Fundamentals of
Measurement Technology
(2)
Prof,Wang Boxiong
In a finite interval of time,a periodic signal x(t) can
be represented by its Fourier series when it
complies with the Dirichlet conditions,
where
n＝ 0,1,2,3,……
T= the period
ω0= the angular frequency or circular frequency
ω0= 2π/T
an(including a0 and bn) are called Fourier
coefficients.
2.2.4 Frequency representation of periodic signals
1 00
0 )s inc o s(2)(
n nn
tnbtnaatx(2.12)
2/ 2/ 0c o s)(2 T Tn t d tntxTa? (2.13)
2/ 2/ 0s in)(2 T Tn t d tntxTb? (2.14)
Fourier coefficients an and bn (functions of
nω0):
an,even function of n or nω0,a-n = an.
bn,odd function of n or nω0,b-n = -bn.
Dirichlet conditions:
x(t) must be absolutely integrable,
x(t) possesses a finite number of maxima and
minima and finite number of discontinuities in
any finite interval,
2.2.4 Frequency representation of periodic signals
dttx )(
Rewrite Eq,(2.12),
where
An,amplitude of signal’s frequency
component
φn,phase-shift
2.2.4 Frequency representation of periodic signals
1 0
0 )c o s (2)(
n nn
tnAatx (2.15)
,2,1
)(
22
n
a
ba r c t g
baA
n
n
n
nnn
(2.16)
,2,1s inc o s?
n
Ab
Aa
nnn
nnn
(2.17)
nnnn AA
a0/2 is the constant-value or the d.c,component of
a periodic signal,
The term for n=1 is referred to as the fundamental
(component),or as the first harmonic
component.
The component for n=N is referred to as the
Nth harmonic component,
The representation of a periodic signal in the
form of Eq,(2.15) is referred to as the
Fourier series representation:
An,amplitude of the nth harmonic component
φn,phase shift of the nth harmonic component
2.2.4 Frequency representation of periodic signals
The plots of the amplitude An and the phase
φn versus signal’s angular frequency ω0 are
called amplitude spectrum plot and phase
spectrum plot respectively,
The frequency spectrum is displayed
graphically by a number of discrete vertical
lines representing the amplitude An and the
phase φn of the analyzed signal respectively,
The frequency spectrum of a periodic signal
is a discrete one,
2.2.4 Frequency representation of periodic signals
Example 1.
Find the Fourier series of the periodic
square wave signal x(t) shown in Fig,2.11,
2.2.4 Frequency representation of periodic signals
Fig,2.11 Periodic square wave signal
Solution,Within one period,signal x(t) can be
expressed as
According to Eqs,(2.13) and (2.14)
2.2.4 Frequency representation of periodic signals
2
0,1
0
2
,1
)( T
t
tT
tx
2/ 2/ 0 0c o s)(2 T Tn t d tntxTa?
6,4,2,0
,5,3,1,
4
c os1
2
)c os(
1
c os
12
s i ns i n)1(
2
s i n)(
2
2/
00
0
0
2/0
0
2/
0
0
0
2/
0
2/
2/
0
n
n
n
n
n
tn
n
tn
nT
t d tnt d tn
T
t d tntx
T
b
T
T
T
T
T
T
n
The Fourier series expression of the square wave
signal
2.2.4 Frequency representation of periodic signals
)5s i n513s i n31( s i n4)( 000 ttttx
Fig,2.12 Frequency spectrum of periodic square wave signal
Fourier series can be used to approximate a
signal,
2.2.4 Frequency representation of periodic signals
Fig,2.13 Approximations of a square wave signal using sums of
partial terms of Fourier series
For an odd function x(t):
2.2.4 Frequency representation of periodic signals
)()( txtx
),2,1(s i n)(4
0
2/
0 0
nt d tntx
T
b
a
T
n
n
(2.18)
),2,1(
i n t e g e r,
2
)12(
n
ismm
bA
n
nn
）（
(2.19)
For an even function x(t)
2.2.4 Frequency representation of periodic signals
)()( txtx
)2,1,0(c os)(4
0
2/
0 0
nt d tntxTa
b
T
n
n
(2.20)
)2,1(in t e g e r, nismmaA
n
nn
）（
(2.21)
Fig,2,14 Examples of even functions,the functions are
symmetrical about the ordinate
Euler’s formula:
Substitute Eq,(2.22) into Eq,(2.12),
Let
2.2.4 Frequency representation of periodic signals
)(
2
s in
)(
2
1c o s
tjtj
tjtj
ee
j
t
eet
(2.22)
1
0 00 )(
2
1)(
2
1
2)( n
tjn
nn
tjn
nn ejbaejba
atx
3,2,1
2
)(
2
1
)(
2
1
0
0
n
a
C
jbaC
jbaC
nnn
nnn
(2.23)
Or
2.2.4 Frequency representation of periodic signals
3,2,1)(
11
0 00
neCeCCtx
n
tjn
n
n
tjn
n
(2.24)
,2,1,0)( 0
neCtx
n
tjn
n
(2.25)
),2,1,0(
)(
1
)s i n) ( c o s(
1
s i n)(c o s)(
1
2/
2/
2/
2/
00
2/
2/
0
2/
2/
0
0
n
dtetx
T
dttnjtntx
T
t d tntxjt d tntx
T
C
T
T
tjn
T
T
T
T
T
T
n
(2.26)
Writing
where |Cn| and φn are the amplitude and
phase of the complex coefficient
respectively,
2.2.4 Frequency representation of periodic signals
nnjnn CjCeCC n ImRe
(2.27)
nnn CCC 22 ImRe
(2.28)
n
n
n C
Ca r c t g
Re
Im
(2.29)
2.2.4 Frequency representation of periodic signals
Fig,2.15 Two graphic representations of spectra of periodic signals
The spectrum of a periodic signal
has the following three features:
1,The spectrum is a discrete spectrum;
2,The spectral lines appear only at the
fundamental frequency and the harmonic
frequencies;
3,The amplitude of harmonic component
decreases with the increase in its
frequency,the higher the harmonic
frequency,the lower the amplitude is,
2.2.4 Frequency representation of periodic signals
Example 2.
