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barcode lib ssrs With h l ( n ) being in cascade with g(n), the overall frequency response becomes in Software
With h l ( n ) being in cascade with g(n), the overall frequency response becomes Code 128 Code Set B Decoder In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Paint ANSI/AIM Code 128 In None Using Barcode encoder for Software Control to generate, create Code 128C image in Software applications. ~ ( e j " = H I(eiW)[Hz(ei") ) Code 128 Code Set A Recognizer In None Using Barcode reader for Software Control to read, scan read, scan image in Software applications. Generating Code 128 Code Set A In Visual C#.NET Using Barcode creator for .NET framework Control to generate, create Code128 image in VS .NET applications. + H 3 ( e j wH4(eJW)1 ) Code 128 Code Set A Creator In VS .NET Using Barcode creator for ASP.NET Control to generate, create ANSI/AIM Code 128 image in ASP.NET applications. Draw Code 128C In .NET Framework Using Barcode encoder for VS .NET Control to generate, create Code 128 Code Set C image in VS .NET applications. FOURIER ANALYSIS
Code 128 Code Set C Printer In VB.NET Using Barcode creator for .NET Control to generate, create Code 128B image in Visual Studio .NET applications. Print DataMatrix In None Using Barcode drawer for Software Control to generate, create DataMatrix image in Software applications. (b) The frequency responses of the systems in this interconnection are
Drawing Barcode In None Using Barcode creation for Software Control to generate, create bar code image in Software applications. Painting ANSI/AIM Code 39 In None Using Barcode printer for Software Control to generate, create Code 3 of 9 image in Software applications. [CHAP. 2
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Create Bar Code In None Using Barcode encoder for Online Control to generate, create bar code image in Online applications. UPC  13 Generation In ObjectiveC Using Barcode creator for iPhone Control to generate, create EAN13 image in iPhone applications. This filter may be viewed as a summation of a lowpass filter, a bandpass filter, and a highpass filter. Because both a bandpass filter and a highpass filter may be synthesized using a parallel connection of lowpass filters, we may proceed as follows. First, we put an allpass filter H3(eJW) A3 in parallel with a lowpass filter with a cutoff = frequency 02 and a gain of Az  A3. This parallel network has a frequency response Recognize Code 128C In C# Using Barcode recognizer for .NET framework Control to read, scan read, scan image in .NET framework applications. EAN13 Creator In Visual Basic .NET Using Barcode drawer for VS .NET Control to generate, create EAN 13 image in .NET framework applications. To produce the correct magnitude over the lower band, Iwl iwl, we add a third lowpass filter in parallel with the other two. This filter has a cutoff frequency of wl and a gain of A I  A2. Two linear shiftinvariant systems are connected in a feedback network as illustrated in the figure below. Assuming that the overall system is stable, so that H(ej") exists, show that the frequency response of this feedback network is . Y (ej") F (ei") J" H(e )  X(ejm) 1  F(ej")G(ej") To analyze this network, we begin by noting that
CHAP. 21
FOURIER ANALYSIS
which, in the frequency domain, becomes
W (el") = X ( e J W ) G ( e J " ) Y( e l W ) Because
y ( e j w )= F ( e J w ) w ( e l " ) then Solving for ~ ( r i " yields ) Y ( e l w )= F ( e j w ) [ x ( e J w ) G ( e I w ) Y( e l w ) ] Therefore, the frequency response is
The DiscreteTime Fourier Transform
A linear shiftinvariant system is described by the LCCDE
Find the value of b so that IH(eJm)l is equal to I at w = 0, and find the haypower point (i.e., the is frequency at which I H(ejW)I2 equal to onehalf of its peak value, which occurs at w = 0). The frequency response of the system described by this difference equation is
Because
l~(ej")= l~
 0.5ejw)( 1  0 . 5 e j w ) 1.25  cos w
I H ( e J w ) lwill be equal to 1 at o = 0 if
h2  1.25  1 This will be true when b = f0.5. To find the halfpower point, we want to find the frequency for which I H ( ~ ' " ) =~ I 0.25 = 0.5 1 .25  cos o
This occurs when cos o = 0.75 or o = 0.2317.
Consider the system defined by the difference equation
where a and b are real, and la I < 1. Find the relationship between a and h that must exist if the frequency response is to have a constant magnitude for all w , that is, I H ( ~ ~ =) 1 * I FOURIER ANALYSIS
[CHAP. 2
Assuming that this relationship is satisfied, find the output of the system when a = and
x(n) = (;)"u(n) The frequency response of the LSI system described by this difference equation is
The squared magnitude is
1 + h2 2b cos w ( b + ejo)(h + eJ") I H ( ~ ' "=I ~ ) I + a2  2a cos w ( 1  aeJU)(l aeju) Therefore, it follows that I H(e'")12 = 1 if and only if b = a. if x(n) = ( i ) " u ( n )Y ( e j U is given by , ) With a = and h = Using the DTFT pair
given in Table 2 1, and using the linearity and delay properties of the DTFT, we have
= What we observe from this example is that although I H (eJU)l 1, the nonlinear phase has a significant effect on the values of the input sequence. Show that the group delay of a linear shiftinvariant system with a frequency response H(ejw)may be expressed as H R ( ~ ~ , ) GR ( d w ) H I (ejW)G(ejw) t h ( ~= ) I H(ejw)I2
where HR(eJw) Hl(ejw)are the real and imaginary parts of H(ejw),respectively, and G R ( e j Wand and ) G r ( e j w ) the real and imaginary parts of the DTFT of nh(n). are In terms of magnitude and phase, the frequency response is
Note that if we take the logarithm of H(eJW), have an explicit expression for the phase we
Differentiating with respect tow, we have
Equating the imaginary parts of both sides of this equation yields
CHAP. 21
If we define
FOURIER ANALYSIS
d H(eJ") = Hk(ei") dw
+j~;(ej") where HA(eJW) the derivative of the real part of H(eiw) and H;(ejW)is the derivative of the imaginary part, the is group delay may be written as

