ee261_hw1

ee261_hw1 - EE 261 The Fourier Transform and its...

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EE 261 The Fourier Transform and its Applications Fall 2011 Problem Set One Due Wednesday, October 5 1. Some practice combining simple signals. (5 points each) The (scaled) triangle function with a parameter a > 0 is Λ a ( t ) = Λ( t/a ) = ( 1 - 1 a | t | , | t | ≤ a 0 , | t | > a The graph is ! a a 1 The parameter a speciﬁes the width, namely 2 a . Alternately, a determines the slopes of the sides: the left side has slope 1 /a and the right side has slope - 1 /a . We can further modify Λ a by scaling the height and shifting horizontally, forming b Λ a ( t - c ). The slopes of the sides of the scaled function are then ± b/a . The graph is: b Λ a ( t c ) b c acc + a 1

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Express each of the following as a sum of two shifted, scaled triangle functions b 1 Λ a 1 ( t - c 1 )+ b 2 Λ a 2 ( t - c 2 ). Think of the sum as ‘left-triangle’ plus a ‘right-triangle’ (‘right’ meaning to the right, not having an angle of 90 ). For part (d), the values x 1 , x 2 and x 3 cannot be arbitrary. Rather, to be able to express the plot as the sum of two Λ’s they must satisfy a relationship that you should determine. (Why do this? It’s superposition waiting to happen – the possibility of operating on a complicated signal by operating on simpler constituents, and adding up the results. Here you are called upon to ﬁnd the constituents. ) ! 2 0 1 2 0 ! 2 5 4 (a) (b) 1 1 3 7 6 ( c ) ( x 1 , 0) ( x 2 ,y 2 ) ( x 3 , 0) ( d ) 2. Creating periodic functions. (5 points each) Let f ( t ) be a function, deﬁned for all t , and let T > 0. Deﬁne g ( t ) = X n = -∞ f ( t - nT ) . (a) Provided the sum converges, show that g ( t ) is periodic with period T . One sometimes says that g ( t ) is the periodization of f ( t ). (Later we’ll learn to express such a sum as a convolution.) (b) Let f ( t ) = Λ 1 / 2 ( t ). Sketch the periodizations g ( t ) of f ( t ) for T = 1 / 2, T = 3 / 4, T = 1, T = 2. (c) If a function f ( t ) is already periodic, is it equal to its own periodization? Explain. 3. Adding periodic functions. (5 points each) 2
(a) Let f ( x ) = sin(2 πmx )+sin(2 πnx ) where n and m are positive integers. Is f ( x ) periodic? If so, what is its period? (b) Let g ( x ) = sin(2 πpx ) + sin(2 πqx ) where p and q are positive rational numbers (say p = m/r and q = n/s , as fractions in lowest terms). Is g ( x ) periodic? If so, what is its period?

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ee261_hw1 - EE 261 The Fourier Transform and its...

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