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

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern