Formula Sheet EE224 Final Frequency and Period 2 f 2 T f 1 T Time Delay Phase Shift 0 2 f 0 t Popular values Deg Rad Cos 0 0 1 3 30 6 2 2 45 4 2 Laws of Exponents 60 3 1 90 2 0 t e jx e jy e j x y e jx e jxy y 1 e jx jx e Polar to Rectangular x r cos y r sin Rectangular to Polar r x y tan 2 2 1 sin 0 1 3 2 1 2 2 Tan sin cos 0 3 2 2 1 1 3 undefined Basic Trigonometric Identities cos 2 sin 2 1 cos 2 cos 2 sin 2 y x Phasor Addition Rule A series of sinusoids with the same frequency can be added up using complex amplitude and phasors sin 2 2 cos sin sin sin cos cos sin cos cos cos sin sin N x t Ak cos 0t k Euler s Formula j t e 0 cos 0t j sin 0t k 1 A cos 0t N Ae Ak e j e j k k 1 Values of complex exponentials e j 0 e j 2 k 1 k is integer e j 2 k e j e j 2 j e j 2 j e j 1 e j z N re j cos j sin N N cos N j sin N Values of Sines and Cosines sin 2 cos sin cos 2 cos cos sin sin cos 2 k cos k is integer sin k 0 k is integer cos 2 k 1 k is integer j 0t cos 0t j sin 0t Inverse Euler s Formula 1 j t j t cos 0t e 0 e 0 2 1 j 0t j 0t e e sin 0t 2j Complex Numbers z1 x1 jy1 r1e j 1 z2 x2 jy2 r2 e j 2 z1 z2 x1 x2 j y1 y2 z1 z2 x1 x2 j y1 y2 z1 z2 x1 x2 y1 y2 j x1 y2 x2 y1 r1r2 e j 1 2 z1 x1 jy1 r1e j 1 z1 1 x1 jy1 x y 2 1 2 1 r1 11e j 1 Formula Sheet EE224 Final Continuous Fourier Series T 1 0 a0 x t dt T0 0 T 1 0 x t e j 0 kt dt T0 0 Discrete Time Fourier Series 1 N0 an x k e j 0 kn N 0 k 0 ak Simple Integrals 1 at at e dt a e e at at te dt a 2 at 1 Procedure for Finding Multiple Roots of zN c 1 Write z N r N e jN 2 Write c as c e j e j 2 k k is integer 3 Equate and solve for magnitude and angle separately r N e jN c e j e j 2 k 4 Magnitude r c 1 N 5 Angle N 2 k 1 N 2 k Magnitudes are the same angles are equally spaced around circle every 2 N radians Digital Frequency 2 f fs 0 2 l 0 2 l l is an integer Reconstruction D to C converter y t y n p t nTs n Discrete time signals Delta function n 0 1 n n 0 0 Unit step n 0 1 u n n 0 0 Linearity Scaling and superposition hold Time invariance response doesn t change with time Discrete time Convolution y n k h k x n k x k h n k k Delta function properties h n n h n h n n n0 h n n0 Frequency Response DTFT H e j h k e j k k Properties of DTFT 1 Digital spectra repeat every 2 H e j H e j H e j H e j 2 Conjugate symmetry H e j H e j Cascaded LTI Systems Time h n h1 n h2 n Frequency H e j k H1 e j k H 2 e j k LTI Sinusoidal System Response If x n Ae j e j n n e j n Then y n A H e j e j H e j System Function for Running Average L 1 sin L 2 j L 1 2 H e j 1 e j k e L sin 2 k 0 Formula Sheet EE224 Final Fourier Transform Pairs Time domain x t to Frequency Domain X j FT 1 e at u t Re a 0 a j FT 1 ebt u t Re b 0 b j FT 2a a t e Re a 0 2 a 2 FT 1 te at u t Re a 0 2 a j sin bt t t t td u t 1 e j 0t A cos 0t cos 0t j FT FT 2 FT 2 0 FT FT 0 kt k t nT 1 e j td n u b u b FT FT ae FT FT sin 0t k sin T 2 2 FT u t 12 T u t 12 T FT A e j 0 e j 0 0 0 j 0 0 2 a k FT 2 T 0 k k 1 j k 2 k T Formula Sheet EE224 Fourier Transform Properties Property Time domain x t to Frequency Domain X j Name FT Linearity ax t by t aX j bY j Conjugation x t Time reversal x t Scaling x at Delay x t td Modulation x t e j 0t x t cos 0t Differentiation in time dk x t dt k FT j Differentiation in Frequency Convolution jtx t FT d X j d FT X j FT X j FT FT FT FT x h t d x t p t Duality X jt x t dt 2 e j td X j X j 0 FT Multiplication Parseval s Theorem Symmetry 1 X j a a 1 2 X j 0 X j 0 k X j X j H j FT 1 X j P j 2 FT 2 x FT 1 2 X j d 2 x t real FT X j X j x t imag FT X j X j