PS-6-2009-Solutions

# PS-6-2009-Solutions - EE 261 The Fourier Transform and its...

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Unformatted text preview: EE 261 The Fourier Transform and its Applications Fall 2009 Solutions to Problem Set Six 1. (10 points) Downconversion A common problem in radio engineering is ‘downconver- sion to baseband’. Consider a signal f ( t ) whose spectrum F f ( s ) satisfies F f ( s ) = 0 , | s − s | ≥ B. To downconvert F f ( s ) to baseband means to move the spectrum so that it is centered around 0. Devise a strategy to downconvert using convolution with an appropriate III and a single ideal low pass filter. What is the new signal in terms of the old? (Note you can assume s > 2 B .) Solution: First we must assume that s > 2 B just as for the sampling theorem, otherwise we may get aliasing artifacts. Now consider III s where III s ( s ) = ∞ summationdisplay n = −∞ δ ( s − ns ) we have ( F f ∗ III s )( s ) = ∞ summationdisplay n = −∞ F f ( s − ns ) Then, we apply an ideal low pass filter Π( s 2 B ) to ( F f ∗ III s )( s ). Finally, we get the new signal in the frequency domain as F g ( s ) = F f ( s + s ). By the shift theorem, we get the time domain expression g ( t ) g ( t ) = e − 2 πis t f ( t ) . It should be noted that systems that involve sampling frequently do this, such as most commercial MRI scanners. 1 2. (25 points) The spectrum of an FM signal: Bessel functions return! As you no doubt know, ‘FM’ stands for ‘Frequency Modulation,’ a way of transmitting radio signals. The transmitted signal is of the form x ( t ) = A cos 2 πf ( t ) where f ( t ) = ν c t + k integraldisplay t −∞ m ( t ) dt. Here ν c is the carrier frequency , k is a constant known as the frequency modulation index , and m ( t ) is the function with the information, and doing the modulating. The instantaneous frequency is the derivative of f ( t ), f ′ ( t ) = ν c + km ( t ) , which shows you in what sense the transmitted signal x ( t ) is modulated about the carrier frequency. You set your receiver to ν c . For this problem we’ll take the case where we modulate a pure tone, x ( t ) = A cos(2 πν c t + k sin 2 πν t ) . Here’s a plot for 0 ≤ t ≤ 2 with A = 1, ν c = 5, k = 2, and ν = 20. An interesting article by John Chowning, a pioneer in ‘computer music’ a Stanford, about the use of FM in simulating musical tones can be found at http://users.ece.gatech.edu/~mcclella/2025/labs-s05/Chowning.pdf 2 It’s also posted on the course web site. What is the spectrum? Remember Bessel functions, introduced in an earlier problem? The answer depends on these. Let’s recall: The Bessel equation of order n is x 2 y ′′ + xy ′ + ( x 2 − n 2 ) y = 0 . A solution of the equation is the Bessel function of the first kind of order n , given by the integral J n ( x ) = 1 2 π integraldisplay 2 π cos( x sin θ − nθ ) dθ ....
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## This note was uploaded on 11/28/2009 for the course EE 261 at Stanford.

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PS-6-2009-Solutions - EE 261 The Fourier Transform and its...

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