Find the frequency spectrum of the periodic
sequence of rectangular pulses (also called
periodic gate function) shown in Fig,2.16,
2.2.4 Frequency representation of periodic signals
Fig,2.16 Periodic sequence of rectangular pulses
Solution,From Eq,(2.26),we have
2.2.4 Frequency representation of periodic signals
,2,1,0
2
2
s i n
2
s i n
2
1
1
)(
1
0
0
0
0
2/
2/
0
2/
2/
2/
2/
0
0
0
n
n
n
T
n
n
T
jn
e
T
dte
T
dtetx
T
C
tjn
tjn
T
T
tjn
n
Substituting ω0=2π/T into the above
equation,we have
Defining
then Eq.(2.36) changes to
So
2.2.4 Frequency representation of periodic signals
,2,1,0,
s in
n
T
n
T
n
T
C n
(2.36)
x
xxc d e f s in)(s in? (2.37)
,2,1,0,2s i ns i n 0 nncTTncTC n
(2.38)
n
tjn
n
tjn
n eT
nc
TeCtx 00 s in)(
(2.39)
Fig,2.17 Frequency spectrum of periodic sequence of rectangular pulses
(T=4τ)
2.2.4 Frequency representation of periodic signals
Under the limit of certain errors,the
frequency range of 0≤ ω≤2π/τ is usually
called the bandwidth of the periodic sequence
of rectangular pulses,and is designated as
ΔC,
2.2.4 Frequency representation of periodic signals
1C
(2.40)
Fig,2.18 Frequency spectra for signals of different pulse widths
2.2.4 Frequency representation of periodic signals
Fig,2.19 Frequency spectra for signals of different periods
When the period becomes larger,the
spectral line spacing decreases,If the period
increases infinitely,i.e,if T→∞,the original
periodic signal will now become a
nonperiodic signal,Then the spectral lines
will become denser and denser,and the line
spacing tends to zero,making the whole
spectral lines form a continuous spectrum,
2.2.4 Frequency representation of periodic signals
The power of a periodic signal x(t) is
Substituting Eq,(2.15) into Eq,(2.41) yields
Expanding Eq,(2.41) results in,
– The first term,dc power of the signal x(t);
– The second term,a sum of the powers of all
harmonic components,
2.2.5,Power of a periodic signal
2/ 2/ 2 )(1 TT dttxTP
(2.41)
2/
2/
2
1
0
0 c o s
2
1 T
T n nn dttnA
a
TP
(2.42)
2
1
2
02/
2/
2
2
1
2)(
1
n
n
T
T A
adttx
TP
(2.43)
Further,Eq,(2.43) can also be written as
which is called Parseval’s theorem.
Power spectrum of a periodic signal x(t),
where Pn denotes the power spectrum
related to the nth harmonic component,
Properties of power spectrum,
– Pn is nonnegative,
– Pn is an even function of n.
– Pn doesn’t change with time shifting τ.
2.2.5,Power of a periodic signal
n
n
n
n
T
T CCCdttxTP
2
1
22
0
2/
2/
2 2)(1
(2.44)
,2,1,0,2 nCP nn
(2.45)
Define
The Fourier frequency spectrum Cn can be
expressed by the Fourier power spectrum
and the phase spectrum as
or
2.2.5,Power of a periodic signal
,2,1,0, nPp nn
(2.46)
,2,1,0 nePeCC nn jnjnn (2.47)
,2,1,0 nepC njnn?
(2.48)
Assuming that x(t) is a periodic function
on the interval (-T/2,T/2),
Substituting Eq,(2.50) into Eq,(2.49),we
get
When T→∞,the interval (-T/2,T/2) →( - ∞,
∞),Δω=ω0=2π/T →∞,And nω0 becomes a
continuous one,ω.
2.2.6 Frequency domain representation of nonperiodic signas
2.2.6.1 Fourier transform and continuous frequency spectra
n
tjn
n eCtx 0)(
(2.49)
2/ 2/ 0)(1 T T tjnn dtetxTC?
(2.50)
n
tjnT
T
tjn edtetx
Ttx 00
2/
2/ )(
1)((2.51)
Write
then Eq,(2.52) becomes
X(ω) is called the Fourier transform of x(t),
and x(t) is called the inverse Fourier
transform of X(ω).
