1
Chapter 2
Signals and Spectra
2
Introduction
? Basic signal properties(dc,rms,dBm,
and power)
? Fourier transform and spectra
? Linear systems and linear distortion
? Bandlimited signal and sampling
? Discrete Fourier transform
? Bandwidth of signal
3
2.1 Properties of signal and Noise
(Properties of Physical Waveform)
? the waveform has significant nonzero values over a
composite time interval that is finite
? The spectrum of the waveform has significant values
over a composite frequency interval that is finite
? The waveform is a continuous function of time
? The waveform has a finite peak value
? The waveform has only real values,That is,at any
time,it cannot have a complex value a+bj,where b is
nonzero,
4
2.1 Properties of signal and Noise
5
2.1 Properties of signal and Noise
? Time average operator
? ? ? ? 1)-(2 1lim 2/ 2/ dtT T TT ???? ???
? Periodic waveform with period T0
3)-(2 t a llf o r )()( 0Ttt ?? ??
? Time average operator for periodic waveform
? ? ? ? 4)-(2 1 )2/( )2/(
0
dtT aT aT? ? ?? ???
6
2.1 Properties of signal and Noise
? Dc value
5)-(2 )(1lim 2/ 2/ dttTW T TTdc ????? ?
? Instantaneous Power
6)-(2 )()()( titvtp ?
7)-(2 )()()( titvtpP ??
? Average Power
12)-(2 )(
)( 2222
r m sr m sr m sr m s IVRIR
VRti
R
tv
P ?????
c ir c ui t
i ( t)
v ( t)
? Theorem,If a load is resistive,the average power is,
? Where R is value of the resistive load
7
2.1 Properties of signal and Noise
? Rms Value
)(2 tW rm s ??
? Periodic waveform with period T0
)()( 0Ttt ?? ??
? Time average operator for periodic waveform
? ? ? ?dt
T
aT
aT?
?
?? ???
)2/(
)2/(
0
1
8
2.1 Properties of signal and Noise
? Example 2-1(p37)
9
2.1 Properties of signal and Noise
? Average normalized power
?????? 2/ 2/ 22 )(1lim)( T TT dttTtP ??? total normalized energy
????? 2/ 2/ 2 )(lim T TT dttE ?
? Power waveform ω(t) is a power waveform if
and only if the average normalized power P is
finite and nonzero (i.e,0<P<∞)
? Energy waveform ω(t) is a energy waveform
if and only if the average total normalized
energy E is finite and nonzero (i.e,0<E<∞)
10
2.1 Properties of signal and Noise
? Decibel(dB) this is a base 10 logarithmic measure of
power ratios,for a circuit,
???
?
???
??
???
?
???
??
inp o w er
o u tp o w er l o g10l o g10
a ver a g e
a ver a g e
P
PdB
in
out
?
?
?
?
?
?
?
?
?
ininr m s
o u to u tr m s
RV
RV
dB
/
/
l o g10 2
_
2
_
???
?
???
??
???
?
???
??
???
?
???
??
???
?
???
??
?
?
?
?
in
out
inr m s
outr m s
out
in
inr m s
outr m s
R
R
I
IdB
R
R
V
VdB l o g10l o g20o r l o g10l o g20
10/10 dB
in
o u t
P
P ??
? In Engineering Practice,
???
?
???
??
???
?
???
??
?
?
?
?
inr m s
outr m s
inr m s
outr m s
I
I
V
VdB l o g20 l o g20
11
2.1 Properties of signal and Noise
? Decibel signal-to-noise ratio is
???
?
???
??
n o i s e
s i g n a l
dB P
PNS l o g10)/(
??
?
?
?
??
?
?
?
?
)(
)(
log10
2
2
tn
ts
???
?
???
??
?
?
n o i serm s
si g n a lrm s
V
Vl o g20
12
2.1 Properties of signal and Noise
? Decibel measure may also be used to indicate
absolute levels of power with respect to some reference
level, for example,when 1mW reference level is used
)( w a t t s ) l e v e lp o w e r a c t u a ll o g (1030
10
( w a t t s ) l e v e lp o w e r a c t u a l
l o g10 3
??
?
?
?
?
?
??
?d B m
? When 1W reference level is used,the dB level is denoted dBW,
? when 1kW reference level is used,the dB level is denoted dBk
? When 1 millivolt rms level across a 75 Ohm load is used
as a reference,dB level is called dBmV,and is defined as
)10l o g (20 3?? rm sVd B m V
13
2.1 Properties of signal and Noise
? Phasors A complex number c is said to
be phasor if it is used to represent a
sinusoidal waveform,That is
)c o s (||)( 0 ctct ??? ??
Where c=x+jy=|c|ejΦ
25sin(2π500t+450) could be denoted by the
phasor 04525 ??
10cos(ωt+350) could be denoted by the
phasor
03510 ?
14
2.2 Fourier Transform and
Spectra
? The Fourier Transform of a waveform
ω(t) is,
? ? ?? ??? dtettfW ftj ??? 2)()]([)( FWhere f is the frequency parameter with units of
Hz(1/s),W(f) is also called a two-sided spectrum
of ω(t),It should be clear that the spectrum of a
voltage(or current)waveform is obtained by a
mathematical calculation and that it does not
appear physically in an actual circuit,f is just a
parameter that determines which point of the
spectral function is to be evaluated,
15
2.2 Fourier Transform and
Spectra
? Parseval’s theorem
?? ? ??? ?? ? dffWfWdttt )()()()( *21*21 ??
)]([)()],([)( 2211 tfWtfW ?? FF ??
where
when
)()()( 21 ttt ??? ??
Rayleigh’s energy theorem is obtained,
?? ? ??? ?? ?? dffWdttE 221 )()(?
Energy spectrum density is,
2)()( fWf ?E
16
2.2 Fourier Transform and
Spectra
? Dirac delta function
? Definition 1 )0()()( ??? ?? ?
?? dttt
? Definition 2 both 1)( ???
?? dtt?
and
??
?
?
???
00
0)(
x
xx? need to be satisfied
? Definition 3
???? ?? dyex xyj 2)(?
Sifting property,
)()()( 00 xdxxxx ??? ?? ????
