signal Space Analysis
Example 5.1: Binary phase shift keying (BPSK) modulation transmits the
signal s1 (t) = cos(2fc t), 0 t T , to send a 1 bit and the
signal s2 (t) = cos(2fc t), 0 t T , to send a 0 bit. Find the set of
orthonormal basis functions and coefficients {sij } for this modulation.

Solution: There is
ponly one basis function for ps1 (t) and s2 (t) ,
(t) = 2/T cos(2fc t) where the 2/T is needed for
normalization. The coefficients are then given by
p p
s1 = T /2 and s2 = T /2.
The coefficients {sij }is denoted as a vector si = (si1 , . . . , siN ) 2 RN which
is called the signal constellation point corresponding to the signal si (t) .
The signal constellation consists of all constellation points {s1 , . . . , sM } .
The representation of si (t) in terms of its constellation point si 2 RN is
called its signal space representation and the vector space containing the
constellation is called the signal space.
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 Signal Space Analysis
A two-dimensional signal space is illustrated in Figure 5.3, where we show
si 2 R2 with the ith axis of R2 corresponding to the basis function i (t) .

With this signal space

representation we can analyze the
infinite-dimensional functions si (t)
as vectors si in finite-dimensional
vector space R2 .
Signal space representations for
common modulation techniques
like MPSK and MQAM are two-
dimensional (corresponding to
the in-phase and quadrature
basis functions).

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 Signal Space Analysis
The length of a vector in RN is defined as
v
uN
uX
ksi k = t s2ij .
j=1

The distance between two signal constellation points si and sk is thus

v s
uN Z T
uX
ksi sk k = t [sij skj ]2 = [si (t) sk (t)]2 dt,
j=1 0

Finally, the inner product < si (t), sk (t) > between two real signals si (t)
and sk (t) on the interval [0,T] is
Z T
< si (t), sk (t) >= si (t)sk (t)dt.
0

Similarly, the inner product < si , sk > between two real vectors is
Z T
< si , sk >= si sTk = si (t)sk (t)dt =< si (t), sk (t) >
0
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 Receiver Structure and Sufficient Statistics
Here we would like to convert the received signal r(t) over each time
interval into a vector, as it allows us to work in finite-dimensional vector
space to estimate the transmitted signal.
For this conversion, consider the receiver structure shown in Figure 5.4,
where Z T Z T
sij = si (t) j (t)dt, and nj = n(t) j (t)dt.
0 0

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 Receiver Structure and Sufficient Statistics
N
X N
X
We can rewrite r(t) as (sij + nj ) j (t) + nr (t) = rj j (t) + nr (t),
j=1 j=1
PN
where rj = sij + nj and nr (t) = n(t) j=1 nj j (t) denotes the
remainder noise, which is the component of the noise orthogonal to the
signal space.
If we can show that the optimal detection of the transmitted signal
constellation point si given received signal r(t) does not make use of the
remainder noise nr (t) , then the receiver can make its estimate m of the
transmitted message mi as a function of r = (r1 , . . . , rN ) alone.
Here r = (r1 , . . . , rN ) is a sufficient statistic for r(t) in the optimal
detection of the transmitted messages.
The remainder noise nr (t) should not help in detecting the transmitted
signal si (t) since its projection onto the signal space is zero. This is
illustrated in Figure 5.5

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 Receiver Structure and Sufficient Statistics

From the figure, it appears that projecting r + nr onto r will not

compromise the detection of which constellation si was transmitted,
since nr lies in a space orthogonal to the space where si lie

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 Receiver Structure and Sufficient Statistics
Since n(t) is a Gaussian random process, if we condition on the
transmitted signal si (t) then the channel output r(t) = si (t) + n(t) is also
a Gaussian random process and r = (r1 , . . . , rN ) is a Gaussian random
vector.
Since rj = sij + nj , conditioned on a transmitted constellation si
we have that
rj |si = E[rj |si ] = E[sij + nj |sij ] = sij

since n(t) has zero mean, and

Moreover,

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 Receiver Structure and Sufficient Statistics

Conditioned on the transmitted constellation si , rj is a

Gauss-distributed random variable that is independent of rk ,,
k 6= j and has mean sij and variance N0 /2 .
The conditional distribution of r is given by

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 Receiver Structure and Sufficient Statistics
We now discuss the receiver design criterion and show it is not affected
by discarding nr (t) .
The goal of the receiver design is to minimize the probability of error in
detecting the transmitted message mi given received signal r(t).
To minimize , we can
maximize p(m = mi |r(t)) , which is equivalent to maximize p(si sent|r(t))
p(si sent|r(t)) can be found as follows

Since r is a sufficient statistic for the received signal r(t), we call r the
received vector associated with r(t).
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Decision Regions and the Maximum Likelihood Decision
We know, given a received vector r, the optimal receiver selects m
= mi
corresponding to the constellation that satisfies
p(si sent|r) > p(sj sent|r), j 6= i
Let us define a set of decisions regions (Z1 , . . . , ZM ) that are subsets of the
signal space RN by
Zi = (r : p(si sent|r) > p(sj sent|r), 8j 6= i).
Clearly these regions do not overlap and they partition the signal space
assuming there is no r 2 RN for which p(si sent|r) = p(sj sent|r) .

