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CHAPTER 13 The Digital Processor
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Fig. 361
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Thus, from inspection of Fig. 358, we see that the value of each OUTPUT SAMPLE is the algebraic SUM of the present value of the input sample and the past value of the previous sample. With this in mind suppose, rst, that the INPUT to Fig. 358 is the sampled form of a relatively LOW FREQUENCY analog signal, such as is illustrated in Fig. 362 which shows a portion of one cycle.
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Fig. 362
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In Fig. 362 we label, for purposes of explanation, just TWO of the many samples that would be fed into the circuit of Fig. 358. We ve labeled the chosen samples 1 and 2, as shown. The two successive samples are the usual T seconds apart. (In the above, for the indicated T, about 20 samples would be taken in each half cycle.) The point we wish to make in connection with Fig. 362 is as follows. Note that (in accordance with the discussion given with Fig. 361) when sample 2 appears at the INPUT to the circuit of Fig. 358 the algebraic SUM of samples 1 and 2 appears at the OUTPUT of the circuit. The important point to note is that the sum of samples 1 and 2 is very closely twice the value of sample 2 alone*; this is because the amplitude of a low-frequency signal changes slowly with time. This is the basic reason why the output magnitude, in Fig. 359, is highest for low values of ! r !=!s . Now suppose the input to Fig. 358 is the sampled form of a relatively HIGH FREQUENCY analog signal, such as is illustrated in Fig. 363 (same peak value, same T, as in Fig. 362).
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Fig. 363
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In Fig. 363 note that we cannot say that the algebraic sum of samples 1 and 2 is very closely equal to twice the value of 2 alone (as we could for the low-frequency case of
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* As you can from Fig. 362, this is closely true for samples taken near the maximum value of the analog wave, while increasingly less so for samples taken farther from the maximum.
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CHAPTER 13 The Digital Processor
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Fig. 362), and this e ect is increasingly great between samples taken farther from the maximum value. This is why the magnitude of the output decreases with increasing analog frequency (Fig. 359), and it is due to the fact that the amplitude of the high-frequency analog wave changes rapidly with time. Now consider Fig. 364. The gure is based upon the requirements of the basic sampling theorem (section 13.1), and illustrates the condition in which r has the maximum allowable value of r 0:5. In this condition the HIGHEST ANALOG FREQUENCY that can be sampled and allowed to enter a digital lter system is equal to one-half the sampling frequency the system uses. Thus two samples per cycle must be taken of the highest permitted frequency component of the analog signal; this is the condition shown in Fig. 364. From the gure, note that the algebraic sum of samples 1 and 2 is always zero; that is, the output of the lter of Fig. 358 is zero for the highest allowable frequency analog signal (as we see in Fig. 359).
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Fig. 364
Fig. 365
Problem 319 Rework the foregoing example for b0 1:0 and b1 1:0, as shown in Fig. 365. Problem 320 Here we wish to sketch the magnitude of the frequency response curve of the following low-pass digital lter.
CHAPTER 13 The Digital Processor
For the above, nd the values of jH r j for r 0; 0:1; 0:2; 0:3; 0:4; 0:5, then sketch the curve of jH r j versus r, from r 0 to r 0:5. Problem 321 Digital lters can be of the band-pass and band-elimination types as well as the lowpass and high-pass types. To illustrate this, consider Fig. 366, which is drawn in the Direct Form II con guration of Fig. 356 in section 13.8 (where, in this particular case, a1 and b1 are both equal to zero).
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