17
2.2 Fourier Transform and
Spectra
? Unit step function
??
?
?
??
00
01)(
t
ttu
? Relationship between u(t) and δ(t)
)()( tudxxt ?? ?? ?
and
)()( t
dt
tdu ??
18
2.2 Fourier Transform and Spectra
? Some useful pulses
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
??
Tt
TtTt
T
t
x
x
xSa
Tt
Tt
T
t
0
/||1
)(
sin
)(
2/0
2/1
)(
?
? Fig.2-6 p56
19
2.3 Power Spectral Density and
Autocorrelation Function
? Power Spectral Density(PSD)
The normalized power of a waveform will
now be related to its frequency domain
description by the use of a function
called the PSD,The PSD is very useful in
describing how the power content of
signals and noise is affected by filters
and other devices in communication
system,
20
2.3 Power Spectral Density and
Autocorrelation Function
? The average normalized power in time domain
description is,
?? ? ??????? ?? dttTdttTP TTT TT )(1lim)(1lim 22/ 2/ 2 ??
Where ωT(t)= ω(t)Π(t/T) is the truncated version
of the waveform,By the use of Parseval’s
theorem,we have,
??
?
?? ??
?
???? ??
?
?
?
??
?
?
?
?? 65)-(2
)(
l i m)(
1
l i m
2
2
df
T
fW
dffW
T
P
TT
21
2.3 Power Spectral Density and
Autocorrelation Function
? For a deterministic power waveform,the
PSD is
6 6 )-(2 )(l i m)(
2
T
fWf
T ??
??P
? The normalized average power can be
rewrite as,
6 7 )-(2 )()(2 ?????? dfftP ?? P
22
2.3 Power Spectral Density and
Autocorrelation Function
? Autocorrelation Function
68)-(2 )()(1l i m)()()( 2/ 2/???? ???? T TT dtttTttR ????????
? For a real waveform the autocorrelation
function is,
? The PSD and the autocorrelation function are
Fourier transform pairs,
6 9 )-(2 )()( fR ?? ? P?
? The PSD can be obtained by either of the
following two methods,
1) Direct method,
2) InDirect method,? ?
)()( ??? Rf F?P
T
fWf
T
2)(
lim)(
??
??P
23
2.3 Power Spectral Density and
Autocorrelation Function
? The total average normalized power for the
waveform ω(t) can be evaluated by using any
of the four techniques embedded in the
following equation,
)0()()( 22 ??? RWdfftP r m s ???? ? ? ?? P? Other equivalent ways of modeling the
waveform are use of a Taylor series expansion
about a point a,that is,
?
?
?
??
0
)(
)(! )()(
n
n
n
atn at ??
? Where
n
n
n
dt
tda )()()( ?? ?
24
2.4 orthogonal series representation
of signals and noise
? Orthogonal function
Function φn(t) and φm(t) are said to be orthogonal with
respect to each other over the interval a<t<b if n is not
equal to m,the condition is satisfied,
77)-(2n m 0)()( * ??? ba mn dttt ?? Furthermore,if the functions in the set {φ
n(t) } are orthogonal,then they also satisfy the relation
7 8,7 9 )-(2 mn mn 10 0)()( * ??
?
?
???
?
?
?
?
?
??
?
?
? ?
?
???
mnmnn
n
b
a mn KnmK
nmdttt ????
if the constants K are all equal to 1,the φn(t) are said to
be orthonormal functions
25
2.4 orthogonal series representation
of signals and noise
? Orthogonal Series
ω(t) can be represented over the interval(a,b) by
the series,
8 3 )-(2 )()( ??
n nn
tat ??Where the orthogonal coefficients are given by,
84)-(2 )()( *?? ba nn dttta ??
And the range of n is over the integer values that
correspond to the subscripts that were used to
denote the orthogonal function in the complete
orthogonal set,
26
2.4 orthogonal series representation
of signals and noise
? The orthogonal series is very useful in representing a
signal,noise,or signal-noise combination,
27
2.5 Fourier Series
? A physical waveform may be represented over the
interval a<t<a+T0 by the complex exponential Fourier
series,
( 2, 8 8 ) )( 0?
?
???
?
n
tjn
n ect
??
? Where the complex(phasor) Fourier coefficients are,
( 2, 8 9 ) )(1 0 0
0
? ? ?? Taa tjnn dtetTc ??
? And
000 /22 Tf ??? ??
28
2.5 Fourier Series
? line spectrum for periodic waveforms
? the spectrum of the waveform ω(t) with
period T0 is,
?
?
???
??
n
n nffcfW )()( 0?
Where f0=1/T0 and cn are the phasor Fourier
coefficients of the waveform as given by
? ? ?? 0 0)(1
0
Ta
a
tjn
n dtetTc
??
29
2.5 Fourier Series
? If ω(t) is a periodic function with period T0 and is
represented by,
?? ?
???
?
???
???
n
tjn
n
n
ecnTtht 0)()( 0 ??
? Where
)(
)2/|(|
0
)()( 0
e l s e w h e r e
Tttth ?
?
?
?? ?
? Then the Fourier coefficients are given by
)( 00 nfHfc n ?
30
2.5 Fourier Series
? For a periodic waveform ω(t),the
normalized power is given by,
( 2, 1 2 4 ) )( 22 ?
?
???
??
n
nctP ??
? Where the {cn} are the complex Fourier
coefficients for the waveform
31
2.5 Fourier Series
? Power spectral density for Periodic waveform
? For a periodic waveform,the PSD is given
by,
( 2, 1 2 6 ) )()( 02?
?
???
??
n
n nffcf ??P
? Where T0=1/f0 is the period of the
waveform and the {cn} are the
corresponding Fourier coefficients for
the waveform
32
2.6 Linear Systems
? An electronic filter or system is linear when
superposition holds,
? ? ? ? ? ?)()()()()( 22112211 txLatxLatxatxaLty ????Where y(t) is the
output and
x(t)=a1x1(t)+a1
x1(t) is the
input,as
shown in the
following Fig,
Fig 2-14
33
2.6 Linear Systems
? The impulse response is the solution to the
differential equation when the forcing fuction
is a Dirac delta function,that is y(t)=h(t) when
x(t)=δ(t),In physical networks,the impulse
response has to be causal,That is,h(t)=0 for
t<0,
An linear time-invariant system can be
expressed by convolution operation,
)(*)()()()( thtxdthxty ??? ? ? ?? ???
