Notes lectures wireless

c Prof. Giorgio Taricco c
Politecnico di Torino
8/9/2019 Notes lectures wireless
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2 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts
Section Outline
1 Basic concepts Model of a digital communication system Band-pass signalling
Problem set 1 Probability Gaussian random variables Complex Gaussian random variables Signal spaces
Problem set 2
Reference books in Wireless Communications
S. Benedetto and E. Biglieri, Principles of Digital Transmission: With Wireless Applications . Kluwer.
A. Goldsmith, Wireless Communications . Cambridge University Press.
U. Madhow, Fundamentals of Digital Communication. Cambridge
University Press. A. Molisch, Wireless Communications . Wiley.
J. Proakis and M. Salehi, Digital Communications (4th Edition). McGraw-Hill.
T. Rappaport, Wireless Communications: Principles and Practice (2nd Edition). Prentice-Hall.
D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cambridge University Press.
B i
Reference books in Wireless Communications
S. Benedetto and E. Biglieri, Principles of Digital Transmission: With Wireless Applications . Kluwer.
A. Goldsmith, Wireless Communications . Cambridge University Press.
U. Madhow, Fundamentals of Digital Communication. Cambridge
University Press. A. Molisch, Wireless Communications . Wiley.
J. Proakis and M. Salehi, Digital Communications (4th Edition). McGraw-Hill.
T. Rappaport, Wireless Communications: Principles and Practice (2nd Edition). Prentice-Hall.
D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cambridge University Press.
B i t M d l f di it l i ti t
Model of a digital communication system
CHANNEL MODEL The main subject of this course is the study of digital communications over a transmission channel.
To this purpose, it is useful to characterize the model of a digital communication system in order to get acquainted with its different constituent parts.
TOP LEVEL CLASSIFICATION
The model can be divided into three sections, as illustrated in
the following picture: 1 The user section 2 The interface section 3 The channel section
Basic concepts Model of a digital communication system
TX ENCODER MODULATOR
C H A
NN E L
C H A
NN E L
C H A
NN E L
D D W
ENCODER
Implements source encoding to limit the amount of transmitted data (for example, voice can be encoded at 4 kbit/s or sent at 64 kbit/s with conventional telephony).
Implements channel encoding to limit the effect of channel disturbances
MODULATOR
Converts the digital signal into a waveform to be transmitted over the channel
CHANNEL
Reproduces the transmitted waveform at the receiver
Its operation is affected by frequency distortion, fading, additive noise, and other disturbances
ENCODER
MODULATOR
CHANNEL
ENCODER
MODULATOR
CHANNEL
DEMODULATOR
Converts the received waveform into a sequence of samples to be processed by the decoder
DECODER
Implements channel decoding to limit the effect of the errors introduced by the channel
Implements source decoding
DEMODULATOR
Converts the received waveform into a sequence of samples to be processed by the decoder
DECODER
Implements channel decoding to limit the effect of the errors introduced by the channel
Implements source decoding
Basic concepts Band-pass signalling
A band-pass signal has spectral components in a limited range
of frequencies f ∈ (−f 2, −f 1) ∪ (f 1, f 2) provided that 0 < f 1 < f 2.
A certain frequency in the range (f 1, f 2) (usually the middle frequency) is called carrier frequency and denoted by f c.
The signal bandwidth is Bx = f 2 − f 1.
f
Bx
− f 1 f 1
− f 2 f 2
− f c f c
It is often convenient to represent band-pass signals as equivalent complex signals with low-pass frequency spectrum (i.e., including the zero frequency).
The analytic signal
A real band-pass signal x(t) can be mapped to a complex analytic signal x(t) by passing through a linear filter with transfer function 2u(f ) = 2 · 1f>0:
x(t) 2u(f ) x(t)
The indicator function 1A = 1 if A is true, 0 otherwise.
Summarizing:
The analytic signal is a complex representation of a real signal.
It is used to simplify the analysis of modulated signals. It generalizes the concept of phasor used in electronics.
The basic properties of the analytic signal derive from the Fourier transform.
The analytic signal (cont.)
If x(t) is a real signal, then its Fourier transform is a Hermitian function since:
X (f )∗ = ∞
= X (−f )
positive frequency (or negative frequency) part.
Since:
X (f ) = 2u(f )X (f ) = X (f ) + sgn(f )X (f ) X (f ) + j X (f ),
applying F −1 yields:
x(t) = x(t) + j x(t).
The signal x(t) is called the Hilbert transform of x(t):
x(t) x(t) ∗ 1
Hilbert transform (cont.)
Here, the Cauchy principal part of the integral has been taken, namely,
lim ε→0,T →∞
we have
Therefore,
B d i lli
Band-pass signalling
Assume that x(t) is a zero-mean stationary band-pass random process with bandwidth Bx and carrier frequency f c so that its power density spectrum is nonzero over the frequencies
f
We define the baseband complex envelope of x(t) as
x(t) = x(t) e− j 2πf ct
The complex envelope is sometimes called baseband equivalent signal.
Th l l ˜(t)
The complex envelope x(t)
Then, we derive the autocorrelation function and the power density spectrum of x(t):
Rx(τ ) = E
x(t + τ ) e− j 2πf c(t+τ )x∗(t) e j 2πf ct
= Rx(τ ) e− j 2πf cτ
=⇒ Gx(f ) = Gx(f + f c).
