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374bhw12 - HW#12 Phys374Spring 2007 Due before class Friday...

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HW#12 —Phys374—Spring 2007 Prof. Ted Jacobson Due before class, Friday, May 4, 2007 Room 4115, (301)405-6020 www.physics.umd.edu/grt/taj/374b/ [email protected] 1. Sampling Theorem Exact reconstruction of a continuous-time signal from its discrete-time sam- ples is possible if the signal is band-limited and the sampling frequency is greater than twice the signal bandwidth. Consider a signal f ( t ) whose Fourier transform ˜ f ( ω ) is zero for | ω | > Ω, f ( t ) = Ω - Ω ˜ f ( ω ) e - iωt dω. (1) This is called a band-limited signal. Evaluating (1) at the discrete times t = nt s , where the sampling time t s is defined by t s = π/ Ω, yields f ( nt s ) = Ω - Ω ˜ f ( ω ) e - inπω/ Ω dω. (2) The right hand side of (2) is recognized as 2Ω times the n th coefficient in the Fourier series for ˜ f ( ω ). Being limited to the finite range - Ω < ω < Ω, the function ˜ f ( ω ) is determined by its Fourier series coefficients, and therefore by the discrete “samples” f ( nt s ). The sample values thus determine f ( t ) via (1). The sampling frequency 1 /t s = Ω is twice the bandwidth Ω / 2 π .
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