Lesson_5 - Elements of Electrical Engineering Dr. Fred J....

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Elements of Electrical Engineering Dr. Fred J. Taylor, Professor Lesson Title: Fourier Series Lesson Number: 5 (Sections 3.5-3.8) Introduction In Chapter 3, it was established that a continuous-time signal x(t), having a fundamental period T 0 , has a Fourier series signal representation given by: ( 29 ( 29 -∞ = = k kt T j k e a t x ) / 2 ( 0 π (Synthesis Equation) 1. It was also established that this representation is valid only for periodic signals. The Fourier coefficients a k , found in Equation 1, are computed using the following production rules: ( 29 ( 29 ( 29 (DC) harmonic k ; 1 (DC) harmonic 0 ; 1 th / 2 0 0 th 0 0 0 0 0 0 dt e t x T a dt t x T a kt T j T k T - = = (Analysis Equation) 2. or ( 29 ( 29 ( 29 (DC) harmonic k ; 1 (DC) harmonic 0 ; 1 th / 2 2 / 2 / 0 th 2 / 2 / 0 0 0 0 0 0 0 dt e t x T a dt t x T a kt T j T T k T T - - - = = (Analysis Equation) 3. The resulting Fourier series representation of x(t) is seen to be defined in terms of (in general) complex coefficients a k and complex basis functions v k (t)=e j2 π kt/T 0 , k {- , }. The coefficient a k corresponds to the complex amplitude and phase of the k th harmonic. In addition it was stated that the basis functions are orthogonal. The graphical display of the Fourier coefficients defines the signal’s spectrum. Example Zero-mean Gaussian noise is added to a sinusoidal signal to define a signal x(t). Technically, because of the noise, the signal x(t) is a-periodic. A mathematician would not proceed any further, knowing that a Fourier series solution does not exists. However, using a computer the Fourier series, defined by Equations 1 and 2, can be machine calculated and displayed. The resulting display defines the signal’s approximate spectrum (engineers will always take a good approximation that can be easily obtained (using a computer) over nothing at all. The results are shown in Figure 1. Fourier Series 1
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Dr. Fred J. Taylor, Professor x(t)=cos( ϖ 0 t)+n(t) Magnitude spectrum - ϖ 0 0 ϖ 0 Phase spectrum - ϖ 0 0 ϖ 0 Real spectrum - ϖ 0 0 ϖ 0 Imaginary spectrum - ϖ 0 0 ϖ 0 Figure 1: Spectrum of a cosine in noise. An alternative form of the Fourier series is called the trigonometric Fourier series which is defined below (synthesis equation followed by analysis equation): ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 dt t n t x T b dt t n t x T a value DC or average dt t x T a T f f t n b t n a a t x T n T n T n n n n 0 0 0 0 0 0 0 0 0 0 1 0 1 0 sin 2 ; cos 2 ) ( 1 / 1 ; 2 ; ) sin ) cos ϖ π = = = = = + + = = 2 π - π
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Lesson_5 - Elements of Electrical Engineering Dr. Fred J....

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