HWVII - ECE 3250 HOMEWORK ASSIGNMENT VII Fall 2009 1. In...

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Unformatted text preview: ECE 3250 HOMEWORK ASSIGNMENT VII Fall 2009 1. In the handouts (specifically, Chapter 5 of the monograph), I asserted that l 2 is a vector space and, in fact, an inner-product space. Here’s how to prove it. (a) Show that for any complex numbers a and b , we have | a + b | 2 = | a | 2 + | b | 2 + 2Re { a b } . Conclude from the fact that | a- b | 2 ≥ 0 that 2Re { a b } ≤ | a | 2 + | b | 2 from which it follows that | a + b | 2 ≤ 2 | a | 2 + 2 | b | 2 . Conclude that l 2 , the set of all complex-valued square-summable discrete-time signals, is closed under addition and scalar multiplication, and is therefore a vector space. (b) Show that for any complex numbers a and b we have | a || b | ≤ | a | 2 + | b | 2 2 . (Suggestion: ( | a | - | b | ) 2 ≥ 0.) Conclude that for any x and y in l 2 , the sequence { x ( k ) y ( k ) : k ∈ Z } is absolutely summable, so the series ∞ X k =-∞ x ( k ) y ( k ) converges, and l 2 is an inner-product space with inner product h x,y i defined by that sum. 2. Let b H 1 and b H 2 be the frequency responses of ideal low-pass and band-pass filters. Specifically, let b H 1 (Ω) = 1 if | Ω | ≤ 3100 π otherwise and b H 2 (Ω) = 1 if 1737 π ≤ | Ω | ≤ 4100 π otherwise . (a) Show that the LTI system you obtain by cascading the two filters (i.e., input x enters the low-pass filter whose output goes through the band-pass filter, leading to system output y ) has a frequency response b H , and then find b H . (b) Write out as a linear combination of cosines and constants the responses of the low-pass filter and the band-pass filter to the A-440 with Fourier series ∞ X k =-∞ 1 k 2 + 1 e jk 880 πt for all t ∈ R . Also find the fundamental period of each output signal. (c) Write out as a linear combination of cosines the response of the system in (a) to the signal in (b). Also find its fundamental period....
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HWVII - ECE 3250 HOMEWORK ASSIGNMENT VII Fall 2009 1. In...

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