Transcript Document

Chapter 2
Signals and Spectra
(All sections, except Section 8, are
covered.)
Physically Realizable Waveform
1. Non zero over finite duration (finite
energy)
2. Non zero over finite frequency range
(physical limitation of media)
3. Continuous in time (finite bandwidth)
4. Finite peak value (physical limitation of
equipment)
5. Real valued (must be observable)
• Power Signal: finite power, infinite
energy
• Energy Signal: finite energy, non-zero
power over limited time
• All physical signals are energy signals.
Nothing can have infinite power.
However, mathematically it is more
convenient to deal with power signals.
We will use power signals to
approximate the behavior of energy
signals over the time intervals of
interest.
T imeaverageoperator:

 limT 
1
T
T
2
  dt

T
2
P eriodicwave form with periodTo : w(t )  w(t  To )
where To is thesmallest positivenumber satisfyingthiscondition.
1
For periodicfunction:  
To
To

2
  dt for any real number .

To

2
DC of periodfunctionw(t ) : Wdc 
1
To
To
2
 w(t ) dt

To
2
t
1 2
DC of w(t ) over a finiteinteralt 2 , t1  
w(t ) dt

t 2  t1 t1
voltage: v(t )
currnet: i (t )
power : p(t)  v(t) i(t)
- p(t)  0 if thecircuit consumespower.
- p(t) 0 if thecircuit generatespower.
averagepower : P  p(t )  v(t )i (t )
Root - mean - square (RMS) of w(t ) : Wrms 
For a resistor: P 
v 2 (t )
R
w2 (t )
2
Vrms
2
 i (t ) R 
 I rms
R  Vrms I rms
R
2
i(t)
R
+ v(t) -
Averagenormalizedpower is thepower given to1  resistor
T
1
P  v (t )  limT 
T
2
T
2
1
v
(
t
)
dt

i
(
t
)

lim
T 
T
T
2

2
2
2
2
i
 (t ) dt

T
2
v(t ) and i (t ) are power waveformsif and onlyif P is finiteand non zero.
0 P
T
T
2
v
T otalnormalizedenergy: E  limT 

2
(t ) dt  limT 
T
2
i

2
2
(t ) dt
T
2
v(t ) and i (t ) are energy waveforms if and onlyif E is finiteand non zero.
0 E
P
 averagepower out 
  10 log out
Decibel gain (dB) : dB  10 log
 averagepower in 
 Pin
power in
power out
System
Decibel signal to noise ration(S/N)dB :
 Psignal
(S/N) dB  10 log
 Pnoise
 s 2 (t )

  10 log 2
 n (t )


signal, s(t)
System
noise, n(t)

  20 log Vrms signal
V

 rms noise







The phasor is a complex number that carries the amplitude and phase angle
information of a sinusoidal function. It does not include the angular frequency.
Euler’s identify :
e  j  co s  j sin 
 
co s  Re e j
Given
w(t )  A co s(wo t   )


 ARee
e 

 ReAe e

 ARe e j ( wo t  )
jwo t
j
j
jwo t
magnitudeand phaseangleinformation
P
PhasorT ransforma
tion: A cos(wot   )  Ae j  x  jy
wherex  A cos and y  A sin  .
(Wemay use thefollowingtwo notationsexchangeably : Ae j  A )
P 1
InversePhasorT ransforma
tion: Ae j  A cos(wot   )

2
2
1 y 
 or x  jy  A cos(wot   ) where A  x  y and   tan .
x 

P 1
Fourier Transform
Signal is a measurable, physical quantity which carries
information. In time, it is quantified as w(t).
Sometimes it is convenient to view through its
frequency components.
Fourier Transform (FT) is a mathematical tool to
identify the presence of frequency component for any
wave form.

W ( f )  F w(t)   w(t )e  j 2ft dt

whereF  denotestheFT of  and f is thefrequency(Hz).

