2 Power Spectal Density (PSD) and Energy Spectral Density (ESD)
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Digital Communications
power of this signal are calculated using (1.75)
and (1.76) as follows:
∞
E=
A2 e − 2αt dt =
0
A2
2α
considered, which is infinite. Therefore, this
signal having infinite energy, is classified as
a power signal.
A2 1 − e − 2αT
0 1.2.2 Autocorrelation Function and
T
T ∞ 2α
T
0
Spectral Density
(1.81)
ESD and PSD are employed to describe the disHence, this is an energy signal. However, if tribution of, respectively, the signal energy and
we assume A = 1 and α = 0 in (1.80), then
the signal power over the frequency spectrum.
ESD/PSD is defined as the Fourier transform
E ∞, P = 1
u t = lim s t
of the autocorrelation function of an energy/
α 0
A 1
(1.82) power signal. [4][5][6][7]
T
1
∞T
A2 e − 2αt dt = lim
P = lim
which shows that a unit step function is a power
signal.
Now consider the following signal:
t
st =
−1 4
t ≥ t0 > 0
0
elsewhere
(1.83)
Its energy and power are found to be
∞
t
E=
−1 2
dt = 2 t
t0
1
P = lim
T ∞ T
= lim 2
T
∞
t0 + T
t
∞
t0
−1 2
∞
0
∞
−∞
∞
(1.84)
dt
−∞
s∗ t s t + τ dt
s t s∗ t − τ dt, − ∞ < τ < ∞
(1.87)
t0
T + t0 − t0
T
0
(1.85)
Since this function is periodic with a period
T0 = 1/f0, its power can be determined by integration over one period:
T0
Autocorrelation of a function s(t) provides a
measure of the degree of similarity between
the signal s(t) and a delayed replica of itself.
The autocorrelation function of an energy signal s(t) is defined by
=
sT0 t = Acos w0 t + θ
1
T0
Energy Signals
Rs τ =
This signal is neither a power signal nor an
energy signal.
Finally we consider a cosine function with
amplitude A:
P=
1.2.2.1
A2 cos2 w0 t + θ dt =
A2
2
(1.86)
The energy of this signal is given by the
product of its power and the number of periods
Note that τ, that varies in the interval (−∞,∞),
measures the time shift between s(t) and its
delayed replica. The sign of the time shift τ shows
whether the delayed version of s(t) leads or lags
s(t). If the time shift τ becomes zero, then s(t) will
overlap with itself and the resemblence between
them will be perfect. Consequently, the value of
the autocorrelation function at the origion is equal
to the energy of the signal:
∞
E = Rs 0 =
2
st
dt
(1.88)
−∞
It is logical to expect that the resemblence
between a signal and its delayed replica to
decrease with increasing values of τ. The properties of the autocorrelation function of a real
valued energy signal may be listed as follows:
17
Signal Analysis
a. Symmetry about τ = 0:
the convolution of r(t) and s(t), given by
(1.47), shows that
Rs τ = Rs − τ
(1.89)
b. Maximum value occurs at τ = 0:
Rs τ ≤ Rs 0 = E,
Rs τ = Rss τ = s t
τ
(1.90)
c. Autocorrelation and ESD are related to each
other by the Fourier transform:
Rs τ
Ψs f
(1.91)
where ESD is defined in (1.14) as the variation of the energy E of s(t) with frequency.
One can prove the relationship given by
(1.91) by taking the Fourier transform of
the autocorrelation function of an energy
signal, given by (1.87):
Ψ s f = ℑ Rs τ =
∞
=
=
=
Rs τ e
∞
−∞
s∗ t dt
−∞
∞
−∞
e − jwτ dτ
−∞
∞
∞
∞
−∞
s∗ t e jwt dt
dτ
s∗ t s t + τ dt
∞
s v e − jwv dv
−∞
S∗
f
S f
2
(1.92)
Example
1.13 Convolution
Versus
Correlation.
In order to discover the similarity between the
correlation and convolution, we first define the
cross correlation of two energy signals r(t) and
s(t) as follows:
Rrs τ =
∞
−∞
∗
r t s t − τ dt =
∞
−∞
(1.94)
s∗ − t
For real-valued functions with even symmetry, that is, for s(t) = s(−t), auto and cross
correlation functions are the same as the
convolution:
Rrs τ = r t
st
Rs τ = s t
st
1.2.2.2
for s − t = s t
(1.95)
Power Signals
The autocorrelation function of a power signal
s(t) is defined by
s t + τ e − jwτ dτ
−∞
= S f
− jwτ
s∗ − t
Rrs τ = r t
∗
s t r t + τ dt
(1.93)
which reduces to (1.87) for Rs τ = Rss τ .
