1 / 7

Basic signals

Complex exponentials serve as essential building blocks in representing various types of signals. The response of Linear Time-Invariant (LTI) systems to these basic signals is particularly straightforward, facilitating analysis and synthesis. The fundamental period and frequency play critical roles in the Fourier Series representation of continuous-time periodic signals. This includes exploration of Fourier coefficients, even and odd functions, and properties that influence signal representation. For example, given the continuous-time signal x(t) = cos(4πt) + 2sin(8πt), we can analyze its Fourier series components with clarity.

glyn
Télécharger la présentation

Basic signals

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. Basic signals • Why use complex exponentials? • Because they are useful building blocks which can be used to represent large and useful classes of signals • Response of LTI systems to these basic signals is particularly simple and useful. ak H(jk0) ej0 t akejk0 t h(t) Complex scaling factor

  2. Fourier Series Representation of CT Periodic Signals x(t) = x(t+T) for all t • Smallest such T is the fundamental period • 0 = 2/T is the fundamental frequency ejt periodic with period T   = kw0  • Periodic with period T • {ak} are the Fourier (series) coefficients • k = 0 DC • k = ±1 first harmonic • k = ±2 second harmonic

  3. CT Fourier Series Pair (Synthesis Equation) (Analysis Equation)

  4. Examples • x(t) = cos(4t) + 2sin(8t) • Periodic square wave (from lecture) • Periodic impulse train

  5. Even and Odd Functions • Even functions are defined by the property: f(x) = f(-x) • Odd functions are defined by the property: f(x) = -f(-x)

  6. Some notes on CT Fourier series • For k = 0, a0 = sT x(t) dt  mean value over the period(DC term) • If x(t) is real, ak = a*-k • If x(t) is even, ak = a-k • If x(t) is odd, ak = -a-k • If x(t) is real and even, ak = a*-k = a-k  ak’s will only have real terms • If x(t) is real and odd, ak = a*-k = -a-k  ak’s will only have odd terms

  7. The CT Fourier Transform Pair x(t) ↔ X(j)

More Related