Taking the Fourier transform of both sides,
)(/)()()()()( fXfYfHfXfHfY ???
34
2.6 Linear Systems
? If a sinusoidal testing signal within the
frequency band of interest is x(t)=Acosω0t for
a linear system,then the output of system is,
))(c o s (|)(|)( 000 fHtfHAty ??? ?
? If the input is a periodic signal with a spectrum
give by
? then the spectrum of periodic output signal is,
?
?
???
??
n
n nffcfX )()( 0?
?
?
???
??
n
n nffnfHcfY )()()( 00 ?
? The relationship between the PSD is,
)()()( 2 ffHf xy PP ?
35
2.6 Linear Systems
? Distortionless Transmission
? If the communication channel is distortionless,
then the channel output is just proportional to
a delayed version of the input,
dfTjd efAXfYTtAxty ?2)()()()( ?????
? Thus,for distortionless,we require that the
transfer function of the channel be given by,
1 4 9 )-(2 )( )(2 fjfTj AeAefH d ?? ?? ?
? where θ(f)=-2πfTd is a linear function of delay
time Td,
36
2.6 Linear Systems
? To have no distortion at the output of a linear
time-invariant system,two requirement must
be satisfied,
? 1) the amplitude response is flat,That is
|H(f)|=constant=A;
? 2) the phase response is a linear function of
frequency,that is,θ(f)=-2πfTd
? When the first condition is satisfied,there is
no amplitude distortion; when the second
condition is satisfied,there is no phase
distortion,
37
2.7 Bandlimited signals and noise
Bandlimited Waveforms
A waveform ω(t) is said to be Bandlimited
to B hertz if W(f)=0 for |f|≥B
Time-limited Waveforms
A waveform ω(t) is said to be time limited
to B hertz if ω(t) =0 for |t|≥T
Theorem
An absolutely bandlimited waveform
cannot be absolutely time limited and
vice versa,
38
2.7 Bandlimited signals and noise
(sampling theorem)
? Any physical waveform may be represented over the
interval -∞<t< ∞ by,
??
??? ?
??
n ss
ssn
fntf
fntfat
))/((
) ) }/((s i n {)(
?
??
? ??? ? ?? ))/(( ) ) }/((s i n {)(
ss
sssn
fntf
fntftfa
?
??
1 6 0 / 1 6 7 )-(2 )()/( )/(2? ??? ??? dfefWfna sfnffsn ??
? Where,
? And fs is a parameter that is assigned some convenient
value greater than zero,Furthermore if ω(t) is
bandlimited to B hertz,and fs>=2B,then,
? That is,for fs>=2B,the orthogonal series coefficients are
simply the values of the waveform that are obtained
when the waveform is sampled every 1/fs seconds
39
2.7 Bandlimited signals and noise
(sampling theorem)
? Proof,
))/((
) ) ]/((s i n [)(
ss
ss
n fntf
fntft
?
??
?
??
nmnmn Kdttt ??? ????? )()( *
??? ?????? dfffdttt mnmn )()()()( ** ????
)/(21)]([)( sfnfj
ss
nn ef
f
ftf
??? ?? ??
?
?
???
??? F
nm
s
f
f
ffmnj
s
mn
f
dfe
f
dfff
s
s
s ?
??
? 11
)()(
2/
2/
)/)((2
2
*
???
?
?
??
?
??
)/(
)(
)()(
)()(
)/(2
*
*
s
fnfj
n
nsn
fn
dfefW
dfffW
dtttfa
s
?
?
??
?
?
??
??
??
?
??
?
?
??
?
??
40
2.7 Bandlimited signals and noise
? It is obvious that the minimum sampling rate allowed to
reconstruct a bandlimited waveform without error is
given by,(fs)min=2B this is called the Nyquist frequency
41
2.7 Bandlimited signals and noise
? How to reproduce a bandlimited waveform by using N
sample value
?
?
?
?
Nn
nn
nn tat
1
1
)()( ??
))/((
))}/((s i n {)(
ss
ss
n fntf
fntft
?
??
?
??
?where
?each samples value is multiplied by the appropriate
(sinx/x) function,and these weighted (sinx/x) functions are
summed to give the original waveform,The minimum
number of sample values that are needed to reconstruct
the waveform is,
N=T0/(1/fs)=fsT0>=2BT0
42
2.7 Bandlimited signals and noise
Impulse sampling and Digital Signal Processing(DSP)
Another useful orthogonal series is the impulse-sampled
series obtained when the sinx/x orthogonal functions of
the sampling theorem are replaced by an orthogonal set
of delta function,The impulse-sampled series is also
identical to the impulse-sampled waveform ωs(t),both
can be obtained by multiplying the unsampled
waveform by a unit-weight impulse train,
tjn
n sn ssn ss
se
TtnTtnTnTttt
??????? ??? ?
???
?
???
?
???
????? 1)()()()()()(
)1732( )(1)( ??? ?
?
???n
s
s
s nffWTfW
Taking the Fourier transform of both sides we get,
43
2.7 Bandlimited signals and noise
? The spectrum of the impulse sampled signal is the
spectrum of the unsampled signal that is repeated
every fsHz,where fs is the sampling frequency,
44
2.7 Bandlimited signals and noise
? Note,this technique of impulse sampling may be used
to translate the spectrum of a signal to another
frequency band that is centered on some harmonic of
the sampling frequency,
? If fs>=2B,the replicated spectra do not overlap,thus
ω(t) is reproduced from ωs(t) by passing ωs(t)
through an ideal low-pass filter that has a cutoff
frequency of fc=fs/2,where fs>=2B
? If fs<2B,the spectra of ωs(t) will consist of overlapped,
replicated spectra of ω(t),The spectral overlap or tail
inversion,is called aliasing or spectral folding,
45
2.7 Bandlimited signals and noise
? Fig,2-19
46
2.7 Bandlimited signals and noise
? Dimensionality theorem
? When BT0 is large,a real waveform may be
completely specified by
N=2BT0 (2.174)
? independent pieces of information that will
describe the waveform over a T0 interval,N is
said to be the number of dimensions required
to specify the waveform,and B is the absolute
bandwidth of the waveform,
47
2.7 Bandlimited signals and noise
? The demensionality theorem imply that the
information which can be conveyed by a bandlimited
waveform or a bandlimited communication system is
proportional to the product of the bandwidth of that
system and the time allowed for transmission of the
information
? The demensionality theorem has profound
implications in the design and performance of all
types of communication systems,
? Application,1) it can be used to calculate the number
of storage locations required to represent a waveform;
2) is used to give a lower bound for the bandwidth of
digital signals,
48
2.8 Discrete Fourier Transform
? Definition,the discrete Fourier transform is
defined by,
Where n=0,1,2,…,N -1,and the inverse discrete
Fourier transform is defined by,
Where k=0,1,2,…,N -1
Matlab notation are X=fft(x) and x=ifft(X)
respectively,
??
