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Unformatted text preview: n 1% of Pf . 3 .5-6 ( a) T he F ourier series for t he periodic signal in Fig. 3.10b is given in Exercise E3.6.
Verify Parseval's theorem for this series, given t hat ~~=7r2
~n2 6 n =l ( b) I f j (t) i s approximated by the first N t erms in this series, find N so t hat the
power of t he error signal is less t han 10% of P f' 3 .5-7 T he signal f (t) in Fig. 3.18 is approximated by t he first 2 N + 1 t erms (from n = - N
t o N ) in its exponential Fourier series given in Exercise E3.1O. Determine t he value
of N if this ( 2N + I )-term Fourier series power is t o be no less t han 99.75% of the
power of j (t).
3 .6-1 C ontinuous-Time Signal Analysis:
T he F ourierTransform
I n C hapter 3, we succeeded i n r epresenting p eriodic s ignals a s a s um o f (everlasting) sinusoids o r e xponentials. I n t his c hapter we e xtend t his s pectral r epresentation t o a periodic s ignals. 4.1 Aperiodic Signal Representation by Fourier Integral A pplying a l imiting p rocess, we now show t hat a n a periodic s ignal c an b e
e xpressed as a c ontinuous s um ( integral) o f e verlasting e xponentials. T o r epresent a n
a periodic signal I (t) s uch a s t he o ne d epicted i n Fig. 4.1a b y e verlasting e xponential
signals, l et us c onstruct a n ew p eriodic s ignal I To (t) f ormed by r epeating t he s ignal
I (t) a t i ntervals o f To s econds, as i llustrated in Fig. 4.1b. T he p eriod To is m ade
long e nough t o a void overlap b etween t he r epeating pulses. T he p eriodic s ignal
!To (t) c an b e r epresented b y a n e xponential F ourier series. I f we l et To - 00, t he
p ulses in t he p eriodic s ignal r epeat a fter a n i nfinite interval a nd, t herefore Find the response of an LTIC system with transfer function lim h o(t) = I (t) T o-co H(8) _
- 82 + 8
28 +3 T hus, t he F ourier series r epresenting I To (t) will also r epresent I (t) in t he l imit
To - 00. T he e xponential F ourier s eries for I To (t) is given by t o the periodic input shown in Fig. 3.7b.
00 I To(t) = L D nejnwot (4.1) n =-oo where
a nd 27r Wo= - To 235 (4.2b) 236 4 Continuous-Time Signal Analysis: T he F ourier Transform 4.1 A periodic Signal R epresentation b y Fourier Integral .
(a) , 237 7nvelope
- F(oo) T, / (a) t_ Dn,. (b)
. ..•• ' r' I T [I [11 / Envelope "r' ,\,/ -'- r 0 1111 ~' ]-, ' 1 F(oo)
T ( b)
•T " r o-- t- Fig. 4 .2
Fig. 4 .1 C onstruction o f a p eriodic s ignal b y p eriodic e xtension o f O bserve t hat i ntegrating fro (t) over (-1f, 1f) is t he s ame as i ntegrating f (t) over
( -00,00). T herefore, Eq. (4.2a) c an b e e xpressed as 1 C hange i n t he F ourier s pectrum w hen t he p eriod b ut a s we shall see, t hese a re t he classic characteristics of a very familiar p henomenon·t
S ubstitution o f Eq. (4.4) in Eq. (4.1) yields f(t)e-jnwot dt ~ F(nwo) jnwot f To (t) = 00 Dn = -1
To To i n F ig. 4.1 is d oubled. f (t). ~ - r,--e n =-oo (4.2c) (4.5) 0 - 00 I t is i nteresting t o see how t he n ature o f t he s pectrum c hanges as To increases.
To u nderstand t h...
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