Fourier transform pair:
2.2.6.1 Fourier transform and continuous frequency spectra
dedtetx
edtetxdtx
tjtj
tjtj
)(
2
1
)(
2
)( (2.52)
dtetxX tj )()(
(2.53)
deXtx tj)(2 1)( (2.54)
)()(?Xtx?
(2.55)
The sufficient condition for the existence
of a Fourier transform for a nonperiodic
function x(t) is that x(t) is absolutely
integrable on the interval (-∞,+∞):
2.2.6.1 Fourier transform and continuous frequency spectra
dttx )(
Since ω=2πf,Eqs,(2.52) and (2.54)
become
and
The Fourier transform pair is
correspondingly written as
X(f) or X(ω) is called the continuous
frequency spectrum of x(t).
2.2.6.1 Fourier transform and continuous frequency spectra
dtetxfX ftj?2)()( (2.56)
dfefXtx ftj?2)()(
(2.57)
)()( fXtx? (2.58)
)()()( fjefXfX
(2.59)
Example 5.
Fig,2.23 shows a rectangular pulse gT(t)
(also called gate function or window
function),Find its frequency spectrum,
2.2.6.1 Fourier transform and continuous frequency spectra
o t h e r w i s e
Tt
tg T
,0
2,1)(
Fig,2.23 Rectangular pulse
Solution,
2.2.6.1 Fourier transform and continuous frequency spectra
2
s i n
2
2
s i n
1
1
)()(
2/2/
2/
2/
2/
2/
T
cT
T
T
T
ee
j
j
e
dte
dtetgG
TjTj
T
T
tj
T
T
tj
tj
TT
(2.60)
The amplitude spectrum is thus
and the phase spectrum is
2.2.6.1 Fourier transform and continuous frequency spectra
2s in TcTG T
(2.61)
0
2
s in,
0
2
s in,0
)(
T
c
T
c
(2.61)
Fig,2.24 Spectrum of GT(jω)
The rectangular pulse and the sinc
function are a pair of Fourier
transform,which we write as
2.2.6.1 Fourier transform and continuous frequency spectra
)(s in)(?ctr e c t?
(2.62)
The energy associated with x(t) is
defined as
Using Eq,(2.54),we get the energy
2.2.6.2 The energy spectrum
dttxE )(2
(2.63)
dXX
ddtetxX
dtdeXtx
dttxE
tj
tj
)()(
2
1
)()(
2
1
)(
2
1
)(
)(
2
(2.64)
For a real signal x(t),X(-ω)= X*(ω),where
X*(ω) is the complex conjugate of X(ω),Thus
Eq,(2.64) becomes
Finally
which is known as the Parseval equation or
energy equality.
2.2.6.2 The energy spectrum
dX
dXX
dXXE
2
*
)(
2
1
)()(
2
1
)()(
2
1
dXdttxE 22 )(2 1)(
(2.65)
The energy within x(t)
where S(ω)=|X (ω)|2/π is called the energy
density spectrum of x(t),or simply the
energy spectrum,S(ω) is an even function
of ω.
2.2.6.2 The energy spectrum
0
0
2
2
)(
)(
1
)(
2
1
dS
dX
dXE
(2.66)
2.2.6.2 The energy spectrum
Fig,2.27 Energy spectrum and the energy of a gate function
1,Symmetry (Duality)
If
then
Proof,because
then
Replacing t by ω,we have
2.2.6.3 Properties of the Fourier transform
)()(?Xtx?
)(2)( xtX
(2.67)
deXtx tj)(2 1)(
deXtx tj)()(2
deXx j)()(2
Finally,replace ω’ by t,
If x(t) is an even function,x(t)= x(-t),then
2.2.6.3 Properties of the Fourier transform
)()()(2 jtXFdtetXx tj
)(2)( xjtX
)(2)( xjtXF?
2,Linearity
If
then
a and b are constants,
2.2.6.3 Properties of the Fourier transform
)()( 11?Xtx?
)()( 22?Xtx?
)()()()( 2121 bXaXtbxtax (2.68)
3,Scaling property
If
then for a real constant a,
2.2.6.3 Properties of the Fourier transform
)()(?Xtx?
aXaatx
1)( (2.69)
2.2.6.3 Properties of the Fourier transform
Fig,2.29 Scaling of the gate function gT(t) (a=3)
4,Odevity
Assuming x(t) is a real function of t,from
Eq,(2.53),then
The real part and the imaginary part of
X(ω)are
2.2.6.3 Properties of the Fourier transform
)(
)(Im)(Re
s i n)(c o s)(
)()(
j
tj
eX
XjX
t d ttxjt d ttx
dtetxX
t d ttxX
t d ttxX
s in)()(Im
c o s)()(Re (2.70)
The modulus and the phase of the
frequency spectrum are calculated as
Eq,(2.70) shows that,ReX(ω)= ReX(-ω)
and ImX(ω)= -ImX(-ω).
From Eq,(2.71),we obtain,|X(ω)|= |X(-ω)|
and φ(ω)=-φ(-ω).
2.2.6.3 Properties of the Fourier transform
)(Re
)(Im
)(
)(Im)(Re)(
22
X
X
a r c t g
XXX (2.71)
If x(t) is a real and even function of time,
i.e.,x(t)= x(-t),then
X(ω) is an real and even function of ω.
