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= tan' = 0.2952~ + 0.59031) Consider the following system consisting of an ideal D/C converter, a linear timeinvariant filter, and an ideal C P converter. The continuoustime system h , ( t ) is an ideal lowpass filter with a frequency response
Hdf) = (a) If T I = T2 =
(b) If T I =
(a) x
find an expression relating the output y ( n ) to the input x(n). and T2 = find y(n) when
I f 1 5 1OkH.z otherwise
(a) When TI = T2,this system behaves as a linear shiftinvariant discretetime system with a frequency response Because H,(jS2) = 1 for 152) < 2 n .
lo4, h(n) = 6(n) Therefore, y(n) = x(n). Another way to analyze this system is to note that the output of the D/C converter, x,(t), is bandlimited to f = 5 kHz. Because Ha(f )is an ideal lowpass filter with a cutoff frequency 10 kHz, yo(!) = xo(t). Therefore, this system is equivalent to the one shown below. Because an ideal D/C converter followed by an ideal D/C converter is the identity system, y(n) = x(n). (b) When T I T2,this system is, in general, no longer a linear shiftinvariant system. However, we may analyze this system in the frequency domain as follows. First, note that the DTFTofx(n) is as illustrated in the following figure: SAMPLING
[CHAP. 3
Thus, the output of the D/C converter is a bandlimited signal that has a Fourier transform as shown in the following figure: The analog lowpass filter removes all frequencies in x a ( t ) above 10 kHz to produce a signal y,(t) that has a Fourier transform as shown below. Because the highest frequency in y a ( t ) is 10 kHz, the Nyquist rate is 20 kHz. However, the sampling frequency of the C/D converter is 10 kHz, so y a ( t ) will be aliased. The DTFT of y ( n ) is related to Y a ( j Q ) as follows: Summing the shifted and scaled transforms yields
Sample Rate Conversion
3.21 Suppose that a discretetime sequence x ( n ) is bandlimited so that
This sequence is then sampled to form the sequence
where N is an integer. Find the largest value for N for which x ( n ) may be uniquely recovered from y ( n ) . The easiest way to view this problem is as illustrated below.
Converting x ( n ) into a continuoustime signal with an ideal D/C converter with a sampling frequency f produces , a continuoustime signal x a ( t ) that is bandlimited to fo = 0.3 . f s / 2 . Therefore, xa(r) may be sampled, without CHAP. 31
SAMPLING
aliasing, if we use a sampling frequency fsf 2 2 fo = 0 . 3 f,, or
Therefore, if T,' = 3T,, y ( n ) = x,(3nTs) = x ( 3 n ) and x ( n ) may be uniquely recovered from y(n). Thus, N = 3.
Consider the following system: Assume that X a ( f ) = 0 for If ( > l / T s and that
How is the output of the discretetime system, y ( n ) , related to the input signal x a ( t ) In this system, the bandlimited signalx,(t) is sampled, without aliasing, to produce the sampled signal x ( n ) = x,(nT,). Upsampling x ( n ) by a factor of L , and filtering with an ideal low.pass filter with a cutoff frequency w, = n / L , produces the signal that is, a signal that is sampled with a sampling frequency L f,. However, because the lowpass filter has linear phase with a group delay of one sample, the interpolated upsampled signal is delayed by 1. Therefore, the output of the lowpass filter is u ( n ) = w ( n  1 ) = x, Downsampling by L then produces the output
Thus, y ( n ) corresponds to samples of x,(c  to) where ro = T , / L .
Consider the system shown in the figure below.
Assume that the input is bandlimited, X a ( j i 2 ) = 0 for ( i 2 ( > 2 n . 1000.
(a) What constraints must be placed on M, T I ,and T2 in order for y a ( t ) to be equal to x,(t) (b) If f , = f2 = 20 kHz and M = 4, find an expression for ya(r) in terms of x , ( t ) . SAMPLING
(a) Suppose that x,(r) has a Fourier transform as shown in the figure below.
[CHAP. 3
Because y ( n ) = x ( M n ) = x , ( n M T I ) , in order to prevent x ( n ) from being aliased, it is necessary that

