中科院信號(hào)與系統(tǒng)總結(jié)

本文格式為Word版，下載可任意編輯——中科院信號(hào)與系統(tǒng)總結(jié)

周期信號(hào)與非周期信號(hào)

連續(xù)時(shí)間信號(hào)：f(t)?f(t?kT)k?0,?1,?2,??????離散時(shí)間信號(hào)：x(n)?x(n?kn)k?0,?1,?2,??????

ej?0t?ej?0(t?T0)T0?2??0

ej?0n?ej?0(n?N)能量信號(hào)和功率信號(hào)連續(xù)時(shí)間信號(hào)

N?2??0k為整數(shù)

E??|f(t)|2dt

???1T1P??T2|f(t)|2dt（周期信號(hào)）P?lim?2T|f(t)|2dt（非周期信號(hào)）

T??T?T22離散時(shí)間信號(hào)

TE?n????|x(n)|?2

NN1122P?|x(n)|（周期信號(hào)）P?limx(n)（非周期信號(hào)）??N??2N?12N?1n??Nn??N1、能量信號(hào)：E有限0?E??，P?0；2、功率信號(hào)：P有限0?P??，P??；

3、若E??，P??，則該信號(hào)既不是能量信號(hào)也不是功率信號(hào)；4、一般周期信號(hào)是功率信號(hào)。線性系統(tǒng)

若x1(t)?y1(t)，x2(t)?y2(t)，則a1x1(t)?a2x2(t)?a1y1(t)?a2y2(t)若x1(n)?y1(n)，x2(n)?y2(n)，則a1x1(n)?a2x2(n)?a1y1(n)?a2y2(n)

時(shí)不變系統(tǒng)

若x(t)?y(t)，則x(t?t0)?y(t?t0)若x(n)?y(n)，則x(n?n0)?y(n?t0)

系統(tǒng)時(shí)不變性：

1電路分析：元件的參數(shù)值是否隨時(shí)間而變化2方程分析：系數(shù)是否隨時(shí)間而變

3輸入輸出分析：輸入鼓舞信號(hào)有時(shí)移，輸出響應(yīng)信號(hào)也同樣有時(shí)移

直流信號(hào)：f(t)?K(???t???)實(shí)指數(shù)信號(hào)：f(t)?Keat(a為實(shí)數(shù))復(fù)指數(shù)信號(hào)：f(t)?Kest(s???j?)正弦信號(hào)：f(t)?Ksin(?t??)鐘形信號(hào)（高斯信號(hào)）：f(t)?Eet2?()當(dāng)??0時(shí)是★Sa(0)?1Sa(t)|t?n??0★偶函數(shù)：Sa(?t)?Sa(t)★?2??的周期信號(hào)?sint抽樣信號(hào)：Sa(t)?t???Sa(t)dt?????1單位階躍信號(hào)：u(t)???0符號(hào)函數(shù)信號(hào)：sgn(t)??0??Sa(t)dt??Sa(t)dt?0?2★sinc(t)?sin?t?tt?0t?0t?0t?0★u(t)?1[sgn(t)?1]2?1??1★sgn(t)?2u(t)?1單位斜坡信號(hào)：f(t)?tu(t)★df(t)?u(t)dt?t??u(?)d??f(t)??1門函數(shù)信號(hào)：g(t)????0三角脈沖信號(hào)：??2其它?t??2★g(t)?u(t?)?u(t?)22???t1????f(t)???1?t???

???t?00?t??★f(t)?(1?|t|?)[u(t??)?u(t??)]

單位沖擊函數(shù)?(t)的性質(zhì)一般定義?t?0???(t)?????0t?0????(t)dt?1?????泛函定義???A?(t)?(t)dt?A?(0)du(t)???dtksink??(t)?lim[Sa(kt)]?limk???k???tt??單位沖擊函數(shù)與單u(t)?位階躍函數(shù)的關(guān)系相乘運(yùn)算時(shí)間位移運(yùn)算反褶運(yùn)算時(shí)間尺度變換卷積運(yùn)算?(?)d??(t)?f(t)?(t)?f(0)?(t)????f(t)?(t)dt?f(0)f(t)?(t?t0)?f(t0)?(t?t0)????f(t)?(t?t0)dt?f(t0)?(t)??(?t)?(t)是t的偶函數(shù)?(at)?11b?(t)?(at?b)??(t?)|a||a|a?(t)??(t)??(t)?(t?t1)??(t?t2)??(t?t1?t2)f(t)??(t)?f(t)f(t)??(t?t0)?f(t?t0)設(shè)f(t)?0有n個(gè)互不相等的實(shí)根t1,t2,?,tn，則有?(t)的復(fù)合函數(shù)?[f(t)]的性質(zhì)?[f(t)]??1?(t?ti)'|f(t)|i?1in其中f'(ti)表示f(t)在t?ti處的導(dǎo)數(shù)，且f'(ti)?0(i?1,2,?,n)

單位沖擊偶函數(shù)?'(t)的性質(zhì)一般定義泛函定義?'(t)?d?(t)dt??????f(t)?'(t)dt??['df(t)]??f'(0)dtt?0積分性質(zhì)反褶運(yùn)算相乘運(yùn)算時(shí)間位移運(yùn)算????(t)dt?0?'(t)???'(?t)?(t)是t的奇函數(shù)f(t)?'(t)?f(0)?'(t)?f'(0)?(t)?????f(t)?'(t)dt??f'(0)f(t)?'(t?t0)?f(t0)?'(t?t0)?f'(t0)?(t?t0)???f(t)?'(t?t0)dt??f'(t0)導(dǎo)數(shù)運(yùn)算時(shí)間尺度變換f(t)?''(t)?f(0)?''(t)?2f'(0)?'(t)?f''(0)?(t)?'(at)?11'?(t)|a|a11(n)?(t)|a|an?(n)(at)?當(dāng)a??1時(shí)，?(n)(?t)?(?1)n?(n)(t)1b?'(at?b)?2?'(t?)a?0aa1b?'(at?b)??2?'(t?)a?0aa注：離散單位脈沖函數(shù)有?(an)??(n)卷積運(yùn)算

f(t)??'(t)?df(t)dt

卷積的性質(zhì)卷積定義交換律分派率結(jié)合律奇異信號(hào)卷積特性f1(t)?f2(t)????f1(?)f2(t??)d?????f2(?)f1(t??)d?f1(t)?f2(t)?f2(t)?f1(t)f1(t)?[f2(t)?f3(t)]?f1(t)?f2(t)?f1(t)?f3(t)[f1(t)?f2(t)]?f3(t)?f1(t)?[f2(t)?f3(t)]??f(t)??(t)?f(t)f(t)??(t?t0)?f(t?t0)f(t)??'(t)?f'(t)f(t)??(n)(t)?f(n)(t)f(t)??(n)(t?t0)?f(n)(t?t0)f(t)?u(t)??f(?)d???tu(t)?u(t)?tu(t)u(t?a)?u(t?b)?(t?a?b)u(t?a?b)tmu(t)?tnu(t)?m!n!tm?n?1u(t)(m?n?1)!e?tu(t)?e?tu(t)?e?tu(t)e?1tu(t)?e?2tu(t)?延時(shí)性質(zhì)微分與積分性質(zhì)1(e?1t?e?2t)u(t)（?1??2）?1??2若f1(t)?f2(t)?f(t)，則f1(t?t1)?f2(t?t2)?f(t?t1?t2)df(t)df(t)d[f