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hw2.425.09 - ECE 425 Digital Signal Processing Fall 2009...

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ECE 425 Digital Signal Processing September 10, 2009 Fall 2009 Homework 2 Assigned: September 10, 2009 Due: Friday, September 18, by 4:00pm in the 425 lock-box on the 2nd floor of Phillips Goals & concepts: Interpretation and use of the DTFT. Thinking about what domain (time or frequency) is appropriate for a particular problem. 1. Ideal low-pass filters in discrete time and cascades of ideal LPFs. An ideal low-pass filter with zero delay is specified in the frequency domain: (a) Find the impulse response of the ideal low-pass filter (don’t just state it; show your work). Is this filter causal? Now, three zero-delay filters are connected in cascade; these filters are defined below on the range . (b) Find the impulse response and the frequency response of the cascaded system. (c) If you write the I/O relationship for the cascaded system in terms of the three individual impulse responses and compare it with your result from (b), you get an interesting result pertaining to sinc functions. What is it? H lp e j ω ˆ ( ) 1 0 = ω ˆ ω c < ω c ω ˆ π < h lp n ( ) ω ˆ π < H 1 e j ω ˆ ( ) 1 0 = ω ˆ π 3 < π 3 ω ˆ π < H 2 e j ω ˆ ( ) 1 0 = π 2 ω ˆ 2 π 3 < < else H 3 e j ω ˆ ( ) 1 0 = 5 π 6 ω ˆ π < < ω ˆ 5 π 6
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2 2. Modulating the ideal low-pass filter. An ideal low-pass filter with zero delay has frequency response on (a) A new filter is defined by the equation .
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