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lecture21

# lecture21 - ECE 5510 Random Processes Lecture Notes Fall...

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ECE 5510: Random Processes Lecture Notes Fall 2009 Lecture 21 Today: (1) LTI Filtering, continued; (2) Discrete-Time Filter- ing HW 9 due Thu at 5pm in HW locker. HW 10: posted today, due Thursday Dec. 10. OH today from 12:15-1:15pm. Reading: Today: 11.2, 11.3, 11.6. next class: 12.1, 12.2. If you have not presented a discussion item, please sign up with me for one of the final three classes. 1 LTI Filtering of WSS Signals Continued from Lecture 20. 1.1 Addition of r.p.s Figure 1: Continuous-time filtering with the addition of noise. If Z ( t ) = Y ( t )+ N ( t ) for two WSS r.p.s (which are uncorrelated with each other), then we also have that S Z ( f ) = S Y ( f )+ S N ( f ). A typical example is when noise is added into a system at a receiver, onto a signal Y ( t ) which is already a r.p. 1.2 Partial Fraction Expansion Lets say you come up with a PSD S Y ( f ) that is a product of frac- tions. Eg, S Y ( f ) = 1 (2 πf ) 2 + α 2 · 1 (2 πf ) 2 + β 2 This can equivalently be written as a sum of two different fractions. You can write it as: S Y ( f ) = A (2 πf ) 2 + α 2 + B (2 πf ) 2 + β 2

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ECE 5510 Fall 2009 2 X(t) Y(t) R C + + - - Figure 2: An RC filter, with input X ( t ) and output Y ( t ). Where you use partial fraction expansion (PFE) to find A and B . You should look in an another textbook for the formal definition of PFE. I use the “thumb method” to find A and B . The thumb method is: 1. Pull all constants in the numerators out front. The numerator should just be 1. 2. Go to the first fraction. What do you need (2 πf ) 2 to equal to make the denominator 0? Here, it is α 2 . 3. Put your thumb on the first fraction, and plug in the value from (2.) for (2 πf ) 2 in the second fraction. The value that you get is A . In this case, A = 1 - α 2 + β 2 . 4. Repeat for each fraction. In the second case, B = 1 α 2 - β 2 Thus S Y ( f ) = 1 α 2 + β 2 bracketleftbigg 1 (2 πf ) 2 + α 2 1 (2 πf ) 2 + β 2 bracketrightbigg 1.3 Discussion of RC Filters
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lecture21 - ECE 5510 Random Processes Lecture Notes Fall...

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