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TELE4653 Digital Modulation & Coding
Fundamentals
Wei Zhang
w.zhang@unsw.edu.au
School of Electrical Engineering and Telecommunications
The University of New South Wales
Outline
Introduction to Communications
Lowpass (LP) and Bandpass (BP) Signals
Signal Space Concepts
Expansion of BP Signals
TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.1/23
Modulation
The information signal is a low frequency (baseband) signal.
Examples: speech, sound, AM/FM radio
The spectrum of the channel is at high frequencies.
Therefore, the information signal should be translated to a
higher frequency signal that matches the spectral
characteristics of the communication channel. This is the
modulation process in which the baseband signal is turned
into a bandpass modulated signal.
TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.3/23
Properties of FT (1)
Linearity Property:
If g(t) ⇔ G(f), then
c1g1(t) + c2g2(t) ⇔ c1G1(f) + c2G2(f).
Dilation Property:
If g(t) ⇔ G(f), then
g(at) ⇔ 1
|a|G(
f
a
)
.
TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.4/23
Properties of FT (2)
Conjugation Rule:
If g(t) ⇔ G(f), then
g∗(t) ⇔ G∗(−f).
Duality Property:
If g(t) ⇔ G(f), then
G(t) ⇔ g(−f).
TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.5/23
Properties of FT (3)
Time Shifting Property:
If g(t) ⇔ G(f), then
g(t − t0) ⇔ G(f) exp(−j2πft0).
Frequency Shifting Property:
If g(t) ⇔ G(f), then
exp(j2πfct)g(t) ⇔ G(f − fc).
TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.6/23
Properties of FT (4)
Modulation Theorem:
Let g1(t) ⇔ G1(f) and g2(t) ⇔ G2(f). Then
g1(t)g2(t) ⇔ G1(f) ? G2(f),
where G1(f) ? G2(f) =∫∞−∞ G1(λ)G2(f − λ)dλ.
Convolution Theorem:
Let g1(t) ⇔ G1(f) and g2(t) ⇔ G2(f). Then
g1(t) ? g2(t) ⇔ G1(f)G2(f),
where g1(t) ? g2(t) =∫∞−∞ g1(τ)g2(t − τ)dτ .
TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.7/23
Properties of FT (5)
Correlation Theorem:
Let g1(t) ⇔ G1(f) and g2(t) ⇔ G2(f). Then∫ ∞
−∞g1(τ)g∗2(t − τ)dτ ⇔ G1(f)G∗
2(f).
Rayleigh’s Energy Theorem:
Let g1(t) ⇔ G1(f) and g2(t) ⇔ G2(f). Then∫ ∞
−∞|g(t)|2dt =
∫ ∞
−∞|G(f)|2df.
Note that in the above formula, it is “=”, not “⇔”.
TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.8/23
Lowpass Signals
A lowpass, or baseband, signal is a signal whose spectrum
is located around the zero frequency.
The bandwidth of a real LP signal is W .
TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.9/23
Bandpass Signals
A bandpass signal is a real signal whose spectrum is
located around some frequency ±f0 which is far from zero.
Due to the symmetry of the spectrum, X+(f) has all the
information that is necessary to reconstruct X(f).
X(f) = X+(f) + X−(f) = X+(f) + X∗+(f) (1)
TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.10/23
Bandpass Signals
Denote x+(t) the analytic signal of BP signal x(t). Then,
x+(t) = F−1[X+(f)] = F−1[X(f)u−1(f)] (2)
= x(t) ? F−1[u−1(f)] = x(t) ?
(
1
2δ(t) + j
1
2πt
)
(3)
=1
2x(t) +
j
2x̂(t), (4)
where in (2) the unit step signal u−1(f) is used, in (3)
Convolution Property is used, and in (4) x̂(t) = 1πt ? x(t) is the
Hilbert transform of x(t).
For details of Fourier Transform, please refer to Tables on pp.
18-19 in textbook.
TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.11/23
Bandpass Signals
Define xl(t) the lowpass equivalent of x(t) whose spectrum is
given by 2X+(f + f0), i.e., Xl(f) = 2X+(f + f0). Then,
xl(t) = F−1 [Xl(f)] = F−1 [2X+(f + f0)]
= 2x+(t)e−j2πf0t
= [x(t) + jx̂(t)] e−j2πf0t −−−−using(4) (5)
Alternatively, we can write
x(t) = <[xl(t)ej2πf0t]. (6)
It expresses any BP signals in terms of its LP equivalent.
TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.12/23
Bandpass Signals
We can continue to write
xl(t) = [x(t) cos(2πf0t) + x̂(t) sin(2πf0t)]
+ j [x̂(t) cos(2πf0t) − x(t) sin(2πf0(t))] . (7)
For simplicity, we write xl(t) = xi(t) + jxq(t), where
xi(t) = x(t) cos(2πf0t) + x̂(t) sin(2πf0t)] (8)
xq(t) = x̂(t) cos(2πf0t) − x(t) sin(2πf0(t)) (9)
Solving above equations for x(t) and x̂(t) gives
x(t) = xi(t) cos(2πf0(t)) − xq(t) sin(2πf0(t)) (10)
x̂(t) = xq(t) cos(2πf0(t)) + xi(t) sin(2πf0(t)) (11)TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.13/23
Bandpass Signals
If we define the envelope and phase of x(t), denoted by rx(t)
and θx(t), respectively, by
rx(t) =√
x2i (t) + x2
q(t) (12)
θx(t) = arctanxq(t)
xi(t)(13)
we have xl(t) = xi(t) + jxq(t) = rx(t)ejθx(t).
