tele4653 l1

TELE4653 Digital Modulation & Coding

Fundamentals

Wei Zhang

[email protected]

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.


Properties of FT (1)

Linearity Property:

If g(t) ⇔ G(f), then

c1g1(t) + c2g2(t) ⇔ c1G1(f) + c2G2(f).

Dilation Property:


g(at) ⇔ 1

|a|G(

f

a

)

.



Conjugation Rule:


g∗(t) ⇔ G∗(−f).

Duality Property:


G(t) ⇔ g(−f).



Time Shifting Property:


g(t − t0) ⇔ G(f) exp(−j2πft0).

Frequency Shifting Property:


exp(j2πfct)g(t) ⇔ G(f − fc).



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 .



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 “⇔”.


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 .


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)


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.


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.


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)


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.


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


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


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)


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)


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

]

.


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.


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 l1

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