?
?? 1
0
)/2()()( N
k
nkNjekxnX ?
??
?
?? 1
0
)/2()(1)( N
n
nkNjenX
Nkx
?
49
2.8 Discrete Fourier Transform-
windowing,sampling and periodic sample genernation
50
2.8 Discrete Fourier Transform-Using
the DFT to compute the Contimuous Fourier Transform
? The windowed waveform,denoted by the
subscript w,is,
? ?????? ????? ??? TTttttw )2/()(e l s e w h e r e t 0 Tt0 )()( ???
Tr
o c t Con F T
? thus
?
??
?
?
?
?
??
??
???
??
?????? ??
??
1
0
)/2(
/
/
,/,
2
0
2
)(|)(
)()()(
N
k
nkNj
Tnfw
NTtdt
Tnftkt
ftjTftj
ww
tetkfW
dtetdtetfW
???
??
??
?
??
Because of then the
relationship between the CFT and DFT is,?
?
?
?? 1
0
)/2()()( N
k
nkNjekxnX ?
)(|)( / ntXfW Tnfw ???
51
2.8 Discrete Fourier Transform
? and it can modified to give spectral values over
the entire fundamental range of –fs/2<f<fs/2,
thus,for positive frequencies we use,
N / 2n0 )(|)( / ???? ntXfW Tnfw ?
? and for negative frequencies we use,
NnN / 2 )(|)( /)( ????? ntXfW TNnfw ?
? Note the fact that if one is not careful,the DFT
may give significant errors when it is used to
approximate the CFT,The factor that cause the
errors may be categorized into three basic effects,
leakage,aliasing,and the picket-fence effect,
52
2.8 Discrete Fourier Transform
? The leakage effects is caused by windowing in the
time domain,
? Because the spectrum of a sampled waveform consists
of replicating the spectrum of the un-sampled
waveform about harmonics of the sampling frequency,
if fs<2B,where fs=1/Δt,and B is the highest significant
frequency component in the unsampled waveform,
Aliasing errors will occur,
? The picket-fence effect occurs because the N-point
DFT cannot resolve the spectral components any
closer than the spacing Δf=1/T,Δf can be decreased by
increasing T,
53
2.9 Bandwidth of Signals
? The spectral width of signals and noise in
communication systems is a very important concept,
the reason is,
? More and more users are being assigned to
increasingly crowded RF bands,so that the spectral
width required for each one needs to be considered
carefully,
? Second,the spectral width is important from the
equipment design viewpoint,since the circuits need to
have enough bandwidth to accommodate the signal
and reject the noise,
54
2.9 Bandwidth of Signals
? In engineering definitions,the bandwidth is taken to
be the width of a positive frequency band,In other
words,the bandwidth is,
f2-f1
Where f2>f1>=0
? For baseband signals,f1=0;
? For bandpass signals,f1>>0,and the bandwidth f2-f1
encompasses the carrier frequency fc,
? When we increase the signaling speed of a signal
(decrease T),the spectrum gets wider,Consequently,
for engineering definitions of bandwidth,we require
the bandwidth to vary as 1/T
55
2.9 Bandwidth of Signals
? 1) absolute bandwidth is f2-f1,where the spectrum is
zero outside the interval f1<f<f2 along the positive
frequency axis
? 2) 3dB bandwidth (or half-power bandwidth) is f2-f1,
where for frequencies inside the band f1<f<f2,the
magnitude spectra,say,|H(f)|,fall no lower than 1/√2
times the maximum value of |H(f)|,and the maximum
value occurs at a frequency inside the band,
? 3) equivalent noise bandwidth is the width of a
fictitious rectangular spectrum such that the power in
that rectangular band is equal to the power associated
with the actual spectrum over positive frequencies,
That is,if f0 is the frequency at which the magitude
spectrum has maximum,
56
2.9 Bandwidth of Signals
? 4) Null-to-null bandwidth (or zero-crossing bandwidth)
is f2-f1,where f2 is the first null in the envelope of the
magnitude spectrum above f0,for bandpass systems,f1 is
the first null in the envelope of the magnitude spectrum
below f0,where f0 is the frequency where the magnitude
spectrum is a maximum,For baseband systems,f1 is
usually zero,
? 5) bounded spectrum bandwidth is f2-f1 such that outside
the band f1<f<f2,the PSD must be down by at least a
certain amount,say,50dB,below the maximum value of
the power spectral density,
? 6) power bandwidth is f2-f1,where f1<f<f2 defines the
frequency band in which 99% of the total power resides,
)192-2( |)(|
|)(|
1
=B
|)(|=p o w er act u al=|)(|B =p o w er eq u i v al en t
∫
∫
∞
0
2
2
0
∞
0
22
0eq
dffH
fH
dffHfH
eq
57
2.9 Bandwidth of Signals
? FCC bandwidth is an authorized bandwidth parameter
assigned by the FCC to specify the spectrum allowed in
communication systems,This is not an engineering
definition,
58
2.9 Bandwidth of Signals
Example2-18 Bandwidths for a BPSK signal (p106)
59
Homework
? Problems,
2.7,2.10,2.44,2.45,2.65,2.89,2.91,2.92
? Reading
§ 2-1,§ 2-7,§ 2-9 in detail
? Extend reading the others
Chapter 2
Signals and Spectra
2
Introduction
? Basic signal properties(dc,rms,dBm,
and power)
? Fourier transform and spectra
? Linear systems and linear distortion
? Bandlimited signal and sampling
? Discrete Fourier transform
? Bandwidth of signal
3
2.1 Properties of signal and Noise
(Properties of Physical Waveform)
? the waveform has significant nonzero values over a
composite time interval that is finite
? The spectrum of the waveform has significant values
over a composite frequency interval that is finite
? The waveform is a continuous function of time
? The waveform has a finite peak value
? The waveform has only real values,That is,at any
time,it cannot have a complex value a+bj,where b is
nonzero,
4
2.1 Properties of signal and Noise
5
2.1 Properties of signal and Noise
? Time average operator
? ? ? ? 1)-(2 1lim 2/ 2/ dtT T TT ???? ???