Then, the power density spectrum of x(t) is nonzero over thefrequencies f ∈ (−Bx/2, Bx/2), i.e., it is a baseband signal with bandwidth Bx/2.
In-phase and quadrature components
The real and imaginary parts of x(t) = xc(t) + j xs(t) arecalled in-phase and quadrature components of the signal.
They can be expressed in terms of the signal itself and of its Hilbert transform:
xc(t) =
R e[x(t)e− j 2πf ct] = x(t) cos(2πf ct) + x(t) sin(2πf ct)
xs(t) = I m[x(t)e− j 2πf ct] = x(t) cos(2πf ct) − x(t) sin(2πf ct)
The previous relationships can be inverted and yield:
x(t) = Re[x(t)e j 2πf ct] = xc(t) cos(2πf ct) − xs(t) sin(2πf ct)
x(t) = I m[x(t)e j 2πf ct] = xs(t) cos(2πf ct) + xc(t) sin(2πf ct)
x(t) consists in the following operation:
x(t) → x(t) cos(2πf ct).
The modulation of a couple of real signals xc(t) and xs(t)consists in the following operation:
[xc(t), xs(t)] → xc(t) cos(2πf ct) − xs(t) sin(2πf ct).
In the analytic signal domain, modulation can be represented as follows:
x(t) → x(t) = x(t)e +j 2πf ct.
x(t) → x(t) = x(t)e− j 2πf ct.
Correspondingly, in the real signal domain, demodulation can be represented by:
xc(t) = x(t) cos(2πf ct) + x(t) sin(2πf ct)
xs(t) = x(t) cos(2πf ct) − x(t) sin(2πf ct)
Frequency down-conversion = demodulation (cont.)
In a real system, demodulation can be implemented by observing that
MULTIPLICATION BY IN-PHASE CARRIER
= x(t) + x(t) cos(4πf ct + 2φ)
In other words, multiplication of the signal x(t) cos(2πf ct + φ) by the phase-coherent sinusoid 2 cos(2πf ct + φ) returns the superposition of
the modulating signal x(t);another modulated signal with carrier frequency 2f c.
Low-pass filtering with bandwidth Bx eliminates the modulated signal with carrier frequency 2f c.
Demodulator
The following picture illustrates the block diagram of a demodulator with input:
x(t) = xc(t) cos(2πf ct) − xs(t) sin(2πf ct).
x(t)
2 cos(2πf ct)
A probability space consists of three parts:
1 A set of all possible outcomes. 2 A set of events, F , which are sets of outcomes.
3 A probability function P : F → [0, 1], assigning a probability to every event.
The probability function is a normalized measure:
Basic concepts Probability
Topics on Probability (cont.)
1 A weighted sum of the probabilities of the outcomes in an
event E , if they are finite or countable:
P (E ) = ω∈E
In this case, outcomes are also events and have nonzero
probabilities. 2 An integral of the probability density function (pdf) over the
event E , if the outcomes are uncountable:
P (E ) = ω∈E dµ(ω).
In this case, outcomes are not events. Some technical assumptions on F are made in order that integration can be carried out (σ-algebra assumption).
P () = 1,
that is, the probability of the set of all possible outcomes is 1. Given two events A, B, we can build the union A ∪ B and the intersection A ∩ B:
A ∪ B = set of outcomes in A or in B.
A
and allows to define the conditional probability as
P (A | B) = P (A ∩ B)
P (B) .
Commonly, we write P (A, B) ≡ P (A ∩ B). Conditional probabilities satisfy the Bayes’ rule:
P (A | B) = P (A, B)
P (B) =
P (B) .
A useful result is the total probability law:
If the events Bi, i = 1, 2, . . . form a partition of (i.e., iBi = and Bi ∩ B j = ∅ for i = j), then:
P (A) =
P (A | Bi)P (Bi).
By the total probability law, one can obtain the conditional probabilities P (Bi | A) from the P (A | Bi):
.
Topics on Probability (cont.)
The above result finds application in the design of digital communication receivers where the event A represents the
received signal and the events Bi represent all the possible transmitted data in a given framework.
The probability of the union has also some special properties.
For illustration, we interpret events as two-dimensional regions and their probabilities as the areas of the regions.
A
B

P (A) ≤ P (A ∪ B) P (B) ≤ P (A ∪ B) P (A ∪ B) ≤ P (A) + P (B)
= P (A) + P (B) − P (A ∩ B)
The inequalities derive from the fact that the area of the union is always greater than or equal to the areas of each event.
p y ( )
Moreover, the sum of the areas is equal to the area of the union plus that of the intersection, which is counted twice. This yields the last inequality.
The previous results can be generalized to the case of m events:
Lower and upper union bounds
Given a set of events {A1, . . . , Am}, the following inequalities hold:
max 1≤i≤m
A discrete real random variable X is characterized by its probability distribution
pX (xn) = P (X = xn)
for n = 1, 2, . . . , N (where N may become infinity).
The expectation operator E[·] is defined by
E[φ(X )] = n φ(xn) pX (xn)
for an arbitrary function φ(·).
Random variables (cont.)
( )
Since the expected value of every constant is the constant itself, we obtain by definition:
E[1] = n
pX (xn) = 1.
The mean of X is µX = E[X ] =
n xn pX (xn).
The second moment of X is µ (2) X = E[X 2] =
n x2
n pX (xn).
The variance of X is σ2X = E[(X − µX )2] = µ (2) X − µ2X .
The square root of the variance is called standard deviation.
Random variables (cont.)
( )
Continuous random variables are characterized by a pdf f X (x) defining the expectation operator as
E[φ(X )] =
φ(x)f X (x)dx,
where I is the support of the random variable, i.e., the set of values where f X (x) > 0.
Again, the expected value of every constant is the constant itself, so that:
E[1] = I f X (x)dx = 1.
This property holds for all pdf’s.
A complex random variable Z = X + j Y has mean
µZ = E[X ] + j E[Y ]
and variance
2
Y )
Basic concepts Gaussian random variables
Gaussian random variables
We will be particularly concerned with Gaussian random variables whose distribution is given by
f X (x) = 1
√ 2πσ2 e−(x−µ)2/(2σ2)
and denoted by N (µ, σ2).
The parameters µ and σ are the mean and the standard deviation of a Gaussian random variable with distribution N (µ, σ2).
Gaussian random variables (cont.)
We will often be interested in calculating the probability
P ( N (µ, σ2) > x), i.e., the probability that a Gaussian random variable with mean µ and standard deviation σ exceeds the real value x.
These probability can be calculated by using the function
Q(x) (referred to as Q-function), which is the countercumulative probability distribution function of the normalized Gaussian random variable N (0, 1).
The Q-function is defined as:
Q(x) P ( N (0, 1) > x) = ∞ x
1√ 2π
By using the Q-function we can see that
P ( N (µ, σ2) > x) = P ( N (0, σ2) > x − µ)
= P N (0, 1) > x − µ
σ = Q
x − µ
.
The Q-function cannot be calculated in terms of elementaryfunctions (such as exp, ln, and trigonometric functions).
Q(x) ≈ e−x2/2.
approximations, which are plotted as the red ( e−x 2/2√
2πx ) and
green (e−x2/2) dashed curves, respectively.
We can see that the approximation of the red curve is better than 10% for x ≥ 3.
0.2
0.4
0.6
0.8
1.0
Basic concepts Complex Gaussian random variables
Complex Gaussian random variables
We will also consider complex Gaussian random variables with a special property, namely, that of having zero mean and independent and identically distributed (iid) real and
imaginary parts.
In other words, if Z = X + j Y , we assume that X ∼ N (0, σ2/2) and Y ∼ N (0, σ2/2), where X and Y are statistically independent.
Complex Gaussian random variables (cont.)
expressions:
= 1√
2/σ2 .
The concept can be extended to vectors of complex Gaussian random variables.
Complex Gaussian random variables (cont.)
If z = (Z 1, . . . , Z n)T is a vector of complex Gaussian random
variables with zero mean and covariance matrix Σz = E[zzH ], then the pdf of z can be expressed as follows:
f z(z) = det(πΣz)−1e−z H Σ −1 z z.
In the special case of iid components of z, corresponding to Σz = σ2I n, the pdf simplifies to
f z(z) = (πσ2)−ne−z 2/σ2 ,
which is the product of the individual (marginal) pdf’s of the Z i’s: (πσ2)−1e−|zi|2/σ2 .
Basic concepts Signal spaces
Signal spaces are linear (or vector) spaces built upon the concept of Hilbert space, i.e., finite or infinite-dimensional complete inner product spaces.
The elements of a signal space are real or complex signals x(t)
defined over a support interval I , for example I = (0, T ).
The inner product of two elements (signals) x and y is defined as
(x, y) = I x(t)y∗(t)dt. (3)
x (x, x)1/2. (4)
Signal spaces (cont.)
I d i f h C h S h i li
| (x, y)
.
If |(x, y)| = x · y, then the two signals are proportional, i.e., y(t) = αx(t) for some α ∈ C.
A signal x(t) ∈ H ⊂ L2( I ) if x is finite.
The squared norm of a signal x(t) is the signal energy:
E (x) (x, x) =
A finite-dimensional signal space H ⊂ L2( I ) is identified
through a base [ψn(t)]N n=1. The elements of a base have the following orthogonality property:
(ψm, ψn) = 1 if m = n and 0 otherwise.
A signal x(t) ∈ H ⊂ L2( I ) can be represented by the expansion
x(t) = N
n=1
Th ffi i t i thi i b l l t d b
The coefficients xn in this expansion can be calculated by
xn = (x, ψn) = I x(t)ψn(t)∗dt.
In many cases, a signal space H is defined as the set of all possible linear combinations of a set of signals:
H = L(s1, . . . , sM ) = {x(t) = α1s1(t) + · · · + αM sM (t),
∀ (α1, . . . , αM ) ∈ C M }.
(s1, . . . , sM ) is called linear span of s1(t), . . . , sM (t).
In general, the signal set {s1, . . . , sM } is not a base, but a base can be found by using the Gram-Schmidt algorithm.
Th G S h idt l ith fi d b (ψ )N f
The Gram-Schmidt algorithm finds a base (ψn)N n=1 of
H = L(s1, . . . , sM ) by the following set of iterative equations:
For k = 1, . . . , n :
(sk, ψi)ψi (projection step)
At every projection step such that dk = 0 the corresponding ψk is not assigned and not accounted for in the remaining
steps.
The number of signals in the base is the number of dimensions of H.
The Gram Schmidt algorithm works since at every step the
The Gram-Schmidt algorithm works since, at every step, the signal dk(t) is orthogonal to all previously generated signals
ψi(t), i = 1, . . . , k − 1. In fact,
(dk, ψi) =
The projection of a signal x(t) ∈ H over the subspace
Y = L(ψ1, . . . , ψN ) ⊂ H is a signal xY (t) with the followingproperties:
It can be expressed through the base of Y as follows:
xY (t) = N
xY = arg min y∈Y
x − y.
Proof The previous property can be proved by assuming y(t) =
N n=1 ynψn(t). Then,
x
− y
= x2 − 2Re[(x, y)] + y2
= x2 − 2 N
The minimum of x − y 2
is obtained by minimizingδ n |yn|2 − 2Re[(x, ynψn)] for all n = 1, . . . , N .
Since (x, ynψn) = y∗n(x, ψn), we have
δ n =
= |yn|2 − 2|yn||(x, ψn)| cos[∠(x, ψn) − ∠yn]
Therefore, for a given |yn|, the minimum is attained when the argument of the cosine is 0.
In that case, δ n = |yn| 2
− 2|yn||(x, ψn)| and the optimumphase of yn is ∠yn = ∠(x, ψn).
y n|
is straightforward and gives |yn| = |(x, ψn)|. In this case, δ n = −|(x, ψn)|2.
Summarizing, the yn minimizing x − y2 over y ∈ Y is:
yn = |yn|e j∠yn = |(x, ψn)|e j∠(x,ψn) = (x, ψn).
The minimum is:
min y∈Y
n=1
An alternative approach to the Gram-Schmidt algorithm is
based on standard linear algebra methods. It can be noticed that the GS algorithm leads to expressions of the orthogonal base signals of the following type:
ψi(t) = j≤i
C ijs j(t).
These equations can be written in matrix form as follows:
ψ = Cs,
where ψ = (ψ1(t), . . . , ψN (t))T and s = (s1(t), . . . , sM (t))T .
C is a lower triangular matrix.
(ψ,ψT ) = (Cs, sT C T ) = C (s, sT )C T .
The lhs is the identity matrix I N since (ψi, ψ j) = δ ij.
The rhs can be written as C ΣsC T where the matrix Σs is the Gram matrix of the signals in s(t).
The elements of Σs are (Σs)ij = (si, s j).
If a signal si(t) is linearly dependent from the signals
s1(t), . . . , si−1(t), the corresponding row and column of Σs
must be eliminated because linear combinations of the previous row and columns.
A reduced Cholesky factorization can be applied to Σs leading
E ΣsE T = LLT
where the N × M matrix E removes the redundant rows and columns and L is a square nonsingular lower triangular matrix.
The previous result leads to C = L−1E . As an example, if we have four signals but only s3 is a linear combination of s1, s2,
E = 1 0 0 0
0 1 0 0 0 0 0 1
The matrix product E (s1, s2, s3, s4)T = (s1, s2, s4)T
eliminates s3.
Basic concepts Problem set 2
Problem set 2
1 Calculate the mean and variance of the discrete random
variable X with probability distribution pX (1) = 0.5, pX (2) = 0.25, pX (4) = 0.25.
2 Calculate the mean and variance of the continuous random variable X with probability distribution f X (x) = e−x
1x>0.
3 Calculate the mean and variance of the continuous randomvariable X with probability distribution f X (x) = 0.5 · 1|x|<1.
4 Calculate the probability P (X > a) for a Gaussian random variable X ∼ N (µ, σ2).
5 Show that the variance identity holds: σ2
X = E[X 2]
− E[X ]2.
6 Assume support interval I = (0, 1) from now on. Let H be the linear span of cos(2πt) and sin(2πt). Determine if the signal cos(2πt + π/4) belongs to H.
Problem set 2 (cont.)
7 Given the signals s1(t) = u(t) − u(t − 0.6),
s2(t) = u(t − 0.4) − u(t − 1), and s3(t) = u(t) − u(t − 1),apply the Gram-Schmidt algorithm to find a base of L(s1, s2, s3) [u(t) = 0 for t < 0 and 1 for t > 0 is the unit step function].
8 Given the signals s1(t) = cos(2πt) and s2(t) = sin(3πt), apply
the Gram-Schmidt algorithm to find a base of L(s1, s2).
9 Check Schwarz’s inequality for the signals s1(t) = u(t) − u(t − 0.6), s2(t) = u(t − 0.4) − u(t − 1), and s3(t) = u(t) − u(t − 1).
10 Let Y = L(s1 = sin(πt), s2 = cos(3πt)). Find the projection of x(t) = u(t) − u(t − 1) over Y and calculate x − xY 2
(notice that s1 and s2 are orthogonal).
11 Apply the matrix GS algorithm to find the matrix C determining the orthogonal base to the set of signals
s1 = 10<t<.36, s2 = 1.36<t<1, s3 = 10<t<1, s4 = 10<t<.5.
Digital modulations over the AWGN channel
Outline
Digital modulations over the AWGN channel
Section Outline
Additive White Gaussian Noise (AWGN) Channel Linear digital modulation Digital receiver design Baseband digital modulation Band-pass digital modulation Signal detection Error probability Standard digital modulations Problem set 3
Power density spectrum of digital modulated signalsComparison of digital modulations Problem set 4
Digital modulations over the AWGN channel Additive White Gaussian Noise (AWGN) Channel
AWGN channel
Thi h l d l i ifi d b th ti
y(t) = Ax(t) + z(t) (6)
x(t) and y(t) are the channel input and output signals.
z(t) is the zero-mean additive white Gaussian noise process. It has autocorrelation function and power density spectrum:
Rz(τ ) = E[z(t + τ )z(t)] = N 02 δ (τ ),
Gz(f ) = N 0
Digital modulations over the AWGN channel Linear digital modulation
Linear modulations
x(t;a) = N
n=1
anψn(t) (7)
where
N is the number of dimensions of the modulation scheme. The vector a = (an)N n=1 represents a modulation symbol vector and is taken from a finite set A = {α1, . . . ,αM }. A is called modulation alphabet or signal constellation. ψn(t) is the nth shaping pulse of the modulated signal. We assume that each ψn(t) = 0 only for t ∈ (0, T ). We also assume that (ψm, ψn) = δ mn = 1 if m = n and 0 otherwise (Kronecker’s delta).