If such points exist then the signal space is partitioned with decision regions
by arbitrarily assigning such points to either decision region Zi or Zj .
Once the signal space has been partitioned by decision regions, then for a
received vector r 2 Zi the optimal receiver outputs the message
estimate m
= mi .

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Decision Regions and the Maximum Likelihood Decision
Figure 5.6 shows a two-dimensional signal space with four decision
regions Z1 , . . . , Z4 corresponding to four constellations s1 , . . . , s4 .

The received vector r lies

in region Z1 , so the
receiver will output the
message as the best

message estimate given

received vector r.

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Decision Regions and the Maximum Likelihood Decision
We now examine the decision regions in more detail. By Bayes rule,

To minimize error probability, the receiver output m = mi corresponds

to the constellation that maximizes p(si |r) , i.e. must satisfy

Assuming equally likely messages (p(si ) = 1/M ) , the receiver output

= mi corresponding to the constellation that satisfies
m

arg max p(r|si ), i = 1, . . . , M.

si

Let us define the likelihood function associated with our receiver as

L(si ) = p(r|si ).
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Decision Regions and the Maximum Likelihood Decision
Given a received vector r, a maximum likelihood receiver outputs
corresponding to the constellation that maximizes .
Since maximizing is equivalent to maximizing the log likelihood
function, defined as
Using then yields
1
kr si k2
N0

Thus, the log likelihood function depends only on the distance

between the received vector r and the constellation point

The maximum likelihood receiver is implemented using the structure

shown in Figure 5.4. First r is computed from r(t), and then the signal
constellation closest to r is determined as the constellation point
satisfying

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Decision Regions and the Maximum Likelihood Decision

This is determined from the decision region that contains r, where

is defined by

Finally, the estimated constellation is mapped to the estimated

message , which is output from the receiver.
An alternate receiver structure is shown in Figure 5.7.
This structure makes use of a bank of filters matched to each of the
different basis function. We call a filter with impulse response
the matched filter to the signal , so Figure
5.7 is also called a matched filter receiver.

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Decision Regions and the Maximum Likelihood Decision
If a given input signal is passed through a filter matched to that signal,
the output SNR is maximized.

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Decision Regions and the Maximum Likelihood Decision

Example 5.2: For BPSK modulation, find decision regions Z1 and Z2

corresponding to constellations s1 = A and s2 = A .

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 Error Probability and the Union Bound
We now analyze the error probability associated with the maximum
likelihood receiver structure. For equally likely messages p(mi sent) = 1/M
we have

We illustrate this error probability calculation in Figure 5.8, where

the constellation points s1 , . . . , s8 are equally spaced around a circle
with minimum separation dmin .

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 Error Probability and the Union Bound
The probability of correct reception assuming the first symbol is sent,
p(r 2 Z1 |m1 sent) corresponds to the probability p(r = s1 + n|s1 ) that
when noise is added to the transmitted constellation , the resulting
vector r = s1 + n remains in the Z1 region shown by the shaded area.

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 Error Probability and the Union Bound
Since we cannot solve for this error probability in closed form, we now
investigate the union bound on error probability, which yields a closed
form expression that is a function of the distance between signal
constellation points.
Let denote the event given that the constellation
point was sent.
If the event occurs, then the constellation will be decoded in error
since the transmitted constellation is not the closest constellation point
to the received vector r.
However, event does not necessarily imply that will be decoded
instead of , since there may be another constellation point with

The constellation is decoded correctly if

Thus
(5.35)

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 Error Probability and the Union Bound
Let us now consider p(Aik ) more closely. We have

(5.36)
i.e. the probability of error equals the probability that the noise n is
closer to the vector si sk than to the origin.
This probability does not depend on the entire noise component n: it
only depends on the projection of n onto the line connecting the origin
and the point si sk , as shown in Figure 5.9.
The event Aik occurs if n is closer to si sk than to zero, i.e. if n > dik /2 ,
where dik = ksi sk k equals the distance between constellation points
si and sk . Thus,

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 Error Probability and the Union Bound

So we can get

where

The union bound

(5.40)

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 Error Probability and the Union Bound
Defining the minimum distance of the constellation as dmin = mini,k dik ,
we can simplify the union bound with the looser bound

dmin
Pe (M 1)Q p . (5.43)
2N0
2
By using Q(z) p1 e z /2 , the above inequality can be further
z 2
simplified as

