1 / 8

Fourier Series

Fourier Series. Course Outline. Roberts, ch. 1-3. Time domain analysis (lectures 1-10) Signals and systems in continuous and discrete time Convolution: finding system response in time domain Frequency domain analysis (lectures 11-16) Fourier series Fourier transforms

tanya-oconnor
Télécharger la présentation

Fourier Series

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Fourier Series

  2. Course Outline Roberts, ch. 1-3 • Time domain analysis (lectures 1-10) Signals and systems in continuous and discrete time Convolution: finding system response in time domain • Frequency domain analysis (lectures 11-16) Fourier series Fourier transforms Frequency responses of systems • Generalized frequency domain analysis (lectures 17-26) Laplace and z transforms of signals Tests for system stability Transfer functions of linear time-invariant systems Roberts, ch. 4-7 Roberts, ch. 9-12

  3. Periodic Signals • For some positive constant T0 f(t)is periodic if f(t) = f(t + T0) for all values of t (-, ) Smallest value of T0 is the period of f(t) • A periodic signal f(t) Unchanged when time-shifted by one period May be generated by periodically extending one period Area under f(t) over any interval of duration equal to the period is same; e.g., integrating from 0 to T0 would give the same value as integrating from –T0/2 to T0 /2

  4. Sinusoids • Fundamental f1(t) = C1 cos(2 p f0 t + q1) Fundamental frequency in Hertz is f0 Fundamental frequency in rad/s is w0 = 2 p f0 • Harmonic fn(t) = Cn cos(2 p n f0 t + qn) Frequency, n f0, is nth harmonic of f0 • Magnitude/phase and Cartesian representations Cn cos(n w0 t + qn) =Cn cos(qn) cos(n w0 t) - Cn sin(qn) sin(n w0 t) =an cos(n w0 t) + bn sin(n w0 t)

  5. Fourier Series • General representationof a periodic signal • Fourier seriescoefficients • Compact Fourierseries

  6. Existence of the Fourier Series • Existence • Convergence for all t • Finite number of maxima and minima in one period of f(t) • What about periodic extensions of

  7. Fundamental period T0 = 2 Fundamental frequency f0 = 1/T0 = 1/2 Hz w0 = 2p/T0 = p rad/s f(t) A -1 0 1 -A Example #1

  8. Fundamental period T0 = 2p Fundamental frequency f0 = 1/T0 = 1/(2p) Hz w0 = 2p/T0 = 1 rad/s f(t) 1 -p/2 -p -2p p/2 p 2p Example #2

More Related