Linear modulations (cont.)
The signal x(t;a) allows us to send one symbol vector every T time units so that T is called symbol time, symbol interval, or signalling interval.
The signal x(t;a) belongs to the Hilbert space H generated by all possible linear combinations of the base signals (ψn)N
n=1.
Linear modulations (cont.)
yH(t) = N
n=1
ynψn(t) = N
n=1
(Aan + zn)ψn(t). (9)
In this expression we find the nth received signal and noise
components: yn = (y, ψn) =
z(t)ψn(t)dt
Here, zn, is a Gaussian random variable.
A receiver calculating the vector y = (y1, . . . , yN ) from y(t) is called correlation receiver.
Digital modulations over the AWGN channel Digital receiver design
Receiver design
The goal of a digital receiver is to recover the transmittedsymbol vector a from the received signal y(t).
The correlation receiver projects y(t) on H to obtain (9) and outputs the coefficients yn = Aan + zn for n = 1, . . . , N .
In the absence of noise, the correlation receiver outputs a scaled version (by the channel gain A) of the transmitted symbol vector.
When noise is present, the receiver guesses which symbol vector from
A was transmitted with the goal of minimizing
the error probability. This process is called detection or decision.
Receiver design (cont.)
The correlation receiver can be interpreted as a matched filter by observing that:
yn =
T
where we defined the impulse response of the matched filter
as hn(t) = ψn(T − t).
Sequential receiver design
symbol times.
x(t;a0, . . . ,aL) = L
y(t) = A
Sequential receiver design (cont.)
The matched filter structure lends itself to a sequential implementation accounting for the transmission of successive
modulated symbols in time.
The bank of matched filters receiver is illustrated as follows:
y(t) ψ1(T
The output of the nth matched filter at time t = (i + 1)T is given by:
=
∞ −∞
+ T
The previous expression holds since the shaping pulses are orthogonal and time-limited to the signalling interval (0, T ). Therefore,
T
0 ψm(t2 + iT − jT )ψn(t2)dt2 = δ m,nδ i,j.
This is just an extension of the digital receiver operation described over the first signalling interval (0, T ).
75 c Prof Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Baseband digital modulation
Baseband digital modulation
If the symbols an and the shaping pulses ψn(t) are real,
( ) x(t;a) in (7) represents a baseband digitally modulated
signal. A simple example of digital modulation is the binary antipodal modulation with alphabet A = {±1}.
If a = (+1, +1, −1, −1, −1, +1, −1, +1, +1, −1), the signal is
illustrated as follows. x(t)
+1 +1 −1 −1 −1 +1 −1 +1 +1 −1
g(t)
T
Digital modulations over the AWGN channel Baseband digital modulation
Baseband digital modulation (cont.)
x(t)
t
+1 +1 −1 −1 −1 +1 −1 +1 +1 −1
Baseband digital modulations are represented in a one-dimensional space.
Digital modulations over the AWGN channel Band-pass digital modulation
Band-pass digital modulation
x(t; a) = a · g(t),
where g(t) is a real baseband signal of bandwidth lower than f c and we assume that a
∈ A =
x(t; a) = Re
= Re(a) · [g(t) cos(2πf ct)] + I m(a) · [−g(t) sin(2πf ct)],
Band-pass digital modulation (cont.)
The signal x(t; a) can be interpreted as a two-dimensional linear modulation.
In fact, it can be represented as a linear combination of:
ψ1(t) = g(t) cos(2πf ct)
ψ2(t) = −g(t) sin(2πf ct)
with coefficients Re(a) and I m(a).
Now, assume that the bandwidth of g(t) is Bg < f c, i.e., its Fourier transform G(f ) =
F [g(t)] is equal to 0 for every
f ≥ f c. Then, G2(f ) F [g(t)2] = G(f ) ∗ G(f ) has bandwidth 2Bg < 2f c.
Band-pass digital modulation (cont.)
Using the above result,
g(t)2 sin(4πf ct)dt = 0.
Thus, if g2 = 2, the signals (ψ1, ψ2) are orthogonal and form the base of a two-dimensional signal space H provided that the bandwidth of g(t) is smaller than the carrier frequency f c.
Digital modulations over the AWGN channel Signal detection
Detection of transmitted symbols
The receiver outputs an estimate of the transmitted symbol a
∈ (0, T ).
The first stage of the receiver converts y(t) into the vector
y = Aa + z
We define a generic decision rule (or detection rule):
a(y) = (a1(y), . . . , aN (y)). (11)
a(y) maps H
.
The decision rule can be optimized according to some goodness criterion.
Typically, the goal is minimizing the error probability.
Optimum digital receiver
where
P (αm) is the a priori probability of transmitting αm. P (e | αm) is the probability of error conditioned on the transmission of αm.
