tele4653 l1
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TELE4653 Digital Modulation & Coding
Fundamentals
Wei Zhang
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