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Unformatted text preview: EE 261 The Fourier Transform and its Applications Fall 2009 Problem Set Six Due Wednesday, November 4 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 .) 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...
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