Mathematics Tables

Home , Exsecant, Hyperbolic functions, Versine

MATHEMATICS TABLES

Department of Applied Mathematics Naval Postgraduate School TABLE OF CONTENTS

Derivatives and Differentials...... 1 Integrals of Elementary Forms...... 2 Integrals Involving au + b ...... 3 Integrals Involving u2 a2 ...... 4 Integrals Involving √u±2 a2, a> 0 ...... 4 Integrals Involving √a2 ± u2, a> 0 ...... 5 Integrals Involving Trigonometric− Functions ...... 6 Integrals Involving Exponential Functions ...... 7 Miscellaneous Integrals ...... 7 Wallis’ Formulas ...... 8 Gamma Function ...... 8 Laplace Transforms ...... 9 Probability and Statistics ...... 11 Discrete Probability Functions ...... 12 Standard Normal CDF and Table ...... 12 Continuous Probability Functions ...... 13 Fourier Series ...... 13 Separation of Variables ...... 15 Bessel Functions ...... 16 Legendre Polynomials ...... 17 Trigonometric Functions in a Right Triangle ...... 18 Trigonometric Functions of Special Angles ...... 18 Complex Exponential Forms ...... 18 Relations among the Functions ...... 19 Sums and Multiples of Angles ...... 19 Miscellaneous Relations ...... 20 Inverse Trigonometric Functions ...... 20 Side-Angle Relations in Plane Triangles ...... 21 Hyperbolic Functions ...... 22 First-Order Ordinary Differential Equations ...... 23 Linear Second-Order Ordinary Differential Equations ...... 24 Vector Algebra ...... 26 Vector Analysis ...... 27 Fundamentals of Signals and Noise ...... 29 Version 2.7, July 2010

Initial typesetting: Carroll Wilde Graphics: David Canright Editing: Elle Zimmerman, David Canright

Cover: in spherical coordinates (ρ, φ, θ), shell surface is ρ(φ, θ) = ekθ sin φ, 1 1+√5 where k = π ln G and G = 2 is the Golden Ratio. 1

FOREWORD

This booklet provides convenient access to formulas and other data that are frequently used in mathematics courses. If a more comprehensive reference is needed, see, for example, the STANDARD MATHEMATICAL TABLES published by the Chemical Rubber Company, Cleveland, Ohio.

DIFFERENTIALS AND DERIVATIVES

A. Letters u and v denote independent variables or functions of an independent variable; letters a and n denote constants.

B. To obtain a derivative, divide both members of the given formula for the diﬀerential by du or by dx. 1. d(a)=0 2. d(au)= a du 3. d(u + v)= du + dv 4. d(uv)= u dv + v du

u v du u dv n n 1 5. d = − 6. d(u )= nu − du v v2 7. d(eu)= eudu 8. d(au)= au ln a du du du 9. d(ln u)= 10. d(log u)= u a u ln a 11. d(sin u) = cos u du 12. d(cos u)= sin u du − 13. d(tan u) = sec2 u du 14. d(cot u)= csc2 u du − 15. d(sec u) = sec u tan u du 16. d(csc u)= csc u cot u du − du du 17. d(arcsin u)= 18. d(arccos u)= √1 u2 −√1 u2 − − du 19. d(arctan u)= 20. d(sinh u) = cosh u du 1+ u2 du 21. d(cosh u) = sinh u du 22. d(arcsinh u)= √u2 +1 du 23. d(arccosh u)= , u> 1 √u2 1 − 24. (Diﬀerentiation of Integrals) If f is continuous, then u d( f(t) dt)= f(u) du. Za 25. (Chain Rule) If y = f(u) and u = g(x), then dy dy du = . dx du dx 2

INTEGRALS

A. Letters u and v denote functions of an independent variable such as x; letters a, b, m, and n denote constants.

B. Generally, each formula gives only one antiderivative. To ﬁnd an expression for all antiderivatives, add a constant of integration.

ELEMENTARY FORMS

1. a du = au Z 2. af(u) du = a f(u) du Z Z 3. (f(u)+ g(u)) du = f(u) du + g(u) du Z Z Z 4. u dv = u v v du (Integration by parts) − Z Z un+1 5. un du = , n = 1 n +1 6 − Z du 6. = ln u u | | Z 7. eudu = eu Z au 8. audu = , a> 0 ln a Z 9. cos u du = sin u Z 10. sin u du = cos u − Z 11. sec2 u du = tan u Z 12. csc2 u du = cot u − Z 13. sec u tan u du = sec u Z 14. csc u cot u du = csc u − Z 3

1 (au + b)n+1 22. (au + b)n du = , n = 1 a n +1 6 − Z du 1 23. = ln au + b au + b a | | Z u du u b 24. = ln au + b au + b a − a2 | | Z u du b 1 25. = + ln au + b (au + b)2 a2(au + b) a2 | | Z du 1 u 26. = ln u(au + b) b au + b Z 2(3 au 2 b) 3 27. u√au + b du = − (au + b) 2 15a2 Z u du 2(au 2b) 28. = − √au + b √au + b 3a2 Z 2 2 2 2 2(8b 12abu + 15a u ) 3 29. u √au + b du = − (au + b) 2 105a3 Z u2 du 2(8b2 4abu +3a2u2) 30. = − √au + b √au + b 15a3 Z 4