Conversely, w(t )  F - 1 W ( f )   W ( f )e j 2ft df

whereF - 1  denotestheinverseFT of .
w(t) andW(f) are thusa uniquely matchedpair under theFT .
w(t)  W(f)
Example:
 e t
w(t)  
0
t 0
t0

W(f)   e e
-t -j 2 πft
0
-( 1 2 πf)t 
e
dt 
1  j 2πf
0
1

1  j 2πf
Note: It is in general difficult to evaluate the FT integrations
for arbitrary functions. There are certain well known functions
used in the FT along with the properties of the FT.
Properties of Fourier
Transform
If w(t) is real, W(-f) = W*(f).
1.
2. Linearity:
a1w1(t)+a2w2(t)  a1W1(f) + a2 W2(f)
3. Time delay: w(t – T) = W(f) e-j2fT
4. Frequency Translation:
w(t) ej2fot  W(f – fo)
5. Convolution: w1(t)X w2(t)  W1(f)W2(f)
6. Multiplication: w1(t)w2(t)  W1(f)X W2(f)
Note:
* is complex conjugate.
X
is convolution integral.
Parseval’s Theorem

 w (t )w
1

2

* (t )dt   W1 ( f )W2 * ( f )df


If w1 (t )  w2 (t )  w(t ),


w(t ) dt 
2

 W ( f ) df  E.
2

E is theenergyof w(t ).
Energyspectraldensity(ESD) : E(f)  W ( f )

T otalnormalizedenergy: E 
 E(f ) df
2

2
(unit  joules per hertz)
Dirac Delta Function

Dirac delta function, ( x), satisfies  w( x) ( x)dx  w(0) for all functionw(x)

continuousat x  0. T hisis a definitionof  ( x). Alternatively, we can define ( x)
based on thefollowing two conditions(together).

0

  ( x)dx  1 AND  ( x)  

x0
x0

Shiftingproperty:
 w( x) ( x  x )dx  w( x ).
o

o
Unit Step Function
1
Unit step function,u ( x) : u ( x)  
0
x
x0
x0
du(x)
T herefore,it is true thatu ( x)    ( )d and
 δ ( x).
dx

More Commonly Used
Functions

1

t
 
Rectangular function,Π    
 T  0

T
t
2
T
t 
2
 t
 t  1 
t T
T riangularfunction,    T
T  0 t T

sin x
Sa() function: Sa( x) 
x
Spectrum of Sine Wave
Couch, Digital and Analog Communication Systems, Seventh Edition
©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 2–8 Waveform and spectrum of a switched sinusoid.
Spectrum of Truncated Sine Wave
Example: Double Exponential
w(t )  e
t




t

t
W ( f )   e  e  j 2ft dt


0
t
  e  e  j 2ft dt   e e  j 2ft dt

0


1
1

W( f ) 
 j 2f
e
1

 t   j 2f 



0
2
1  (2f ) 2
0
1
1

 j 2f
e
1

t   j 2f 



Convolution

w3 (t )  w1 (t )  w2 (t )   w1 ( ) w2 (t   )d

Supposed t is fixed at an arbitrary value.
Within the integration, w2(t-) is a “horizontally flipped about
=0, and move to the right by t version” of w2().
Now, multiply w1() with w2(t-) for each point of .
Then, integrate over -  <  < . The result is w3(t) for this
fixed value of t.
Repeat this process for all values of t, -  < t < .
Example
 1 
t  T 
2 
w1 (t )   
 T 



t
T
w2 (t )  e u (t ).
w3 (t )  w1 (t )  w2 (t )
Power Spectrum Density
 WT ( f ) 2 
 wattsper hertz wherew (t )  W ( f )
Pw ( f )  lim 
T
T
T  

T



normalizedaveragepower P  w (t )   Pw ( f )df
2

1
T  T
Autocorrelation: R w ( )  w(t ) w(t   )  lim
T
2
 w(t )w(t   )dt

Wiener- KhintchineT heorem: R w ( )  Pw ( f )

P  w (t )  W
2
2
rms
  Pw ( f )df  R w (0)

T
2
w(t)
1
t
Example:
-1
1  1  t  1
Given w(t)  
0 elsewhere
Find R w ( ), Pw ( f ), and P.
0
1
w(t+) if  < -2
t
-1- 
0
1- 
w(t+) if  > 2
R w ( )  w(t ) w(t   )
For   2, R w ( )  0.
For   2, R w ( )  0.
-1- 
For 0    2,
1
R w ( ) 
2
1
 1dt  t
1
1
1
t
1- 
0
w(t+) if -2  < 0
1
1
 (1   )  (1)  1   .
2
2
t
For - 2    0,
-1-  0
1- 
1
1
1
1
1
R w ( )   1dt  t 1  (1)  (1   )  1   .
2 1
2
2
w(t+) if 0   2
P  R w (0)  1.
t
-1- 
0 1- 
 
