PS2 - EE 261 The Fourier Transform and its Applications...

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EE 261 The Fourier Transform and its Applications Fall 2011 Problem Set Two Due Wednesday, October 12, 2011 1. (30 points) Convolution, Autocorrelation and Fourier Series Recall the convolution of two functions f ( t ) and g ( t ) of period 1 is defined by ( f * g )( t ) Z 1 0 f ( τ ) g ( t - τ ) dτ . (a) (5) Show that f * g is periodic of period 1. (b) (10) Suppose f ( t ) and g ( t ) have Fourier series f ( t ) = X n = -∞ a n e 2 πint g ( t ) = X n = -∞ b n e 2 πint , respectively. Find the Fourier series of ( f * g )( t ) and explain why this implies that f * g = g * f . Let f ( x ) be a real, periodic function of period 1. The autocorrelation of f with itself is the function ( f ? f )( x ) = Z 1 0 f ( y ) f ( y + x ) dy . (c) (5) Show that f ? f is also periodic of period 1. (d) (10) If f ( x ) = X n = -∞ ˆ f ( n ) e 2 πinx show that the Fourier series of ( f ? f )( x ) is ( f ? f )( x ) = X n = -∞ | ˆ f ( n ) | 2 e 2 πinx . See the supplementary problems for more on the autocorrelation function. 2. (25 points) The Dirichlet Problem, Convolution, and the Poisson Kernel 1
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When modeling physical phenomena by partial differential equations it is frequently neces- sary to solve a boundary value problem . One of the most famous and important of these is associated with Laplace’s equation: Δ u = 2 u ∂x 2 + 2 u ∂y 2 = 0 , where u ( x,y ) is defined on a region R in the plane. The operator Δ = 2 2 + 2 2 is called the Laplacian and a real-valued function u ( ) satisfying Δ u = 0 is called harmonic . The Dirichlet problem for Laplace’s equation is this: Given a function f ( ) defined on the boundary of a region R , find a function u ( ) defined on R that is harmonic in R and equal to f ( ) on the boundary. Fourier series and convolution combine to solve this problem when R is a disk. As with many problems where circular symmetry is involved, in this case that the functions are defined on a circular disk, it is helpful to introduce polar coordinates ( r,θ ), with x = r cos θ , y = r sin θ . Writing U ( ) = u ( r cos θ,r sin θ ), so regarding u as a function of r and θ , Laplace’s equation becomes 2 U ∂r 2 + 1 r ∂U + 1 r 2 2 U ∂θ 2 = 0 . (You need not derive this.) (a) (5) Let { c n } , n = 0 , 1 ,... be a bounded sequence of complex numbers, let r < 1 and define u ( ) by the series u ( ) = Re ( c 0 + 2 X n =1 c n r n e inθ ) . From the assumption that the coefficients are bounded, and comparison with a geometric series, it can be shown that the series converges, but you need not do this. Show that u ( ) is a harmonic function. [Hint: Superposition in action – the Laplacian of the sum is the sum of the Laplacians, and the c n comes out, too. Also, the derivative of the real part of something is the real part of the derivative of the thing.] (b) (5) Suppose that f ( θ ) is a real-valued, continuous, periodic function of period 2 π and let f ( θ ) = X n = -∞ c n e inθ be its Fourier series. Now form the harmonic function u ( ) as above, with these coefficients c n . This solves the Dirichlet problem of finding a harmonic function on the unit disk x 2 + y 2 < 1 with boundary values f ( θ ) on the unit circle x 2 + y 2 = 1; precisely, lim r 1 u ( ) = f ( θ ) .
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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.

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PS2 - EE 261 The Fourier Transform and its Applications...

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