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lecture_03

# lecture_03 - 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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1 1 1 1 1 Massachusetts Institute of Technology Department of Mechanical Engineering 2.161 Signal Processing - Continuous and Discrete Fall Term 2008 Lecture 3 1 Reading: Class handout: Frequency Domain Methods The Fourier Series and Transform In Lecture 2 we looked at the response of LTI continuous systems to sinusoidal inputs of the form u ( t ) = A sin(Ω t + φ ) and saw that the system was characterized by the frequency response function H ( j Ω). In signal processing work, linear filters are usually specified by a desired frequency re- sponse function. (We will see that often the magnitude function | H ( ) | alone is used to specify a filter). The following figure shows the four basic forms of ideal linear filters: | H ( j W ) | l o w - p a s s f i l t e r | H ( j W ) | h i g h - p a s s f i l t e r - W c 0 W c W - W c 0 W c W | H ( j W ) | | H ( j W ) | b a n d - s t o p f i l t e r b a n d - p a s s f i l t e r - W c 2 - W c 1 0 W c 1 W c 2 W - W c 0 W c W In this lecture we generalize the response of LTI systems to non-sinusoidal inputs. We do this using Fourier methods. 1 copyright c D.Rowell 2008 3–1
T t t T T 2 Periodic Input Functions - The Fourier Series In general, a periodic function is a function that satisfies the relationship: x ( t ) = x ( t + T ) for all t , or x ( t ) = x ( t + nT ) for n = ± 1 , ± 2 , ± 3 , . . . , where T is defined as the period . Some examples of periodic functions is shown below. x 3 ( t ) x 2 ( t ) x 1 ( t ) t 0 0 The fundamental angular frequency Ω 0 (in radians/second) of a periodic waveform is defined directly from the period 2 π Ω 0 = , T and the fundamental frequency F 0 (in Hz) is 1 F 0 = T so that Ω 0 = 2 πF 0 . Any periodic function with period T is also be periodic at intervals of nT for any positive integer n . Similarly any waveform with a period of T/n is periodic at intervals of T seconds. Two waveforms whose periods, or frequencies, are related by a simple integer ratio are said to be harmonically related . If two harmonically related functions are summed together to produce a new function g ( t ) = x 1 ( t ) + x 2 ( t ), then g ( t ) will be periodic with a period defined by the longest period of the two components. In general, when harmonically related waveforms are added together the resulting function is also periodic with a repetition period equal to the fundamental period. 3–2

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Example 1 A family of waveforms g N ( t ) ( N = 1 , 2 . . . 5) is formed by adding together the first N of up to five component functions, that is N g N ( t ) = x n ( t ) 1 < N 5 n =1 where x 1 ( t ) = 1 x 2 ( t ) = sin (2 πt ) 1 x 3 ( t ) = sin (6 πt ) 3 1 x 4 ( t ) = sin (10 πt ) 5 1 x 5 ( t ) = sin (14 πt ) . 7 The first term is a constant, and the four sinusoidal components are harmonically related, with a fundamental frequency of Ω 0 = 2 π rad/s and a fundamental period of T = 2 π/ Ω 0 = 1 second. (The constant term may be considered to be periodic with any arbitrary period, but is commonly considered to have a frequency of zero rad/s.) The figure below shows the evolution of the function
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