# 信号与系统课件第3章周期信号的傅里叶级数表示资料_图文

CHAPTER 3 FOURIER SERIES REPRESENTATION OF PERIODIC SIGNALS 3.0 INTRODUCTION ? Representation of continuous-time and discretetime periodic signals ― Fourier series（傅立叶 级数）. ? Use Fourier methods to analyze and understand signals and LTI systems. 3.1 THE RESPONSE OF LTI SYSTEMS TO COMPLEX EXPONENTIALS ? Important concept ― signal decomposition ? basic signals: possess two properties 1. The set of basic signals can be used to construct a broad and useful class of signals. 2. It should be convenient for us to represent the response of an LTI system to any signal constructed as a linear combination of the basic signals. ? complex exponential signals st e in continuous time: in discrete time: zn Eigenfunction（特征函数） ?Defining two quantities: H(s) and H(z) H (s) ? ? h(? )e ?? ?? ? s? d? H ( z) ? k ? ?? ?k h [ k ] z ? ?? ? H(s) or H(z) is in general a function of the complex variable s or z . ?Eigenfunction( of the system): an input signal for which the system output is a constant times the input. e ? H (s)e st st z ? H ( z) z n n For a specific value of sk or zk , H (sk ) or H ( zk ) : eigenvalue . （特征值） (1) Continuous time LTI system x(t)=est y (t ) = x(t ) * h(t ) = = h(t) ò +? - ? s (t- t ) y(t)=H(s)est x(t - t ) h(t ) d t st ? ? 蝌 - ? +? e +? h (t ) d t = e h ( t ) e- s t d t = e st H ( s ) H ( s) = ò - ? h(t )e- st d t ( system function ) (2) Discrete time LTI system x[n]=zn h[n] k= - ? y[n]=H(z)zn x[n - k ]h[k ] n ? k - ? y[n] = x[n]* h[n] = = ? +? 邋z k= - ? +? ( n- k ) h[k ] = z z - k h[k ] = z n H ( z) H ( z) = k= - ? ? +? h[k ]z - k ( system function ) (3) Input as a combination of Complex Exponentials Continuous time LTI system: N ì ? sk t ? x ( t ) = a e ? ? k ? k= 1 ? í N ? sk t ? y ( t ) = a H ( s ) e ? ? k k ? ? k= 1 ? Discrete time LTI system: N ì ? ? x[n] = ? ak zk n ? ? k= 1 ? í N ? n ? y [ n ] = a H ( z ) z ? ? k k k ? ? k = 1 ? 3.2 FOURIER SERIES REPRESENTATION OF CONTINUOUS-TIME PERIODIC SIGNALS 3.2.1 Complex Exponential Fourier Series（指数型傅立叶级数） Given periodic x(t) with fundamental period T , its complex exponential Fourier series is : ?? x(t ) ? k ? ?? ?a e k jk?0 t ?0 ? 2? / T where the coefficients ak is generally a complex function of k?0 . The signals in the set e jk?0 t , k ? 0, ? 1, ? 2,? are harmonically related complex exponentials. ? ? （基波分量） （一次谐波分量） a? N e ? jN?0 t : the Nth harmonic components（ N次谐波分量） a?1e ? j?0 t: fundamental : components or the first harmoni