Using (6), we have
x(t) = <[rx(t)ejθx(t)ej2πf0t]
= rx(t) cos(2πf0(t) + θx(t)). (14)
TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.14/23
Mod/Demod of BP Signals
FIGURE 2.1-5 (a) is a modulator given by Eq. (6).
FIGURE 2.1-5(b) is a modulator given by Eq. (10).
FIGURE 2.1-5(c) is a general representation for a modulator.
FIGURE 2.1-6 (a) is a demodulator given by Eq. (5).
FIGURE 2.1-6(b) is a demodulator given by Eq. (7).
FIGURE 2.1-6(c) is a general representation for a demodulator.
TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.15/23
Vector Space Concepts
For n-dimensional vectors v1 and v2,
Inner product: 〈v1,v2〉 =∑n
i=1 v1iv∗2i = v
H2 v1
Orthogonal: 〈v1,v2〉 = 0
Norm: ‖v‖ =√∑n
i=1 |vi|2
Triangle inequality: ‖v1 + v2‖ ≤ ‖v1‖ + ‖v2‖ with equality if
v1 = av2 for some positive real scalar a
Cauchy-Schwarz inequality: |〈v1,v2〉| ≤ ‖v1‖ · ‖v2‖ with
equality if v1 = av2 for some complex scalar a
TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.16/23
Signal Space Concepts
For two complex-valued signals x1(t) and x2(t),
Inner product: 〈x1(t), x2(t)〉 =∫∞−∞ x1(t)x
∗2(t)dt
Orthogonal: 〈x1(t), x2(t)〉 = 0
Norm: ‖x(t)‖ =(
∫∞−∞ |x(t)|2dt
)1/2=
√Ex
Triangle inequality: ‖x1(t) + x2(t)‖ ≤ ‖x1(t)‖ + ‖x2(t)‖
Cauchy-Schwarz inequality:
|〈x1(t), x2(t)〉| ≤ ‖x1(t)‖ · ‖x2(t)‖ =√
Ex1Ex2
TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.17/23
Orthogonal Expansion of Signals
To construct a set of orthonormal waveforms from signals
sm(t),m = 1, 2, · · · ,K, we use Gram-Schmidt procedure:
1. φ1 = s1(t)√E1
2. φk(t) = γk(t)√Ek
for k = 2, · · · ,K,
where
γk(t) = sk(t) −k−1∑
i=1
ckiφi(t) (15)
cki = 〈sk(t), φi(t)〉 =
∫ ∞
−∞sk(t)φ
∗i (t)dt (16)
Ek =
∫ ∞
−∞γ2
k(t)dt (17)
TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.18/23
Orthogonal Expansion of Signals
Once we have constructed the set of orthonormal waveforms
{φn(t)} (m = 1, 2, · · · ,M ), we may write
sm(t) =
N∑
n=1
smnφn(t), m = 1, 2, · · · ,M (18)
TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.19/23
BP and LP Orthonormal Basis
Suppose that {φnl(t)} constitutes an orthonormal basis for the
set of LP signals {sml(t)}. We have
sm(t) = <{sml(t)ej2πf0t}, m = 1, 2, · · · ,M (19)
= <{(
N∑
n=1
smlnφnl(t)
)
ej2πf0t
}
(20)
=N∑
n=1
{
<[
smln
(
φnl(t)ej2πf0t
)]}
(21)
Define φn(t) =√
2<[
φnl(t)ej2πf0t
]
and φn(t) =
−√
2=[
φnl(t)ej2πf0t
]
.
TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.20/23
BP and LP Orthonormal Basis
Define
φn(t) =√
2<[
φnl(t)ej2πf0t
]
(22)
φ̃n(t) = −√
2=[
φnl(t)ej2πf0t
]
. (23)
Substituting (22)-(23) into (21), we may have
sm(t) =N∑
n=1
[
s(r)mln
2φn(t) +
s(i)mln
2φ̃n(t)
]
(24)
where we have assumed that smln = s(r)mln + js
(i)mln.
Eq. (24) shows how a BP signal can be expanded in terms of the
basis used for expansion of its LP signal.
TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.21/23
Gaussian RV
The density function of a Gaussian RV X is
fX(x) =1
√
2πσ2X
exp
{
−(x − µX)2
2σ2X
}
.
For a special case when µX = 0 and σ2X = 1, it is called
normalized Gaussian RV.
Q-function, defined as
Q(x) =1√2π
∫ ∞
xexp(−s2/2)ds.
Q-function can be viewed as the tail probability of the
normalized Gaussian RV.TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.22/23
Random Process
The random process X(t) is viewed as RV in term of time.
At a fixed tk, X(tk) is a RV.
Autocorrelation of the random process is
RX(t, s) = E[X(t)X∗(s)].
Wide-sense stationary requires: 1) the mean of the random
process is a constant independent of time, and 2) the
autocorrelation E[X(t)X∗(t − τ)] = RX(τ) of the random
process only depends upon the time difference τ , for all t
and τ .
TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.23/23