? Periodic waveform with period T0
3)-(2 t a llf o r )()( 0Ttt ?? ??
? Time average operator for periodic waveform
? ? ? ? 4)-(2 1 )2/( )2/(
0
dtT aT aT? ? ?? ???
6
2.1 Properties of signal and Noise
? Dc value
5)-(2 )(1lim 2/ 2/ dttTW T TTdc ????? ?
? Instantaneous Power
6)-(2 )()()( titvtp ?
7)-(2 )()()( titvtpP ??
? Average Power
12)-(2 )(
)( 2222
r m sr m sr m sr m s IVRIR
VRti
R
tv
P ?????
c ir c ui t
i ( t)
v ( t)
? Theorem,If a load is resistive,the average power is,
? Where R is value of the resistive load
7
2.1 Properties of signal and Noise
? Rms Value
)(2 tW rm s ??
? Periodic waveform with period T0
)()( 0Ttt ?? ??
? Time average operator for periodic waveform
? ? ? ?dt
T
aT
aT?
?
?? ???
)2/(
)2/(
0
1
8
2.1 Properties of signal and Noise
? Example 2-1(p37)
9
2.1 Properties of signal and Noise
? Average normalized power
?????? 2/ 2/ 22 )(1lim)( T TT dttTtP ??? total normalized energy
????? 2/ 2/ 2 )(lim T TT dttE ?
? Power waveform ω(t) is a power waveform if
and only if the average normalized power P is
finite and nonzero (i.e,0<P<∞)
? Energy waveform ω(t) is a energy waveform
if and only if the average total normalized
energy E is finite and nonzero (i.e,0<E<∞)
10
2.1 Properties of signal and Noise
? Decibel(dB) this is a base 10 logarithmic measure of
power ratios,for a circuit,
???
?
???
??
???
?
???
??
inp o w er
o u tp o w er l o g10l o g10
a ver a g e
a ver a g e
P
PdB
in
out
?
?
?
?
?
?
?
?
?
ininr m s
o u to u tr m s
RV
RV
dB
/
/
l o g10 2
_
2
_
???
?
???
??
???
?
???
??
???
?
???
??
???
?
???
??
?
?
?
?
in
out
inr m s
outr m s
out
in
inr m s
outr m s
R
R
I
IdB
R
R
V
VdB l o g10l o g20o r l o g10l o g20
10/10 dB
in
o u t
P
P ??
? In Engineering Practice,
???
?
???
??
???
?
???
??
?
?
?
?
inr m s
outr m s
inr m s
outr m s
I
I
V
VdB l o g20 l o g20
11
2.1 Properties of signal and Noise
? Decibel signal-to-noise ratio is
???
?
???
??
n o i s e
s i g n a l
dB P
PNS l o g10)/(
??
?
?
?
??
?
?
?
?
)(
)(
log10
2
2
tn
ts
???
?
???
??
?
?
n o i serm s
si g n a lrm s
V
Vl o g20
12
2.1 Properties of signal and Noise
? Decibel measure may also be used to indicate
absolute levels of power with respect to some reference
level, for example,when 1mW reference level is used
)( w a t t s ) l e v e lp o w e r a c t u a ll o g (1030
10
( w a t t s ) l e v e lp o w e r a c t u a l
l o g10 3
??
?
?
?
?
?
??
?d B m
? When 1W reference level is used,the dB level is denoted dBW,
? when 1kW reference level is used,the dB level is denoted dBk
? When 1 millivolt rms level across a 75 Ohm load is used
as a reference,dB level is called dBmV,and is defined as
)10l o g (20 3?? rm sVd B m V
13
2.1 Properties of signal and Noise
? Phasors A complex number c is said to
be phasor if it is used to represent a
sinusoidal waveform,That is
)c o s (||)( 0 ctct ??? ??
Where c=x+jy=|c|ejΦ
25sin(2π500t+450) could be denoted by the
phasor 04525 ??
10cos(ωt+350) could be denoted by the
phasor
03510 ?
14
2.2 Fourier Transform and
Spectra
? The Fourier Transform of a waveform
ω(t) is,
? ? ?? ??? dtettfW ftj ??? 2)()]([)( FWhere f is the frequency parameter with units of
Hz(1/s),W(f) is also called a two-sided spectrum
of ω(t),It should be clear that the spectrum of a
voltage(or current)waveform is obtained by a
mathematical calculation and that it does not
appear physically in an actual circuit,f is just a
parameter that determines which point of the
spectral function is to be evaluated,
15
2.2 Fourier Transform and
Spectra
? Parseval’s theorem
?? ? ??? ?? ? dffWfWdttt )()()()( *21*21 ??
)]([)()],([)( 2211 tfWtfW ?? FF ??
where
when
)()()( 21 ttt ??? ??
Rayleigh’s energy theorem is obtained,
?? ? ??? ?? ?? dffWdttE 221 )()(?
Energy spectrum density is,
2)()( fWf ?E
16
2.2 Fourier Transform and
Spectra
? Dirac delta function
? Definition 1 )0()()( ??? ?? ?
?? dttt
? Definition 2 both 1)( ???
?? dtt?
and
??
?
?
???
00
0)(
x
xx? need to be satisfied
? Definition 3
???? ?? dyex xyj 2)(?
Sifting property,
)()()( 00 xdxxxx ??? ?? ????
