barcode generator for ssrs Find the overall frequency response of this system. in Software

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Find the overall frequency response of this system.
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Assuming bandlimited inputs with X , ( J R ) = 0 for ( R (> R s / 2 , the output Y a ( j R )is related to the input X,(jS2) as follows: ya(jQ) = H J j R ) X ( J Q )
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Because the frequency response of the discrete-time system is
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Consider the system shown in Fig. 3-9 for implementing a continuous-time system in terms of a discretetime system. Assuming that the input signals x,(t) are bandlimited so that X,( f ) = 0 for ( f( > 10 kHz, find the discrete-time system that produces the output r,(f) = Iflx,(f> 20005Ifli8000 otherwise
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For bandlimited inputs, the system in Fig. 3-9 is a linear shift-invariant system with an effective frequency response equal to
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The system that we would like to realize has a frequency response
Ha(jQ) =
40001~ I 1 5 I sooh 1 n
otherwise
SAMPLING
[CHAP. 3
If we assume a sampling frequency f = 20 kHz, the frequency response of the discrete-time system should be ,
0 . 2 ~ 101 5 0 . 8 ~ 5 otherwise
where T, = 1/20000.
Diagrammed in the figure below is a hybrid digital-analog network.
) The discrete-time system H ( e J W is a low-pass filter ~ ( e " )=
I 4 5 oo
else
and the analog system Hhpf( ) is a high-pass filter with a frequency response as shown below. f
t "'"'
The input xa(r) is bandlimited to 4 kHz, and the sampling frequencies of the ideal C P and DIC converters are 10 kHz. Find the value for A and q that will result in perfect reconstruction of x a ( t ) ,
Because x a ( t ) is bandlimited to 4 kHz, the upper branch of this hybrid system acts as an ideal analog low-pass filter with a frequency response
Because the analog network is a high-pass filter with a cutoff frequency of 4 kHz, and
& ( t ) will be equal to x a ( t ) provided that A = 1 and
A digital sequence x ( n ) is to be transmitted across a linear time-invariant bandlimited channel as illustrated
in the figure below.
CHAP. 31
SAMPLING
Transmitter
Receiver
The transmitter is a D/C converter, and the receiver simply samples the received waveform y,(r):
Assume that the channel may be modeled as an ideal low-pass filter with a cutoff frequency of 4 kHz:
(a) Assuming an ideal C P and D/C, and perfect synchronization between the transmitter and receiver,
what values of T, (if any) will guarantee that y(n) = x(n)
(b) Suppose that the D/C is nonideal. Specifically, suppose that x(n) is first converted to an impulse train and then a zero-order hold is used to perform the "interpolation" between the sample values. In other words, the impulse response of the interpolating filter is a pulse of duration T,:
M t )
1 = 0
051iT'
otherwise
Because the received sequence y(n) will no longer be equal to x(n), in order to improve the performance of the receiver, the received samples are processed with a digital filter as shown below.
Find the frequency response of the filter that should be used to filter y(n).
(a) The output of the D/C converter is a bandlimited signal x a ( t ) with a Fourier transform that is equal to zero for If 1 > f s / 2 . Because x a ( t ) is passed through a bandlimited channel that rejects all frequencies greater than 4 kHz, in order for there to be no distortion at the receiver, it is necessary that
Thus, the C/D and D/C converters must operate at a rate less than 8 kHz.
(b) In order to get the maximum amount of data through the channel per unit of time, we will let T, be the minimum sampling period,
T = s
When the reconstruction filter in the D/C converter is a zero-order hold, the frequency response of the discretetime system that relates the input sequence x ( n ) to the reconstructed sequence y ( n ) is
SAMPLING
[CHAP. 3
where
otherwise
Therefore, and the discrete-time filter for processing y(n) to remove the distortion introduced by the zero-order hold should approximate the response
Consider the following system for processing a continuous-time signal with a discrete-time system:
The frequency response of the discrete-time filter is
Iff, = 2 kHz and x,(t) = sin(lOOO~t), the output y , ( t ) . find
Sampling n,(r) = sin(10007rr) with a sampling frequency f, = 2000 produces the discrete-time sequence x(n) = x,(nT,) = sin(10007rnTs) = sin This sequence is then filtered with the discrete-time filter
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