M 1 d2min
Pe p exp . (5.44)
4N0

Finally, is sometimes approximated as the probability of error

associated with constellations at the minimum distance multiplied
by the number of neighbors at this distance

dmin
Pe = Mdmin Q p . (5.45)
2N0
This approximation is called the nearest neighbor approximation to Pe .
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 Error Probability and the Union Bound
Example 5.3:

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 Error Probability and the Union Bound

Note that for binary modulation where M = 2, there is only one way to
make an error and dmin is the distance between the two signal
constellation points, so the bound (5.43) is exact:

dmin
Pb = Q p (5.46)
2N0
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 Error Probability and the Union Bound
The minimum distance squared in (5.44) and (5.46) is typically
proportional to the SNR of the received signal. Thus, error probability is
reduced by increasing the received signal power.
Recall that Pe is the probability of a symbol (message) error
Pe = p(m 6= mi |mi sent) , where mi corresponds to a message with log2 M
bits.
However, system designers are typically more interested in the bit error
probability, also called the bit error rate (BER), than in the symbol error
probability, since bit errors drive the performance of higher layer
networking protocols and end-to-end performance.
So we would like to design the mapping of the M possible bit sequences to
messages mi , i = 1, . . . , M so that a symbol error associated with an
adjacent decision region, which is the most likely way to make an error,
corresponds to only one bit error.
With such a mapping, assuming that mistaking a signal constellation for a
constellation other than its nearest neighbors has a very low probability,
we can make the approximation Pb Pe
log2 M
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 Passband Modulation Principles
The goal of modulation is to send bits at a high data rate while
minimizing the probability of data corruption.
In general, modulated carrier signals encode information in the
amplitude (t) , frequency f(t), or phase (t) of a carrier signal. Thus, the
modulated signal can be represented as

where and is the phase offset of the carrier. This

representation combines frequency and phase modulation into angle
modulation.
We can rewrite the above expression as

where is called the in-phase component of s(t) and

is called its quadrature component.
Therefore,
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 Amplitude and Phase Modulation
In amplitude and phase modulation the information bit stream is encoded
in the amplitude and/or phase of the transmitted signal.

Specifically, over a time interval of Ts , K = log2 M bits are encoded into

the amplitude and/or phase of the transmitted signal s(t), 0 t < Ts .
The transmitted signal over this period s(t) = sI (t) cos(2fc t) sQ (t) sin(2fc t)
can be written in terms of its signal space representation as
s(t) = si1 1 (t) + si2 2 (t) with basis functions 1 (t) = g(t) cos(2fc t + 0 )
and 2 (t) = g(t) sin(2fc t + 0 ) here g(t) is a shaping pulse.
These in-phase and quadrature signal components are baseband signals
with spectral characteristics determined by the pulse shape g(t).
In particular, their bandwidth B equals the bandwidth of g(t), and the
transmitted signal s(t) is a passband signal with center frequency fc and
passband bandwidth 2B. In practice we take B = Kg /Ts where Kg
depends on the pulse shape.

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 Amplitude and Phase Modulation
Since the pulse shape g(t) is fixed, the signal constellation for amplitude
and phase modulation is defined based on the constellation point:
(si1 , si2 ) 2 R2 , i = 1, . . . , M . The complex baseband representation of
s(t) is
s(t) = R{x(t)e 0 ej(2fc t) }
where x(t) = sI (t) + jsQ (t) = (si1 + jsi2 )g(t) .
The constellation point si = (si1 , si2 ) is called the symbol associated
with the log2 M bits and Ts is called the symbol time. The bit rate for
this modulation is K bits per symbol or R = log2 M/Ts bits per second.
There are three main types of amplitude/phase modulation:

Pulse Amplitude Modulation (MPAM): information encoded in

amplitude only.
Phase Shift Keying (MPSK): information encoded in phase only.
Quadrature Amplitude Modulation (MQAM): information
encoded in both amplitude and phase.

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 Amplitude and Phase Modulation
The number of bits per symbol K = log2 M , signal constellation
(si1 , si2 ) 2 R2 , i = 1, . . . , M, and choice of pulse shape g(t) determines
the digital modulation design. The pulse shape g(t) is designed to
improve spectral efficiency and combat ISI.
Amplitude and phase modulation over a given symbol period can be
generated using the modulator structure shown in Figure 5.10.
Demodulation over each symbol period is performed using the
demodulation structure of Figure 5.11, which is equivalent to the
structure of Figure 5.7 for 1 (t) = g(t) cos(2fc t + ) and
2 (t) = g(t) sin(2fc t + )
Typically the receiver includes some additional circuitry for carrier
phase recovery that matches the carrier phase at the receiver to the
carrier phase 0 at the transmitter, which is called coherent detection.
The receiver structure also assumes that the sampling function every Ts
seconds is synchronized to the start of the symbol period, which is called
synchronization or timing recovery.
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 Amplitude and Phase Modulation