17
2.2 Fourier Transform and
Spectra
? Unit step function
??
?
?
??
00
01)(
t
ttu
? Relationship between u(t) and δ(t)
)()( tudxxt ?? ?? ?
and
)()( t
dt
tdu ??
18
2.2 Fourier Transform and Spectra
? Some useful pulses
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
??
Tt
TtTt
T
t
x
x
xSa
Tt
Tt
T
t
0
/||1
)(
sin
)(
2/0
2/1
)(
?
? Fig.2-6 p56
19
2.3 Power Spectral Density and
Autocorrelation Function
? Power Spectral Density(PSD)
The normalized power of a waveform will
now be related to its frequency domain
description by the use of a function
called the PSD,The PSD is very useful in
describing how the power content of
signals and noise is affected by filters
and other devices in communication
system,
20
2.3 Power Spectral Density and
Autocorrelation Function
? The average normalized power in time domain
description is,
?? ? ??????? ?? dttTdttTP TTT TT )(1lim)(1lim 22/ 2/ 2 ??
Where ωT(t)= ω(t)Π(t/T) is the truncated version
of the waveform,By the use of Parseval’s
theorem,we have,
??
?
?? ??
?
???? ??
?
?
?
??
?
?
?
?? 65)-(2
)(
l i m)(
1
l i m
2
2
df
T
fW
dffW
T
P
TT
21
2.3 Power Spectral Density and
Autocorrelation Function
? For a deterministic power waveform,the
PSD is
6 6 )-(2 )(l i m)(
2
T
fWf
T ??
??P
? The normalized average power can be
rewrite as,
6 7 )-(2 )()(2 ?????? dfftP ?? P
22
2.3 Power Spectral Density and
Autocorrelation Function
? Autocorrelation Function
68)-(2 )()(1l i m)()()( 2/ 2/???? ???? T TT dtttTttR ????????
? For a real waveform the autocorrelation
function is,
? The PSD and the autocorrelation function are
Fourier transform pairs,
6 9 )-(2 )()( fR ?? ? P?
? The PSD can be obtained by either of the
following two methods,
1) Direct method,
2) InDirect method,? ?
)()( ??? Rf F?P
T
fWf
T
2)(
lim)(
??
??P
23
2.3 Power Spectral Density and
Autocorrelation Function
? The total average normalized power for the
waveform ω(t) can be evaluated by using any
of the four techniques embedded in the
following equation,
)0()()( 22 ??? RWdfftP r m s ???? ? ? ?? P? Other equivalent ways of modeling the
waveform are use of a Taylor series expansion
about a point a,that is,
?
?
?
??
0
)(
)(! )()(
n
n
n
atn at ??
? Where
n
n
n
dt
tda )()()( ?? ?
24
2.4 orthogonal series representation
of signals and noise
? Orthogonal function
Function φn(t) and φm(t) are said to be orthogonal with
respect to each other over the interval a<t<b if n is not
equal to m,the condition is satisfied,
77)-(2n m 0)()( * ??? ba mn dttt ?? Furthermore,if the functions in the set {φ
n(t) } are orthogonal,then they also satisfy the relation
7 8,7 9 )-(2 mn mn 10 0)()( * ??
?
?
???
?
?
?
?
?
??
?
?
? ?
?
???
mnmnn
n
b
a mn KnmK
nmdttt ????
if the constants K are all equal to 1,the φn(t) are said to
be orthonormal functions
25
2.4 orthogonal series representation
of signals and noise
? Orthogonal Series
ω(t) can be represented over the interval(a,b) by
the series,
8 3 )-(2 )()( ??
n nn
tat ??Where the orthogonal coefficients are given by,
84)-(2 )()( *?? ba nn dttta ??
And the range of n is over the integer values that
correspond to the subscripts that were used to
denote the orthogonal function in the complete
orthogonal set,
26
2.4 orthogonal series representation
of signals and noise
? The orthogonal series is very useful in representing a
signal,noise,or signal-noise combination,
27
2.5 Fourier Series
? A physical waveform may be represented over the
interval a<t<a+T0 by the complex exponential Fourier
series,
( 2, 8 8 ) )( 0?
?
???
?
n
tjn
n ect
??
? Where the complex(phasor) Fourier coefficients are,
( 2, 8 9 ) )(1 0 0
0
? ? ?? Taa tjnn dtetTc ??
? And
000 /22 Tf ??? ??
28
2.5 Fourier Series
? line spectrum for periodic waveforms
? the spectrum of the waveform ω(t) with
period T0 is,
?
?
???
??
n
n nffcfW )()( 0?
Where f0=1/T0 and cn are the phasor Fourier
coefficients of the waveform as given by
? ? ?? 0 0)(1
0
Ta
a
tjn
n dtetTc
??
29
2.5 Fourier Series
? If ω(t) is a periodic function with period T0 and is
represented by,
?? ?
???
?
???
???
n
tjn
n
n
ecnTtht 0)()( 0 ??
? Where
)(
)2/|(|
0
)()( 0
e l s e w h e r e
Tttth ?
?
?
?? ?
? Then the Fourier coefficients are given by
)( 00 nfHfc n ?
30
2.5 Fourier Series
? For a periodic waveform ω(t),the
normalized power is given by,
( 2, 1 2 4 ) )( 22 ?
?
???
??
n
nctP ??
? Where the {cn} are the complex Fourier
coefficients for the waveform
31
2.5 Fourier Series
? Power spectral density for Periodic waveform
? For a periodic waveform,the PSD is given
by,
( 2, 1 2 6 ) )()( 02?
?
???
??
n
n nffcf ??P
? Where T0=1/f0 is the period of the
waveform and the {cn} are the
corresponding Fourier coefficients for
the waveform
32
2.6 Linear Systems
? An electronic filter or system is linear when
superposition holds,
? ? ? ? ? ?)()()()()( 22112211 txLatxLatxatxaLty ????Where y(t) is the
output and
x(t)=a1x1(t)+a1
x1(t) is the
input,as
shown in the
following Fig,
Fig 2-14
33
2.6 Linear Systems
? The impulse response is the solution to the
differential equation when the forcing fuction
is a Dirac delta function,that is y(t)=h(t) when
x(t)=δ(t),In physical networks,the impulse
response has to be causal,That is,h(t)=0 for
t<0,
An linear time-invariant system can be
expressed by convolution operation,
)(*)()()()( thtxdthxty ??? ? ? ?? ???
