清華信號(hào)與系統(tǒng)chapter3

1、3.4 非周期信號(hào)的頻譜分析變換推導(dǎo)變換定義2006-11-113.4.0本節(jié)內(nèi)容變換推導(dǎo)變換定義3.4.0本節(jié)內(nèi)容2006-11-133.4.0本節(jié)內(nèi)容信號(hào)幅度趨于零，但是總能量還是存在的。2006-11-143.4.1變換推導(dǎo)周期信號(hào) 非周期信號(hào)周期:T1譜線間隔: 2 01T1離散譜線譜線長(zhǎng)度: F (n1)諧波分量級(jí)數(shù)連續(xù)譜線0頻譜密度變換2006-11-153.4.1變換推導(dǎo)從周期信號(hào)的離散頻譜到非周期信號(hào)的連續(xù)頻譜2006-11-163.4.1變換推導(dǎo)從周期信號(hào)的離散頻譜到非周期信號(hào)的連續(xù)頻譜2006-11-173.4.1變換推導(dǎo)設(shè)周期信號(hào)f (t)和它的頻譜如下圖所示F n1 t

2、f (t) jnF ne指數(shù)形式的級(jí)數(shù)：11nT121f (t)e jn1tF (n ) T1 0頻譜：T1T1122006-11-183.4.1變換推導(dǎo)2 F (n )T12F (n )T jn t1f (t)edt111T112非周期信號(hào)重復(fù)周期T1 重復(fù)頻率1 譜線間隔n1 頻率微分d周期信號(hào)離散譜線n 連續(xù)頻率1譜線長(zhǎng)度F (n ) 012 F n1 F (), F ( j)頻率密度Spectrum Densitiy12006-11-193.4.1變換推導(dǎo)2F (n) F (lim1 01lim F (nT1 )T111頻譜密度函數(shù)(頻譜密度)f (t)e jtdt2006-11-11

3、03.4.1變換推導(dǎo)T1 1F () limjn1t dt f (t)e jtdtf (t)e2T變換T 12 )n1 F (ne tf (t) njn1111n1 (n1) d級(jí) 數(shù)展開(kāi)逆變換F (n1) F ()1n1 2 1 2F ()e jtdf (t) 逆變換2006-11-1113.4.2變換定義正變換) Ff (t)e jtdtF (f (t)逆變換 1 2)F ()e jtdf (t) F1F (2006-11-1123.4.2變換定義F () F () e j ( )頻譜函數(shù)幅度頻譜相位頻譜F () 頻率的連續(xù)函數(shù)離散頻譜的包絡(luò)線() 當(dāng)函數(shù)f (t)為實(shí)函數(shù)的時(shí)候，則有)

4、f (t)e jtdt F *()F (故 F () 是的偶函數(shù),()是的奇函數(shù)2006-11-1133.4.2變換定義 1 2 1 2F () e jt ( )d)ed tj逆變換的三角函數(shù)形式f (t)F (1F () cost ()d2 j 2F () sint ()d1F () cost ()df (t) 2當(dāng)f(t)是實(shí)函數(shù)，傅逆變換可以簡(jiǎn)化成為：1F cost ()d02006-11-1143.4.2變換定義變換的條件：函數(shù)在- 期間滿足絕對(duì)可積；函數(shù)在任意有限期間中的最大最小值點(diǎn)的個(gè)數(shù)有限；函數(shù)在任意有限期間中的不連續(xù)點(diǎn)的個(gè)數(shù)有限；f (t) dt 變換存在的充分條件2006-11-1153.4.2變換定義定義了奇異函數(shù)（沖擊函數(shù)），可以定義一些函數(shù)的周期函數(shù)變換：階躍函數(shù)符號(hào)函數(shù)F u(t) (t) 不滿足絕對(duì)可積條件1j2006-11-1163.4.2變換定義廣義變換：構(gòu)造函數(shù)序列 fn (t)逼近函數(shù)f (t)f (t) lim fn (t)n而fn (t)滿足絕對(duì)可積條件，并且 fn (t)的變換序列Fn ()是極限收斂的，則變換F ()為：定義f (t)的F () lim Fn ()n2006-11-1173.4.x小結(jié)