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# lecture_10 - MIT OpenCourseWare http/ocw.mit.edu 2.161...

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MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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t f ( t ) 1 Massachusetts Institute of Technology Department of Mechanical Engineering 2.161 Signal Processing - Continuous and Discrete Fall Term 2008 Lecture 10 1 Reading: Class Handout: Sampling and the Discrete Fourier Transform Proakis & Manolakis (4th Ed.) Secs. 6.1 6.3, Sec. 7.1 Oppenheim, Schafer & Buck (2nd Ed.) Secs. 4.1 4.3, Secs. 8.1 8.5 The Sampling Theorem Given a set of samples { f n } and its generating function f ( t ), an important question to ask is whether the sample set uniquely defines the function that generated it? In other words, given { f n } can we unambiguously reconstruct f ( t )? The answer is clearly no, as shown below, where there are obviously many functions that will generate the given set of samples. In fact there are an infinity of candidate functions that will generate the same sample set. The Nyquist sampling theorem places restrictions on the candidate functions and, if satisfied, will uniquely define the function that generated a given set of samples. The theorem may be stated in many equivalent ways, we present three of them here to illustrate different aspects of the theorem: A function f ( t ), sampled at equal intervals Δ T , can not be unambiguously reconstructed from its sample set { f n } unless it is known a-priori that f ( t ) contains no spectral energy at or above a frequency of π/ Δ T radians/s. In order to uniquely represent a function f ( t ) by a set of samples, the sampling interval Δ T must be suﬃciently small to capture more than two samples per cycle of the highest frequency component present in f ( t ). There is only one function f ( t ) that is band-limited to below π/ Δ T radians/s that is satisfied by a given set of samples { f n } . 1 copyright c D.Rowell 2008 10–1
�� Note that the sampling rate, F s = 1 / Δ T , must be greater than twice the highest cyclic frequency F max in f ( t ). Thus if the frequency content of f ( t ) is limited to Ω max radians/s (or F max cycles/s) the sampling interval Δ T must be chosen so that π Δ T < Ω max or equivalently 1 Δ T < 2 F max The minimum sampling rate to satisfy the sampling theorem F N = Ω max samples/s is known as the Nyquist rate . 1.1 Aliasing Consider a sinusoid f ( t ) = A sin( at + φ ) sampled at intervals Δ T , so that the sample set is { f n } = { A sin( an Δ T + φ ) } , and noting that sin( t ) = sin ( t + 2 ) for any integer k , 2 πm f n = A sin( an Δ T + φ ) = A sin a + n Δ T + φ Δ T where m is an integer, giving the following important result: Given a sampling interval of Δ T , sinusoidal components with an angular frequency a and a

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