Formula Sheet EE224 Final Frequency and Period22/;1/fTf Tωππ== = Time Delay/Phase Shift 002tftφφωπφω=− =−=− Laws of Exponents: ()()1;;yjx yjx jy jx jxy jxjxee e e e ee+−=== Polar to Rectangular cos ; sinxr yrθθ== Rectangular to Polar ()22 1;tan /rxy yxθ−=+ = Phasor Addition Rule A series of sinusoids with the same frequency can be added up using complex amplitude and phasors: () ( )()0101coscoskNkkkNjjkkxt A tAtAe A eφφωφωφ===+=+=∑∑ Values of complex exponentials: ()()()02j+2j/2 -j/21integeree;e ; 1cos sincos sinjjkkjjjNNNjee kisejje ezre jNjNπθπθπππ πφφφφφ−=====−=−===+=+ Values of Sines and Cosines: ()() ()( ) () ( ) ()()()()()2sin cos ;sin cos2cos cos ;sin sincos 2 cos ; integersin 0; integercos 2 1; integerkkiskkiskkisππθθθθθθθ θθπ θππ⎛⎞+= = −⎜⎟⎝⎠−= −=−+=== Popular values Deg. Rad. Cos sin Tan=sin/cos 0 0 1 0 0 30 π/6 32 12 13 45 π/4 22 22 1 60 π/3 12 32 3 90 π/2 0 1 undefined Basic Trigonometric Identities ()()( ) () ()( ) () ()()()2222cos sin 1cos 2 cos sinsin 2 2cos sinsin sin cos cos sincos cos cos sin sinθθθθθθθθαβαβαβαβαβαβ+==−=±= ±±= ∓ Euler’s Formula ()()()()()()000000cos sincos sinjtjtetjtetjtωθωθωθωθωθωθ+−+=++ +=+− + Inverse Euler’s Formula ()() ()()()() ()()0000001cos21sin2jt jtjt jtteeteejωθ ωθωθ ωθωθωθ+−++−++= ++= − Complex Numbers ()()()()()()()12121111 11 2 2 2212 12 1212 12 1 21 2 12 12 12 2112111111111112211;jjjjjzxjyrezxjyrezz xx jyyzz xx jyyzz xx yy jxy xyrrezxjyrexjyzrexyφφφφφφ+−∗−−−=+ = =+ =+= + + +−= − + −•= − + +==− =−==+Formula Sheet EE224 Final Continuous Fourier Series ()()0000000011TTjktkaxtdtTaxtedtTω−==∫∫ Discrete-Time Fourier Series []00001NjknnkaxkeNω−==∑ Simple Integrals ()211at atatatedt eaete dt ata==−∫∫ Procedure for Finding Multiple Roots of Nzc=: 1. Write NNjNzreφ= 2. Write c as 2; integerjjkce e kisθπ 3. Equate and solve for magnitude and angle separately: 2NjN j jkre ceeφθπ= 4. Magnitude: 1/ Nrc= 5. Angle: ()122NNk kφθπ φ θπ=+ ⇒= + Magnitudes are the same, angles are equally spaced around circle, every 2π/N radians Digital Frequency 2ˆsffπω= 00ˆˆ2; 2llωπω π+−+; l is an integer Reconstruction, D to C converter ()[]()snytynptnT∞=−∞=−∑ Discrete-time signals Delta function []1000nnnδ=⎧=⎨≠⎩ Unit step []1000nunn≥⎧=⎨<⎩ Linearity: Scaling and superposition hold Time-invariance: response doesn’t change with time Discrete-time Convolution [] [][ ] [][ ]kkyn hkxn k xkhn k∞∞=−∞ =−∞=−= −∑∑ Delta function properties: [][][][][][]00hn n hn hn n n hn nδδ∗= ∗−=− Frequency Response (DTFT) ()[]ˆˆjjkkHe hkeωω∞−=−∞=∑ Properties of DTFT 1. Digital spectra repeat every 2π 2. Conjugate symmetry ()()ˆˆjjHe H eωω−∗= ()() ()()ˆˆ ˆ ˆjj j jHe He He Heωωω ω−−=∠=−∠ Cascaded LTI Systems Time: [][][]12*hn h n h n= Frequency:()()()ˆˆˆ12jk jk jkHe H e H eωωω= LTI Sinusoidal System Response If []ˆjjnxn Ae e nφω=−∞ < < ∞ Then []()()()ˆˆˆjjHejjnyn AHe e eωφωω∠+= System Function for Running Average: ()()()()1ˆ1/2ˆ10ˆsin / 2ˆsin /2LjLjjkLkLHe e eωωωωω−−−−===∑Formula Sheet EE224 Final Fourier Transform Pairs Time-domain, x(t) to Frequency-Domain: X(jω) ()1(Re 0)FTateut aajω−>↔+ ()1(Re 0)FTbteu t bbjω−> ↔− ()222Re 0FTataeaaω−>↔+ ()()21(Re 0)FTatte u t aajω−>↔+ ()()()1122sin / 2/2FTTut T ut Tωω⎡⎤+−− ↔⎣⎦ ()()()sinFTbbbtuutωωω ωωπ↔+−−⎡⎤⎣⎦ ()1FTtδ↔ ()dFTjtdtt eωδ−−↔ () ( )1FTutjπδ ωω↔+ ()12FTπδω↔ ()002FTjteωπδω ω↔− () () ()0 00cosFTjjAt Ae eθθωθ π δωω δωω−⎡⎤+↔ −++⎣⎦ () ( ) ( )0 00cosFTtωπδωωδωω↔−++⎡⎤⎣⎦ () ( ) ( )0 00sinFTtjωπδωωδωω↔−−++⎡⎤⎣⎦ ()002FTjktk kkkae a kωπδω ω∞∞=−∞ =−∞↔−∑∑ ()22FTn ktnT kTTππδδω∞ ∞=−∞ =−∞⎛⎞−↔ −⎜⎟⎝⎠∑∑Formula Sheet EE224 Fourier Transform Properties Property Name Time-domain, x(t) to Frequency-Domain: X(jω) Linearity () () () ()FTax t by t aX j bY jωω+↔ + Conjugation () ( )FTxtXjω∗∗↔− Time-reversal () ( )FTxtXjω−↔− Scaling () ()1/FTxat X j aaω↔ Delay () ()dFTjtdxtt e Xjωω−−↔ Modulation () ( )()() ( ) ()()()()0010002cosFTjtFTxte X jxt t X j X jωωωωωωωω↔−⎡⎤↔−++⎣⎦ Differentiation in time () ( ) ( )kFTkkdxtjXjdtωω↔ Differentiation in Frequency () ()FTdjtx t X jdωω−↔ Convolution ()( ) ()()FTxht d X j H jτττ ω ω∞−∞−↔∫ Multiplication () () () ()12FTxtpt X j Pjωωπ↔∗ Duality () ( )2FTXjt xπω↔− Parseval’s Theorem () ()2212FTxtdt Xj dωωπ∞∞−∞ −∞↔∫∫ Symmetry () ( ) ( )() ( ) ( )FTFTxt real X j X jxtimag X j X
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