If x(t) is a real and odd function of time,
i.e.,x(t)= x(-t),then
X(ω) is now an imaginary and odd function
of ω,
2.2.6.3 Properties of the Fourier transform
0 c o s)(2c o s)()(Re)( t d ttxt d ttxXX
0 s i n)(2s i n)()(Im)( t d ttxjt d ttxjXjX
Calculate the Fourier transform of x(-t)
Let τ=-t,then
Since ReX(ω) is an even function of ω,then
2.2.6.3 Properties of the Fourier transform
dtetxtxF tj?)()(
)(
)()(
)()()(
)(
X
dex
dextxF
j
j
)(
)(Im)(Re
)(Im)(Re)(
*?
X
XjX
XjXX
Finally,we get
called the reversal of Fourier transform,
Note,all the above conclusions hold for
the condition when x(t) is a real function of
time t,
If x(t) is an imaginary function of t,then
2.2.6.3 Properties of the Fourier transform
)()()( * XXtx
(2.72)
)()(,)()(
)(Im)(Im),(Re)(Re
XX
XXXX (2.73)
)()()( * XXtx (2.74)
5,Time shifting
If
then
2.2.6.3 Properties of the Fourier transform
)()(?Xtx?
0)()( 0 tjeXttx
(2.75)
Example 8,
Find the frequency spectrum of a rectangular
pulse with amplitude A and width T,centered
on some position t0≠0 (Fig,2.30)
2.2.6.3 Properties of the Fourier transform
Fig,2.30 Rectangular pulse with a time shift t0
Solution:
and can be considered as if it is formed by a
rectangular pulse centered on the origin of
coordinates shifted to the position t0,Its
Fourier transform,as a result of Eqs,(2.59)
and (2.75),is
Hence,its amplitude and phase spectra are
2.2.6.3 Properties of the Fourier transform
TttA r e c t)t(x 0
02)(s in)( ftjefTcATfX
0)(s i n,2
0)(s i n,2
)(
)(s i n)(
0
0
fTcft
fTcft
f
fTcATfX
2.2.6.3 Properties of the Fourier transform
Fig,2.31 Amplitude and phase spectra of rectangular pulse function
with a time delay t0
6,Frequency Shifting-Modulation
If
then
2.2.6.3 Properties of the Fourier transform
)()(?Xtx?
)()( 00 Xetx tj
(2.76)
Consider,the multiplication of a time
function x(t) by a sinusoid cosω0t,
x(t),the modulating signal
Cosω0t,the carrier or modulated signal
Similarly
2.2.6.3 Properties of the Fourier transform
)()(
2
1
)(
2
1
)(
2
1
2
)(c o s)(
00
0
00
00
XX
etxFetxF
ee
txFttxF
tjtj
tjtj
)()(21]s i n)([ 000 XXjttxF
(2.78)
2.2.6.3 Properties of the Fourier transform
Fig,2.32 Frequency spectrum of x(t)cosω0t
7,Convolution
Time Convolution
If
then
2.2.6.3 Properties of the Fourier transform
)()(?Xtx?
)()(?Hth?
)()()()( HXthtx
(2.79)
Proof:
By definition
Its Fourier transform is
From the time shifting property,
Substituting it in the above yields
2.2.6.3 Properties of the Fourier transform
dthxthtx )()()()(
ddtethx
dtdthxethtxF
tj
tj
)()(
)()()()(
jtj eHdteth
)()(
)()(
)()(
)()()()(
XH
dexH
deHxthtxF
j
j
Frequency Convolution
If
and
then
2.2.6.3 Properties of the Fourier transform
)()(?Xtx?
)()(?Hth?
)()(2 1)()( HXthtx
(2.81)
2.2.6.3 Properties of the Fourier transform
Fig,2.34 Graphical illustration of convolution computational procedure
Example 10.
Find the frequency spectrum of the triangular
pulse defined by
Solution,Using the convolution of two
identical rectangular pulses,
T,the width of the rectangular pulse
,amplitude
2.2.6.3 Properties of the Fourier transform
ot he r w i s e,
Tt,
T
t
A)t(x
0
1
AT
tgTAtx T?)(1
Convolute the two rectangular pulse
functions x1(t) (Fig,2.35(a)),
2.2.6.3 Properties of the Fourier transform
Fig,2.35 Obtaining the frequency spectrum of a triangular pulse by
use of the convolution theorem
From Eq,(2.62)
the frequency spectrum of the signal
is
2.2.6.3 Properties of the Fourier transform
2s in TcTtg T?
tgAT T
2s i n TcTTA?
2
s in
2
s in
)()(
)()(
)()(
2
2
11
11
T
cAT
T
cT
T
A
XX
txtxF
txFX
8,Frequency Differentiation and
Integration
If
then
Furthermore,we get
and
where
If x(0)=0,then
2.2.6.3 Properties of the Fourier transform
)()(?Xtx?
d
dXtj t x )()( (2.91)
n
nn
d
Xdtxjt
)()()( (2.92)
dXtxjttx )()(1)()0( (2.93)
dXx )(21)0(
dXjttx )()( (2.94)
Measurement Technology
(2)
Prof,Wang Boxiong
In a finite interval of time,a periodic signal x(t) can
be represented by its Fourier series when it
complies with the Dirichlet conditions,
where
n＝ 0,1,2,3,……
T= the period
ω0= the angular frequency or circular frequency
ω0= 2π/T
an(including a0 and bn) are called Fourier
coefficients.