Comparison of (1.93) with the expression for
1
∞T
Rs τ = lim
T
T 2
−T 2
s∗ t s t + τ dt
(1.96)
If the power signal is periodic with a period
T0, then the autocorrelation can be computed by
integration over a single period:
Rs τ =
1
T0
T0 2
− T0 2
s∗ t s t + τ dt
(1.97)
The value of the autocorrelation function at
the origin is equal to the power of s(t):
1
∞T
P = Rs 0 = lim
T
T 2
st
2
dt
(1.98)
−T 2
The properties of the autocorrelation function of a real-valued power signal s(t) are given
by
a. Symmetry about τ = 0:
Rs τ = Rs − τ
(1.99)
b. Maximum value occurs at τ = 0:
Rs τ ≤ Rs 0 = P,
τ
(1.100)
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Digital Communications
c. Autocorrelation and power spectral density
(PSD) are related to each other by Fourier
transform:
Rs τ
Gs f
(1.101)
PSD, the variation of the power P of s(t)
with frequency, is given by (1.66) and
(1.68) for respectively periodic and aperiodic power signals.
Based on (1.87), (1.92), (1.96) and (1.101),
the PSD and ESD of a signal may be obtained
either by taking the magnitude-square of
its Fourier transform or the Fourier transform
of the autocorrelation function (see
Figure 1.11):
s(t)
⇓
⇔ S(f)
⇓
Rs(τ) ⇔ |S(f)|2
Figure 1.11 Two Alternative Approaches for
Obtaining the ESD/PSD of a Signal s(t).
Example 1.14 PSD and ESD of a Rectangular Pulse.
To have a better feeling about the two alternative approaches shown in Figure 1.11 to determine the PSD/ESD, consider a rectangular
pulse s(t) and its Fourier transform S(f ) (see
(1.15) and (1.16)):
s t = AΠ t T
S f = ATsin c fT
(1.102)
which is an energy signal, with energy A2 T.
The autocorrelation function and the ESD of
s(t) are related to each other by (1.50), (1.52)
and (1.95):
Rs τ = A2 T Λ τ T
Ψs f = A2 T 2 sinc2 fT
(1.103)
Noting that Rs 0 = A2 T denotes the energy
of s(t), Ψs f = S f 2 clearly represents the
ESD, since the area under it is equal to A2T.
The corresponding PSD is given by
Gs f =
Ψs f
S f
=
T
T
2
= A2 T sinc2 fT
(1.104)
The integration of (1.104) with frequency
gives A2, which denotes the signal power over
the finite pulse duration T.
1.3 Random Signals
In a digital communication system, one of the
M symbols is transmitted, in a finite symbol
duration, using a pre-defined M-ary alphabet.
These symbols are unknown to the receiver
when they are transmitted. At the receiver,
these signals bear a random character in
AWGN, fading or shadowing channels. Since
a random signal can not be predicted before
reception, a receiver processes the received
random signal in order to estimate the transmitted symbol. In view of the above, the performance of telecommunication systems can be
evaluated by characterizing the random signals
statistically.
This section aims to provide a short introduction to random signals and processes encountered in telecommunication systems. The
reader may refer to Appendix F for further
details about the random signals and their characterization. [3][9]
1.3.1 Random Variables
Given a sample space S and elements s S, we
define a function X(s) whose domain is S and
whose range is a set of numbers on the real line
(see Figure 1.12). The function X(s) is called a
random variable (rv). For example, if we consider coin flipping with outcomes head
(H) and tail (T), the sample space is defined
19
Signal Analysis
c. The cdf of a continuous rv X is a nondecreasing and smooth function of X.
Therefore, for x2 ≥ x1, P x1 < X ≤ x2 =
Sample space S
s
X(s)
Real line
Range: set of real numbers
Domain: S
Figure 1.12 Definition of a Random Variable.
by S = {H, T} and the rv may be assumed to be
X(s) = 1 for s = H and −1 for s = T. Note that a
rv, which is unknown and unpredictable
beforehand, is known completely once it
occurs. For example, one does not know the
outcome before flipping a coin. However, once
the coin is flipped, the outcome (head or tail)
is known.