Taking the Fourier transform of both sides,
)(/)()()()()( fXfYfHfXfHfY ???
34
2.6 Linear Systems
? If a sinusoidal testing signal within the
frequency band of interest is x(t)=Acosω0t for
a linear system,then the output of system is,
))(c o s (|)(|)( 000 fHtfHAty ??? ?
? If the input is a periodic signal with a spectrum
give by
? then the spectrum of periodic output signal is,
?
?
???
??
n
n nffcfX )()( 0?
?
?
???
??
n
n nffnfHcfY )()()( 00 ?
? The relationship between the PSD is,
)()()( 2 ffHf xy PP ?
35
2.6 Linear Systems
? Distortionless Transmission
? If the communication channel is distortionless,
then the channel output is just proportional to
a delayed version of the input,
dfTjd efAXfYTtAxty ?2)()()()( ?????
? Thus,for distortionless,we require that the
transfer function of the channel be given by,
1 4 9 )-(2 )( )(2 fjfTj AeAefH d ?? ?? ?
? where θ(f)=-2πfTd is a linear function of delay
time Td,
36
2.6 Linear Systems
? To have no distortion at the output of a linear
time-invariant system,two requirement must
be satisfied,
? 1) the amplitude response is flat,That is
|H(f)|=constant=A;
? 2) the phase response is a linear function of
frequency,that is,θ(f)=-2πfTd
? When the first condition is satisfied,there is
no amplitude distortion; when the second
condition is satisfied,there is no phase
distortion,
37
2.7 Bandlimited signals and noise
Bandlimited Waveforms
A waveform ω(t) is said to be Bandlimited
to B hertz if W(f)=0 for |f|≥B
Time-limited Waveforms
A waveform ω(t) is said to be time limited
to B hertz if ω(t) =0 for |t|≥T
Theorem
An absolutely bandlimited waveform
cannot be absolutely time limited and
vice versa,
38
2.7 Bandlimited signals and noise
(sampling theorem)
? Any physical waveform may be represented over the
interval -∞<t< ∞ by,
??
??? ?
??
n ss
ssn
fntf
fntfat
))/((
) ) }/((s i n {)(
?
??
? ??? ? ?? ))/(( ) ) }/((s i n {)(
ss
sssn
fntf
fntftfa
?
??
1 6 0 / 1 6 7 )-(2 )()/( )/(2? ??? ??? dfefWfna sfnffsn ??
? Where,
? And fs is a parameter that is assigned some convenient
value greater than zero,Furthermore if ω(t) is
bandlimited to B hertz,and fs>=2B,then,
? That is,for fs>=2B,the orthogonal series coefficients are
simply the values of the waveform that are obtained
when the waveform is sampled every 1/fs seconds
39
2.7 Bandlimited signals and noise
(sampling theorem)
? Proof,
))/((
) ) ]/((s i n [)(
ss
ss
n fntf
fntft
?
??
?
??
nmnmn Kdttt ??? ????? )()( *
??? ?????? dfffdttt mnmn )()()()( ** ????
)/(21)]([)( sfnfj
ss
nn ef
f
ftf
??? ?? ??
?
?
???
??? F
nm
s
f
f
ffmnj
s
mn
f
dfe
f
dfff
s
s
s ?
??
? 11
)()(
2/
2/
)/)((2
2
*
???
?
?
??
?
??
)/(
)(
)()(
)()(
)/(2
*
*
s
fnfj
n
nsn
fn
dfefW
dfffW
dtttfa
s
?
?
??
?
?
??
??
??
?
??
?
?
??
?
??
40
2.7 Bandlimited signals and noise
? It is obvious that the minimum sampling rate allowed to
reconstruct a bandlimited waveform without error is
given by,(fs)min=2B this is called the Nyquist frequency
41
2.7 Bandlimited signals and noise
? How to reproduce a bandlimited waveform by using N
sample value
?
?
?
?
Nn
nn
nn tat
1
1
)()( ??
))/((
))}/((s i n {)(
ss
ss
n fntf
fntft
?
??
?
??
?where
?each samples value is multiplied by the appropriate
(sinx/x) function,and these weighted (sinx/x) functions are
summed to give the original waveform,The minimum
number of sample values that are needed to reconstruct
the waveform is,
N=T0/(1/fs)=fsT0>=2BT0
42
2.7 Bandlimited signals and noise
Impulse sampling and Digital Signal Processing(DSP)
Another useful orthogonal series is the impulse-sampled
series obtained when the sinx/x orthogonal functions of
the sampling theorem are replaced by an orthogonal set
of delta function,The impulse-sampled series is also
identical to the impulse-sampled waveform ωs(t),both
can be obtained by multiplying the unsampled
waveform by a unit-weight impulse train,
tjn
n sn ssn ss
se
TtnTtnTnTttt
??????? ??? ?
???
?
???
?
???
????? 1)()()()()()(
)1732( )(1)( ??? ?
?
???n
s
s
s nffWTfW
Taking the Fourier transform of both sides we get,
43
2.7 Bandlimited signals and noise
? The spectrum of the impulse sampled signal is the
spectrum of the unsampled signal that is repeated
every fsHz,where fs is the sampling frequency,
44
2.7 Bandlimited signals and noise
? Note,this technique of impulse sampling may be used
to translate the spectrum of a signal to another
frequency band that is centered on some harmonic of
the sampling frequency,
? If fs>=2B,the replicated spectra do not overlap,thus
ω(t) is reproduced from ωs(t) by passing ωs(t)
through an ideal low-pass filter that has a cutoff
frequency of fc=fs/2,where fs>=2B
? If fs<2B,the spectra of ωs(t) will consist of overlapped,
replicated spectra of ω(t),The spectral overlap or tail
inversion,is called aliasing or spectral folding,
45
2.7 Bandlimited signals and noise
? Fig,2-19
46
2.7 Bandlimited signals and noise
? Dimensionality theorem
? When BT0 is large,a real waveform may be
completely specified by
N=2BT0 (2.174)
? independent pieces of information that will
describe the waveform over a T0 interval,N is
said to be the number of dimensions required
to specify the waveform,and B is the absolute
bandwidth of the waveform,
47
2.7 Bandlimited signals and noise
? The demensionality theorem imply that the
information which can be conveyed by a bandlimited
waveform or a bandlimited communication system is
proportional to the product of the bandwidth of that
system and the time allowed for transmission of the
information
? The demensionality theorem has profound
implications in the design and performance of all
types of communication systems,
? Application,1) it can be used to calculate the number
of storage locations required to represent a waveform;
2) is used to give a lower bound for the bandwidth of
digital signals,
48
2.8 Discrete Fourier Transform
? Definition,the discrete Fourier transform is
defined by,
Where n=0,1,2,…,N -1,and the inverse discrete
Fourier transform is defined by,
Where k=0,1,2,…,N -1
Matlab notation are X=fft(x) and x=ifft(X)
respectively,
??