We notice that
P (e | αm) = P a(y = Aαm + z) = αm,
i.e., the probability that the decision rule returns a symbol different from the transmitted one.
Optimum digital receiver (cont.)
It is plain to see that minimizing the (average) error probability is equivalent to maximizing the (average) probability of correct decision P (c) since P (c) = 1 − P (e).
Let us define
The pdf of y given the transmitted symbol α: f (y
| α).
Notice that there is a one-to-one correspondence between the
set of decision regions and the decision rule.
Since the decision rule a(y) is a well defined (i.e., single-valued) function for all y ∈ H = RN (the signal space), the decision regions do not intersect and their union fills H itself:
M m=1
Now, we can write the probability of correct decision as a f f h b b l ( ) f ( ) d h
function of the a priori probabilities P (αm), f (y
| α), and the
decision rule. Since a(y) = αm for y ∈ Rm, we get:
P (c) = M m=1
P (αm)P (
Maximizing P (c) requires to maximize the integrand in (13), which can be accomplished by selecting the symbol α ∈ A
aopt(y) = arg max α∈A P (α)f (y | α). (14)
,
the optimum decision rule is equivalent to maximizing the a posteriori probability P (αm
| y).
Thus, the optimum decision rule is called maximum a-posteriori (MAP) decision.
When transmitted symbols are equiprobable, i.e., P ( ) 1/M h MAP l d i
P (αm) = 1/M , the MAP rule reduces to a maximum
likelihood (ML) rule:
f (y | αm)
This name comes from the name of the functions f (y | αm) (likelihood functions in radar theory).
The decision regions can be represented as follows:
R m = {y : P (αm)f (y | αm) > P (αn)f (y | αn) ∀ n = m} MAP
{y : f (y | αm) > f (y | αn) ∀ n = m} ML
89 P f Gi i T i WIRELESS COMMUNICATIONS
Special case: The AWGN channel
Proposition. The additive noise components of an AWGN
channel are iid Gaussian random variables with zero mean andvariance N 0/2.
Proof. We have, by definition,
zn = T
0 z(t)ψn(t)dt
since the additive noise random process has zero mean.
90 P f Gi i T i WIRELESS COMMUNICATIONS
Special case: The AWGN channel (cont.)
Moreover,
=
= T
0
T
0
=
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In other words, different components of the noise vector z are uncorrelated (and hence independent since Gaussian), and
h h i N /2
each one has variance N 0/2.
As a result, the conditional pdf of the received vector y is
f (y | α) = f z(y − Aα)
= (πN 0)−N/2e−y−Aα2/N 0 . (15)
It is worth noting that the joint pdf (15) depends only on the distance of the received signal from the transmitted one scaled by the channel gain A.
Using (15), the logarithms of the likelihood functions are readily obtained as follows:
ln f (y | αm) = −N
2 ln(πN 0) − 1
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Special case: The AWGN channel (cont.) Since these functions depend on a distance, they are called decision metrics.
Th MAP d ML d i i l b d i f
The MAP and ML decision rules can be expressed in terms of decision metrics for the AWGN channel as follows:
a(y) =
MAP
arg minαm y − Aαm2 ML
As a result, the ML decision rule for the AWGN channel is often referred to as minimum distance decision.
The decision regions on the AWGN channel can be t d f ll
Rm =
{y : y − Aαm2 − N 0 ln P (αm) < y − Aαn2 − N 0 ln P (αn) ∀ n = m} MAP
{y : y − Aαm2 < y − Aαn2 ∀ n = m} ML
Special case: The AWGN channel (cont.) Here is an example of minimum distance decision regions (aka Voronoi regions):
Here is another (asymmetric) example of minimum distance decision regions:
Error probability
Using the optimum decision rule (14) and assuming that αm
has been transmitted, we can see that the symbol decision is
.
= m.
Notice that all the pairwise error events contain the conditioning clause “| αm”. This clause is equivalent to the assumption that αm was transmitted.
Digital modulations over the AWGN channel Error probability
Error probability (cont.)
Thus, the error probability, conditioned on the transmission of αm, is given by
αm
The above expression of the error probability is too complex
Pairwise Error Probability
P (αm → αn)

can be calculated very simply!
Thus, lower and upper bounds are used to approximate P (e|αm).
Error probability (cont.) Applying the bounds (1) to the conditional probabilities P (e|αm), we obtain
M m=1
P (αm) n=m
Assuming the MAP decision rule over the AWGN channel, andletting A = 1, the PEPs are given by
Pairwise error probability
+ N 0 ln[P (αm)/P (αn)]√ 2N 0 αm −αn .
(17)
Equation (17) is based on the Q-function (2) and will bederived in detail in a problem.
Error probability of binary modulations
The inequalities (16) yield the exact error probability in the case of for binary modulations (M = 2):
P (e) = P (α1)P (α1 → α2) + P (α2)P (α2 → α1)
= P (α1)Qα1 −α2 2
+ N 0 ln[P (α1)/P (α2)]√ 2N 0 α1 −α2
+ P (α2)Q
.
Error probability of binary modulations (cont.)
With equiprobable signals i e P (αm) = 1/M inequalities
With equiprobable signals, i.e., P (αm) 1/M , inequalities (16) yield:
1
M
M
≤ 1
M
High SNR approximation
In most situations, one is mostly interested to the high SNR (and then low N 0) case.
Since the Q function decreases very quickly, we can keep in the bounds only the terms with minimum distance:
dmin min m=n
and disregard the others which are very small.
