hw07su08 - GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of...

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GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL and COMPUTER ENGINEERING ECE 2025 Summer 2008 Problem Set #7 Assigned: Week of 30-Jun-08 Due Date: 7/8-Jul-08 Please check the T-square “Announcements” daily. All official course announcements will be posted there. ALL of the STARRED problems should be turned in for grading. A solution to the starred problems will be posted on WebCT. Your homework is due at the start of your recitation section. No late homework will be accepted. PROBLEM 7.1 : In this class, we’ve tried to build ideas slowly over time, introducing complex ideas by showing the simplest cases first. When we discussed the z -transform of FIR filters, we noted that when you take the inverse z -transform of an impulse response like H ( z ) = 1 - z - 1 , you get a system function like h [ n ] = δ [ n ] - δ [ n - 1] . Unfortunately, this leads some people to believe that H ( z ) = 1 1 - z - 1 has an inverse z -transform given by h [ n ] = 1 δ [ n ] - δ [ n - 1] . ←- NOT TRUE!!! BAD! AWFUL! Such an expression, with a delta function in the denominator , is a mathematical abomination. It should be cast into the fires of Mount Doom. The correct inverse z -transform, of course, is the unit step function h [ n ] = u [ n ] . TRUE AND GOOD Notice that 1 / ( δ [ n ] - δ [ n - 1]) yields total nonsense for most n ; for n < 0 or n > 1, you get 1 / 0 = which is completely crazy. Why might people think the equation above labeled “NOT TRUE!!! BAD! AWFUL!” above could at all be reasonable? What would possess them, year after year, to write such things 1 / ( δ [ n ] - δ [ n - 1]) as answers on quizzes, even when we jump up and down in class repeatedly telling them not to? Well, recall that inverse z -transforms are linear, and hence they commute over addition: Z - 1 { F ( z ) + G ( z ) } = Z - 1 { F ( z ) } + Z - 1 { G ( z ) } . ←- TRUE, GOOD, AND WONDERFUL Alas, this seems to lead some people to think that inverse z -transforms commute over division, as in Z - 1 F ( z ) G ( n ) = Z - 1 { f ( n ) } Z - 1 { g ( n ) } , ←- NOT TRUE!!! BAD! AWFUL!
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