158 cheatsheet

158 cheatsheet - TABLE 2.3 FOURIER TRANSFORM PAIRS Sequence...

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Unformatted text preview: TABLE 2.3 FOURIER TRANSFORM PAIRS Sequence Fourier Transform 1. 8[n] 1 2. 6[n — no] e_j“’"0 3. 1 (—00 < n < 00) Z 21r6(w + 23k) k=—OC 4. a“u[n] (lal < 1) 1 — ae'f‘” 5. u[n] 1 _:_J.w + Z 115(w+21rk) 1 =-oo 6. (n +1)a"u[n] (m < 1) (1 _ MW); r" sinwpoz + 1) 1 7‘ sincup “[n] ("I < 1) 1 —- 2r cos cope—1w + rze‘fl‘“ sinwcn ‘0 _ 1. lwl < 60c, 8' 1n: X(e})—{O. wc<lw|57r _ 1. 0 E n 5 M Sin[w(M+ 1)/2] — 'wM/Z 9' fin] ‘ {0, otherwise sin(w/2) e I TABLE 2.2 FOURIER TRANSFORM THEOREMS Sequence Fourier Transform 14"] X (8%”) fin] Y(e"”) 1. ax[n] + by[nj aX(ej“) + bY(ej“’) 2. x[n — nd] (:14 an integer) e‘fw’m X(ei“’) 3. ejw0”x[n] X(ej(w_w°)) 4. x[——n] X(e_j“’) X*(ei“’) if x[n] real. 5. nx[n] jdX(e’w) da) 6‘ x[n] * y[n] X(ej“’)Y(ej“’) 7- xln];v[n] Parseval’s theorem: )1 8. Z |x[n]I2= |X(eiw)|2dw n=—oo —7r oo 1 " . — — X(e19)Y(eI(w-9))de 27: _n on time n. TABLE 3.1 10. 11. 12. 13. _—_ . 5[n] . u[n] Sequence . —u[—n — 1] .5[n——m] . a"u[n] —a”u[-n # 1] . na”u [n] —na"u[—n — I] . [cos won]u[n] [sin won]u[n] [r’1 cos won]u[n] [r’1 Sin won]u[n] { a”, 0. 0 g n 5 N — 1, otherwise SOME COMMON z-TRANSFORM PAIRS Transform 1 1 1— z“ 1 1— z‘] 2-”! 1 1— az‘l l—az—l az~1 (1 — az'l)2 azfil (1 —az‘1)2 1 a [cos wok—1 1 — [Zoos wok—1 + z—2 [sin we]?1 1 — [2cos wok—1 + z—2 1 — [r cos wok—1 1 — [Zr (:03 wok—1 +r21”2 [r sin cook—1 1 — [ZrCOSw0]Z_1 +r2r2 1—a~z‘~ 1 — az‘l ROC All 2 lz|>1 IZ|<1 All 2 except 0 (if m > 0) or 00 (if m < 0) lzl > MI IZI < 10! [21> Ial Izl < I!!! Iz| > 1 |z| >1 Izl > r |z[>r Izl>0 mi mu — 21.54311 fin-2) ...
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158 cheatsheet - TABLE 2.3 FOURIER TRANSFORM PAIRS Sequence...

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