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 Pulse Amplitude Modulation (MPAM)
We will start by looking at the simplest form of linear modulation, one-
dimensional MPAM, which has no quadrature component (si2 = 0)
For MPAM all of the information is encoded into the signal amplitude
The transmitted signal over one symbol time is given by

where Ai = (2i 1 M )d, i = 1, 2, . . . , M defines the signal constellation,

parameterized by the distance d which is typically a function of the signal
energy, and g(t) is the pulse shape
The minimum distance between constellation points is
dmin = min |Ai Aj | = 2d
i,j

The amplitude of the transmitted signal takes on M different values,

which implies that each pulse conveys log2 M = K bits per symbol
time Ts .

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 Pulse Amplitude Modulation (MPAM)
Over each symbol period the MPAM signal associated with the ith
constellation has energy

Assuming equally likely symbols, the average energy is

M
1 X 2
Es = A
M i=1 i
The constellation mapping is usually done by Gray encoding, where the
messages associated with signal amplitudes that are adjacent to each other
differ by one bit value, as illustrated in Figure 5.12
With this encoding method, if noise causes the demodulation process to
mistake one symbol for an adjacent one (the most likely type of error),
this results in only a single bit error in the sequence of K bits. Gray codes
can be designed for MPSK and square MQAM constellations, but not
rectangular MQAM.
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 Pulse Amplitude Modulation (MPAM)

Example 5.4:

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 Pulse Amplitude Modulation (MPAM)
The decision regions Zi , i = 1, . . . , M associated with the pulse amplitude
Ai = (2i 1 M )d for M = 4 and M = 8 are shown in Figure 5.13.
Mathematically, for any M, these decision regions are defined by

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 Pulse Amplitude Modulation (MPAM)
MPAM has only a single basis function 1 (t) = g(t) cos(2fc t) . Thus,
the coherent demodulator of Figure 5.11 for MPAM reduces to the
demodulator shown in Figure 5.14, where the multithreshold device
maps x to a decision region Zi and outputs the corresponding bit
sequence m = mi = {b1 , . . . , bK }.

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 Phase Shift Keying (MPSK)
For MPSK all of the information is encoded in the phase of the
transmitted signal. Thus, the transmitted signal over one symbol time is
given by

The constellation points or symbols are given by

The minimum distance between constellation points is dmin = 2A sin(/M)

where A is typically a function of the signal energy.
2PSK is often referred to as binary PSK or BPSK, while 4PSK is often
called quadrature phase shift keying (QPSK).
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 Phase Shift Keying (MPSK)
All possible transmitted signals si (t) have equal energy:
Z Ts
E si = s2i (t)dt = A2
0
As for MPAM, constellation mapping is usually done by Gray encoding,
where the messages associated with signal phases that are adjacent to each
other differ by one bit value, as illustrated in Figure 5.15.

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 Phase Shift Keying (MPSK)
The decision regions Zi , i = 1, . . . , M , associated with MPSK for M = 8
are shown in Figure 5.16. If we represent r = rej 2 R2 in polar
coordinates then these decision regions for any M are defined by

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 Phase Shift Keying (MPSK)
BPSK has only a single basis function and, since
there is only a single bit transmitted per symbol time , the bit
time Tb = Ts .
The coherent demodulator of Figure 5.11 for BPSK reduces to the
demodulator shown in Figure 5.17, where the threshold device maps x to
the positive or negative half of the real line, and outputs the
corresponding bit value.

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 Quadrature Amplitude Modulation (MQAM)
For MQAM, the information bits are encoded in both the amplitude
and phase of the transmitted signal.
Thus, whereas both MPAM and MPSK have one degree of freedom
in which to encode the information bits (amplitude or phase),
MQAM has two degrees of freedom.
As a result, MQAM is more spectrally-efficient than MPAM and
MPSK, in that it can encode the most number of bits per symbol for
a given average energy.
The transmitted signal is given by

The energy in si (t) is

Z Ts
E si = s2i (t)dt = A2i ,
0

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 Quadrature Amplitude Modulation (MQAM)
the same as for MPAM. The distance between any pair of symbols in the
signal constellation is

For square signal constellations, where si1 and si2 take values on
(2i 1 L)d, i = 1, 2, . . . , L = 2l , the minimum distance between signal
points reduces to dmin = 2d , the same as for MPAM.
MQAM with square constellations of size L2 is equivalent to MPAM
modulation with constellations of size L on each of the in-phase and
quadrature signal components.
Common square constellations are 4QAM and 16QAM, which are
shown in Figure 5.18 below.
These square constellations have M = 22l = L2 constellation points,
which are used to send 2l bits/symbol, or l bits per dimension.