To be conservative, we use the upper bound to P (e) and obtain this approximation:
P (e) ≈ N minQ dmin√ 2N 0 (19)
where N min = 1 M
m
.
Standard plots of P (e)
In most cases, the error probability is plotted in log-log
graphics where the abscissa is the energy ratio E b/N 0 expressed in dB on a linear scale; the ordinate is the error probability in logarithmic scale.
E b is the energy per information bit. For the uncodedmodulations considered, we have:
E b = E s log2 M
E s = m
Bit error probability
In some applications, it is better considering the bit error probability P b(e) than the symbol error probability P (e).
Typically, modulation symbols are assigned to bit vectors (bit mapping), so that a symbol error corresponds to having a received bit vector different from the transmitted one.
The bit error probability is the average number of errors in the
received bit vector divided by the vector size:
P b(e) = E[N b]
log2 M ,
where N b
denotes the number of bit errors. Of course, P b(e) depends on the bit mapping.
Assuming high SNR, most errors occur between minimum distance symbols.
Bit error probability (cont.) For some modulations it is possible to select a bit mapping such that all minimum distance symbols differ in their corresponding bit vectors at only one position (Gray
00 01 11 10
In this case, with high probability, every symbol error corresponds to 1 bit error (out of log2 M transmitted bits).
In general, we have N b = 0 with probability 1 − P (e) and N b ≥ 1 with probability P (e). If the probability that N b > 1 is very small, then
P b(e) ≈ 0 · (1 − P (e)) + 1 · P (e)
log2 M =
P (e)
log2 M .
Digital modulations over the AWGN channel Standard digital modulations
PAM = Pulse Amplitude Modulation
The alphabet of M -PAM is = (2m M 1) M m=1.
• • • • • • • • −
P (e) = 2 M − 1
M
QAM = Quadrature Amplitude Modulation
= (2m √
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
QAM = Quadrature Amplitude Modulation (cont.)
P (e)
PSK = Phase Shift Keying
The alphabet of M -PSK is A = {√ E se j (2m−1)π/M }M
m=1.
•
••
•
•
• •
•
PSK = Phase Shift Keying (cont.)
The error probability of M -PSK is:
P (e) ≈ 2 Q
Orthogonal modulations
The alphabet of an orthogonal modulation consists of M vectors in RM with a single nonzero coordinate equal to
√ E s.
For example, a quaternary orthogonal modulation is represented by the following four signals:
α1 = (
E s, 0, 0),
E s).
The error probability of an M -ary orthogonal modulation is:
P (e) ≈ (M − 1)Q
. (23)
Orthogonal modulations (cont.)
1 Pulse position modulation (M -PPM): Given the signal pulse ψ(t), the modulated signals are:
xm(t) = √
M ψ(M t − (m − 1)T ), (24)
i.e., ψ(t) is contracted in time to (0,T/M ) and shifted by (m − 1)T /M .
2 Frequency shift keying (M -FSK):
xm(t) = √
where f c · T and f · T are integer numbers.
Asymptotic comparison of digital modulations
Consider two digital modulation schemes with approximate union bounds to the error probability
for i = 1, 2.
The asymptotic behavior of the error probability (when E b/N 0 is very large) is dominated by the Q-function term and can be approximated by
P (e) ≈ e−(βi/2)E b/N 0
for i = 1, 2 (we used Q(x) ≈ exp(−x2/2)).
If β 1 > β 2, the first modulation is better than the second since its error probability is smaller at the same E b/N 0.
Problem set 3
2 Derive the union bound approximation (19) along with the
expression of N min by keeping only those terms from the upper bound
1
M
M
P (αm → αn)
corresponding to minimum distance errors, i.e., such that αm −αn = dmin.
3 Derive the error probability in (20).
4
Derive the error probability in (21). 5 Derive the error probability in (22).
6 Check the orthogonality of the signals in (24) and (25).
Digital modulations over the AWGN channel Problem set 3
Problem set 3 (cont.) 7 Derive the error probability in (23).
8 Calculate the error probability of the 32-QAM modulation characterized by the following signal set:
−5
−3
−1
+1
+3
+5
9 Find the error probability of the binary modulation whose signals are
s1(t) = 10<t<0.8, s2(t) = 10.4<t<1, T = 1.
Hint: it is not necessary to represent the signals in a normalized signal space, only the average energy and the
distance are required.
10 Consider the quaternary modulation obtained by using the following four signals:
s1(t) = A[u(t) − u(t − T )]
s2(t) = A[u(t) − u(t − T /4) + u(t − T /2) − u(t − 3T /4)]
s3(t) = A[u(t) − u(t − T /4) − u(t − T /2) + u(t − 3T /4)]
s4(t) = A[u(t) − 2u(t − T /2) + u(t − T )]
Calculate i) the average energy per bit E b, ii) the minimum distance d2
min, and iii) the average symbol error probability (high-SNR approximation) in the form αQ(
βE b/N 0).
11 Calculate the error probability of a 4-PSK signal set assuming that the receiver has a constant phase offset that rotates the
decision regions b an angle
decision regions by an angle α. 12 Calculate the error probability of an octonary signal set whose
signals are located over two concentric circles with rays 1 and√ 0.5 +
√ 1.5. The signals are equally spaced over each circle
and have a phase offset of π/4 radians between the corresponding signals over different circles.
13 Calculate the error probability of the digital modulation based on the following four signals:
sm(t) = sin 5π2T t − (m − 1) T 5 1|t−mT/5|≤T/5
for m = 1, 2, 3, 4.
