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# lect5 - MAGNITUDE CHARACTERISTICS FOR REAL RATIONAL...

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MAGNITUDE CHARACTERISTICS FOR REAL RATIONAL TRANSFER FUNCTIONS In this section we will restrict ourselves to causal, stable systems ) ( z H . Recall that such systems always possess a frequency response, i.e. the unit circle is always included in the ROC. Now, by the inversion formula (1.9) with , 1 = r it follows that - = p p w w w p d e e H n h n j j ) ( 2 1 ) ( . (5.1) Suppose now that we do not know ) ( z H , but only the frequency response, ) ( w j e H . That is, we know ) ( z H only on the unit circle. Formula (5.1) implies that this knowledge is enough to determine ) ( n h . Then, once ) ( n h is obtained, we can take its z-transform to obtain ) ( z H . The conclusion is that The transfer function, ) ( z H , of an LTI system is completely determined by its frequency response, ) ( w j e H , provided the frequency response exists. This is a remarkable fact, in that the entire z-domain function, ) ( z H can be obtained from its values taken only on a circle within its region of convergence. In general, both the magnitude and phase of ) ( w j e H are required to completely determine ) ( z H , but the magnitude alone gives us a good deal of information in itself. To investigate this we restrict ourselves to real rational ) ( z H .

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) ( z H is called a real rational transfer function if it is rational, and if it is real for all real values of z . Equivalently, it is real rational if it is rational, and the coefficients of its numerator and denominator polynomials are real. If ) ( z H is real rational it can always be written in the form ) ( ) )( ( ) ( ) )( ( ) ( 2 1 2 1 N M p z p z p z z z z z z z K z H - - - - - - = (5.2) in which K is a real constant, and if there are any complex poles or zeros, their complex conjugates must also be poles or zeros, respectively. This complex conjugate pair property follows from the fact that a polynomial with real coefficients must have roots that are either real, or occur in complex conjugate pairs. From the realness of ) ( z H it easily follows that ) ( ) ( w w j j e H e H - = . Now, consider the function ) / 1 ( ) ( z H z H . Clearly this function is well defined on the unit circle, since ) ( z H is, and note that 2 ) ( ) ( ) ( ) ( ) ( ) / 1 ( ) ( w w w w w w j j j j j e z e H e H e H e H e H z H z H j = = = - = . Thus, w w j e z j z H z H e H = = ) / 1 ( ) ( ) ( 2 . (5.3)
For convenience, let us introduce some (non-standard) teminology in calling the product, ) / 1 ( ) ( z H z H , the reciprocal product (RP). Consider now the RP of the function i z z - . We have { } ( 29 2 1 2 1 2 1 1 i i i i i z z z z z z z z z z RP + + - = - - = - . This can be re-written as ( 29 2 2 1 1 i i i i z w z z z z z + - = - - where the complex variable, w , is related to z by + = z z w 1 2 1 . It follows from this that the RP for the general rational system in

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lect5 - MAGNITUDE CHARACTERISTICS FOR REAL RATIONAL...

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