R w ( ) is a triangular function: R w ( )   
2
2






Pw ( f )  F R w ( )  2 Sa 2f
P  R w (0)  1
 
R w ( )   
2
P  R w (0)  1
1

-2
2
Pw ( f )  2Sa 2f 
2
2
f
-1.5 -1.0 -0.5
0
0.5
1.0
1.5
Orthogonal Series
Representation
In general,signals and noise are difficult to be represented by closed form mathematic
al functions.
A convenientway torepresentsignals and noise is theuse of orthogonalseries. Orthogonalseries
is a methodof representing signals and noise as elementsin an abstract vectorspace.
Definition. T wo functions,φn(t) and φm(t) are orthogonalon theintervala,b if
b
  (t)
n
m
* (t )dt  0.
a
1 n  m
Definition.  nm is Kroneckerdelta function with  nm  
0 n  m
Definition. A set of functions,φi(t), i  1,2,3,...is mutuallyorthogonalon theintervala,b if
b
K n
0
 n (t)m * (t )dt  
a
orthonorma
l.
nm
nm
 K n nm . If K n  1 for all n, φi(t), i  1,2,3,...is mutually


Example.Show that e jnwo t , n  1,2,3,... are orthogonalover theinterval a  t  a  To ,
T o 1/f o , wo  2πfo , and n is an integer.
If n  m,
a  To

e
jnwo t  jmw o t
e
dt 
a
a  To

a
e j(n  m)w o t
j(n  m)w o t
e
dt 
j (n  m) wo
a  To
a


e j(n  m)w o a e j(n  m) 2  1

0
j (n  m) wo
since e j(n  m) 2  cos2 (n  m)  j sin 2 (n  m)  1 for n  m.
If n  m,
a  To
e
jnwo t  jnwo t
e
a
dt 
a  To
 1 dt  T .
o
a
T herefore,
a  To

a
T
e jnwo t e  jmwo t dt   o
0
nm
 To nm . T hus, K n  To for all n, and e jnwo t , n  1,2,3,... is mutually
nm

 1 jnw t

orthonorma
l. Note that
e o , n  1,2,3,... is mutuallyorthonorma
l.
 To


Examples of Orthogonal Functions
Sinusoids
Polynomials
Square Waves
Example. Consider w1 (t ) and w2 (t ), where w1 (t ) and w2 (t ) haveFourier tranformsin non - overlapping
intervalsin thefrequencydomain. T henw1 (t ) and w2 (t ) are orthogonal.
*

 

j 2ft
j 2st
w
(
t
)
w
*
(
t
)
dt

W
(
f
)
e
df
W
(
s
)
e
ds
  2
 dt
 1 2
  1
  



 


   W ( f )W
1
2
* ( s )e j 2ft e  j 2st dfdsdt
  
  j 2 ( f  s ) t 
   W1 ( f )W2 * ( s )   e
dtdfds
 
 

 
 

  W ( f )W
1
2
 

* ( s ) ( f  s )dfds (because  e j 2 ( f  s )t dt  1 if f  s, and 0 otherwise.)



 W ( f )W
1

2
* ( f )df  0 (by thenon - overlapping assumption).
T heorem.A waveformw(t) can be represented overint erval(a, b)by w(t )   a n n (t).
n
Note: T he theoremimplies thatw(t) belongs to a certain type of functionswheren (t ), n  1,2,3,...
can representw(t) precisely. In reality,in theminimum we need n (t ), n  1,2,3,...such thatw(t )   a n n (t)
n
with sufficientaccuracy. Under this assumptionn (t ), n  1,2,3,...is said to be a orthogonalbasis set for
w(t). Alternatively, we say w(t) belongs to a vectorspace spannedby thebasis set n (t ), n  1,2,3,....
A specificfunction,w(t), is a pointa1 , a2 , a3 , a4 ,... in thevectorspace is defined by thebasis set n (t ), n  1,2,3,....
Whatare theorthogonalcoefficients an , n  1,2,3,...?
b
 w(t)
a
b
n
* (t )dt  
a


 a m m (t) n * (t )dt   a m  m (t)n * (t )dt   a m K m mn  an K n
m
m
m

a
b
b
T herefore,an K n  
a
1
w(t)n * (t )dt which impliesan 
Kn
b
P arseval's T heorem(another ersion).
v

a
b
 w(t)
n
* (t )dt.
a
2
w ( t ) dt   a n . (P rovethisfor yourself.)
2
2
n
Generationof w(t ) fromn (t ), n  1,2,3,

Is therea way tofind an , n  1,2,3,  from w(t ) in a similar way?
 Yes. We will see later.
Fourier Series
(pages 71 – 78 not covered)
Any periodfunctionw(t) with finiteenergy can be represented by a complexFourier Series.


