4 FOURIER SERIES(latest) part1 - Excellent does not an...

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Excellent does not an accident, but it comes through a hard work!!
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Excellent does not an accident, but it comes through a hard work!! Introduction on Fourier Series Introduction on Fourier Series The ability to analyze waveforms of various types is an important engineering skill. Fourier analysis provides a set of mathematical tools which enable the engineer to break down a wave into its various frequency components. It is then possible to predict the effect of a particular waveform may have from knowledge of the effects of its individual frequency components. Often an engineer finds it useful to think of a signal in terms of its frequency components rather than in terms of its time domain representation.
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Excellent does not an accident, but it comes through a hard work!! LESSON OUTCOMES LESSON OUTCOMES 2 T The oscillation of an undamped spring-mass system around the equilibrium is a sine wave .
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Excellent does not an accident, but it comes through a hard work!! 4.1 Periodic functions
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Excellent does not an accident, but it comes through a hard work!! PERIODIC FUNCTIONS PERIODIC FUNCTIONS A function is said to be periodic if its image values are repeated at regular intervals in its domain. Thus the graph of a periodic function can be divided into ‘vertical strips’ that are replicas of each other. The interval between two successive replicas is called the period of the function. A function is periodic with period T if, for all its domain values , f t f t t ( ) ( ) f t T f t For example, The function has periods since t t f sin ..., , 6 , 4 , 2 sin( 2 ), sin( 4 ), sin( 6 ), ..., all equal to sin . t t t t + + + Illustrating the cosine wave's fundamental relationship to the circle.
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Excellent does not an accident, but it comes through a hard work!! PERIODIC FUNCTIONS PERIODIC FUNCTIONS A periodic function has a basic shape which is repeated over and over again. The fundamental range is the time (or sometimes distance) over which the basic shape is defined. The length of the fundamental range is called the period . Example: Periodic function Note: Throughout this chapter, we will use/ express general periodic function as: ( ) be a function in the interval 2 2 and ( ) be periodic with period, . So ( ) ( ) f t T T t f t T f t T f t
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Excellent does not an accident, but it comes through a hard work!! PERIODIC FUNCTIONS PERIODIC FUNCTIONS We consider the waveforms of and For these waveforms, is the angular frequency. Then, Let us introduce phase angle, , then The quantity is called the time displacement . The sine and cosine functions form a class of functions known as sinusoids or harmonics . t A y cos t A y sin 2 T The period 2 1 T f The frequency + + t A t A y cos ) cos( + + t A t A y sin ) sin( t A y sin
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Excellent does not an accident, but it comes through a hard work!!
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