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Fig. 3-8. (a) The magnitude of the frequency response of a zero-order
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hold compared to the ideal reconstruction filter. (b)The ideal reconstruction compensation filter.
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DISCRETE-TIME PROCESSING OF ANALOG SIGNALS
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One of the important applications of A D and D/A converters is the processing of analog signals with a discretetime system. In the ideal case, the overall system, shown in Fig. 3-9, consists of the cascade of a C/D converter, a discrete-time system, and a D/C converter. Thus, we are assuming that the sampled signal is not quantized and that an ideal low-pass filter is used for the reconstruction filter in the D/C converter. Because the input x a ( t ) and the output ya(t) are analog signals, the overall system corresponds to a continuous-time system. To analyze this system, note that the C/D converter produces the discrete-time signal x ( n ) , which has a DTFT given by
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If the discrete-time system is linear and shift-invariant with a frequency response H ( e j W ) ,
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Fig. 3-9. Processing an analog signal using a discrete-time system.
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Finally, the D/C converter produces the continuous-time signal y,(t) from the samples y ( n ) as follows:
~ ~= ( t C0 y ( n )sin n (( -- nT,)/ Ts ~ tnT,)/Ts
n = - ~
Either using Eq. (3.7) or by taking the DTFT directly, in the frequency domain this relationship becomes
If x,(t) is bandlimited with X , ( j Q ) = 0 for IQI > T I T , , the low-pass filter H , ( j Q ) eliminates all terms in the sum except the first one, and
Therefore, the overall system behaves as a linear time-invariant continuous-time system with an effective frequency response n H(ejnK) lQl I H,(jQ) = 1 0 otherwise TS Just as a continuous-time system may be implemented in terms of a discrete-time system, it is also possible to implement a discrete-time system in terms of a continuous-time system as illustrated Fig. 3-10. The signal x,(t) is related to the sequence values x ( n ) as follows:
Fig. 3-10. Processing a discrete-time signal using a continuous-time system.
Because x,(t) is bandlimited, y,(t) is also bandlimited and may be represented in terms of its samples as follows:
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The relationship between the Fourier transform of x a ( t )and the DTFT of x ( n ) is
X a ( j n )=
cx(ejaTs)
Is21 < T, otherwise
and the relationship between the Fourier transforms of x, (t ) and ya (t) is
n Y a ( j n )= ( ~ ~ ( j ~ r ( j" I W< Ts otherwise
Therefore, and the frequency response of the equivalent discrete-time system is
3.5 SAMPLE RATE CONVERSION
In many practical applications of digital signal processing, one is faced with the problem of changing the sampling rate of a signal. The process of converting a signal from one rate to another is called sample rate conversion. There are two ways that sample rate conversion may be done. First, the sampled signal may be converted back into an analog signal and then resampled. Alternatively, the signal may be resampled in the digital domain. This approach has the advantage of not introducing additional distortion in passing the signal through an additional D/A and A D converter. In this section, we describe how sample rate conversion may be performed digitally.
3.5.1 Sample Rate Reduction by an Integer Factor
Suppose that we would like to reduce the sampling rate by an integer factor, M. With a new sampling period T,' = MT,, the resampled signal is
Therefore, reducing the sampling rate by an integer factor M may be accomplished by taking every Mth sample of x(n). The system for performing this operation, called adown-sampler, is shown in Fig. 3-1 l(a). Down-sampling generally results in aliasing. Specifically, recall that the DTFT of x ( n ) = x,(nT,) is
Similarly, the DTFT of x&) = x(n M ) = x,(n M T,) is
Note that the summation index r in the expression for Xd(ejo)may be expressed as
r=i+kM
CHAP. 31
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Fig. 3-11. (a)Down-samplingby an integer factor M . ( b )Decimation by a factor of M, where H ( e j U )is a low-pass filter with a cutoff frequency
where -oo < k < oo and 0 5 i 5 M - 1. Therefore, X d ( e J Wmay be expressed as )
The term inside the square brackets is
Thus, the relationship between x ( e j w ) and X d ( e j w )is
xd (ejw) = _
C ~ ( ~ i i w - 2 n k l l1M
Therefore, in order to prevent aliasing, x ( n ) should be filtered prior to down-sampling with a low-pass filter = that has a cutoff frequency o,. n / M . The cascade of a low-pass filter with a down-sampler illustrated in Fig. 3- 11(b) is called a decimator.
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