1連續(xù)傅里葉變換F(jro)=J?f(t)e-j?tdt2連續(xù)拉普拉斯變換(單迦F(s)=J8f(t)e-stdt0-f(t)=JjfF(s)estds2叼O_j83離散Z變換彈邊)F(z)=Xf(k)z-kk=0f(k)— JF(z)zk-1dz,k>02兀jL4離散傅里葉變換F(ej0)—藝f(k)e-j%f(t)=丄卜F(j?)ej(Btd?2冗-8k—-8f(k)—+J2兀27F(ej0)ej0kd0T線性af.(t)+bf2(t)分aF1(jw)+bF2(jw)線性af1(t)+bf2(t)OaF1(s)+bF2(s)線性af1(k)+bf2(k)OaF1(z)+bF2(z)線性af,(k)+bf2(k)OaF,(ej0)+bF?(ej0)時移1212f(t土t0)分e土jWoF(jW)時移f(t土10)Oe土叫F(s)時移f(k土m)oz±m(xù)F(z)(雙邊)時移f(k±m(xù))oe±j亦F(ej°)頻移e土W(t)分F(j(w+w0))頻移e土sotf(t)OF(s+s0)頻移e±jw0kf(k)OF(e+jw0z)(尺度變換)頻移e±胛0f(k)OF(ej(°+0°))尺度變換b1jWwf(at+b) F(j)1a1 a尺度變換1 -bf(at+b)O easF(亠)1a1 a尺度變換akf(k)OFL)a尺度變換/(nW—M(k/n)OF(e“0)0反轉f(-t)導F(-jW)反轉f(-t)OF(-s)反轉f(-k)oF(z-1)(僅限雙邊)反轉f(-k)OF(e-j°)時域卷積f1(t)*f2(t)分F1(jw)F2(jw)時域卷積f1(t)*f2(t)OF1(s)F2(s)時域卷積f1(t)*f2(t)oF1(z)F2(z)時域卷積f1(k)*f2(k)OF1(ej0)F2(ej9)頻域卷積f(t)f2(t)匕十F1(jw)*F2(jw)時域f'(t)OsF(s)-f(0)時域f(k-1)oz-1F(z)+f(-1)f(k-2)Oz-2F(z)+z-1f(-1)+f(-2)f(k+1)ozF(z)-zf(0)f(k+2)Oz2F(z)-z2f(0)-zf(1)頻域卷積f1(k)f2(k)o*JF(e/W)F.(ejW-0))d屮2兀時域微分廣(t<f(")(t)宀jwF(jw/(jw)nF(jw)微分f〃(t)Os2F(s)-sy(0-)-y'(0-)差分時域差分f(k)-f(k-1)O(1-ej°)F(ej°)頻域微分/ dF(jw)/dnF(jw)f(t)(-jt)nf(t)O 廠丿dw1 dwnS域微分丿， ./dnF(s)f(t)F(-t)nf(t)O-F'(s)'1 dsnZ域微分dF(z)f(k)O-z；dz頻域微分dF(e./9)kf(k)Oj dg時域積分Jtf(x)dx,f(-8)=0OF(jw)+兀F(0)5(w)-8 jw時域積分J/ F(s) f(-1)(0)J f(x)dxO +J■」-8 s s部分求和f(k)*￡(k)—為f(i)o亠z-1i—-8時域累加為f(k)oF(e)+兀F(ejo)另5(0-2rat)k--8 1-ej0 k--8