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 Quadrature Amplitude Modulation (MQAM)

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 Frequency Modulation
Frequency modulation encodes information bits into the frequency of the
transmitted signal. Specifically, each symbol time K = log2 M bits are
encoded into the frequency of the transmitted signal s(t), 0 t < Ts ,
resulting in a transmitted signal si (t) = A cos(2fi t + i ) where i is the
index of the ith message corresponding to the log2 M bits and i is the
phase associated with the ith carrier.
Since frequency modulation encodes information in the signal frequency,
the transmitted signal s(t) has a constant envelope A. Because the signal is
constant envelope, nonlinear amplifiers can be used with high power
efficiency, and the modulated signal is less sensitive to amplitude
distortion introduced by the channel or the hardware.
Frequency modulation over a given symbol period can be generated using
the modulator structure shown in Figure 5.22. Demodulation over each
symbol period is performed using the demodulation structure of Figure
5.23.

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 Frequency Modulation

2017/3/26 Lecture 3: Digital Modulation & Detection 54

 Frequency Modulation

Note that the

demodulator of
Figure 5.23 requires
that the jth carrier
signal be matched in
phase to the jth
carrier signal at the
transmitter, similar to
the coherent phase
reference requirement
in amplitude and
phase modulation.

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 Frequency Shift Keying (FSK)
In MFSK the modulated signal is given

si (t) = A cos[2fc t + 2i fc t + i ], 0 t < Ts ,

where i = (2i 1 M ), i = 1, 2, . . . , M = 2K. The minimum frequency

separation between FSK carriers is thus 2 fc .
p
MFSK consists of M basis functions i (t) = 2/Ts cos[2fc t + 2i fc t + i ]
Over a given symbol time only one basis function is transmitted through
the channel.
A simple way to generate the MFSK signal is as shown in Figure 5.22
where M oscillators are operating at the different frequencies
fi = fc + i fc and the modulator switches between these different
oscillators each symbol time Ts .
However, with this implementation there will be a discontinuous phase
transition at the switching times due to phase offsets between the
oscillators.
This discontinuous phase leads to a spectral broadening, which is
undesirable.
2017/3/26 Lecture 3: Digital Modulation & Detection 56
 Frequency Shift Keying (FSK)
For binary signaling the structure can be simplified to that shown in
Figure 5.24, where the decision device outputs a 1 bit if its input is greater
than zero and a 0 bit if its input is less than zero.

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 Pulse Shaping
For amplitude and phase modulation the bandwidth of the baseband and
passband modulated signal is a function of the bandwidth of the pulse
shape g(t).
If g(t) is a rectangular pulse of width Ts , then the envelope of the signal
is constant. However, a rectangular pulse has very high spectral
sidelobes, which means that signals must use a larger bandwidth to
eliminate some of the adjacent channel sidelobe energy.
Pulse shaping is a method to reduce sidelobe energy relative to a
rectangular pulse, however the shaping must be done in such a way that
intersymbol interference (ISI) between pulses in the received signal is not
introduced.
To avoid ISI between samples of the received pulses, the effective
received pulse shape p(t) = g(t) c(t) g ( t)must satisfy the Nyquist
criterion, which requires the pulse equals zero at the ideal sampling
point associated with past or future symbols:
(Since the channel model is AWGN, c(t) = (0) = 1 and p(t) = g(t) g ( t) )

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 Pulse Shaping
(
p0 = p(0), k=0
p(kTs ) =
0, k 6= 0
In the frequency domain this translates to
1
X
P (f + l/Ts ) = p0 Ts .
l= 1
The following pulse shapes all satisfy the Nyquist criterion.
p
Rectangular pulses: g(t) = 2/Ts , 0 t Ts , which yields the
triangular effective pulse shape. This pulse shape leads to constant envelope
signals in MPSK, but has lousy spectral properties due to its high side lobes.
Cosine pulses: p(t) = sin(t/Ts ), 0 t Ts . Cosine pulses are mostly
used in MSK modulation, where the quadrature branch of the PSK
modulation has its pulse shifted by Ts /2 . This leads to a constant amplitude
modulation with side lobe energy that is 10 dB lower than that of rectangular
pulses.

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 Pulse Shaping
Raised Cosine Pulses: These pulses are designed in the frequency
domain according to the desired spectral properties. Thus, the pulse p(t)
is first specified relative to its Fourier Transform:

where is defined as the rolloff factor, which determines the rate of

spectral rolloff, as shown in Figure 5.26. Setting = 0 yields a rectangular
pulse. The pulse p(t) in the time domain corresponding to P(f) is

The time and frequency domain properties of the Raised Cosine pulse
are shown in Figures 5.26-5.27.