A rv is characterized by its probability density function (pdf ) or cumulative distribution
function (cdf ), which are interrelated. The
cdf FX(x) of a rv X is defined by
FX x = P X ≤ x , − ∞ < x < ∞
(1.105)
which specifies the probability with which the
rv X is less than or equal to a real number x. The
pdf of a rv X is defined as the derivative of
the cdf.
fX x =
d
FX x , − ∞ < x < ∞
dx
fX u du
(1.107)
In view of (1.105)−(1.107), a rv is characterized by the following properties:
a. fX x ≥ 0 and the area under fX(x) is always
equal to unity: FX ∞ =
∞
−∞
fX u du = 1.
b. 0 ≤ FX(x) ≤ 1 since FX ∞ = 1 and F − ∞ =
−∞
−∞
fX u du = 0.
d. When a rv X is discrete or mixed, its cdf is
still a non-decreasing function of X but contains discontinuities.
A rv is characterized by its moments. The
n-th moment of a rv X is defined as
E Xn =
∞
−∞
x n fX x dx
(1.108)
where E[.] denotes the expectation. The rv’s
encountered in telecommunication systems
are mostly characterized by their first two
moments, namely the mean (expected) value
mX and the variance σ 2X :
∞
mX = E X =
−∞
xfX x dx
var X = σ 2X = E X − mX
2
(1.109)
= E X 2 − m2X
When X is discrete or of a mixed type, the pdf
contains impulses at the points of discontinuity
of FX(x). In such cases, the discrete part of fX(x)
may be expressed as
N
P X = xi δ x − xi
fX x =
x
−∞
fX u du = FX x2 − FX x1 ≥ 0.
x1
(1.106)
Conversely, the cdf is defined as the integral
of the pdf:
FX x =
x2
(1.110)
i=1
where the rv X may assume one of the N values,
x1, x2,…, xN at the discontinuities. For
example, in case of coin flipping, two outcomes
may be represented as x1 for head and x2 for tail.
Then, P(X = x1) = p ≤ 1 shows the probability of
head, while P(X = x2) = 1 − p denotes the probability of tail.
Mean value and the variance of a discrete rv
is found by inserting (1.110) into (1.109):
N
mX = E X =
N
σ 2X
x2i P
=
i=1
xi P X = xi
i=1
X
= xi − m2X
(1.111)
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Digital Communications
For the special case of equally-likely probabilities, that is, P X = xi = 1 N, then (1.111)
simplifies to the well-known expression
mX = E X =
σ 2X
1 N
xi
N i=1
1 N
=
xi − mX
N i=1
2
1 N 2
=
x − m2X
N i=1 i
(1.112)
Now consider two rv’s X1 and X2, each of
which may be continuous, discrete or mixed.
The joint cdf is defined as
FX1 X2 x1 , x2 = P X1 ≤ x1 , X2 ≤ x2
x1
x2
−∞
−∞
=
fX1 X2 u1 , u2 du2 du1
FX1 X2 − ∞ , − ∞ = 0
FX1 X2 ∞, ∞ = 1
(1.113)
and is related to the joint pdf as follows:
fX1 X2 x1 , x2 =
∂2
FX X x1 , x2
∂x1 ∂x2 1 2
(1.114)
The marginal pdf’s are found as follows:
following implications: If the experiments
result in mutually exclusive outcomes, then
the probability of an outcome in one experiment is independent of an outcome in any other
experiment. The joint pdf’s (cdf’s) may then be
written as the product of the pdf’s (cdf’s) corresponding to each outcome.
Example 1.15 Averaging.
