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2d barcode generator vb.net By the definition of h[n] [Eq. (2.30)] we have in Visual Studio .NET
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which are sketched in Fig. 626(b).We see that the system is a lowpass FIR filter with linear phase. 6.42. Consider a causal discretetime 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. 627(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. 627(b). We see that the system is a bandpass FIR filter with linear phase. CHAP. 61 FOURIER ANALYSIS OF DISCRETETIME SIGNALS AND SYSTEMS
Fig. 627 SIMULATION
6.43. Consider the RC lowpass filter shown in Fig. 628(a)with RC = 1
Construct a discretetime 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 discretetime 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 discretetime 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 DISCRETETIME SIGNALS AND SYSTEMS [CHAP. 6
Fig. 628 Simulation of an RC filter by the impulse invariance method.
Then, taking the ztransform of Eq. (6.175), the system function Hd(z) of the discretetime filter is given by eTsz, from which we obtain the difference equation describing the discretetime filter as
y [ n ]  e  T s y [ n  1) = x [ n ] (6.176) from which the discretetime filter that simulates the RC filter is shown in Fig. 628(b). By Eq. (5.40) Then
By Eqs. (6.34) and (6.81) From Eq. (6.149) CHAP. 61 FOURIER ANALYSIS O F DISCRETETIME 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 discretetime filter for T, = 1 and T, = 0.1 are plotted in Fig. 629. 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 discretetime system to simulate the continuoustime system is known as the impulseinuariance method. Fig. 629 6.44. By applying the impulseinvariance method, determine the frequency response H d ( f l ) of the discretetime system to simulate the continuoustime LTI system with the system function Using the partialfraction expansion, we have
Thus, by Table 31 the impulse response of the continuoustime system is
h c ( t )= ( e  t  e  " ) u ( t ) Let hd[nl be the impulse response of the discretetime system. Then, by Eq. (6.177) (6.177) )4n] h d [ n ]= h,(nT,) = (e"'5  e'"'j
FOURIER ANALYSIS O F DISCRETETIME SIGNALS AND SYSTEMS
[CHAP. 6
and the system function of the discretetime system is given by
Thus, the frequency response H d ( f l )of the discretetime 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 continuoustime LTI system is given by
(6.179) then the impulseinvariance method yields the corresponding discretetime system with the system function H,( z given by 6.45. A differentiator is a continuoustime LTI system with the system function [Eq. (3.2011 A discretetime 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 discretetime system. ( b ) Find the frequency response H d ( f l ) of the discretetime system and plot its magnitude and phase responses. ( a ) Let H , ( z ) be the system function of the discretetime system. Then, from Eqs. (6.182) and (6.183)we have

