2d barcode generator vb.net ( a ) Using the DFT, estimate the Fourier spectrum X ( w ) of the continuous-time signal in VS .NET

Generate QR in VS .NET ( a ) Using the DFT, estimate the Fourier spectrum X ( w ) of the continuous-time signal

6.61. ( a ) Using the DFT, estimate the Fourier spectrum X ( w ) of the continuous-time signal
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Assume that the total recording time of x ( t ) is T, = 10 s and the highest frequency of x ( t ) is w , = 100 rad/s. ( b ) Let X [ k ] be the DFT of the sampled sequence of x ( t ) . Compare the values of X[O], X [ l ] , and X [ 1 0 ] with the values of X(O), X ( A w ) , and X ( 1 0 A w ) .
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FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS
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From Eq. (6.241)
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Thus, choosing N = 320, we obtain
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=0.625 rad
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Then from Eqs. (6.244), (6.249), and (1.921, we have
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X [ k ]=
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A t x ( n A t ) e-j(2T/N'nk
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which is the estimate of X ( k A o ) . Setting k = 0 , k = 1 , and k = 10 in Eq. (6.250), we have
From Table 5-2
x(l0Ao)
= X(6.25) =
+ j6.25
- 0,158e-11.412
Even though x ( t ) is not band-limited, we see that X [ k l offers a quite good approxima-. tion to X ( w ) for the frequency range we specified.
FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS [CHAP. 6
Supplementary Problems
Find the discrete Fourier series for each of the following periodic sequences: (a) x[n] = cos(0,l~n) (b) x[n] = sin(0.l.rrn) (c) x[n] = 2cos(1.6~n) sin(2.47rn) +
(a) x[n] = $ e j n o n
+1 ze
R0 =~0 . 1 ~ ~
Find the discrete Fourier series for the sequence x[n] shown in Fig. 6-40.
Fig. 6-40
Find the trigonometric form of the shown in Fig. 6-7 in Prob. 6.3. 3 Tr Tr Am. x[n] = - - cos-n - sin-n 2 2 2
discrete Fourier series for the periodic sequence x[n]
1 -cos r n 2
Find the Fourier transform of each of the following sequences: (a) x[nl= al"l,la1 < 1 (6) x[n] = sin(flon), IRoI < 7r (c) x[nl= u[ -n - 11
(a) X(fl)=
1-a2 1-2acosfl+a2
CHAP. 61 FOURIER ANALYSIS O F DISCRETE-TIME SIGNALS AND SYSTEMS
Find the Fourier transform of the sequence x[n] shown in Fig. 6-41
Am. X(R)
=j2(sin
+ 2 sin 2 R + 3 sin 3R)
Fig. 6-41
Find the inverse Fourier transform of each of the following Fourier transforms: X ( R ) = cos(2R) (6) X(R) =j R
(a) x [ n l = f8[n - 21 + 3 [ n + 21
Consider the sequence y[n] given by n even n odd Express y(R) in terms of X(R).
Ans.
Y(R) = $X(R) + $x(R
- 7)
(a) Find y[n 1 = x[n] * x[n]. ( b ) Find the Fourier transform Y(0) of y[n].
(a) y[n] =
In15 5 In1 > 5
Verify Parseval's theorem [Eq. (6.66)] for the discrete-time Fourier transform, that is,
Hint:
Proceed in a manner similar to that for solving Prob. 5.38.
FOURIER ANALYSIS O F DISCRETE-TIME SIGNALS AND SYSTEMS
[CHAP. 6
A causal discrete-time LTI system is described by
y [ n ] - i y [ n - 11
+ i y [ n - 21 = x [ n ]
where x [ n ] and y[nl are the input and output of the system, respectively.
( a ) Determine the frequency response H ( R ) of the system. ( b ) Find the impulse response h [ n ] of the system. ( c ) Find y[nl if x[nl = ( i ) " u [ n l .
Consider a causal discrete-time LTI system with frequency response
H ( R ) = Re{ H ( R ) ) + j I m { H ( R ) )= A ( R ) +j B ( R ) ( a ) Show that the impulse response h [ n ] of the system can be obtained in terms of A ( R ) or B ( R ) alone. ( b ) Find H ( R ) and h [ n ] if
( a ) Hint: Process in a manner similar to that for Prob. 5.49. ( b ) Ans. H ( R ) = 1 + ePin, h [ n ]= ~ [ n+]S[n - 11
Find the impulse response h [ n ] of the ideal discrete-time HPF with cutoff frequency R , (0 < R , < r )shown in Fig. 6-42. sin R,n Am. h [ n ]= S [ n ]- Tn
Fig. 6-42
Show that if H L P F ( z is the system function of a discrete-time low-pass filter, then the ) ) discrete-time system whose system function H ( z ) is given by H ( z ) = H L P F ( - z is a high-pass filter.
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