PS-4-2009

PS-4-2009 - EE 261 The Fourier Transform and its...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

This note was uploaded on 11/28/2009 for the course EE 261 at Stanford.

Page1 / 6

PS-4-2009 - EE 261 The Fourier Transform and its...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online