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Fig. 357
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Note that the actual input to the DF is a sampled sinusoidal wave of unit amplitude, x nT sin !nT. Then note that, corresponding to x nT , the output of the DF is y nT B sin !nT  , where B and  are both functions of frequency !. Thus the frequency response of a DF is expressed in terms of the manner in which B and  vary with the frequency !. (If we wish, B and  can be measured at the output of the D/A circuit, as shown.) Consider now the following. In eq. (573), section 13.3, the variable z is de ned to be a complex quantity z A j!T , where A is a real variable assumed to be always large enough to assure that the value of F z does not become in nitely great as n ! 1. Now consider the SPECIAL CASE that arises for the speci c value A 1. For this particular condition we have z  j!T cos !T j sin !T which is the equation, in the complex plane, for a purely SINUSOIDAL WAVE of UNIT AMPLITUDE. Therefore the equation for a SAMPLED unit-amplitude sinusoidal wave at the INPUT to the DF in Fig. 357 is X z cos !nT j sin !nT  j!nT zn
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* See section 9.5, Fig. 186, and discussion following the gure.
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CHAPTER 13 The Digital Processor
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which produces Y z at the OUTPUT of the DF, a corresponding sampled sinusoidal wave of amplitude B and phase angle ; thus Y z B j !nT  B j  j!nT B j zn in which we made use of the basic relationship a b a b . Thus, letting H z denote the TRANSFER FUNCTION of the lter, we have that H z Y z =X z B j from which we may infer, correctly, the IMPORTANT PRACTICAL RULE that If H z is the transfer function of a DT processor, then, to nd the steadystate sinusoidal frequency response of the processor, simply set z  j!T in H z . Equation (607) thus gives the AMPLITUDE and PHASE ANGLE, B and , of the output sinusoidal wave produced by the input unit reference signal sin !t in the test setup of Fig. 357. In connection with the above procedures it s convenient to use the following notation. Let f input analog frequency (Hz), or 2f ! input analog frequency (rad/sec); now let fs constant sampling frequency, where (see eq. (91) in Chap. 5) fs T 1. Thus, T hence, !T ! 2=!s 2r where r !=!s ; that is, r ratio of analog frequency to xed sampling frequency. Thus we can, if we wish, state the foregoing rule as: To nd the sinusoidal frequency response of a DT processor, set z  j2r in H z , where r !=!s . In our work we ll basically make use of eq. (606) in section 13.8. With this in mind, note that if a processor is purely non-recursive (does not use feedback)* then all of the a coe cients in eq. (606) are equal to zero, and thus, for a non-recursive case, eq. (606) becomes H z b0 b1 z 1 b2 z 2 bp z p 608 1 2 2 fs 2fs !s 607
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As the equation shows, an actual calculation of H z will require nding the sum of a number of complex numbers. We recall, however, that this will require that the numbers rst be put into the rectangular a jb form; thus we ll need to make use of the famous Euler relationship  j2r cos 2r j sin 2r 609
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Another important point to note is as follows. In Fig. 357 let !h denote the HIGHEST FREQUENCY COMPONENT of importance present in the input analog signal. Then, in order to satisfy the basic requirement of the sampling theorem, it has to be true that !s 2!h ; thus, for ! !h and !s 2!h , we have that r !h =2!h 0:5
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