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A signal x(t) is called a band-limited signal if
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Thus, for a band-limited signal, it is natural to define o, as the bandwidth.
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We call a set of signals {*,Jt)} orthogonal on an interval ( a , b) if any two signals ql,,(t) and q k ( t ) in the set satisfy the condition
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where * denotes the complex conjugate and a + 0. Show that the set of complex exponentials {ejk"o': k = 0, f 1, f 2,. . . ) is orthogonal on any interval over a period To, where To = 2.rr/oU.
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For any t o we have
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= since eJm2" 1. When m
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we have eJm"o'lm=o 1 and =
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FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS
[CHAP. 5
Thus, from Eqs. ( 5 . 9 6 ) and ( 5 . 9 7 ) we conclude that
which shows that the set {eJkwo': k To.
= 0, +_
1, f 2,. . . ) is orthogonal on any interval over a period
Using the orthogonality condition (5.98), derive Eq. ( 5 . 5 ) for the complex Fourier coefficients.
From Eq. ( 5 . 4 )
Multiplying both sides of this equation by e-imwo' and integrating the result from to to ( t o + To),we obtain
Then by Eq. ( 5 . 9 8 ) Eq. ( 5 . 9 9 ) reduces to
Changing index m to k, we obtain Eq. (5.51, that is,
We shall mostly use the following two special cases for Eq. (5.101): to = 0 and to = - T 0 / 2 , respectively. That is,
Derive the trigonometric Fourier series Eq. ( 5 . 8 ) from the complex exponential Fourier series Eq. (5.4).
Rearranging the summation in Eq. ( 5 . 4 ) as
and using Euler's formulas
CHAP. 51
FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS
we have
~ ( t=)c ,
Setting
+ C [ ( c k+ c P k )cos kwOt + j(ck - C - , ) sin k ~ , t ]
k- 1
(5.103)
-5Eq. (5.103) becomes
+c-, =ak
j ( c k - c - ~ = bk )
a0 ~ ( t=) - +
C ( a kcos kwot + bk sin k w , , t )
Determine the complex exponential Fourier series representation for each of the following signals:
( a ) x ( t ) = cos w,t (6) x ( t ) = sin w,t
( c ) x ( t ) = cos 2 t
(dl x ( t ) =
( ++sin 6 t 3 cos4t
(el x ( t ) = sin2t
( a ) Rather than using Eq. ( 5 . 5 ) to evaluate the complex Fourier coefficients c , using Euler's formula, we get
Thus, the complex Fourier coefficients for cos w,t are
( b ) In a similar fashion we have
Thus, the complex Fourier coefficients for sin w,t are
( c ) The fundamental angular frequency w , of x ( t ) is 2. Thus,
FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS
[CHAP. 5
Thus, the complex Fourier coefficients for cos(2t
+ 7r/4)
( d ) By the result from Prob. 1.14 the fundamental period To of x ( t ) is Thus,
and w,, = 21r/T,
= 2.
Again using Euler's formula, we have
Thus, the complex Fourier coefficients for cos 4t
+ sin 6t are
and all other c, = 0. (e) From Prob. 1.16(e) the fundamental period To of x(t) is rr and w,
= 2rr/T,, = 2.
Thus,
Again using Euler's formula, we get
Thus, the complex Fourier coefficients for sin2 t are
- - -I 4
c0=$
c,=-$
and all other c,
= 0.
Consider the periodic square wave
x(t)
shown in Fig. 5-8.
x ( t ).
Determine the complex exponential Fourier series of ( b ) Determine the trigonometric Fourier series of x ( t 1.
CHAP. 51
FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS
Fig. 5-8
( a ) Let
Using Eq. (5.102a), we have
since ooTo= 27r and ePik"= ( - I)&.Thus, ck = 0 k=2m#O
Hence,
and we obtain
1 ej(2m A A " x(t)=-+C - + I)""' 2 ~ ~ , , , ~ - ~ 2 r n + l
( b ) From Eqs. (5.1051, (5.10), and (5.12) we have
FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS
[CHAP. 5
Substituting these values in Eq. (5.81, we get
sin w,l
1 1 + -sin3wot + -sin5w0t + . 3 5
(.5.107)
Consider the periodic square wave
x(t)
shown in Fig. 5-9.
( a ) Determine the complex exponential Fourier series of x ( t 1. ( b ) Determine the trigonometric Fourier series of x ( r ) .
Fig. 5-9
( a ) Let
Using Eq. (5.102b),we have
Thus.
Hence.
c0 =
c,,,, = 0, m
C ~ + I (=- 1 )
(2m I ) a
( 5.108)
CHAP. 51
FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS
and we obtain ( - I ) ej(2m+1)q,t x(t)=-+C 2 ~,,-,Zrn+l (b) From Eqs. (5.108), (5.10), and (5.12) we have
A A "
Substituting these values into Eq. (5.81, we obtain
1 cos wOt- -cos3w0t
+ -cos5wot 5
(5.110)
Note that x(t) is even; thus, x(t) contains only a dc term and cosine terms. Note also that x(t) in Fig. 5-9 can be obtained by shifting x(t) in Fig. 5-8 to the left by T0/4. 5.7.
Consider the periodic square wave
x(t)
shown in Fig. 5-10.
(a) Determine the complex exponential Fourier series of x(t). ( b ) Determine the trigonometric Fourier series of x(t).
Note that x(t) can be expressed as ~ ( t =x1(t) - A ) where x,(t) is shown in Fig. 5-11. Now comparing Fig. 5-11 and Fig. 5-8 in Prob. 5.5, we see that x,(t) is the same square wave of x(t) in Fig. 5-8 except that A becomes 2A.
Fig. 5-10
FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS
[CHAP. 5
Fig. 5-11
Replacing A by 2 A in Eq. (5.106), we have
Thus,
( b ) Similarly, replacing A by 2A in Eq. (5.107), we have
Thus,
1 1 2(sin o,t + -sin3w,,t + -sin 5w,r + - . 3 5
Note that x ( t ) is odd; thus, x(t) contains only sine terms. 5.8.
Consider the periodic impulse train S G , ( t ) shown in Fig. 5-12 and defined by
- 70
2 70
Fig. 5-12
CHAP. 51
FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS
( a ) Determine the complex exponential Fourier series of ST$t). ( 6 ) Determine the trigonometric Fourier series of ST$t). ( a ) Let
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