ComplexFourier Series has a orthogonalbasis set φn(t)  e jnwo t , n  1,2,3,... where wo  2πfo 
2π
To
and To is theperiodof w(t ).
T heorem.For a periodic wave w(t) with finiteenergy,w(t) 

c e
n  


n 0
n 1
n
jnwo t
1
wherecn 
To
To
 w(t) e
Alternatively, w(t)   an cosnwot   bn sin nwot where
T

1 o
w(t )dt
n0

T
To 0
2 o

an   To
and bn   w(t ) sin nwotdt for all n  0.
To 0
2

w
(t
)
cos
nw
tdt
n

1
o
To 
 0
T hisis the well known formof Fourier series in a real vectorspace.
0
 jnwo t
dt.
Properties of Fourier Series
1. If w(t) is real, cn  c* n .
2. If w(t) is real and even,Im[cn ]  0.
3. If w(t) is real and odd, Re[cn ]  0.
1
4. P arseval's theorem:
To
bn
 an

j
 2
2

5. cn  
a0
 a n
b n

j
 2
2
To

w(t) dt 
0
n0
n0
n0
2

c
n  
n
2
.
P roof of P arseval's theorem:
1
To
To

0
1
w(t) dt 
To
2
1

To
To
 w(t)w*(t)dt
0
To

 cne


j 2 nwo t
0 n  


* 1
   cn cm
To
n   m  



 c c 
n   m  


c
n  
n
2
*
n m nm
*  j 2 mw o t
c
dt
 me
m  
To

0
e j 2 wo ( n  m )t dt
Impulse Response
For linear time invariantsystems(without artificialtimedelay), theimpulse responsefunction,h(t),
describes thesystemcompletely.
y (t )  h(t ) if x(t )   (t )
For physicalsystems,h(t)  0, for t  0 (Causality).

For arbitraryx(t), y(t) x(t)  h(t ) 
 x( )h(t   )d

T ransferFunctionH ( f ) : h(t )  H ( f )
Y(f)  X(f)H(f) or equivalently H(f) 
Y( f )
( H(f) is also called thefrequencyresponsefunction.)
X(f)
For a physicalsystem,h(t)is a real function.T hen, H(f) is even and H(f) is odd.
H(f) relatesthemagnitudesand thephaseangles of theinputsand theoutputsat frequency f .
For example,if x(t)  A cos2πft , then y(t)  A H(f) cos2πft  H(f).
Example: RC Filt er
x(t): input voltage
y (t): output voltage
x(t )  Ri(t )  y (t ) and i (t )  C
dy(t )
dt
dy(t )
 y (t )  x(t )
dt
T aketheFourier T ransformof both sides of theaboveequation.
RC
RC( j 2f )Y ( f )  Y ( f )  X ( f )
Y( f )
1
H( f ) 

X ( F ) 1  ( j 2RC) f
 1  t
 e o
 o
 h(t )  
 0


t0
t0
where o  RC (timeconstant ).
P ower T ransferFunct ion: Gh ( f )  H(f) 
2
1
 f 
1   
 fo 
2
with f o  1 / 2πRC.
Gh ( f )  0.5 if f  f o .
f o is called the3-dB frequencybecause
10log Gh(fo )  10log 0.5  3.
At f o , thepower of y(t) is a half of (or 3 dB less than)x(t).
Distortion
Communication systemis said to be distortionless, if
y (t )  Ax(t  Td )
A : amplitudegain,
Td : timedelay ( A and Td are constant.)
Y ( f )  AX ( f )e  j 2fTd
H ( f )  Ae j 2fTd
 H ( f )  A and H(f)   2Td  f
For communication system to be distortionless :
1. H ( f ) is constantfor all f .
2. H(f) is a linear functionof f .
Note: Sections 2.7 and 2.9 will be covered briefly.
Section 2.8 will not be covered.
Definition. A waveformw(t) is said to be (absolutely) bandlimited to B herzif
W(f)  0 for f  B.
Definition. : A waveformw(t) is said to be (absolutely) timelimited toT secondsif
w(t)  0 for t  T .
T heorem.A bandlimited waveformcannotbe timelimited,and vice versa.
Sampling
Definition. Samplingis a processof evaluatingsignal w(t) at a discreteset of points,
t1,t2 ,t3 ,t4 , ....in time. Usually,t1  nTs 
n
. (Ts : samplingperiod. f s : samplingfrequency.)
fs
w(t)
t1
t2
t3
t4
t5
t
If w(t) is sufficiently smooth,thenw(t) can be completelyreconstructed from thesamplesw(tn ), n  1,2 ,3, ....
S am plin gTh e ore m
Any physicalwaveform(i.e.,finit eenergysignal) may be represented over theint erval -  t   by

w(t )   an


sinf s t  n / f s 
sinf s t  n / f s 
where an  f s  w(t )
dt.

