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

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

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