信号与系统chapter_3.ppt

Chapter 3 Fourier series representation of periodic signals Why complex exponential? It can be used to construct a broad and useful class of signals The response of an LTI system to it is simple enough in structure to provide us with a convenient representation for the response of the system to any signal constructed as a linear combination of it. 3.2 The Response of LTI Systems to Complex Exponentials Response of an LTI system to a complex exponential is the same complex exponential with only change in amplitude CT: est ? H(s) est DT: zn ? H(z) zn est, zn are eigenfunction of the system;H(s), H(z) are the system’s eigenvalue For a CT LTI system For a DT LTI system Example 3.1 S: y(t)=x(t-3) Discuss the outputs of the system when x(t)=ej2t x(t)=a1es1t+a2es2t+a3es3tes1t→ H(s1)es1tes2t→ H(s2)es2t es3t→ H(s3)es3t y(t)= a1H(s1)es1t+ a2H(s2)es2t + a3H(s3)es3t Example 3.1 S: y(t)=x(t-3) Discuss the outputs of the system when x(t)=cos(4t)+cos(7t) 3.3 Fourier Series representation of CT Periodic Signals Consider a periodic signal, The set of harmonically related complex exponentials: Example 3.2 Alternative form of Fourier Series If x(t) is real x*(t)=x(t) Alternative form of Fourier Series Another form: 3.3.2 Determination of Fourier Series representations of CT Periodic Signals Assume that a given periodic signal x(t) has a FS representation. Find FS coefficients, ak of: Example 3.3 x(t)=sinω0t Example 3.4 Example 3.5 3.4 Convergence of Fourier Series Some comments Some scientists Periodic signals without discontinuities Periodic signals with discontinuities, such as the square wave Fourier Any periodic signal can be represented by a Fourier series Validity of Fourier series representations If x(t) has a Fourier series representation The best approximation using N harmonically related complex exponentials is obtained by truncating the number of Fourier series to N. N→∞, the limit of EN is zero, When a periodic signal