chapter 2. signals and linear systems essentials of communication systems engineering
TRANSCRIPT
Chapter 2. Signals and Linear Systems
Essentials of Communication Systems Engineering
2
Energy-Type and Power-Type Signals Energy content
For any signal x(t), the energy content of the signal is defined by
Power content For any signal x(t), the power content of the signal is defined by
For real signal, is replaced by
A signal is an energy-type signal if and only if Ex is finite
A signal is an power-type signal if and only if Px satisfies
2/
2/
22)(lim)(
T
TTx dttxdttxE
2/
2/
2)(
1lim
T
TTx dttx
TP
2)(tx )(2 tx
xP0
3
Example 2.1.10 The energy content of
Therefore, this signal is not an energy-type signal However, the power of this signal is
Hence, x(t) is a power-type signal and its power is
tfAtx 02cos)(
2/
2/ 0222/
2/
2)2(coslim)(lim
T
TT
T
TTx dttfAdttxE
2
)24cos(82
lim
)24cos(12
1lim
)2(coslim)(1
lim
2
2/
2/
00
22
2/
2/ 0
2
2/
2/ 0222/
2/
2
A
tftf
A
T
TA
dttfA
T
dttfAdttxT
P
T
TT
T
TT
T
TT
T
TTx
2
2A
4
Sinusoidal Signal & Complex Exponential Signal
Sinusoidal signals Definition :
A : Amplitude f0 : Frequency : Phase Period : T0 = 1/f0
Complex exponential signal Definition :
A : Amplitude f0 : Frequency : Phase
tfAtx 02cos)(
)2( 0)( tfjAetx
5
Unit Step, Rectangular & Triangular Signal
Unit step signal Definition
Rectangular pulse Definition
Triangular Signal Definition
00
01)(1 t
ttu
otherwise
tt
0
1)( 2
121
otherwise
tt
tt
t
0
101
011
)(
)(*)()( ttt
6
Sinc & Sign or Signum Signal Sinc signal
Definition
The sinc signal achieves its maximum of 1 at t = 0.
The zeros of the sinc signal are at t = 1, 2, 3,
Sign or Signum signal Definition :
Sign of the independent variable t
Can be expressed as the limit of the signal xn(t) when n
01
0)sin(
)(sinct
tt
tt
00
01
01
)sgn(
t
t
t
t
00
0
0
)(
t
te
te
tx nt
nt
n
7
Impulse or Delta Signal Definition of the impulse signal
1. (t) = 0 for all t 0 and (0) =
2. Properties
1. x(t)(t-t0) = x(t0)(t-t0)
2.
3.
4.
1)(
dtt
)()()( 00 tdtttt
)()(*)( txttx )()(*)( 00 ttxtttx
t
dtu )()(1)()( 1 tu
dt
dt
8
Fourier Series LTI systems
Model of a large number of building blocks in a communication system
Good and accurate models for a large class of communication channels
Some basic components of transmitters and receivers Such as filters, amplifiers, and equalizers
Convolution integral : Input and output relation of an LTI system :
where h(t) : Impulse response of the system.
dxthdtxhty )()()()()(
9
Fourier Series Another approach to analyzing LTI systems
Basic idea Expand the input as a linear combination of some
basic signals whose output can be easily obtained Employ the linearity properties of the system to obtain
the corresponding output
Easier than a direct computation of the convolution integral
Provide better insight into the behavior of LTI systems
10
Fourier SeriesResponse of an LTI system to a complex exponenti
al A complex exponential with the same frequency with a
change in amplitude and phase (p.45,Example 2.1.25)
Which signals can be expanded in terms of complex exponentials? Answer: periodic signals which satisfy Dirichlet condit
ions
11
Fourier Series Dirichlet conditions 1. x(t) is absolutely integrable over its period, i.e.,
2. The number of maxima and minima of x(t) in each perio
d is finite 3. The number of discontinuities of x(t) in each period is fi
nite Fourier series
for some arbitrary (usually, = 0 or )
0
0)(
Tdttx
n
tT
nj
nextx 0
2
)(
00
2
0
)(1 T t
T
nj
n dtetxT
x
20T
12
Fourier Series Observations concerning Fourier series
xn : Fourier-series coefficients of the signal x(t)
Dirichlet conditions are only sufficient conditions for the existence of the Fourier series For some signals that do not satisfy these conditions, we can still find
the Fourier series expansion
The quantity f0 = 1/T0 is called the fundamental frequency
of the signal x(t) The frequencies of the complex exponential signals are multiples of t
his fundamental frequency
The nth multiple of f0 is called the nth harmonic
13
Example 2.2.1 x(t) : Periodic signal depicted in Figure 2.25 and described analytically by
: A given positive constant (pulse width) Determine the Fourier series coefficient .
n
nTttx
0)(
nx
14
Example 2.2.1 Solution
Period of the signal is T0 and
For n = 0, the integration is very simple and yields
000
0
0
2/
2/
2
0
2/
2/
2
0
sinsin1
0][2
11
1)(
1000
0
0
0
T
nc
TT
n
n
neejn
T
Tdte
Tdtetx
Tx T
jnT
jntT
jnT
T
tT
jn
n
00 /Tx
15
Fourier Series for Real Signals Real signal x(t)
The positive and negative coefficients are conjugates |xn| : Even symmetry (|xn| = |x-n| )
xn : Odd symmetry (xn = - x-n) with respect to the n = 0 axis
*
*2
0
2
0
00
00 )(
1)(
1n
T tT
njT t
T
nj
n xdtetxT
dtetxT
x
16
Response of LTI Systems to Periodic Signals If h(t) is the impulse response of the system, that the response
to the exponential ej2f0t is H( f0) ej2f0t (From ex 2.1.5 with A= 0 & )
x(t) , the input to the LTI system, is periodic with period To and has a Fourier-series representation
Response of LTI systems
dtethfH ftj 2)()(
n
tT
nj
nextx 0
2
)(
n
tT
nj
nn
tT
nj
n
n
tT
nj
n
eT
nHxeLx
exLtxLty
00
0
2
0
2
2
][
)()(
dtethfH ftj 2)()(
0
17
Response of LTI Systems to Periodic Signals
If the input to an LTI system is periodic with period To,
then the output is also periodic. The output has a Fourier-series expansion given by
where
n
tT
nj
neyty 0
2
)(
0T
nHxy nn
0T
nHxy nn
0T
nHxy nn
18
Parseval's Relation The power content of a periodic signal is the sum of the po
wer contents of its components in the Fourier-series representation of that signal
The left-hand side of this relation is Px, the power content of the signal
x(t)
|xn|2 is the power content of , the nth harmonic
Parseval's relation says that the power content of the periodic signal is the sum of the power contents of its harmonics
0
2T
nj
nex
nn
Txdttx
T
22
0
0
)(1