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2.7The Modified z-Transform > 2.7The Modified z-Transform - Pg. 39

. The Modified z-Transform 39 Note that delaying a causal sequence always results in an initial zero value. To avoid inappropriate initial values, we rewrite the delay as T d = T - mT , 0 m < 1 m = 1 - T d T (2.33) For example, a time delay of 0.2 s with a sampling period T of 1 s corresponds to m = 0.8 -- that is, a time advance of 0.8 of a sampling period and a time delay of one sampling period. If y -1 (t + mT) is defined as y(t + mT) delayed by one complete sampling period, then, based on (2.32), y d (t) is given by y d ( t ) = y ( t - T + mT ) = y -1 ( t + mT ) We now sample the delayed waveform with sampling period T to obtain y d ( kT ) = y -1 ( kT + mT ) , k = 0 , 1 , 2 , . . . (2.35) (2.36) (2.34) From the delay theorem, we know the z-transform of y -1 (t) Y - 1 ( z ) = z - 1 Y ( z ) We need to determine the effect of the time advance by mT to obtain the z-