In order to model the wireless channel between
base and mobile stations of a mobile radio system, the average signal level received by a
mobile station is measured at points evenly distributed on a circle of radius d0 = 100 meters
from the base station. These measurement
results are rescaled for arbitrary distances in
order to develop a reliable channel path-loss
model. Therefore, sufficiently many shortrange measurements are required in order to
take into account all potential propagation
effects, for example, climate, topography of
the terrain, vegetation. Now consider for simplicity that ten measurements are conducted
in dBm and (1.112) is used to determine the
average signal level and its standard deviation
at d0 = 100 meters:
X = − 41 1 − 45 − 39 6 − 42 − 40 3 − 44
− 48 − 43 − 43 3 − 41 5
∞
−∞
∞
−∞
fX1 X2 x1 , x2 dx1 = fX2 x2
(1.115)
fX1 X2 x1 , x2 dx2 = fX1 x1
The conditional pdf f(xi|xj), which gives the
pdf of xi for a given deterministic value of
the rv xj, has the following properties
f xi xj = fXi Xj xi , xj
fXj xj
F − ∞ xj = 0, F ∞ xj = 1
i
j, i, j = 1, 2
(1.116)
Generalization of (1.113)-(1.116) to more
than two rv’s is straightforward. Statistical
independence of random events has the
mX =
1 N
xi = − 42 78 dBm
N i=1
1 N 2
σX =
x − m2X
N i=1 i
1
2
= 2 35 dBm
(1.117)
Consequently, the average signal level at a
distance of d0 = 100 meters may be modeled
by mX = − 42 78 dBm and a standard deviation
σ X = 2 35 dBm.
1.3.2 Random Processes
Future values of a deterministic signal can be
predicted from its past values. For example,
Acos(wt + Φ) shows a deterministic signal as
21
Signal Analysis
long as A, w and Φ are deterministic and
known; its future values can be determined
exactly using its value at a certain time. However, it describes a random signal if any of A,
w or Φ is random. For example, a noise generator generates a noise with a random amplitude and phase at any instant of time and
these values change randomly with time.
Similarly, amplitude and phase of a multipath
fading signal vary randomly not only at a
given instant of time but also with time.
Hence, a random signal changes randomly
not only at an instant of time but also with
time. Therefore, future values of random
signals cannot be predicted by using their
observed past values. [4][6][7]
A rv may be used to describe a random signal
at a given instant of time. Random process
extends the concept of a rv to include also
the time dimension. Such a rv then becomes
a function of the possible outcomes s of a random event (experiment) and time t. In other
words, every outcome s will be associated with
a time (sample) function. The family (ensemble) of all such sample functions is called a random process and denoted as X(s, t). [3][4][9]
A random process does not need to be a function of a deterministic argument. For example,
the terrain height between transmitter and
receiver may be a random process of the distance, since the location and the height of the
obstacles may change randomly. The random
process may be continuous or discrete either
in time or in the value of the rv.
For example, consider L Gaussian noise generators. The time variation of the noise voltages
at the output of each of the noise generators is
called as a sample function, or a realization of
the process. Thus, each of the L sample functions, corresponding to a specific event sj,
changes randomly with time; noise voltages
at any two instants are independent from each
other (see Figure 1.13). The totality of sample
functions is called an ensemble. On the other
hand, at an instant of time tk, the value of the
rv X(s, tk) depends on the event. For a specific
event and time tk, X(sj, tk) = Xj(tk) is a real
X1(t) : sample function
X2(t)
time-averaging
ensemble-averaging
XL(t)
tk
time
Figure 1.13 Random (Gaussian Noise) Process.
number. The properties of X(s,t) is summarized
below: [3]
a. X(S, t) represents a family (ensemble) of
sample functions, where S = s1 , s2 , , sL .
b. X(sj, t) represents a sample function for the
event sj (an outcome of S or a realization of
the process).
c. X(S, tk) is a rv at time tk.
d. X(sj, tk) is a non-random number.
For the sake of convenience, we will denote
a random process by X(t) where the presence of
the event is implicit.
1.3.2.1
Statistical Averages
Empirical determination of the pdf of a random
process is neither practical nor easy since it may
change with time. For example, the pdf of a
Gaussian noise voltage may change with time
as the temperature of the noise generator (a
resistor) increases; then, the noise variance
would be higher. Nevertheless, the mean
and the autocorrelation function adequately
describe the random processes encountered in
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Digital Communications
telecommunication systems. The mean of a random process X(t) at time tk is defined as
∞
E X tk = mX tk =
−∞
xfXk x dx
(1.118)
(1.119)
A random process X(t) is said to be stationary in the strict sense, if none of its statistics
changes with time, but they depend only on
the time difference tk−tj. This implies that the
joint pdf satisfies the following:
f x1 , x2 ,
, xL ; t1 , t2 ,
= f x1 , x2 ,
, tL
, xL ; t1 + τ, t2 + τ,
, tL + τ
(1.120)
Here, (1.120) should be read as the joint pdf
of x1 , x2 , ,xL at times t1 , t2 , , tL . For
example, a strict sense stationary process satisfies the following for L = 1 and 2:
t2 = t1 + τ
= f x1 , x2 ; τ
(1.122)
Note that strict-sense stationarity implies
wide-sense stationarity, but the reverse is not
true. In most applications, the analysis based
on WSS assumption provides sufficiently
accurate results at least over some time
intervals of interest. The autocorrelation function of WSS random processes, which is an
even function of τ, provides a measure of the
degree of correlation between the random values of a process withτ seconds time shift from
each other.
