2d barcode generator vb.net By the definition of h[n] [Eq. (2.30)] we have in Visual Studio .NET

Encode QR in Visual Studio .NET By the definition of h[n] [Eq. (2.30)] we have

By the definition of h[n] [Eq. (2.30)] we have
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h[n]
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which is sketched in Fig. 6-26(a). Note that h[n] satisfies the condition (6.163) with N=3. ( b ) Taking the Fourier transform of Eq. (6.168), we have
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Fig. 6-26
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FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS [CHAP. 6
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By Eq. (1.90), with a = e-jR, we get
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1 1 - e-i3fi 1 e - i 3 f i / 2 ( e j 3 R / 2 - ,- j 3 R / 2 1 H ( R ) = - ~ - ~ - j= n e - i R / 2 ( e j R / 2 - e - i ~ / 2 ) 3 3
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e(n)=
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when H r ( R ) > 0 when H r ( R ) < 0
which are sketched in Fig. 6-26(b).We see that the system is a low-pass FIR filter with linear phase.
6.42. Consider a causal discrete-time FIR filter described by the impulse response
h [ n ] = {2,2,- 2, - 2)
( a ) Sketch the impulse response h [ n ] of the filter.
( b ) Find the frequency response H ( R ) of the filter. (c) Sketch the magnitude response IH(R)I and the phase response 8 ( R ) of the filter.
( a ) The impulse response h [ n ] is sketched in Fig. 6-27(a). Note that h [ n ] satisfies the condition (6.164) with N = 4. ( b ) By definition ( 6 . 2 7 )
where
IH(R)l= IHr(R)l= sln
I ( 1 + ("z")i
sin -
which are sketched in Fig. 6-27(b). We see that the system is a bandpass FIR filter with linear phase.
CHAP. 61 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS
Fig. 6-27
SIMULATION
6.43. Consider the RC low-pass filter shown in Fig. 6-28(a)with RC = 1
Construct a discrete-time filter such that
h d [ n ] = h c ( t ) l ,= n ~ = h c ( n T S ) ,
(6.172)
where h c ( t ) is the impulse response of the RC filter, h,[n] is the impulse response of the discrete-time filter, and T, is a positive number to be chosen as part of the design procedures. Plot the magnitude response IH , ( o ) ) of the RC filter and the magnitude response ( H J w T J of the discrete-time filter for T, = 1 and T, = 0.1. The system function H,(s) of the RC filter is given by (Prob. 3.23)
HJs)
s+ 1
and the impulse response h$) is
h c ( t )= e-'u(t)
By Eq. (6.172) the corresponding h,[nl is given by
h , [ n ] = e - n c u [ n ] = (e-")"u[d
FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS [CHAP. 6
Fig. 6-28 Simulation of an RC filter by the impulse invariance method.
Then, taking the z-transform of Eq. (6.175), the system function Hd(z) of the discretetime filter is given by
-e-Tsz-,
from which we obtain the difference equation describing the discrete-time filter as
y [ n ] - e - T s y [ n - 1) = x [ n ]
(6.176)
from which the discrete-time filter that simulates the RC filter is shown in Fig. 6-28(b). By Eq. (5.40)
Then
By Eqs. (6.34) and (6.81)
From Eq. (6.149)
CHAP. 61 FOURIER ANALYSIS O F DISCRETE-TIME SIGNALS AND SYSTEMS
From T, = 1,
For T, = 0.1,
The magnitude response IHc(w)l of the RC filter and the magnitude response IH,(wq)l of the discrete-time filter for T, = 1 and T, = 0.1 are plotted in Fig. 6-29. Note that the plots are scaled such that the magnitudes at w = 0 are normalized to 1. The method utilized in this problem to construct a discrete-time system to simulate the continuous-time system is known as the impulse-inuariance method.
Fig. 6-29
6.44. By applying the impulse-invariance method, determine the frequency response H d ( f l ) of the discrete-time system to simulate the continuous-time LTI system with the system function
Using the partial-fraction expansion, we have
Thus, by Table 3-1 the impulse response of the continuous-time system is
h c ( t )= ( e - t - e - " ) u ( t )
Let hd[nl be the impulse response of the discrete-time system. Then, by Eq. (6.177)
(6.177) )4n]
h d [ n ]= h,(nT,) = (e-"'5
- e-'"'j
FOURIER ANALYSIS O F DISCRETE-TIME SIGNALS AND SYSTEMS
[CHAP. 6
and the system function of the discrete-time system is given by
Thus, the frequency response H d ( f l )of the discrete-time system is
H d ( f l )= H d ( z ) l , , , , ~ ~ 1 - e - n T s e - ~ n- 1 - e - 2 n ~ ,,-in =
Note that if the system function of a continuous-time LTI system is given by
(6.179)
then the impulse-invariance method yields the corresponding discrete-time system with the system function H,( z given by
6.45. A differentiator is a continuous-time LTI system with the system function [Eq. (3.2011
A discrete-time LTI system is constructed by replacing s in H c ( s ) by the following transformation known as the bilinear transformation:
to simulate the differentiator. Again T, in Eq. ( 6 . 1 8 3 ) is a positive number to be chosen as part of the design procedure.
( a ) Draw a diagram for the discrete-time system. ( b ) Find the frequency response H d ( f l ) of the discrete-time system and plot its magnitude and phase responses. ( a ) Let H , ( z ) be the system function of the discrete-time system. Then, from Eqs. (6.182) and (6.183)we have
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