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FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS
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Derive the harmonic form Fourier series representation (5.15) from the trigonometric Fourier series representation (5.8).
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Rewrite a , cos k w , t
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sin k w , t
and use the trigonometric formula cod A - B) = cos A cos B + sin A sin B.
Show that the mean-square value of a real periodic signal x ( r ) is the sum of the mean-square values of its harmonics.
Hint:
Use Parseval's identity (5.21) for the Fourier series and Eq. (5.168).
Show that if then
Hint:
Repeat the time-differentiation property (5.55).
Using the differentiation technique, find the Fourier transform of the triangular pulse signal shown in Fig. 5-38.
Ar..Ad[
sin( w d / 2 ) wd/2
Fig. 5-38
Find the inverse Fourier transform of
Hint:
Differentiate Eq. (5.155) N times with respect to ( a ) .
CHAP. 51
FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS
Find the inverse Fourier transform of X(w) = Hint: Note that
2 - w 2 + j3w
2 - w2 + j3w = 2 + ( jw12 + j3w
Am. x ( t ) = (e-' - e - 2 ' ) u ( t )
+ j w ) ( 2 +j w )
and apply the technique of partial-fraction expansion.
Verify the frequency differentiation property (5.561, that is,
Hint:
Use definition (5.31) and proceed in a manner similar to Prob. 5.28. the Fourier transform of each of the following signals: x ( t ) = cos wotu(t) x ( t ) = sin wotu(t) x ( t ) = e - " ' ~ ~ ~ w ~ tau>(O ) , t x ( t ) = e-"'sin w,tu(t), a > 0 Use multiplication property (5.59).
Find (a) (b) (c) (dl Hint:
( a ) X ( w ) = -S(w 2
- w o ) + -S(w + w , ) + 2
iw ( jw)' + w i
(dl X(w)=
( a + jo12
Let x ( t ) be a signal with Fourier transform X ( w ) given by
Consider the signal
Find the value of
Hint: Use Parseval's identity (5.64) for the Fourier transform.
FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS
[CHAP. 5
Let x ( t ) be a real signal with the Fourier transform X ( w ) . The analytical signal x + ( t ) associated with x ( t ) is a complex signal defined by
x + ( t )= x ( t ) + j i ( t )
where
. is the Hilbert transform of x(t 1
( a ) Find the Fourier transform X + ( w ) of x + ( t 1. ( 6 ) Find the analytical signal x + ( t ) associated with cos w,t and its Fourier transform X + ( w ) .
Consider a continuous-time LTI system with frequency response H(w). Find the Fourier transform S ( w ) of the unit step response s ( t ) of the system.
Hint: Am.
Use Eq. (2.12) and the integration property (5.57).
S ( w ) = .rrH(O)G(o)+ ( l / j w ) H ( w )
Consider the RC filter shown in Fig. 5-39. Find the frequency response H ( w ) of this filter and discuss the type of filter.
Ans. H(o)= iw , high-pass filter ( l / R C ) +jw
Fig. 5-39
Determine the 99 percent energy containment bandwidth for the signal
Ans.
W,,= 2.3/a radians/second or f ,
= 0.366/a
hertz
The sampling theorem in the frequency domain states that if a real signal x ( t ) is a durationlimited signal. that is,
CHAP. 51
FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS
then its Fourier transform X ( w ) can be uniquely determined from its values X ( n s r / t , ) at a series of equidistant points spaced n / t , apart. In fact, X ( w ) is given by
o t , - n.rr
Verify the above sampling theorem in the frequency domain.
Hint:
Expand x ( t ) in a complex Fourier series and proceed in a manner similar to that for Prob. 5.59.
6
Fourier Analysis of Discrete-Time Signals and Systems
INTRODUCTION
In this chapter we present the Fourier analysis in the context of discrete-time signals (sequences) and systems. The Fourier analysis plays the same fundamental role in discrete time as in continuous time. As we will see, there are many similarities between the techniques of discrete-time Fourier analysis and their continuous-time counterparts, but there are also some important differences.
6.2 DISCRETE FOURIER SERIES
Periodic Sequences:
In Chap. 1 we defined a discrete-time signal (or sequence) x [ n ] to be periodic if there is a positive integer N for which
x [ n +N] = x [ n ]
all n
(6.1)
The fundamental period No of x [ n ] is the smallest positive integer N for which Eq. (6.1) is satisfied. As we saw in Sec. 1.4, the complex exponential sequence where no= 27r/Nu, is a periodic sequence with fundamental period Nu. As we discussed in Sec. 1.4C, one very important distinction between the discrete-time and the continuoustime complex exponential is that the signals el"^' are distinct for distinct values of wO,but the sequences eiR~~", which differ in frequency by a multiple of 2rr, are identical. That is, Let
and more generally,
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