The properties of the autocorrelation function of a real-valued WSS random process are
listed below:
a. An even function of τ:
RX τ = RX − τ
(1.123)
b. Maximum value occurs at τ = 0:
RX τ ≤ RX 0 ,
τ
(1.124)
c. The value of the autocorrelation function
at the origin is equal to the average power
of the signal
RX 0 = P = E X t
2
(1.125)
d. Autocorrelation and PSD are related to each
other by the Fourier transform:
RX τ
f x; t = f x; t + τ
f x1 , x2 ; t1 , t2
E X t = mX , t
RX tj , tk = RX τ , τ ≜ tj − tk
where the pdf of X(tk) is defined over the
ensemble of events at time tk. Evidently, the
mean would be time-invariant if the pdf is
time-invariant. On the other hand, identical
time variations of the mean and the variance
of two random processes do not imply that
these process are equivalents. For example,
one of these processes may be changing faster
than the other, and hence have higher spectral
components. Therefore, autocorrelation functions (and corresponding PSDs) of these processes should also be compared with each
other. The autocorrelation function of a random
process X(t) provides a measure of the degree of
similarity between the rv’s X(tj) and X(tk):
RX tj , tk = E X tj X ∗ tk
mean is a constant and the autocorrelation function depends only on the time difference tk−tj:
GX f
(1.126)
(1.121)
A random process X(t) is said to be wide
sense stationary (WSS), if only its mean and
autocorrelation function are unaffected by a
shift in the time origin. This implies that the
Example 1.16 Poisson Process.
Poisson process is a discrete random process
which describes the number of occurrences
of an event as a function of time. The event
may represent the number of customers
23
Signal Analysis
X(t)
7
6
5
4
3
2
1
0
t1
0
t2
t3
t4
Figure 1.14
arriving to a bank or a supermarket, the failure
of some components in a system, the number
of planes arriving to and departing from an airport, the number of goals scored in a football
game, and so on. A single event of the process,
which consists of counting the number of
occurrences (arrivals) with time, is also
random.
Let the rv k denote the number of occurrences
(arrivals) during a time interval τ = t2 − t1 where
t1 and t2 are arbitrary times with t2 ≥ t1. The
average number of arrivals per unit time is
defined by the arrival rate λ in arrivals/s. The
probability of k arrivals during τ is given by
Pk τ = e − λτ
λτ k
, k = 0, 1, 2,
k
(1.127)
∞
k=0
t6
t7
t8
the number of occurrences during (ti, tj), with
discontinuities at random time instants ti and
tj. As shown in Figure 1.14, this process may
represent, for example, the number of uncoordinated customers entering into a bank office
with random arrival times ti. Figure 1.14 shows
that one customer arrives during (0, t1), no customers arrive during (t2, t3) and two customers
arrive during (t3, t4). Since the time intervals of
any two events do not overlap, the corresponding rv’s are independent. [3]
The first two moments of the rv X(tk) at a specific time tk are found, with the help of (D.1), as
follows:
E X t1 = e − λt1
∞
k
λt1 k − 1
k−1
∞
= λt1 e − λt1
k=1
= λt1
E X 2 t1 = e − λt1
∞
λt1
k
k2
k
δ x− k
(1.128)
= λt1 e − λt1
∞
k
k=1
∞
Noting that
P τ = 1 (see (D.1)), the
k=0 k
area under the pdf given by (1.128) is unity.
Based on (1.127) and (1.128), we now define
a discrete random process X(t), accounting for
k
λt1
k
k=0
λτ
k
t
Poisson Process.
k=0
For example, the probability that no customers arrive to a bank during τ = 1 minute is given
by P0 τ = exp − λτ = 0 905 if one customer
arrives on the average per 10 minutes, that is,
λ = 0.1 arrivals/minutes. The pdf of the number
of arrivals during τ may be written as
fk x = e −λτ
t5
= λt1 e − λt1
k
λt1 k − 1
k−1
∞
k +1
k =0
= λt1 1 + λt1
(1.129)
λt1
k
k
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Digital Communications
It is clear from (1.129) that the first two
moments of the rv, defined as the number of
occurrences during (0, t1), are time-dependent.
