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Unformatted text preview: EE 261 The Fourier Transform and its Applications Fall 2009 Problem Set Four Due Wednesday, October 21 1. (10 points) Eva and Rajiv continue their conversation about convolution: Rajiv: You know, convolution really is a remarkable operation, the way it imparts properties of one function onto the convolution with another. Take periodicity: If f ( t ) is periodic then ( f * g )( t ) is periodic with the same period as f . That was a homework problem last week Eva: Theres a problem with that statement. You want to say that if f ( t ) is a periodic function of period T then ( f * g )( t ) is also periodic of period T . Rajiv: Right. Eva: What if g ( t ) is also periodic, say of period R ? Then doesnt ( f * g )( t ) have two periods, T and R ? Rajiv: I suppose so. Eva: But wouldnt this lead right to a contradiction? I mean, for example, you cant have a function with two periods, can you? Rajiv: I think weve found a fundamental contradiction in mathematics. Eva: Why dont we look at a simple, special case first. What happens if you convolve sin2 t with itself? Rajiv: OK, both functions have period 1 so for the convolution you get a function thats periodic of period 1, no problem. Eva: No, you dont. Something goes wrong. Whats going on? With whom do you agree and why? What do think about that statement If f ( t ) is periodic then f * g is periodic. 2. (10 points) Equivalent width: Still another reciprocal relationship The equivalent width of a signal f ( t ), with f (0) 6 = 0, is the width of a rectangle having height f (0) and area the same as under the graph of f ( t ). Thus W f = 1 f (0) Z - f ( t ) dt. This is a measure for how spread out a signal is. 1 Show that W f W F f = 1. Thus, the equivalent widths of a signal and its Fourier transform are reciprocal. From the Internet Encyclopedia of Science: Equivalent width A measure of the strength of a spectral line. On a plot of intensity against wavelength, a spectral line appears as a curve with a shape defined by the line profile. The equivalent width is the width of a rectangle centered on a spectral line that, on a plot of intensity against wavelength, has the same area as the line. 3. (15 points) Exponential decay and a common differential equation (a) Let a > 0 and let g ( t ) = e- a | t | be the two-sided exponential decay. Find F g ( s ). Hint: Use the one-sided exponential decay and a reversal. (b) Let u ( t ) be the unit step function u ( t ) = ( 1 , t > , t Show that a solution of the differential equation x 00 ( t )- x ( t ) =- 2 e- t u ( t ) is x ( t ) = Z e-| t- | e- d....
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This note was uploaded on 11/28/2009 for the course EE 261 at Stanford.