HWc-sol - F ( s ), but if the function has poles on or to...

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ECE 493 HW #12 – Version 1.0 February 21, 2011 Spring 2011 Univ. of Illinois Due Tues, Feb 22 Prof. Allen Topic of this homework: Harder Laplace transforms Deliverable: Show your work. 1 Semi-periodic functions 1. DeFne the somewhat weird notation: f ( t )) T s n =0 f ( t - nT ) . ±ind an expression for f ( t )) T in terms of the Laplace transform ( L ) of f ( t ). Hint: Write this as a convolution, then take the L of the convolution. Sol: We may write f ( t )) T = f ( t ) s δ ( t - nT ) F ( s ) L b s n =0 e - snT B = F ( s ) 1 - e - sT 2. Describe the poles of f ( t )) T in terms of the poles of f ( t ). Sol: It seems somewhat obvious that the poles of f ( t )) T are those of F ( s ) along with the zeros of 1 - e - sT , which are at ω = n 2 πT . 3. What is the ROC for this example? Sol: σ > 0 is required to assure that the inFnite series converges. One must also consider the poles of
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Unformatted text preview: F ( s ), but if the function has poles on or to the left of the j axis, then there is no further restriction due to F ( s ). 4. Next deFne the even more weird notation: f (( t ) T s n =- f ( t-nT ) . Does this make sense? If so, why, if not why not? Sol: This function is not causal due to the tail of f ( t ) that extends beyond t = 0. I want to think more about this case. I just put one of my feet higher on the physics ladder, but havent yet put the other on the math ladder. 1.1 More on Riemann 1. State the Riemann Hypothesis. Hint: Look this up on Wikipedia. /493/Assignments/HW #12 Version 1.0 ebruary 21, 2011 1...
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