Since the intervals (0, t1) and (t2 − t1) do not
overlap with each other, the rv’s X(t1) and
X(t2) − X(t1) are independent and Poisson distributed with parameters λt1 and λ(t2 − t1). Hence,
E X t1 X t2 − X t1
(1.130)
= λt1 λ t2 − t1 , t1 ≤ t2
The autocorrelation function for t2 ≥ t1 may
then be found using (1.129) and (1.130) as
follows:
(1.131)
Since R t1 , t2 = R t2 , t1 for a real random
process, the autocorrelation function is given by
λt1 + λ2 t1 t2 t1 ≤ t2
T 2
X t dt
(1.133)
−T 2
A random process is said to be ergodic in the
autocorrelation function, if RX(x) is correctly
described through the use of a sample function,
that is, time averaging instead of ensemble
averaging (see (1.119)):
T 2
1
∞T
−T 2
X ∗ t X t + τ dt (1.134)
+ X 2 t1
= λ2 t1 t2 + λt1 , t1 ≤ t2
R t1 , t2 =
T
T
= λt1 λ t2 − t1 + λt1 1 + λt1
λt2 + λ2 t1 t2 t1 ≥ t2
1
∞T
mX = lim
RX τ = lim
R t1 , t2 = E X t1 X t2
= E X t1 X t2 − X t1
A random process is said to be ergodic in the
mean if its mean can be calculated using a sample function, that is, using time averaging
instead of ensemble averaging (see (1.118)):
It is not easy to test whether a random process is ergodic or not. However, in most applications, it is reasonable to assume that time and
ensemble averaging are interchangeable.
Example 1.17 Test for Stationarity.
Consider a random process defined by
X t = A cos wt + Φ
(1.132)
where A and w are constants but Φ is uniformly
distributed in [0, 2π]:
Since the mean of the random process and
the autocorrelation are time dependent, the
Poisson process is not stationary.
1.3.2.2 Time Averaging and Ergodicity
The computation of the mean and the autocorrelation function of a random process by
ensemble averaging is often not practical since
it requires the use of all sample functions. For
the so-called ergodic processes, ensemble averaging may be replaced by time averaging. This
means that the mean and the autocorrelation
function of an ergodic process may be determined by using a single sample function. [3]
[4][5][7][8]
(1.135)
fΦ ϕ =
1
, 0 ≤ ϕ ≤ 2π
2π
(1.136)
In order to determine whether this process is
WSS, we first determine its mean:
A
mX = E X t =
2π
=
−A
xfX
t
x dx
(1.137)
Acos wt + ϕ fΦ ϕ dϕ = 0
0
which is obviously independent of time. Note
that the evaluation of the integral on the first line
requires the knowledge of the pdf of X(t), while
the second integral uses the pdf of Φ. The pdf of
25
Signal Analysis
X(t) is obtained from the pdf of Φ with a variable transformation (see Example F.6):
RZ t, t + τ = E Z ∗ t Z t + τ = RZ τ
L
fX
t
dϕ
dx
x = fΦ ϕ
ϕ = ϕ1
+ fΦ ϕ
dϕ
dx
ℓ=1
ϕ = ϕ2
L
αℓ e − j wt + Φℓ
=E
αk e j w
t + τ + Φk
k=1
L
L
E αℓ αk E e j
= e jwτ
Φk − Φℓ
k=1 ℓ=1
=
1
π A2 − x2
, −A ≤ x ≤ A
where ϕ1 = ϕ2 − π = π 2 − wt are the two roots
of the multi-valued function (1.135). Note that
the pdf is independent of w and time t, accounting for the time-invariance of the mean. Also
note that the pdf peaks at x = ±A.
The autocorrelation of X(t) is given by
RX t1 ,t2 = E X t1 X t2
= A2 E cos wt1 + ϕ cos wt2 + ϕ
1
= A2 E cos w t1 + t2 + 2ϕ + cos w t1 − t2
2
1
= A2 cos wτ , τ = t1 − t2
2
Since the mean is constant and the autocorrelation is a function of the time difference, this
random process is WSS.
Example 1.18 Complex Random Process.
A received signal, described by the following
complex random process
L
αℓ e j wt + Φℓ
L
(1.140)
ℓ=1
represents the sum of L replicas of a complex
carrier signal ejwt scattered from L obstacles
with random channel gains αℓ and phases Φℓ.