連續(xù)傅里葉變換對F(jw)-J8f(t)e-j?tdt—8拉普拉斯變換對(單邊)F(s)=J8f(t)e-stdt0—Z變換對(單邊)F(z)=￡f(k)z-kk0函數(shù)f(t)傅里葉變換F(jTO)函數(shù)f(t)象函數(shù)F(s)函數(shù)f(k),k>0象函數(shù)函數(shù)f(k),k>0象函數(shù)s(ty'112k5(to)S(t)1S(k)1S(k-m),m>0z—m§，(t丫S(n)(t)jTO(j'?)nS'(t)s17z—1s(k-m),m>0-z—mz—1￡(t)丄+兀S(TO)j?￡(t)1ss(k)zz—1k2S(k)72+z(z-1)3t￡(t)j鈕(TO)-丄TO2ts(t)t吒(t)丄fn!s2sn+1ks(k)7(z-1)2(k+1)aks(k)z2(z-a)2e—at￡(t)te—at￡(t),a〉01/1e-ats(t)/te-ats(t)1/1akS(k)kak-1s(k)a+jTO(a+jTO)2s+a(s+a)2z—a(z-a)2COS(QQt)sin叫t)冗[S(TO+TO0)+S(TO_TO0)]j冗[S(TO+TO0)—S(TO—TO。)]cos(0t)s(t)seCkS(k)zz-eakaks(k)azs2+02(z-a)21t-j兀sgn@)sin(0t)s(t)pej0S(k)zk2aks(k)az2+a2zs2+02z-ej0(z—a)3頻域積分吋頻域積分吋(o)t+嚴分i°F(j)dx,F(—8)=0(-jt) —8S域積分f(t) LnaJF們)dnt sZ域積分皿amJ83dnk+m zn肌+1f(0)=limF(z)，f(1)=lim[zF(z)—zf(0)]zT8 zT8對稱F(jt)導2吋(-TO)初值f(0+)=limsF(s),F(s)為真分式ST8初值f(M)=limzMF(z)(右邊信號)，f(M+1)=lim[zM+1F(z)—zf(M)ZT8 zT8帕斯瓦爾E=J81f(t)I2dt=丄J81F(jTO)I2dTO—8 2?！?終值f(8)=limsF(s),s=0在收斂域內stO終值f(8)=lim(z—1)F(z)(右邊信號)zt1帕斯瓦爾為|f(k)|2=—JIF(ej0)I2d02兀2兀k——8信號與系統(tǒng)公式性質一覽 常用連續(xù)傅里葉變換、拉普拉斯變換、Z變換對一覽表111丄?2cosh(Pt)8(t)sak—(-a)k28(k)2azak+(—a)k28(k)2az2z2-a2S2-P2z2-a20j■卑）2k5(??o>sinh(Pt)8(t)Pk(k-1)z(k+1)kz2S2-P2七丄8(k)2(z-1)3弋1匹8(k)2(z-1)3e_atcos(Pt)8(t)jm+ae-atcos(Pt)8(t)s+aak—bk8(k)a—bzak+1一bk+1’ 8(k)a-bz2(z—a)(z—b)(jrn+a)2+P2(s+a)2+P2(z-a)(z-b)e~asin(Pt)8(t)Pe_asin(Pt)8(t)Pcos(Pk)8(k)z(z—cosB)sin(Pk)8(k)zsinP(jm+a)2+P2(s+a)2+P2z2—2zcosP+1z2—2zcosP+1e—°^8(t),a>02a(b01+b1)8(t)b十bscos(Pk+0)8(k)z2cos0—zcos(P—0)z2一2zcosP+1sin(Pk+0)8(k)z2sin0+zsin(P—0)a2+m20 1s2z2-2zcosP+1ttnj2k5'(m)，2兀(j)n5(n)(m)-(^0-b)e-a8(t)a a 1bs+b10s(s+a)akcos(Pk)8(k)z(z—acosP)aksin(Pk)8(k)azsinPz2—2azcosP+a2z2—2azcosP+a2sgn(t)丄jm—[Pt-sin(Pt)]8(t)P31akcosh(Pk)8(k)z(z—acoshP)aksinh(Pk)8(k)azsinhPs2(s2+P2)z2一2azcoshP+a2z2—2azcoshP+a2[-eat,t<0{ ,Q>0)[e-at,t>02m丿a2+m2盂[1-Pt)]sin(Pt)8(t)1(s2+P2)2ak8(k),k>0kln[z-a]止8(k)k!aezf(t)h兀 TCOS(—t),1tl<—T 20,ltl>T〔2皿COS(2)2(—)2—()2222^tsin(Pt)8(t)s(s2+P2)2(lna)k …8(k)k!