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 Pulse Shaping

The tails of this pulse in the time domain decay as 1/t3 (faster than for
the previous pulse shapes), so a mistiming error in sampling leads to a
series of intersymbol interference components that converge.
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 Pulse Shaping

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 Performance Analysis for AWGN Channels
We now consider the performance of the digital modulation techniques discussed
in the previous chapter when used over AWGN channels and channels with flat-
fading.
There are two performance criteria of interest: the probability of error, defined
relative to either symbol or bit errors, and the outage probability, defined as the
probability that the instantaneous signal-to-noise ratio falls below a given
threshold.
Flat fading can cause a dramatic increase in either the average bit-error-rate or
the signal outage probability.
Wireless channels may also exhibit frequency selective fading and Doppler shift.
Frequency-selective fading gives rise to intersymbol interference (ISI), which
causes an irreducible error floor in the received signal.
Doppler causes spectral broadening, which leads to adjacent channel
interference (typically small at reasonable user velocities).
We first define the signal-to-noise power ratio (SNR) and its relation to energy-
per-bit (Eb) and energy per-symbol (Es). We then examine the error probability
on AWGN channels for different modulation techniques as parameterized by
these energy metrics.
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 Signal-to-Noise Power Ratio and Bit/Symbol Energy
In an AWGN channel the modulated signal s(t) = R{u(t)ej2fc t } has noise n(t)
added to it prior to reception. The noise n(t) is a white Gaussian random process
with mean zero and power spectral density N0 /2. The received signal is
thus r(t) = s(t) + n(t) .
Define the received signal-to-noise power ratio (SNR) as the ratio of the received
signal power Pr to the power of the noise within the bandwidth of the
transmitted signal s(t).
Specifically, if the bandwidth of the complex envelope u(t) of s(t) is B then the
bandwidth of the transmitted signal s(t) is 2B. Since the noise n(t) has uniform
power spectral density N0 /2 , the total noise power within the bandwidth 2B
is N = N0 /2 2B = N0 B .
So the received SNR is given by
Pr
SNR =
N0 B
In systems with interference, we often use the received signal-to-interference-
plus-noise power ratio (SINR) in place of SNR for calculating error probability.

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 Signal-to-Noise Power Ratio and Bit/Symbol Energy
If the interference statistics approximate those of Gaussian noise then this is a
reasonable approximation. The received SINR is given by

where PI is the average power of the interference.

The SNR is often expressed in terms of the signal energy per bit Eb or per
symbol Es as

where Ts is the symbol time and Tb is the bit time (for binary modulation
Ts = Tb and Es = Eb ).

For data pulses with Ts = 1/B , e.g. raised cosine pulses with =1, we have
SNR = Es /N0 for multilevel signaling and SNR = Eb /N0 for binary signaling.

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 Performance Analysis for AWGN Channels
The quantities s = Es /N0 and b = Eb /N0 are sometimes called the SNR per
symbol and the SNR per bit, respectively.
For performance specification, we are interested in the bit error probability Pb
as a function of b .
However, for M-ray signaling (e.g. MPAM and MPSK), the bit error probability
depends on both the symbol error probability and the mapping of bits to symbols.
Thus, we typically compute the symbol error probability Ps as a function of
and then obtain Pb as a function of b using an exact or approximate conversion.
s

The approximate conversion typically assumes that the symbol energy is divided
equally among all bits, and that Gray encoding is used so that at reasonable
SNRs, one symbol error corresponds to exactly one bit error. These assumptions
for M-ray signaling lead to the approximations

s Ps
b and Pb
log2 M log2 M

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 Error Probability for BPSK and QPSK
First consider BPSK modulation with coherent detection and perfect recovery of
the carrier frequency and phase. With binary modulation each symbol corresponds
to one bit, so the symbol and bit error rates are the same. The transmitted signal is
s1 (t) = Ag(t) cos(2fc t) to sent a 0 bit and s2 (t) = Ag(t) cos(2fc t)to send a 1 bit.
From (5.46) we have that the probability of error is

dmin
Pb = Q p .
2N0
From Chapter 5, dmin = ks1 s0 k = kA ( A)k = 2A . Let us now relate A
to the energy-per-bit. We have

p
This yields the minimum distance dmin = 2A = 2 Eb . Thus, we have

(6.6)

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 Error Probability for BPSK and QPSK
QPSK modulation consists of BPSK modulation on both the in-phase and
quadrature components of the signal. With perfect phase and carrier recovery,
the received signal components corresponding to each of these branches are
orthogonal. Therefore,
p the bit error probability on each branch is the same as
for BPSK: Pb = Q( 2 b ). The symbol error probability equals the probability
that either branch has a bit error:
p
Ps = 1 [1 Q( 2 b )]2 . (6.7)

Since the symbol energy is split between the in-phase and quadrature branches,
we have s = 2 b . Therefore, we have
p 2 (6.8)
Ps = 1 [1 Q( s )] .