Digital modulations over the AWGN channel Power density spectrum of digital modulated signals
Power density spectrum of digital modulations
The power density spectrum of x(t) = n anψ(t − nT ), where an is a wide-sense stationary sequence with
autocorrelation function Ra(p) E[an+pa∗n] can be expressed
autocorrelation function Ra( p) = E[an+ pa∗n], can be expressed as the product of two terms:
Power density spectrum: Gx(f ) = S a(f ) · Gψ(f )
S
Gψ(f ) 1
T |Ψ(f )|2 (pulse spectrum)
In many circumstances, the bandwidth of a digital signal is approximated by an expression depending only on the signalling interval T .
Shannon bandwidth
A common approximation to the bandwidth of a digital signal
is the Shannon bandwidth:
2T
where N d is the signal space dimension and T is the symbol
interval. It can be shown that this approximation is very good when the number of dimensions is large.
However, even with N d = 1, the bandwidth overhead is
limited for suitably chosen pulses, as illustrated in the following example.
Bandwidth of antipodal signals
Consider a binary PAM signal with iid equiprobable symbols an
∈ {± 1
2sin(πt/T )1t∈(0,T ) sinusoidal pulse
The power density spectrum is given by Gx(f ) = |Ψ(f )|2/T , and (check as homework):
Ψ(f ) = T sinc(f T )e− j πfT square pulse
−√ 2 e− j πfT
sinusoidal pulse
Bandwidth of antipodal signals (cont.)
The following diagram plots the fractional power content
W
−∞ Gx(f )df
versus the normalized bandwidth W T (normalized with respect to the signalling rate 1/T ).
This quantity represents the fraction of power of the digital modulation signal contained in the bandwidth W with respect
to the total power.
Bandwidth of antipodal signals (cont.)
0.8
1
0.2
0.4
0.6
0.8
WT
Bandwidth of antipodal signals (cont.) One way to limit the bandwidth occupation of a digital modulation signal consists of extending the duration of the modulation pulse beyond the signalling interval (0, T ).
When the signalling pulse is limited to the signalling interval
When the signalling pulse is limited to the signalling interval, the modulation signal is called full response. When its duration exceeds T , it is called partial response.
Stretching in time the signalling pulse by a factor β
corresponds to an equivalent stretching in the frequency domain by the inverse of β :
ψ(t) → ψ(t/β ) ⇒ Ψ(f ) → Ψ(βf ) ⇒ η(W ) → η(βW ).
The price to be payed is related to the generation of intersymbol interference.
Digital modulations over the AWGN channel Comparison of digital modulations
Key parameters
The performance of different modulation schemes is described by three system parameters:
1 Error probability (symbol or bit).
2 Spectral efficiency, i.e., the ratio between the bit rate Rb and
Spectral efficiency, i.e., the ratio between the bit rate R and the occupied bandwidth W .
3 The signal-to-noise ratio E b/N 0.
For N d-dimensional signal sets, the occupied bandwidth is approximately equal to the Shannon bandwidth
W sh = N d · 1
2T =
ηb Rb
W sh =
2log2 M
N d .
Spectral efficiency
The spectral efficiency η grows slowly (logarithmically) with the constellation size M and decreases rapidly (linearly) with the number of dimensions N d.
For a fixed M , PAM modulations have higher spectral efficiency than orthogonal modulations. Therefore,
PAM modulations are used in channels with limited bandwidth (bandwidth limited channels) and high power.
Orthogonal modulations are used in channels with limited power (power limited channels) and large bandwidth.
Shannon’s bound
Shannon’s theorem yields the maximum bit rate that can be
sustained with arbitrarily low error probability by an
sustained with arbitrarily low error probability by an N d-dimensional digital modulation with symbol interval T over an AWGN channel:
Rb =
N . (26)
Here, S is the received power, N is the noise power, and S/N is called signal-to-noise ratio.
We assume that the signal bandwidth is W sh = N d/(2T ).
Shannon’s bound (cont.) Since the noise power is N = N 0W sh and the signal power is S = E b/T b = RbE b, (26) can be written as:
Rb ≤ W sh log21 + RbE b WshN0.
≤ g + W shN 0 Since the spectral efficiency is ηb = Rb/W sh, we obtain:
ηb ≤ log2 1 + ηb
N 0 ≥ 2ηb − 1
Shannon’s bound (cont.)
Problem set 4
1 Derive the power density spectrum formula Gx(f ) = S a(f ) · Gψ(f ) for the signal
(t)
where
an is a wide-sense stationary sequence with autocorrelationfunction Ra( p) = E[an+ pa∗n];
S a(f )
Hint: Consider the (randomly delayed and stationary) signal
x(t − Θ), with Θ uniformly distributed in (0, T ), and calculate the Fourier transform of its autocorrelation function to obtain the power density spectrum.
2 Calculate the power density spectrum of the signal (27)
assuming that the symbols an are uncorrelated with mean µa d i 2
a.
3 Calculate the power density spectrum of the signal (27) assuming that the transmitted symbols an have zero mean and correlation Ra(m) = ρ|m| (where ρ
∈ (0, 1)), ψ(t) has
unit energy, and the average signal power is P .
4 Calculate the power density spectrum of (27) assuming that the transmitted symbols are iid and taken from a 4-PSK signal set with probabilities (0.7, 0.1, 0.1, 0.1).

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