EEE203 Final Exam Note Sheets

EEE203 Final Exam Note Sheets - ` Exponential Identities: x...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ` Exponential Identities: x a Ax b = x a + b a xffffff a @b ff =x xb ` aa x a Ay a = xy b wwc a w w w w bw x =p x affff f f b xa ab = x ab 1f ff ff fff xa Trigonometric Identities: 1 1 1 sin x = ffffffffffffffff cos x = fffffffffffffffff tan x = ffffffffffffffff csc x sec x cot x 1ffffffff 1fffffffff 1 csc x = fffffff sec x = ffffffff cot x = fffffffffffffffff sin x cos x tan x 2 sin x + cos 2 x = 1 1 + tan2 x = sec 2 x 1 + cot 2 x = csc 2 x sin x cos x tan x = fffffffffffffffff cot x = fffffffffffffffff cos x sin x 1 1 1 sin x = ffffffffffffffff cos x = fffffffffffffffff tan x = ffffffffffffffff csce x sec xd cot x d e d e ff fff f ff ff sin @x = cos x cos @x x = sin x tan fffff@x x = cot x 2 2 2 d e d e d e csc fffff@x = sec x sec fffff@x x = csc x cot fffff@x x = tan x 2 2 2 ` a ` a ` a sin @x = @sin x cos @x = cos x tan @x = @tan x ` a ` a ` a csc @x = @csc x sec @x = sec x cot @x = @cot x sin x F y = sin x cos y F cos x sin y cos x F y = cos x cos y G sin x sin y b c tan x F tan y tan x F y = fffffffffffffffffffffffffffffffffffffffffffffffff 1 G tan x tan y ` a 2 tan x sin 2x = 2 sin x cos x = fffffffffffffffffffffffffffffff 1 + tan2 x ` a 1 @tan x 2 2 cos 2x = cos 2 x @sin x = 2 cos 2 x @1 = 1 @2 sin x = fffffffffffffffffffffffffffffff 1 + tan2 x ` a 2 tan x tan 2x = ffffffffffffffffffffffffffffffff 1 @tan2 x ` a cot x @tan x cot 2x = ffffffffffffffffffffffffffffffffffffffffffff 2 2 Properties of the Fourier Transform: Property Aperiodic Signal x t y t Linearity Time Shifting Frequency Shifting Conjugation Time Reversal Time & Frequency Scaling Convolution Multiplication Differentiation in Time t Fourier Transform X j X j Y j b b c c b c D b cE `a ` a ` a ax t + by t x t @t o ` a ` a a X j + b Y j e@jt o X j X j @jo X C b c b c e jo t x t xC t ` ` a `a b c b @j c x @a = b c x @t ` a X @j f b c b c x at `a a j 1fff f ff X fffffffff |a| a b c b +1 g c Logarithmic Identities: ` a y y = log b x a x = b log b 1 = 0 log b b = 1 ` a f g ` a ` a x t Cy t x t y t `a X j Y j ` a `a b c b c 1fff fff fZ f X j Y j @j d 2 @1 d ff ` a fff f x t dt Z @1 jX j b b c log b x A = log b x + log b y y log b ` ` a ` a Integration Differentiation in Frequency x t dt ` a ` a c ` a 1 f f ff ffff f X j + X 0 d j ` a ` a xfff f = log b x @log b y y a ` a ` a log b x n = n log b x ` a ` a ` a log c x ` a log b x = log b c Alog c x = ffffffffffffffffffffffff log c b Euler. s Formula: ` a ` a e jx = cos x + jA sin x [ e ix = cos x + i A sin x [ jx ln cos x + jA sin x = jx ln cos x + i A sin x = ix @jx B ` a C ` a C ` a ` a ` a B ` a R S e +e ` a cos x = e e jx = ffffffffffffffffffffffffffffff 2j R S e jx @e@jx ` a ffffffffffffffff fffffffffffffff jx sin x = m e = 2j ` a ` a ` a ` a e@jx = cos @x + jAsin @x = cos x @jA sin x Integrals containing exponentials: 1fff f R e ax dx = e ax a f g 1fff ax 1 f ax x @ ffff R xe dx = e a a 1fff 2xfff f fff ff R x e dx = e ax x 2 @ + 2 a a a 2 ax 1ffffffffffffffffffffffffffffffffffff f @cos 2x 2 ` a 1ffffffffffffffffffffffffffffffffffff + cos 2x 2 cos x = 2 ` a 1ffffffffffffffffffffffffffffffffffff