EEE203 Final Exam Note Sheets

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

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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 [email protected] x = cot x 2 2 2 d e d e d e csc [email protected] = sec x sec [email protected] x = csc x cot [email protected] 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 [email protected] 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 @[email protected] ` a ffffffffffffffff fffffffffffffff jx sin x = m e = 2j ` a ` a ` a ` a [email protected] = 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 = [email protected] = fff; ak = 0, otherwise 2 1 a1 = @[email protected] = 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 @[email protected] 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 @[email protected] 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 @[email protected] @ A 1 ff ff ff ff ff ff ff ff ff ff fff @u @n @1 |z| < 1 1 @[email protected] 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 @ [email protected] ff ff ff1 ff ff ff f ff ff fff ff ff ff ff 1 @ [email protected] @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 @ [email protected] 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 @ [email protected] D cos 0 n u n b cE @A 1 @ 2cos 0 [email protected] + [email protected] 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 [email protected] b |z| >1 D sin 0 n u n b cE @A 1 @ 2cos 0 [email protected] + [email protected] 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 [email protected] |z| >1 D r n cos 0 n u n b cE @A 1 @ 2 r cos 0 [email protected] + [email protected] 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 [email protected] |z| >r D r n sin 0 n u n b cE @A 1 @ 2 r cos 0 [email protected] + [email protected] 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 [email protected] |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 ...
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