# hw3 - EE 261 The Fourier Transform and its Applications...

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 2011 Problem Set Three Due Wednesday, October 20, 2011 1. (5 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. 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. 2. (20 points) Reversals, Shifts and Stretches If f ( t ) is a signal the corresponding reversed signal is defined to be f- ( t ) = f (- t ) . Define the shift operator τ b f and the stretch operator σ a f by ( τ b f )( t ) = f ( t- b ) , ( σ a f )( t ) = f ( at ) . (a) Express f (2 t +3) as σ a ( τ b f ) and as τ b ( σ a f ) for suitable shifts and stretches. Throwing in a reversal (using f- instead of f ), do the same for f (- 2 t + 3). Find the Fourier transform of each. (b) Find the Fourier transforms of the function shown in the graph (a shifted sinc) 1 3. (a) (5 points) Show that Π * Π = Λ using the definition of convolution. (b) (5 points) What about Π a * Π a ? (Use any method you wish.) (c) (10 points) Stephanie and Aditya discuss convolution: Stephanie: You know, I think this problem suggests something general about how convolution spreads out a signal. Aditya: How so? Stephanie: Well, we showed that Π * Π = Λ. I know that Λ has a different shape than Π, but notice that while Π( x ) = 0 for | x | ≥ 1 / 2 we have Λ( x ) = 0 outside the bigger interval | x | ≥ 1....
View Full Document

## This note was uploaded on 01/10/2012 for the course EE 216 taught by Professor Harris,j during the Fall '09 term at Stanford.

### Page1 / 6

hw3 - EE 261 The Fourier Transform and its Applications...

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

View Full Document
Ask a homework question - tutors are online