The union bound (5.40) on Ps for QPSK is

(6.9)

Writing this in terms of yields

p
3Q( s)

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 Error Probability for BPSK and QPSK
The closed form bound (5.44) becomes

(6.11)

Using the fact

p that the minimum distance between constellation points
is dmin = 2A2 , we get the nearest neighbor approximation

Note that with Gray encoding, we can approximate Pb from Ps by Pb Ps /2 ,

since we have 2 bits per symbol.

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 Error Probability for MPSK
The signal constellation for MPSK has

The symbol energy is Es = A2 , so s = A2 /N0 .

From (5.57), for the received vector x = rej represented in polar coordinates, an
error occurs if the ith signal constellation point is transmitted and
2
/ (2(i 1.5)/M, 2(i 0.5)/M )
The joint distribution of r and can be obtained through a bivariate
transformation of the noise n1 and n2 on the in-phase and quadrature branches
[Proakis, Chapter 4.3-2], which yields

Since the error probability depends only on the distribution of , we can integrate
out the dependence on r, yielding

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 Error Probability for MPSK

By symmetry, the probability of error is the same for each constellation point.
Thus, we can obtain Ps from the probability of error assuming the constellation
point s1 = (A, 0) is transmitted, which is

Each point in the MPSK constellation has two nearest neighbors at

distance dmin = 2A sin(/M ) . Thus, the nearest neighbor approximation (5.45)
to Ps is given by

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 Error Probability for MPAM and MQAM
The constellation for MPAM is Ai = (2i 1 M )d, i = 1, 2, . . . , M .
Each of the M-2 inner constellation points of this constellation have two
nearest neighbors at distance 2d.
The probability of making an error when sending one of these inner
constellation points is just the probability that the noise exceeds d in either
direction:
For the outer constellation points there is only one nearest neighbor, so an
error occurs if the noise exceeds d in one direction only:

The probability of error is thus

From (5.54) the average energy per symbol for MPAM is

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 Error Probability for MPAM and MQAM
Thus we can write Ps in terms of the average energy E s as
r !
2(M 1) 6 s
Ps = Q . (6.21)
M M2 1

Consider now MQAM modulation with a square signal constellation of

size M = L2 . This system can be viewed as two MPAM systems with signal
constellations of size L transmitted over the in-phase and quadrature signal
components, each with half the energy of the original MQAM system.
The constellation points in the inphase and quadrature branches take values

The symbol error probability for each branch

p of the MQAM system is thus
given by (6.21) with M replaced by L = M and s equal to the average energy
per symbol in the MQAM constellation:
p r !
2( M 1) 3 s
Ps = p Q .
M M 1
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 Error Probability for MPAM and MQAM
The probability of symbol error for the MQAM system is then

If we take a conservative approach and set the number of nearest neighbors to

be four, we obtain the nearest neighbor approximation
r !
3 s
Ps 4Q
M 1

For nonrectangular constellations, it is relatively straightforward to show that

the probability of symbol error is upper bounded as

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 Error Probability for MPAM and MQAM
The nearest neighbor approximation for nonrectangular constellations is

dmin
Ps Mdmin Q p , (6.26)
2N0
where Mdmin is the largest number of nearest neighbors for any constellation
point in the constellation and dmin is the minimum distance in the
constellation.
The MQAM demodulator requires both amplitude and phase estimates of the
channel so that the decision regions used in detection to estimate the transmitted
bit are not skewed in amplitude or phase.
The channel amplitude is used to scale the decision regions to correspond to the
transmitted symbol: this scaling is called Automatic Gain Control (AGC).
If the channel gain is estimated in error then the AGC improperly scales the
received signal, which can lead to incorrect demodulation even in the absence of
noise.
The channel gain is typically obtained using pilot symbols to estimate the channel
gain at the receiver. However, pilot symbols do not lead to perfect channel
estimates, and the estimation error can lead to bit errors.
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 Error Probability Approximation for Coherent Modulations

Many of the approximations or exact values for Ps derived above for coherent
modulation are in the following form:
p
Ps ( s ) M Q M s , (6.33)

where M and M depend on the type of approximation and the modulation type.

In particular, the nearest neighbor approximation has this form, where M is the
number of nearest neighbors to a constellation at the minimum distance, and M
is a constant that relates minimum distance to average symbol energy.

(6.31) Table 6.1

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 Error Probability for MPAM and MQAM
Performance specifications are generally more concerned with the bit error
probability Pb as a function of the bit energy b .
To convert from Ps to Pb and from s to b , we use the approximations (6.3)
and (6.2), which assume Gray encoding and high SNR.
Using these approximations in (6.33) yields a simple formula for Pb as a
function of b : q
Pb ( b ) =
M Q M b , (6.34)

M = M / log2 M and M = (log2 M )

where M for M and M in (6.33).