2.2.4 Frequency representation of periodic signals
1 00
0 )s inc o s(2)(
n nn
tnbtnaatx(2.12)
2/ 2/ 0c o s)(2 T Tn t d tntxTa? (2.13)
2/ 2/ 0s in)(2 T Tn t d tntxTb? (2.14)
Fourier coefficients an and bn (functions of
nω0):
an,even function of n or nω0,a-n = an.
bn,odd function of n or nω0,b-n = -bn.
Dirichlet conditions:
x(t) must be absolutely integrable,
x(t) possesses a finite number of maxima and
minima and finite number of discontinuities in
any finite interval,
2.2.4 Frequency representation of periodic signals
dttx )(
Rewrite Eq,(2.12),
where
An,amplitude of signal’s frequency
component
φn,phase-shift
2.2.4 Frequency representation of periodic signals
1 0
0 )c o s (2)(
n nn
tnAatx (2.15)
,2,1
)(
22
n
a
ba r c t g
baA
n
n
n
nnn
(2.16)
,2,1s inc o s?
n
Ab
Aa
nnn
nnn
(2.17)
nnnn AA
a0/2 is the constant-value or the d.c,component of
a periodic signal,
The term for n=1 is referred to as the fundamental
(component),or as the first harmonic
component.
The component for n=N is referred to as the
Nth harmonic component,
The representation of a periodic signal in the
form of Eq,(2.15) is referred to as the
Fourier series representation:
An,amplitude of the nth harmonic component
φn,phase shift of the nth harmonic component
2.2.4 Frequency representation of periodic signals
The plots of the amplitude An and the phase
φn versus signal’s angular frequency ω0 are
called amplitude spectrum plot and phase
spectrum plot respectively,
The frequency spectrum is displayed
graphically by a number of discrete vertical
lines representing the amplitude An and the
phase φn of the analyzed signal respectively,
The frequency spectrum of a periodic signal
is a discrete one,
2.2.4 Frequency representation of periodic signals
Example 1.
Find the Fourier series of the periodic
square wave signal x(t) shown in Fig,2.11,
2.2.4 Frequency representation of periodic signals
Fig,2.11 Periodic square wave signal
Solution,Within one period,signal x(t) can be
expressed as
According to Eqs,(2.13) and (2.14)
2.2.4 Frequency representation of periodic signals
2
0,1
0
2
,1
)( T
t
tT
tx
2/ 2/ 0 0c o s)(2 T Tn t d tntxTa?
6,4,2,0
,5,3,1,
4
c os1
2
)c os(
1
c os
12
s i ns i n)1(
2
s i n)(
2
2/
00
0
0
2/0
0
2/
0
0
0
2/
0
2/
2/
0
n
n
n
n
n
tn
n
tn
nT
t d tnt d tn
T
t d tntx
T
b
T
T
T
T
T
T
n
The Fourier series expression of the square wave
signal
2.2.4 Frequency representation of periodic signals
)5s i n513s i n31( s i n4)( 000 ttttx
Fig,2.12 Frequency spectrum of periodic square wave signal
Fourier series can be used to approximate a
signal,
2.2.4 Frequency representation of periodic signals
Fig,2.13 Approximations of a square wave signal using sums of
partial terms of Fourier series
For an odd function x(t):
2.2.4 Frequency representation of periodic signals
)()( txtx
),2,1(s i n)(4
0
2/
0 0
nt d tntx
T
b
a
T
n
n
(2.18)
),2,1(
i n t e g e r,
2
)12(
n
ismm
bA
n
nn
）（
(2.19)
For an even function x(t)
2.2.4 Frequency representation of periodic signals
)()( txtx
)2,1,0(c os)(4
0
2/
0 0
nt d tntxTa
b
T
n
n
(2.20)
)2,1(in t e g e r, nismmaA
n
nn
）（
(2.21)
Fig,2,14 Examples of even functions,the functions are
symmetrical about the ordinate
Euler’s formula:
Substitute Eq,(2.22) into Eq,(2.12),
Let
2.2.4 Frequency representation of periodic signals
)(
2
s in
)(
2
1c o s
tjtj
tjtj
ee
j
t
eet
(2.22)
1
0 00 )(
2
1)(
2
1
2)( n
tjn
nn
tjn
nn ejbaejba
atx
3,2,1
2
)(
2
1
)(
2
1
0
0
n
a
C
jbaC
jbaC
nnn
nnn
(2.23)
Or
2.2.4 Frequency representation of periodic signals
3,2,1)(
11
0 00
neCeCCtx
n
tjn
n
n
tjn
n
(2.24)
,2,1,0)( 0
neCtx
n
tjn
n
(2.25)
),2,1,0(
)(
1
)s i n) ( c o s(
1
s i n)(c o s)(
1
2/
2/
2/
2/
00
2/
2/
0
2/
2/
0
0
n
dtetx
T
dttnjtntx
T
t d tntxjt d tntx
T
C
T
T
tjn
T
T
T
T
T
T
n
(2.26)
Writing
where |Cn| and φn are the amplitude and
phase of the complex coefficient
respectively,
2.2.4 Frequency representation of periodic signals
nnjnn CjCeCC n ImRe
(2.27)
nnn CCC 22 ImRe
(2.28)
n
n
n C
Ca r c t g
Re
Im
(2.29)
2.2.4 Frequency representation of periodic signals
Fig,2.15 Two graphic representations of spectra of periodic signals
The spectrum of a periodic signal
has the following three features:
1,The spectrum is a discrete spectrum;
2,The spectral lines appear only at the
fundamental frequency and the harmonic
frequencies;
3,The amplitude of harmonic component
decreases with the increase in its
frequency,the higher the harmonic
frequency,the lower the amplitude is,
2.2.4 Frequency representation of periodic signals
Example 2.