f @cos 2x 2 ` a tan x = 1 + cos 2x d e d e x+y x @y sin x + sin y = 2 sin ffffffffffffffff cos fffffffffffffffff 2 2 d e d e xffffffffffffff xffffffffffffffff ff + y f @y sin x @sin y = 2 cos sin 2 2 d e d e xfffffffffffffff xfffffffffffffff f+y ff @y cos x + cos y = 2 cos cos 2 2 d e d e xfffffffffffffff xffffffffffffffff f+y f @y cos x @cos y = @2 sin sin 2 2 C a ` a 1B ` fff sin x sin y = cos x @y @cos x + y 2 C a ` a 1B ` fff cos x cos y = cos x @y + cos x + y 2 C a ` a 1B ` sin x cos y = fff sin x + y + sin x @y 2 C a ` a 1B ` sin x cos y = fff sin x + y + sin x @y 2 sin x = 2 ` a tx t d b c j fffffffffX j d X b c Y b c C ^ ^ ^ ^ ^ ^ ^ ^ ^X j = X @j ^ ^ ^ ^ ^ ^ ^ ^ T b cU T b ^ ^ cU ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^e X j = e X @j ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ T b cU T b U^ \ ] c ^ ^m X j = @m X ^ ^ ^ ^ ^ b c ^ ^ b c ^ ^|X j | = |X @j | ^ ^ ^ ^ ^ ^ ^ b c b c ^ ^ ^ X j = @ X @j ^ ^ Z b c Conjugate Symmetry for Real Signals x t Real ` a @j ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ [ Symmetry for Real & Even Signals Symmetry for Real & Odd Signals Even @Odd Decomposition for Real Signals x t real & even x t real & odd x e t = Ev x t `a ` a `a ` a X j real & even X j purely imaginary & odd x t real +1 b c R ` aB ` a S C C +1 x o t = Od x t R ` aB ` a S x t real Parseval. s Relation for Aperiodic Signals Z Basic Fourier Transform Pairs: Signal Fourier Transform +1 +1 @1 b c ` a2 1 2 Z |x t | = ffffffff |X j | d 2 @1 Fourier series b c coefficients if periodic X k = @1 jk ake 0 t c 2 X D b k = @1 ak @k0 c b b ak a1 = 1 ; ak = 0, otherwise 1 a1 = a@1 = fff; ak = 0, otherwise 2 1 a1 = @a@1 = ffffff; ak = 0, otherwise 2j a0 = 1, ak = 0, k 0, for T>0 +1 e j0 t cos0 t sin0 t 2 @0 c b f g 2ff ffff f ` a3 ` ` ` ` Common Algebraic Identities: x + y = x 2 + 2 xy + y 2 x @y = x 2 @2 xy + y 2 x + y = x 3 + 3x 2 y + 3xy 2 + y 3 x @y = x 3 @3x 2 y + 3xy 2 @y 3 x @y = x + y x @y 2 3 2 3 @0 + + 0 D fff b f f c b cE a2 ax ax effaxfff ax ff ff dx = ln x + ffffffffffffff+ ffffffffffffffff+ ffffffffffffffff+ ... R x 1A 1! 2 A 2! 3 A 3! ax ax effaxfff ff ff fffff@effffffffff fffffffff ffffffff fffffffffffffffffff fffff ffaffff e fffff ff ff a n @1 + R n dx = ` R n @1 dx x n @1 x n @1 x ax efffffffffffffffffffffffff 1fff effaxfff ln f A x ln @ f R ffff dx R e ax A x dx = a a x ` a2 a2 a3 a3 @0 @ @0 j `a ` a x t =1 2 Periodic Square Wave X Y ^1 , |t|<T 1 ^ ^ ^ ] `a \ ` a `a x t =^ Tff & x t + T = x t f^ ff ^ ^0 , T < |t| [ Z 1 2 f cE X k = @1 c 2sinffffffffffffffffffffffff b fffff k0 T 1 f fff @k0 k b b c ` c ` a` a x @y = x @y x + xy + y 2 ab 2 c X +1 x 3 + y 3 = x + y x 2 @xy + y 2 a ` ab c X Y ` a\ 1 , |t| < T 1] x t Z 0 , |t| > T [ 1 n = @1 t @nT ` a 2fff + 1 2k fff ff X @ fffffffffffff T k = @1 T 2fffffffffffffffffffff1fff f sinT f X g f0ffffffff1fff kf T f sin k0 T 1 ff T f f sinc fffffff0ffffffff1fff= fffffffffffffffffffffffffffff k 1fff f f ak = for all k T sinffWtfff ffff f ff ff ffff t t u t b ` a `a c P Q b c \ 1 , || < W ] X j = Z 0 , || > W[ Y 1 ` a 1 fffff ff ff + j e@jt 0 ffff1 ffff ffff fffff fff a + j 1 ffffffffffffffff ffff ffff ffff f t @t 0 e @at u t , Re a > 0 ` a P Q ` a te@at u t , Re a >0 ffftff ffff fff @at ff ffffffffff f ` a e n @1 b P Q a + j c2 n @1 ! u t , Re a >0 `a ff ffff ffff ffff ffffff1ffffffff ff b cn a + j Property Linearity Time Shifting Scaling in z domain Properties of the Z @Transform: Signal z @Transform @A ` a x n X z @A ` a x1 n X1 z @A ` a x2 n X2 z @A @A ` a ` a a x1 n + b x 2 n a X 1 z + bX 2 z x n @n0 @ A z@n 0 X z b f g z ff ffff b ` a c ROC R R1 R2 At least the intersection of R1 & R2 R, except possibly add or remove origin R z0 R e j0 n x n z x n a x n n n 0 @A X e@j0 z X z0 @1 @A @A A X a b z c Scaled versions of R, |a|R Inverted R; R @1 Time Reversal Time Expansion Conjugation Convolution First Difference Accumulation X @A Y ^x r , n = rk ^ ^ ^ ^ ^ ] @A \ x k n = ^ for some integer r^ ^ ^ ^ ^ Z 0 , n rk [ x @n @ X z@1 c X zk C` b c Rk 1fff f f xC n @A @A x1 n Cx 2 n @A @ n @A x n @x n @1 X A b X zC ` a ` a X1 z X 2 z c a 1 @z@1 X z ` a R At least the intersection of R1 a& R2 At least the intersection of R and |z v 0 x k @A @A ffff ffff fff f ` a fff1ffff ffff k = @1 1 @z@1 X z At least the intersection of R and |z v 1 ` a Differentiation in z domain Initial Value Theorem dffffffffffffffffffff fX z R dz @A @A ` a If x n = 0 for n < 0, then x 0 = zlim X z Q 1 nx n @z Common z @Transform Pairs: Signal Transform ROC @A n 1 all z @A f ff fff ff ff ff ff ff1 ff ff f ff u n |z| >1 1 @z@1 @ A 1 ff ff ff ff ff ff ff ff ff ff fff @u @n @1 |z| < 1 1 @z@1 b c b c @ A @m n @m z All z, except 0, if m>0 or 1 if m<0 n u n @A @ A f fff ff1ff ff ff f ff ff ff ff ff fff ff 1 @ z@1 ff ff ff1 ff ff ff f ff ff fff ff ff ff ff 1 @ z@1 @1 f fff ff ffzff ff ff ff ff ff ff ff ff fff ff ff ff ff |z| > || |z| < || |z| > || @ n u @n @1 n n u n @A b A 1 @ z@1 c2 @n n u @n @1 D @ ff ff fff ff ff ff ff fff ff ff ff z ff ff ff ff f ff ff b c2 @1 |z| < || cE 1 @ z@1 D cos 0 n u n b cE @A 1 @ 2cos 0 z@1 + z@2 D ff ff ff ff fff ff ff ff ff fff ff ff ff ff fff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff D b cE b cE 1 @ cos 0 z@1 b |z| >1 D sin 0 n u n b cE @A 1 @ 2cos 0 z@1 + z@2 D b cE ff ff fff ff ff ff ff fff ff ff ff ff fff ff ff ff ff fff f ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff D b cE sin 0 z@1 |z| >1 D r n cos 0 n u n b cE @A 1 @ 2 r cos 0 z@1 + z@2 D ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff fff ff ff ff ff fff ff ff ff ff fff ff ff ff ff ff D b cE b cE 1 @ r cos 0 z@1 |z| >r D r n sin 0 n u n b cE @A 1 @ 2 r cos 0 z@1 + z@2 ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff fff ff ff ff ff fff ff ff ff ff fff ff ff ff ff fff ff f D b cE r sin 0 z@1 |z| >r Sampling Theorem: `a `a Let x t be a band @limited sigl with X j = 0 for || > M A Then x t is uniquely determined by its samples x nT , n = 0, F 1, F 2, ..., if s > 2 M 2 where; s = ffffffff T `a Given these samples, we can reconstruct x t by generating a periodic impulse train in which successive impulses have amplitudes that are successive sample valuesAThis impulse train is then processed through an ideal lowpass filter with gain T and cutoff frequency greater than M `a and less than s @ M AThe resulting output signal will exactly equal x t A ` a b c ...
View Full Document

Ask a homework question - tutors are online