+az1(2k)!rcosh*zFejnGtng-m2—為F5(m-n0),0=2—n tn^-m京[sin(卩t)+卩tcos(卩t)]￡(t)s21(z)1-;lQ2 J-1(s2+P2)2,M)k+1zlnL-1J” ,8(k)2k+1

8T(t)=為8(t-nT)n=_g備@)=0乙8 nQ)n=sQ=紐TtCOS(Pt)￡(t)s2-P2b—ba b—bP-bs+(S2+P2)2[0 1e-at+(Q【)e-Pt]8(t)P-a P-a—1 0—(s+a)(s+P)1,|11<8牝(t)=1 20,|t|>一1 2TSa]如］=2$川如、2丿oI2丿[(b°-b$)t+b]]e-Ct(s+a)2b—ba+ba2 b—bP+bP2一[_“1、e「"2、e—e-at+0 P 2 e-Pt(卩-a)(Y-a) (a-p)(丫-卩)b-bv+b/v2+Fi2 e-vt]8(t)(a-v)(卩-丫)s2+b】s+(s+a)(s+P)(s+v)W sin(Wt)WSa(Wt)=('兀 兀tL,,W1,lol<2F(jo)=1 2W0,lo|>一I 2Ae?sin(Pt+0)8(t),其中0=b0二bi(d-jP)Pb.s+b0b-bP+bP2一b-ba+ba2[0 r 2 e-Pt+0 1 2 .te-at(a-P)2 P-ab-bP+ba(2P-a)-0 1 2、p 丿e-at]8(t)(P-a)2b.s2+bs+b10(s+a)2+P2210(s+a)2(s+P)f應)=1-,111<T 20,|t|>L〔2:Sa2網(wǎng)[b?e-at+(b]-2b?a)te-at+1(b°-々a+b尹2)t2e-at]8(t)bs2+bs+bb—bv+bv2[0 1 2 e-Y+Asin(Pt+0)]8(t)v2+P2其中Aej0=(b0二b2P2)土冋PP(v+j0)bs2+bs+b210(s+a)3210(s+v)(s2+P2)f(t)=<丄 二 ！_(t+ ),1t1<T 2 2O,ltl>匚2.丄joe-j「Sa(7)1,lt宀2T 2ltlTf(t)=1 (1- )，丁<lT-T] T 20,ltl>T〔28 o(T+T)? Sin 、 1o2(T-T]) L 4tl<T2o(T-T)1<sin、 rl4」[b0b1v+b2ve-vt+Ae-atsin(Pt+0)]￡(t)(a-v)2+P2其中Aej0=b0二b1(a—jP)土b2(a—jP)2P(Y-a+j0)bs2+bs+b210(s+v)[(s+a)2+P2)]雙邊拉普拉斯變換與雙邊Z變換對一覽表

雙邊拉普拉斯變換對F(s)Jf(t)e-stdt-g雙邊Z變換對F(z)=為f(k)z-kk 函數(shù)象函數(shù)F(s)和收斂域函數(shù)象函數(shù)F(z)和收斂域3(t)1,整個S平面6(k)1,整個Z平面6(n)(t)sn,有限S平面An6伙)zn,1zl>0(z-1)n￡(t)],Re{s}>0s￡(k)z\,1zl>1z—1t￡(t)丄,Re{s}>0s2(k+1)￡(k)z2,1zl>1(z—1)2tn-1 ￡(八1(k+n—1)!￡(k)7nQ I_I、1￡(t)(n-1)!,Re{s}>0sn￡(k)k!