Find the frequency spectrum of the periodic
sequence of rectangular pulses (also called
periodic gate function) shown in Fig,2.16,
2.2.4 Frequency representation of periodic signals
Fig,2.16 Periodic sequence of rectangular pulses
Solution,From Eq,(2.26),we have
2.2.4 Frequency representation of periodic signals
,2,1,0
2
2
s i n
2
s i n
2
1
1
)(
1
0
0
0
0
2/
2/
0
2/
2/
2/
2/
0
0
0
n
n
n
T
n
n
T
jn
e
T
dte
T
dtetx
T
C
tjn
tjn
T
T
tjn
n
Substituting ω0=2π/T into the above
equation,we have
Defining
then Eq.(2.36) changes to
So
2.2.4 Frequency representation of periodic signals
,2,1,0,
s in
n
T
n
T
n
T
C n
(2.36)
x
xxc d e f s in)(s in? (2.37)
,2,1,0,2s i ns i n 0 nncTTncTC n
(2.38)
n
tjn
n
tjn
n eT
nc
TeCtx 00 s in)(
(2.39)
Fig,2.17 Frequency spectrum of periodic sequence of rectangular pulses
(T=4τ)
2.2.4 Frequency representation of periodic signals
Under the limit of certain errors,the
frequency range of 0≤ ω≤2π/τ is usually
called the bandwidth of the periodic sequence
of rectangular pulses,and is designated as
ΔC,
2.2.4 Frequency representation of periodic signals
1C
(2.40)
Fig,2.18 Frequency spectra for signals of different pulse widths
2.2.4 Frequency representation of periodic signals
Fig,2.19 Frequency spectra for signals of different periods
When the period becomes larger,the
spectral line spacing decreases,If the period
increases infinitely,i.e,if T→∞,the original
periodic signal will now become a
nonperiodic signal,Then the spectral lines
will become denser and denser,and the line
spacing tends to zero,making the whole
spectral lines form a continuous spectrum,
2.2.4 Frequency representation of periodic signals
The power of a periodic signal x(t) is
Substituting Eq,(2.15) into Eq,(2.41) yields
Expanding Eq,(2.41) results in,
– The first term,dc power of the signal x(t);
– The second term,a sum of the powers of all
harmonic components,
2.2.5,Power of a periodic signal
2/ 2/ 2 )(1 TT dttxTP
(2.41)
2/
2/
2
1
0
0 c o s
2
1 T
T n nn dttnA
a
TP
(2.42)
2
1
2
02/
2/
2
2
1
2)(
1
n
n
T
T A
adttx
TP
(2.43)
Further,Eq,(2.43) can also be written as
which is called Parseval’s theorem.
Power spectrum of a periodic signal x(t),
where Pn denotes the power spectrum
related to the nth harmonic component,
Properties of power spectrum,
– Pn is nonnegative,
– Pn is an even function of n.
– Pn doesn’t change with time shifting τ.
2.2.5,Power of a periodic signal
n
n
n
n
T
T CCCdttxTP
2
1
22
0
2/
2/
2 2)(1
(2.44)
,2,1,0,2 nCP nn
(2.45)
Define
The Fourier frequency spectrum Cn can be
expressed by the Fourier power spectrum
and the phase spectrum as
or
2.2.5,Power of a periodic signal
,2,1,0, nPp nn
(2.46)
,2,1,0 nePeCC nn jnjnn (2.47)
,2,1,0 nepC njnn?
(2.48)
Assuming that x(t) is a periodic function
on the interval (-T/2,T/2),
Substituting Eq,(2.50) into Eq,(2.49),we
get
When T→∞,the interval (-T/2,T/2) →( - ∞,
∞),Δω=ω0=2π/T →∞,And nω0 becomes a
continuous one,ω.
2.2.6 Frequency domain representation of nonperiodic signas
2.2.6.1 Fourier transform and continuous frequency spectra
n
tjn
n eCtx 0)(
(2.49)
2/ 2/ 0)(1 T T tjnn dtetxTC?
(2.50)
n
tjnT
T
tjn edtetx
Ttx 00
2/
2/ )(
1)((2.51)
Write
then Eq,(2.52) becomes
X(ω) is called the Fourier transform of x(t),
and x(t) is called the inverse Fourier
transform of X(ω).