?
?? 1
0
)/2()()( N
k
nkNjekxnX ?
??
?
?? 1
0
)/2()(1)( N
n
nkNjenX
Nkx
?
49
2.8 Discrete Fourier Transform-
windowing,sampling and periodic sample genernation
50
2.8 Discrete Fourier Transform-Using
the DFT to compute the Contimuous Fourier Transform
? The windowed waveform,denoted by the
subscript w,is,
? ?????? ????? ??? TTttttw )2/()(e l s e w h e r e t 0 Tt0 )()( ???
Tr
o c t Con F T
? thus
?
??
?
?
?
?
??
??
???
??
?????? ??
??
1
0
)/2(
/
/
,/,
2
0
2
)(|)(
)()()(
N
k
nkNj
Tnfw
NTtdt
Tnftkt
ftjTftj
ww
tetkfW
dtetdtetfW
???
??
??
?
??
Because of then the
relationship between the CFT and DFT is,?
?
?
?? 1
0
)/2()()( N
k
nkNjekxnX ?
)(|)( / ntXfW Tnfw ???
51
2.8 Discrete Fourier Transform
? and it can modified to give spectral values over
the entire fundamental range of –fs/2<f<fs/2,
thus,for positive frequencies we use,
N / 2n0 )(|)( / ???? ntXfW Tnfw ?
? and for negative frequencies we use,
NnN / 2 )(|)( /)( ????? ntXfW TNnfw ?
? Note the fact that if one is not careful,the DFT
may give significant errors when it is used to
approximate the CFT,The factor that cause the
errors may be categorized into three basic effects,
leakage,aliasing,and the picket-fence effect,
52
2.8 Discrete Fourier Transform
? The leakage effects is caused by windowing in the
time domain,
? Because the spectrum of a sampled waveform consists
of replicating the spectrum of the un-sampled
waveform about harmonics of the sampling frequency,
if fs<2B,where fs=1/Δt,and B is the highest significant
frequency component in the unsampled waveform,
Aliasing errors will occur,
? The picket-fence effect occurs because the N-point
DFT cannot resolve the spectral components any
closer than the spacing Δf=1/T,Δf can be decreased by
increasing T,
53
2.9 Bandwidth of Signals
? The spectral width of signals and noise in
communication systems is a very important concept,
the reason is,
? More and more users are being assigned to
increasingly crowded RF bands,so that the spectral
width required for each one needs to be considered
carefully,
? Second,the spectral width is important from the
equipment design viewpoint,since the circuits need to
have enough bandwidth to accommodate the signal
and reject the noise,
54
2.9 Bandwidth of Signals
? In engineering definitions,the bandwidth is taken to
be the width of a positive frequency band,In other
words,the bandwidth is,
f2-f1
Where f2>f1>=0
? For baseband signals,f1=0;
? For bandpass signals,f1>>0,and the bandwidth f2-f1
encompasses the carrier frequency fc,
? When we increase the signaling speed of a signal
(decrease T),the spectrum gets wider,Consequently,
for engineering definitions of bandwidth,we require
the bandwidth to vary as 1/T
55
2.9 Bandwidth of Signals
? 1) absolute bandwidth is f2-f1,where the spectrum is
zero outside the interval f1<f<f2 along the positive
frequency axis
? 2) 3dB bandwidth (or half-power bandwidth) is f2-f1,
where for frequencies inside the band f1<f<f2,the
magnitude spectra,say,|H(f)|,fall no lower than 1/√2
times the maximum value of |H(f)|,and the maximum
value occurs at a frequency inside the band,
? 3) equivalent noise bandwidth is the width of a
fictitious rectangular spectrum such that the power in
that rectangular band is equal to the power associated
with the actual spectrum over positive frequencies,
That is,if f0 is the frequency at which the magitude
spectrum has maximum,
56
2.9 Bandwidth of Signals
? 4) Null-to-null bandwidth (or zero-crossing bandwidth)
is f2-f1,where f2 is the first null in the envelope of the
magnitude spectrum above f0,for bandpass systems,f1 is
the first null in the envelope of the magnitude spectrum
below f0,where f0 is the frequency where the magnitude
spectrum is a maximum,For baseband systems,f1 is
usually zero,
? 5) bounded spectrum bandwidth is f2-f1 such that outside
the band f1<f<f2,the PSD must be down by at least a
certain amount,say,50dB,below the maximum value of
the power spectral density,
? 6) power bandwidth is f2-f1,where f1<f<f2 defines the
frequency band in which 99% of the total power resides,
)192-2( |)(|
|)(|
1
=B
|)(|=p o w er act u al=|)(|B =p o w er eq u i v al en t
∫
∫
∞
0
2
2
0
∞
0
22
0eq
dffH
fH
dffHfH
eq
57
2.9 Bandwidth of Signals
? FCC bandwidth is an authorized bandwidth parameter
assigned by the FCC to specify the spectrum allowed in
communication systems,This is not an engineering
definition,
58
2.9 Bandwidth of Signals
Example2-18 Bandwidths for a BPSK signal (p106)
59
Homework
? Problems,
2.7,2.10,2.44,2.45,2.65,2.89,2.91,2.92
? Reading
§ 2-1,§ 2-7,§ 2-9 in detail
? Extend reading the others
