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 π - = = (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 π - - - = = (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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Elements of Electrical Engineering 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 π ϖ ϖ ϖ = = = = = + + = = = 4. Fourier Series 2 π - π
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Elements of Electrical Engineering Dr. Fred J. Taylor, Professor The spectra produced by the exponential and trigonometric Fourier series formulas are identical. Is there, however,a computational advantage of one over the other? In reality, no! In
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