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 Fading and Error Probability
In a fading environment the received signal power varies randomly over distance
or time due to shadowing and/or multipath fading. Thus, in fading s is a random
variables with distribution p s ( ) , and therefore Ps ( s ) is also random.
The performance metric when s is random depends on the rate of change of the
fading.
There are three different performance criteria that can be used to characterize
the random variable Ps :
The outage probability, Pout, defined as the probability that s falls below
a given value corresponding to the maximum allowable Ps .
The average error probability, P s , averaged over the distribution of s .

Combined average error probability and outage, defined as the average

error probability that can be achieved some percentage of time or some
percentage of spatial locations.
The average probability of symbol error applies when the signal fading is on the
order of a symbol time ( Ts Tc ), so that the signal fade level is constant over
roughly one symbol time.
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 Outage Probability due to Fading
If the signal power is changing slowly ( Ts Tc ), then a deep fade will affect
many simultaneous symbols. Thus, fading may lead to large error bursts, which
cannot be corrected for with coding of reasonable complexity.
These error bursts can seriously degrade end-to-end performance. In this case
acceptable performance cannot be guaranteed over all time or, equivalently,
throughout a cell, without drastically increasing transmit power.
Under these circumstances, an outage probability is specified so that the channel
is deemed unusable for some fraction of time or space. Outage and average error
probability are often combined when the channel is modeled as a combination of
fast and slow fading.
Outage Probability: The outage probability relative to 0 is defined as
Z 0

Pout = p( s < 0) = p s ( )d ,
0
where 0 typically specifies the minimum SNR required for acceptable
performance.

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 Outage Probability due to Fading
In Rayleigh fading the outage probability becomes
Z 0
1 s/ s 0/ s
Pout = e d s =1 e .
0 s

Inverting this formula shows that for a given outage probability, the required
average SNR s is
0
s = .
ln(1 Pout )

Average Probability of Error: The average probability of error is used as a

performance metric when Ts Tc . Thus, we can assume that s is roughly
constant over a symbol time.

Then the averaged probability of error is computed by integrating the error

probability in AWGN over the fading distribution:

where Ps ( ) is the probability of symbol error in AWGN with SNR , which can be
approximated by the expressions in Table 6.1
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 Average Error Probability
For a given distribution of the fading amplitude r (i.e. Rayleigh, Rician, log-
normal, etc.), we compute p s ( ) by making the change of variable

p s ( )d = p(r)dr.

For example, in Rayleigh fading the received signal amplitude r has the Rayleigh
distribution
r r 2 /2 2
p(r) = 2
e , r 0,

and the signal power is exponentially distributed with mean 2 2. The SNR per
symbol for a given amplitude r is
r 2 Ts
= ,
2 n2
where n2 = N0 /2 is the PSD of the noise in the in-phase and quadrature branches.
Differentiating both sides of this expression yields
rTs
d = 2
dr.
n
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 Average Error Probability
Then we can have
(6.55)

Since the average SNR per symbol s is just 2

Ts / n,
2 we can rewrite (6.55) as

(6.56)

which is exponential. For binary signaling this reduces to

(6.57)

Integrating (6.6) over the distribution (6.57) yields the following average
probability of error for BPSK in Rayleigh fading.

where the approximation holds for large b .

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 Error Probability for Fading
A similar integration of (6.31) over (6.57) yields the average probability of
error for binary FSK in Rayleigh fading as
" s #
1 b 1
Pb = 1 (6.59)
2 2+ b 2 b
p
If we use the general approximation Ps M Q( M s ) then the average
probability of symbol error in Rayleigh fading can be approximated as
Z 1 p 1 /
Ps M Q M e s d s
0 s
" s #
m 0.5 M s M
= 1
2 1 + 0.5 M s 2 M s

where the last approximation is in the limit of high SNR.

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Combined Outage and Average Error Probability
When the fading environment is a superposition of both fast and slow fading, i.e.
log-normal shadowing and Rayleigh fading, a common performance metric is
combined outage and average error probability, where outage occurs when the
slow fading falls below some target value and the average performance in non-
outage is obtained by averaging over the fast fading. We use the following
notation:

An outage is declared when the received SNR per symbol due to shadowing and
path loss alone, s , falls below a given target value s0 .

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Combined Outage and Average Error Probability
When not in outage s s0 , the average probability of error is obtained by
averaging over the distribution of the fast fading conditioned on the mean SNR:

The criterion used to determine the outage target s0 is typically based on a

given maximum average probability of error, i.e. P s P s0 , where the target s0
must then satisfy

Clearly whenever s > s0 , the average error probability will be below the
target value.

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