f s t  n / f s 
f s t  n / f s 
Note. f s has to sat isfy a certainminimum value.
If w(t) is bandlimited to B hertzand f s  2 B, an  w(n / f s ).
T o reconstruct a bandlimited waveform(to B hertz)without error:  f s min  2 B.
 f s min is known as theNyquist frequency.
Interprettion
a : Bandlimited signals are equivalent to a certainsum of Sinc functions.
1
, w(t)can be compltely
2B
reconstructed from thesamplesw(nTs ), n  1,2,3,.... T hisis a fundamental principlewhich
If w(t) is bandlimited to B hertz,and w(t) is known at intervalsof Ts 
governsthe tranmission of analogsignals throughdigital communication system.
 If w(t) is sufficiently smooth,thenw(t) can be completelyreconstructed from thesamplesw(tn ), n  1,2,3, ....

Let ws(t)  w(t )  (t  nTs ). ws(t) : impulse sampledseries of w(t ).


ws(t)   w(nTs ) (t  nTs )
w(t)


1 jn2f s t
e
T
 s
ws(t)  w(t )
  1 jn2f s t 
Ws(f)  W ( f )  F  e

   Ts


1
 W ( f )  F e jn2f s t
Ts


t2
t3
t4
t5
t4
t5
t
ws(t)


1
 W ( f )     f  nfs 
Ts

1 
 W ( f  nfs )
Ts  
t1
t1
t2
t3
t
w(t)
sampler
(t-nTs)
ws(t)
Low Pass Filter
Ideal cutoff at B
w(t)
If fs < 2B, the sampling rate is insufficient, i.e., there aren’t enough samples
to reconstruct the original waveform.  Aliasing or spectral folding.
 The original waveform cannot be reconstructed without distortion.
Dimensionality Theorem
For a bandlimited waveform with bandwidth B hertz, if the waveform
can be completely specified (i.e., later reconstructed by an ideal
low pass filter) by N=2BTo samples during a time period of To,
then N is the dimension of the wave form.
Conversely, to estimate the bandwidth of a waveform, find a number
N such that N=2BTo is the minimum number of samples needed to reconstruct
the waveform during a time period To. Then B follows.
As To  , any approximation goes to zero.
A slightly modified version of this theorem is the Bandpass Dimensionality
Theorem: Any bandpass waveform (with bandwidth B) can be determined
by N=2BTo samples taken during a period of To.
Data Rate Theorem (Corollary to Dimensionality Theorem)
The maximum number of independent quantities which can be transmitted by
a bandlimited channel (B hertz) during a time period of To is N=2BTo.
Definition. The baud rate of a digital communication system is the rate of
symbols or quantities transmitted per second.
From the Data Rate Theorem, the maximum baud rate of a system with a
bandlimited channel (B hertz) is 2B symbols / second.
Definition. The data rate (or bit rate), R, of a system is the baud rate times the
information content per symbol (H): R= 2BH bits / second
Suppose a source transmits one of M equally likely symbols. The information
content of each symbol: H = log 2 (1/probablity of each symbol) = log2 M
 R= 2Blog2 M
Data rate is (also known as the Channel Capacity) is determined by (1) channel
bandwidth and (2) channel SNR.
Diffe re n De
t fin ition
s of Ban dwidth
(Commonlyused terms)
1. Absolute bandwidth f 2 - f1 : For f2  f or f  f1 , W(f)  0.
2. 3-dB (or half power) bandwidth f 2 - f1 : For any f1  f*  f 2 , H(f* ) 
2
value of H(f) over f 2  f  f1 .
2
(Less useful definitions)
3. Equivalentbandwidth
4. Null to null bandwidth
5. Bounded spectrumbandwidth
6. P ower bandwidth
7. FCC bandwidth
1
of themaximum
2