(n-1)!,Izl>1(z—1)n-￡(-t)丄,Re{s}<0s-￡(-k—1)z\,IzI<1z—1—1￡(-1)—,Re{s}<0s2-(k+1)￡(-k-1)z2,IzI<1(z—1)2tn-1 (八1 d小「J丿a7nQ IrI，1- ￡(-t)(n-1)!,Re{s}<0sn— ￡(-k—1)k!(n-1)!,IzI<1(z—1)ne-at￡(t),Re{s}>Re{-a}s+aak￡(k)z,IzI>IaIz一ate-at￡(t),Re{s}>Re{-a}(s+a)2(n+1)an￡(k)z2,IzI>IaI(z—a)2

2 /八(k+"一1)!一(k)Z" 丨卜i,e一at匕(t)(n-1)!,Re{s}>Re{-a}(s+a)"an￡(k)k!(n-1)!,izl>lal(Z一a)n一e-at￡(-t),Re{s}<Re{-a}s+a—aks(—k-1)z,lzl<lalz-az" i少i”ii- e-at￡(-t)(n-1)!，RC{S}<RC{a}(s+a)"- a"￡(-k-1)k!(n-1)!,lzl<lal(z一a)nCOS(Bt)￡(t)s。,Re{s}>0S2+P2cos(Pk)￡(k)z2-zcosPz2一2zcosP+1sin(Pt)￡(t)PP,Re{s}>0S2+P2sin(Pk)￡(k)zsinPz2一2zcosP+1e-atcos(Pt)￡(t)s+a/ 、P，Re{s}>Re{-a}(s+a)2+P2akcos(Pk)￡(k)z2一zacosPz2一2zacosP+1e-asin(Pt)￡(t)/ 、q，Re{s}>Re{-a}(s+a)2+P2aksin(Pk)￡(k)zasinPz2-2zacosP+1e-aiti,Re{a}>0一2a,Re{a}>Re{s}>Re{-a}s2-a2aiki,|al<1(a2-1)z ll,l丄,lal<lzl<l l(z-a)(az-1) aeaisgn(t),Re{a}>02s,Re{a}>Re{s}>Re{-a}s2-a2aikisgn,lal<1a(z2-z) ,,<,<1,,lal<lzl<ll(z-a)(az-1) a卷積積分一覽表f1(t)*f2(t)=J：小)f(t-)dTf1(t)f2(t)f1(t)* )f1(t)f2(t)f1(t)*f2(t)f(t)f'(t)f(t)8(t)f(t)

f(t)e(t)11f(入)dX—gE(t)E(t)tE(t)e-%(t)￡(t)-(1-e-a)e(t)aE(t)tE(t)丄12(/)2e-We(t)e-aje(t)(e-ait—e-a2t)e(t),a-Ha.a2-a- 1 2e-aE(t)e-atE(t)te-ate(t)te-a,t￡(t)e-aje(t)(a.-a)t—1 1■^-2 e-ait+ e-a2te(t)L(a2-a1)2 (a2-a1)2 Ja2Ha1tE(t)e-atE(t)1+丄e-aE(t)(a2 a2 丿e-Wcos(Pt+0)￡(t)e-C2e(t)cos(Pt+0-cd) cos(0-cd)；' 1ea1t-f ， 「 ea2te(t)丫(a2-a1)2+p2 \'(a2-a1)2+p2D=arctanl 1la2-a1丿te-atE(t)e-aE(t)丄12e-aE(t)2卷積和一覽表f1(t)*f2(t)“卩f(k-i)i=-gf](t)f2(t)ft)*f2(t)f](t)f2(t)ft)*f2(t)f(k)5(k)f(k)f(k)E(k)￡f(i)E(k)E(k)(k+1)E(k)kE(k)E(k)2(k+1)kE(k)akE(k)E(k)1—ak+1E(k),a豐0