barcode lib ssrs Because IH(elw)(= 0 at w = 31714, the sinusoid in x(n) is filtered out, and the output is simply in Software

Generation Code 128C in Software Because IH(elw)(= 0 at w = 31714, the sinusoid in x(n) is filtered out, and the output is simply

Because IH(elw)(= 0 at w = 31714, the sinusoid in x(n) is filtered out, and the output is simply
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Find the magnitude, phase, and group delay of a system that has a unit sample response
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h ( n ) = S(n) - c d ( n - I )
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where ol is real.
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The frequency response of this system is
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Therefore, the magnitude squared is
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The phase, on the other hand, is H[(eJW) gh( w ) = tan-' ---- = tanp' HR(eiw)
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a sin w
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Finally, the group delay may be found by differentiating the phase (see Prob. 2.19). Alternatively, we may note that because this system is the inverse of the one considered in Example 2.2.2, the phase and the group delay are simply the negative of those found in the example. Therefore, we have
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A 90" phase shifter is a system with a frequency response
FOURIER ANALYSIS
[CHAP. 2
Note that the magnitude is constant for all o,and the phase is -n/2 for 0 < o < n and n/2 for -IT < w < 0. Find the unit sample response of this system.
The unit sample response may be found by integration:
Therefore, we have h(n) = which may also be expressed as
n odd n even
Filters
2.6 Let h ( n ) be the unit sample response of a low-pass filter with a cutoff frequency o, What type of filter has a unit sample response g(n) = (- l)"h(n) If a filter with a unit sample response h ( n ) is implemented with a difference equation of the form
how should this difference equation be modified to implement the system that has a unit sample response g ( n ) = (- 1)" h(n)
Given that g(n) = (- l)"h(n), the frequency response G(eJU)is related to the frequency response of the low-pass filter, H(eJU), follows: as
Therefore, G(ejU)is formed by shifting ~ ( e j " ) frequency by 7r. Thus, if the passband of the low-pass filter in is lo[ 5 w,, the passband of G(ejU)will be n - w, < Iwl n . As a result, it follows that g(n) is the unit sample response of a high-pass filter.
If a filter with a unit sample response h(n) may be realized by the difference equation given in Eq. (2.7). the frequency response of the filter is
H(eJw) =
Multiplying h(n) by (-1)"produces a system with a frequency response
CHAP. 2 1
Because eJkn (- I)), =
FOURIER ANALYSIS
and the difference equation becomes
That is. the coefficients a(k) and b(k) fork odd are negated,
Let H ( e J m )be the frequency response of an ideal low-pass filter with a cutoff frequency wc as shown in the figure below.
Assume that the phase is linear, #h(w) = -now. Determine whether or not it is possible to find an input x(n) and a cutoff frequency w, < n that will produce the output
1 = 0
n=0,1,
..., 2 0
otherwise
If X(eJo) is the DTFT of x(n), the output of the low-pass filter will have a DTFT
Therefore, Y ( d o ) must be equal to zero for w,. 5
lo( 5
n. However, the DTFT of y(n) is
which is not zero for w, p l 1 5 n. Therefore, there is no value for w, < n , and no input x(n) that will generate o the given output y(n).
Let h(n) be the unit sample response of an ideal low-pass filter with acutoff frequency wc = n/4. Shown in the figure below is a linear shift-invariant system that is formed from a cascade of a low-pass filter and two modulators. Find the frequency response of the overall system relating the input x(n) to the output y(n).
There are two ways that we may use to find the frequency response of this system. The first is to note that because the input to the low-pass filter is (- I)"x(n), the output of the low-pass filter is
FOURIER ANALYSIS
[CHAP. 2
Therefore. Bringing the term (-1)" inside the summation, and using the fact that ( - I ) " - ~ = (we have
Thus, the unit sample response of the overall system is (- I)"h(n), and the frequency response is
otherwise
Another way to determine the frequency response is to find the response of the system to a complex exponential, x(n) = eln'". Modulating by (-1)" = e-inn produces the sequence
which is the input to the LSI system. Because u(n) is a complex exponential, the response of the system to v(n) is
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