Fourier transform pair:
2.2.6.1 Fourier transform and continuous frequency spectra
dedtetx
edtetxdtx
tjtj
tjtj
)(
2
1
)(
2
)( (2.52)
dtetxX tj )()(
(2.53)
deXtx tj)(2 1)( (2.54)
)()(?Xtx?
(2.55)
The sufficient condition for the existence
of a Fourier transform for a nonperiodic
function x(t) is that x(t) is absolutely
integrable on the interval (-∞,+∞):
2.2.6.1 Fourier transform and continuous frequency spectra
dttx )(
Since ω=2πf,Eqs,(2.52) and (2.54)
become
and
The Fourier transform pair is
correspondingly written as
X(f) or X(ω) is called the continuous
frequency spectrum of x(t).
2.2.6.1 Fourier transform and continuous frequency spectra
dtetxfX ftj?2)()( (2.56)
dfefXtx ftj?2)()(
(2.57)
)()( fXtx? (2.58)
)()()( fjefXfX
(2.59)
Example 5.
Fig,2.23 shows a rectangular pulse gT(t)
(also called gate function or window
function),Find its frequency spectrum,
2.2.6.1 Fourier transform and continuous frequency spectra
o t h e r w i s e
Tt
tg T
,0
2,1)(
Fig,2.23 Rectangular pulse
Solution,
2.2.6.1 Fourier transform and continuous frequency spectra
2
s i n
2
2
s i n
1
1
)()(
2/2/
2/
2/
2/
2/
T
cT
T
T
T
ee
j
j
e
dte
dtetgG
TjTj
T
T
tj
T
T
tj
tj
TT
(2.60)
The amplitude spectrum is thus
and the phase spectrum is
2.2.6.1 Fourier transform and continuous frequency spectra
2s in TcTG T
(2.61)
0
2
s in,
0
2
s in,0
)(
T
c
T
c
(2.61)
Fig,2.24 Spectrum of GT(jω)
The rectangular pulse and the sinc
function are a pair of Fourier
transform,which we write as
2.2.6.1 Fourier transform and continuous frequency spectra
)(s in)(?ctr e c t?
(2.62)
The energy associated with x(t) is
defined as
Using Eq,(2.54),we get the energy
2.2.6.2 The energy spectrum
dttxE )(2
(2.63)
dXX
ddtetxX
dtdeXtx
dttxE
tj
tj
)()(
2
1
)()(
2
1
)(
2
1
)(
)(
2
(2.64)
For a real signal x(t),X(-ω)= X*(ω),where
X*(ω) is the complex conjugate of X(ω),Thus
Eq,(2.64) becomes
Finally
which is known as the Parseval equation or
energy equality.
2.2.6.2 The energy spectrum
dX
dXX
dXXE
2
*
)(
2
1
)()(
2
1
)()(
2
1
dXdttxE 22 )(2 1)(
(2.65)
The energy within x(t)
where S(ω)=|X (ω)|2/π is called the energy
density spectrum of x(t),or simply the
energy spectrum,S(ω) is an even function
of ω.
2.2.6.2 The energy spectrum
0
0
2
2
)(
)(
1
)(
2
1
dS
dX
dXE
(2.66)
2.2.6.2 The energy spectrum
Fig,2.27 Energy spectrum and the energy of a gate function
1,Symmetry (Duality)
If
then
Proof,because
then
Replacing t by ω,we have
2.2.6.3 Properties of the Fourier transform
)()(?Xtx?
)(2)( xtX
(2.67)
deXtx tj)(2 1)(
deXtx tj)()(2
deXx j)()(2
Finally,replace ω’ by t,
If x(t) is an even function,x(t)= x(-t),then
2.2.6.3 Properties of the Fourier transform
)()()(2 jtXFdtetXx tj
)(2)( xjtX
)(2)( xjtXF?
2,Linearity
If
then
a and b are constants,
2.2.6.3 Properties of the Fourier transform
)()( 11?Xtx?
)()( 22?Xtx?
)()()()( 2121 bXaXtbxtax (2.68)
3,Scaling property
If
then for a real constant a,
2.2.6.3 Properties of the Fourier transform
)()(?Xtx?
aXaatx
1)( (2.69)
2.2.6.3 Properties of the Fourier transform
Fig,2.29 Scaling of the gate function gT(t) (a=3)
4,Odevity
Assuming x(t) is a real function of t,from
Eq,(2.53),then
The real part and the imaginary part of
X(ω)are
2.2.6.3 Properties of the Fourier transform
)(
)(Im)(Re
s i n)(c o s)(
)()(
j
tj
eX
XjX
t d ttxjt d ttx
dtetxX
t d ttxX
t d ttxX
s in)()(Im
c o s)()(Re (2.70)
The modulus and the phase of the
frequency spectrum are calculated as
Eq,(2.70) shows that,ReX(ω)= ReX(-ω)
and ImX(ω)= -ImX(-ω).
From Eq,(2.71),we obtain,|X(ω)|= |X(-ω)|
and φ(ω)=-φ(-ω).
2.2.6.3 Properties of the Fourier transform
)(Re
)(Im
)(
)(Im)(Re)(
22
X
X
a r c t g
XXX (2.71)
If x(t) is a real and even function of time,
i.e.,x(t)= x(-t),then
X(ω) is an real and even function of ω.
