barcode generator for ssrs Sample Rate Increase by an Integer Factor in Software

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3.5.2 Sample Rate Increase by an Integer Factor
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Suppose that we would like to increase the sampling rate by an integer factor L. If x a ( t ) is sampled with a sampling frequency f s = I / T,, then
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x ( n ) = xa(nTs)
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To increase the sampling rate by an integer factor L, it is necessary to extract the samples
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from x(n). The samples of x ; ( n ) for values of n that are integer multiples of L are easily extracted from x ( n ) as follows:
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xi(nL) = x(n)
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SAMPLING
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[CHAP. 3
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Shown in Fig. 3-12(a) is an up-sampler that produces the sequence Zi(n) = x(n/L) n = O , f L , f 2 L , ... otherwise
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In other words, the up-sampler expands the time scale by a factor of L by inserting L - 1 zeros between each sample of x(n). In the frequency domain, the up-sampler is described by
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Therefore, X(eJW) simply scaled in frequency. After up-sampling, it is necessary to remove the frequency is scaled images of X,(jQ), except those that are at integer multiples of 2 x . This is accomplished by filtering Z;(n)
Fig. 3-12. (a) Up-sampling by an integer factor L . (b)Interpolation by a factor of L .
with a low-pass filter that has a cutoff frequency of n / L and a gain of L. In the time domain, the low-pass filter interpolates between the samples at integer multiples of L as shown in Fig. 3-13. The cascade of an up-sampler with a low-pass filter shown in Fig. 3-12(b) is called an interpolator. The interpolation process in the frequency domain is illustrated in Fig. 3-14.
Fig. 3-13. (a)The output of the up-sampler. (b) The interpolation between the samples T,(n)that is performed by the low-pass filter.
CHAP. 31
SAMPLING
4n -L
Zn -L
n -L
L = 2n
Fig. 3-14. Frequency domain illustration of the process of interpolation. (a) The continuous-time signal. (b) The DTFT of the sampled signal x ( n ) = x,(nT,). ( c ) The DTFT of the up-sampler output. (d) The ideal low-pass filter to perform the
interpolation. (e) The DTFT of the interpolated signal.
3.5.3 Sample Rate Conversion by a Rational Factor
The cascade of a decimator that reduces the sampling rate by a factor of M with an interpolator that increases the sampling rate by vital factor of L results in a system that changes the sampling rate by a rational factor of L / M . This cascade is illustrated in Fig. 3-15(a). Because the cascade of two low-pass filters with cutoff frequencies n / M and n/L is equivalent to a single low-pass filter with a cutoff frequency
the sample rate converter may be simplified as illustrated in Fig. 3-15(b).
SAMPLING
[CHAP. 3
Fig. 3-15. (a) Cascade of an interpolator and a decimator for changing the sampling rate by a rational factor L I M . (b) A simplified structure that results when the two low-pass ti lters are combined.
EXAMPLE 3.5.1 Suppose that a signal x,,(t) has been sampled with a sampling frequency of 8 kHz and that we would like to derive the discrete-time signal that would have been obtained if x u ( ! ) had been sampled with a sampling frequency of 10 kHz. Thus, we would like to change the sampling rate by a factor of
This may be accomplished by up-sampling x ( n ) by a factor of 5, tiltering the up-sampled signal with a low-pass filter that has a cutoff frequency w, = n / 5 and a gain of 5, and then down-sampling the filtered signal by a Factor of 4.
Solved Problems
AID and DIA Conversion
Consider the discrete-time sequence
Find two different continuous-time signals that would produce this sequence when sampled at a frequency o f f , = 10 Hz. A continuous-time sinusoid
= .%(I) = COS(QOI) cos(2n fat)
that is sampled with a sampling frequency off, results in the discrete-time sequence
However, note that for any integer k , cos 2rr - n Therefore, any sinusoid with a frequency
= cos 2 n fofkf'n)
CHAP. 31
SAMPLING
will produce the same sequence when sampled with a sampling frequency f,. With x(n) = cos(nn/8), we want
fo = i!g fs = 625 Hz
Therefore, two signals that produce the given sequence are x,(t) = cos(1250nf)
.r2(t) = cos(21250nt)
If the Nyquist rate for x a ( t ) is a , , what is the Nyquist rate for each of the following signals that are derived from xa (r )
dxa(t) 7
(b) xa ( 2 t )
(c) ~ , 2 ( t )
(4 x a 0 ) cos(Qot)
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