They are all assumed to be statistically independent of each other and the pdf of Φℓ is given
by (1.136). In view of (1.137), the mean value
of (1.140) is identically equal to zero. Its autocorrelation function is given by
E αℓ 2
= e jwτ
ℓ=1
(1.141)
The random process Z(t) is evidently WSS.
The value of the autocorrelation function at
the origin RZ(0) shows that the received power
is given by the sum of the powers of independent channel gains. The pdf of the received
power can be determined using the pdf’s of
αℓ’s (see for example (F.109)–(F.111) and
Example F.14).
1.3.2.3
(1.139)
Z t =
δ k−ℓ
(1.138)
PSD of a Random Process
Random processes are generally classified as
power signals with a PSD as given by (1.68)
and with the following properties:
a. The PSD is positive real-valued function of
frequency:
GX f ≥ 0
b. The PSD
frequency:
has
even
(1.142)
symmetry
GX f = GX − f
with
(1.143)
c. The PSD and autocorrelation function are
Fourier transform pairs:
RX τ
GX f
(1.144)
d. The average normalized power of a random
process is given by the area under the PSD:
∞
PX = RX 0 =
−∞
GX f df
(1.145)
26
Digital Communications
Example 1.19 Autocorrelation and PSD of a
Random Binary Sequence.
A baseband signal consisting of a binary
sequence can be expressed as follows:
∞
bn p t − nT
s t =A
(1.146)
n= −∞
where bn
− 1, + 1 represents the bit 0 and 1
with equal likelihood, P bn = 1 = P bn = − 1 =
1 2, p(t) denotes the pulse shape, T is the bit
duration and A is the amplitude. Therefore,
(1.146) represents a sequence of pulses with
amplitudes ±1. The infinitely long bit
sequence, which is a power signal, becomes
perfectly random if the bits are independent
from each other. The autocorrelation function
of a random bit sequence is given by a delta
function, since all dibit combinations are
equally likely with probability ẳ:
Rb k = E bn bn + k
=
1
1 ì 1 + 1 × −1 + − 1 × 1 + − 1 × − 1 = 0 k 0
4
1
k=0
Rb k = δ k
where the PSD of the infinitely long random
bit sequence in the second line is flat and given
by (1.23).
The autocorrelation function of s(t) is given
by the expectation of s(t) with a delayed replica
of itself:
Rs τ = E s∗ t s t + τ
∞
= A2
m= −∞n= −∞
E b∗n bm
n= −∞
∞
n+1 T
= A2
=A
1.3.2.4
Noise
The thermal noise voltage n(t) that we encounter in telecommunication systems is a random
process with zero mean and an autocorrelation
function described by Dirac delta function:
δ n− m
p∗ t − nT p t + τ − nT dt
n = − ∞ nT
∞
2
∗
−∞
(1.149)
An alternative approach for determining the
PSD of s(t) is based on the observation that
(1.146) is the convolution of the random bit
sequence and the rectangular pulse Π t T .
Then, in view of the convolution property of
the Fourier transform, given by (1.46), the
PSD of s(t) may be written as the product of
the PSD’s of Π t T and the infinite bit
sequence. The result is evidently identical to
(1.149). Noting that the area under Tsinc2(fT)
in (1.149) is equal to unity, the integration of
Gs(f) gives the total signal power A2, as
expected.
Rn τ = E n t n t + τ =
E p∗ t − nT p t + τ − nT
= A2
Gs f = A2 T sinc2 fT
E n t =0
E p∗ t − nT p t + τ − mT
∞
Rs τ = A2 Rp τ = A2 TΛ t T
(1.147)
Gb f = 1
∞
It is clear from (1.148) that the autocorrelation of a random binary sequence is identical
to that of the pulse p(t). For example, if we
assume p t = Π t T , which is an even function of t, then, in view of (1.50), (1.95) and
(1.104), the autocorrelation function is triangular and the PSD is sinc-squared:
p u p u + τ dt = A2 Rp τ
(1.148)
N0
δτ
2
(1.150)
The formula (1.150) implies that any two different noise samples are uncorrelated with each
other, no matter how close the time difference
between the samples is. Using the Fourier
transform relationship between the autocorrelation function and the PSD, the two-sided noise
PSD is given by
Gn f = ℑτ Rn τ =
N0
2
W Hz
(1.151)