FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS in .NET

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FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS
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( a ) From Eq. (5.137) (Prob. 5.20) we have sin at ~ ( t=)-- X ( w ) ~t Then when a < w,, we have Y ( w )=X ( w ) H ( w )= X ( w ) Thus, sin at y(t) =x(t)= Xt ( b ) When a > w,, we have Y ( w ) = X ( w ) H ( w )= H ( w ) Thus, y(t) =h(t) =
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In case ( a ) , that is, when w, > a , y ( t ) = x ( t ) and the filter does not produce any distortion. In case ( b ) , that is, when w, < a , y ( t ) = h ( t ) and the filter produces distortion.
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5.53. Consider an ideal low-pass filter with frequency response
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The input to this filter is the periodic square wave shown in Fig. 5-27. Find the output
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Setting A
10, T,
= 2,
and w ,
2n/To = n in Eq. (5.107) (Prob. 5 . 9 , we get
rad, the filter passes all harmonic components Since the cutoff frequency o, of the filter is 4 7 ~ of x ( t ) whose angular frequencies are less than 4 n rad and rejects all harmonic components of x ( t ) whose angular frequencies are greater than 477 rad. Therefore, 20 y ( t ) = 5 + -sinnt x
+ -s i n 3 ~ t 3n
5 - 5 4 Consider an ideal low-pass filter with frequency response
The input to this filter is Find the value of o, such that this filter passes exactly one-half of the normalized energy of the input signal x ( t ).
CHAP. 51
FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS
From Eq. (5.155)
Then
The normalized energy of x ( t ) is
Using Parseval's identity (5.641, the normalized energy of y ( t ) is
1 d w, =-/ -- o0 2 - -17 tan-' 4+ 2 2
-E = 2 " 8
from which we obtain
'"c -=
tan - = 1
o, = 2 rad/s
5.55. T h e equivalent bandwidth of a filter with frequency response H ( o ) is defined by
where IH(w)lm, denotes the maximum value of the magnitude spectrum. Consider the low-pass RC filter shown in Fig. 5 - 6 ( a ) .
( a ) Find its 3-dB bandwidth W,. , ( b ) Find its equivalent bandwidth We,.
From Eq. (5.91) the frequency response H ( w ) of the RC filter is given by
H(o)=
l+joRC
l+j(o/o,)
where o, = 1 /RC. Now
The amplitude spectrum lH(w)l is plotted in Fig. 5-6(b). When w IH(o,)l = I / & . Thus, the 3-dB bandwidth of the RC filter is given by
= o, =
1/RC,
FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS
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( 6 ) From Fig. 5-6(b) we see that IH(O)I Rewriting H(w) as
1 is the maximum of the magnitude spectrum.
and using Eq. (5.179), the equivalent bandwidth of the RC filter is given by (Fig. 5-32)
Fig. 5-32 Filter bandwidth.
5.56. The risetime t , of the low-pass RC filter in Fig. 5-6(a) is defined as the time required for a unit step response to go from 10 to 90 percent of its final value. Show that
where f,
W,,,/2.rr
1/2.rrRC is the 3-dB bandwidth (in hertz) of the filter.
From the frequency response H(w) of the RC filter, the impulse response is
Then, from Eq. (2.12) the unit step response d t ) is found to be
Dividing the first equation by the second equation on the right-hand side, we obtain e ( r ~ - r d / R C= 9
CHAP. 51
FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS
Fig. 5-33
which indicates the inverse relationship between bandwidth and risetime.
5.57. Another definition of bandwidth for a signal x ( t ) is the 90 percent energy containment bandwidth W,, defined by
where Ex is the normalized energy content of signal x ( t ) . Find the W , for the following signals:
( a ) x ( t ) = e-"'u(t), a > 0 sin at ( b ) x(t)= (a)
rr l From Eq. (5.155)
~ ( t=)e - " u ( t )
X( W ) = a +jo
From Eq. (1.14)
Now, by Eq. (5.180)
FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS
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from which we get
Thus.
From Eq. (5.137) sin at ~ ( t = --X(w) ) 7~t Using Parseval's identity (5.64), we have =pu(w) =
Iwl < a
Iwl > a
Then, by Eq. (5.180)
from which we get
Ww = 0.9a rad/s
Note that the absolute bandwidth of x(t) is a (radians/second).
5.58. Let x ( t ) be a real-valued band-limited signal specified by [Fig. 5-34(b)]
Let x,(t be defined by
( a ) Sketch x $ t ) for
T, < r/o, and for T, > r/oM.
( b ) Find and sketch the Fourier spectrum X $ o ) of x J r ) for T, < r/oM and for T, > n/w,.
(a) Using Eq. (I.26), we have
The sampled signal x , ( r ) is sketched in Fig. 5-34(c) for Tq< r/w,, and in Fig. 5-34(i) for T, > T / w ~ . The signal x,(t) is called the ideal sampled signal, T, is referred to as the sampling interr.al (or period), and f , = 1/T, is referred to as the sampling rate (or frequency ).
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