If x(t) is a real and odd function of time,
i.e.,x(t)= x(-t),then
X(ω) is now an imaginary and odd function
of ω,
2.2.6.3 Properties of the Fourier transform
0 c o s)(2c o s)()(Re)( t d ttxt d ttxXX
0 s i n)(2s i n)()(Im)( t d ttxjt d ttxjXjX
Calculate the Fourier transform of x(-t)
Let τ=-t,then
Since ReX(ω) is an even function of ω,then
2.2.6.3 Properties of the Fourier transform
dtetxtxF tj?)()(
)(
)()(
)()()(
)(
X
dex
dextxF
j
j
)(
)(Im)(Re
)(Im)(Re)(
*?
X
XjX
XjXX
Finally,we get
called the reversal of Fourier transform,
Note,all the above conclusions hold for
the condition when x(t) is a real function of
time t,
If x(t) is an imaginary function of t,then
2.2.6.3 Properties of the Fourier transform
)()()( * XXtx
(2.72)
)()(,)()(
)(Im)(Im),(Re)(Re
XX
XXXX (2.73)
)()()( * XXtx (2.74)
5,Time shifting
If
then
2.2.6.3 Properties of the Fourier transform
)()(?Xtx?
0)()( 0 tjeXttx
(2.75)
Example 8,
Find the frequency spectrum of a rectangular
pulse with amplitude A and width T,centered
on some position t0≠0 (Fig,2.30)
2.2.6.3 Properties of the Fourier transform
Fig,2.30 Rectangular pulse with a time shift t0
Solution:
and can be considered as if it is formed by a
rectangular pulse centered on the origin of
coordinates shifted to the position t0,Its
Fourier transform,as a result of Eqs,(2.59)
and (2.75),is
Hence,its amplitude and phase spectra are
2.2.6.3 Properties of the Fourier transform
TttA r e c t)t(x 0
02)(s in)( ftjefTcATfX
0)(s i n,2
0)(s i n,2
)(
)(s i n)(
0
0
fTcft
fTcft
f
fTcATfX
2.2.6.3 Properties of the Fourier transform
Fig,2.31 Amplitude and phase spectra of rectangular pulse function
with a time delay t0
6,Frequency Shifting-Modulation
If
then
2.2.6.3 Properties of the Fourier transform
)()(?Xtx?
)()( 00 Xetx tj
(2.76)
Consider,the multiplication of a time
function x(t) by a sinusoid cosω0t,
x(t),the modulating signal
Cosω0t,the carrier or modulated signal
Similarly
2.2.6.3 Properties of the Fourier transform
)()(
2
1
)(
2
1
)(
2
1
2
)(c o s)(
00
0
00
00
XX
etxFetxF
ee
txFttxF
tjtj
tjtj
)()(21]s i n)([ 000 XXjttxF
(2.78)
2.2.6.3 Properties of the Fourier transform
Fig,2.32 Frequency spectrum of x(t)cosω0t
7,Convolution
Time Convolution
If
then
2.2.6.3 Properties of the Fourier transform
)()(?Xtx?
)()(?Hth?
)()()()( HXthtx
(2.79)
Proof:
By definition
Its Fourier transform is
From the time shifting property,
Substituting it in the above yields
2.2.6.3 Properties of the Fourier transform
dthxthtx )()()()(
ddtethx
dtdthxethtxF
tj
tj
)()(
)()()()(
jtj eHdteth
)()(
)()(
)()(
)()()()(
XH
dexH
deHxthtxF
j
j
Frequency Convolution
If
and
then
2.2.6.3 Properties of the Fourier transform
)()(?Xtx?
)()(?Hth?
)()(2 1)()( HXthtx
(2.81)
2.2.6.3 Properties of the Fourier transform
Fig,2.34 Graphical illustration of convolution computational procedure
Example 10.
Find the frequency spectrum of the triangular
pulse defined by
Solution,Using the convolution of two
identical rectangular pulses,
T,the width of the rectangular pulse
,amplitude
2.2.6.3 Properties of the Fourier transform
ot he r w i s e,
Tt,
T
t
A)t(x
0
1
AT
tgTAtx T?)(1
Convolute the two rectangular pulse
functions x1(t) (Fig,2.35(a)),
2.2.6.3 Properties of the Fourier transform
Fig,2.35 Obtaining the frequency spectrum of a triangular pulse by
use of the convolution theorem
From Eq,(2.62)
the frequency spectrum of the signal
is
2.2.6.3 Properties of the Fourier transform
2s in TcTtg T?
tgAT T
2s i n TcTTA?
2
s in
2
s in
)()(
)()(
)()(
2
2
11
11
T
cAT
T
cT
T
A
XX
txtxF
txFX
8,Frequency Differentiation and
Integration
If
then
Furthermore,we get
and
where
If x(0)=0,then
2.2.6.3 Properties of the Fourier transform
)()(?Xtx?
d
dXtj t x )()( (2.91)
n
nn
d
Xdtxjt
)()()( (2.92)
dXtxjttx )()(1)()0( (2.93)
dXx )(21)0(
dXjttx )()( (2.94)