INTEGRALS INVOLVING u2 a2 ±

du 1 u 31. = arctan [See 19] u2 + a2 a a Z du 1 u a 32. = ln − u2 a2 2a u + a Z − u du 1 33. = ln u 2 a2 u2 a2 2 | ± | Z ± u2 du a u a 34. = u + ln − u2 a2 2 u + a Z − u2 du u 35. = u a arctan u2 + a2 − a Z du 1 u2 36. = ln u(u2 a2) ±2a2 u2 a2 Z ± ± u du 1 37. = , n =0 (u2 a2)n+1 −2n(u 2 a2)n 6 Z ± ±

INTEGRALS INVOLVING √u2 a2 , a> 0 ±

u du 38. = u2 a2 [See 5] √u2 a2 ± Z ± p 1 2 2 3 39. u u2 a2 du = (u a ) 2 [See 5] ± 3 ± Z pdu 40. = ln u + u2 a2 √u2 a2 | ± | Z ± u p a2 41. u2 a2 du = u2 a2 ln u + u2 a2 ± 2 ± ± 2 | ± | Z p du 1 p u p 42. = ln u√u2 + a2 a a + √u2 + a2 Z du 1 u 43. = arcsec [See 21] u√u2 a2 a a Z − √u2 a2 u 44. − du = u2 a2 a arcsec u − − a Z √u2 + a2 p u 45. du = u2 + a2 + a ln u a + √u2 + a2 Z p

INTEGRALS INVOLVING √u2 a2 , a> 0 (Continued) ±

u2 du u a2 46. = u2 a2 ln u + u2 a2 √u2 a2 2 ± ∓ 2 | ± | Z ± p 2 p 2 u 2 2 3 a u 47. u u2 a2 du = (u a ) 2 u2 a2 ± 4 ± ∓ 8 ± − Z p a4 p ln u + u2 a2 8 | ± | p INTEGRALS INVOLVING √a2 u2 , a> 0 −

du u 48. = arcsin [See 20] √a2 u2 a Z − u a2 u 49. a2 u2 du = a2 u2 + arcsin − 2 − 2 a Z p u du p 50. = a2 u2 [See 5] √a2 u2 − − Z − p 1 2 2 3 51. u a2 u2 du = (a u ) 2 [See 5] − −3 − Z √pa2 u2 u 52. − du = a2 u2 + ln u − a + √a2 u2 Z p − du 1 u 53. = ln u√a2 u2 a a + √a2 u2 Z − − du √ a2 u2 54. = − u2√a2 u2 − a2u Z − √a2 u2 √a2 u2 u 55. − du = − arcsin u2 − u − a Z u2 du u a2 u 56. = a2 u2 + arcsin √a2 u2 − 2 − 2 a Z − p u a2u 57. u2 a2 u2 du = (a2 u2)3/2 + a2 u2 + − − 4 − 8 − Z p a4 u p arcsin 8 a 6

INTEGRALS INVOLVING TRIGONOMETRIC FUNCTIONS

u sin2au 58. sin2 audu = 2 − 4a Z u sin2au 59. cos2 audu = + 2 4a Z cos3 au cos au 60. sin3 audu = 3a − a Z sin au sin3 au 61. cos3 audu = a − 3a Z n 1 n sin − au cos au n 1 n 2 62. sin audu = + − sin − au du, − na n Z n 1 Z n cos − au sin au n 1 n 2 63. cos audu = + − cos − au du, na n Z Z u sin4au 64. sin2 au cos2 audu = 8 − 32a Z du 1 65. = cot au sin2 au −a Z du 1 66. = tan au cos2 au a Z 1 67. tan2 audu = tan au u a − Z 1 68. cot2 audu = cot au u −a − Z 1 1 69. sec3 audu = sec au tan au + ln sec au + tan au 2a 2a | | Z 1 1 70. csc3 audu = csc au cot au + ln csc au cot au −2a 2a | − | Z 1 u 71. u sin audu = sin au cos au a2 − a Z 1 u 72. u cos audu = cos au + sin au a2 a Z 2u a2u2 2 73. u2 sin audu = sin au − cos au a2 − a3 Z 2u a2u2 2 74. u2 cos audu = cos au + − sin au a2 a3 Z 7

INTEGRALS INVOLVING EXPONENTIAL FUNCTIONS

eau 75. ueau du = (au 1) a2 − Z eau 76. u2eau du = (a2u2 2au + 2) a3 − Z n au n au u e n n 1 au 77. u e du = u − e du, n> 1 a − a Z Z eau 78. eau sin budu = (a sin bu b cos bu) a2 + b2 − Z eau 79. eau cos budu = (a cos bu + b sin bu) a2 + b2 Z du 1 80. = u ln(1 + eau) 1+ eau − a Z MISCELLANEOUS INTEGRALS

√1 a2u2 81. arcsin audu = u arcsin au + − a Z √1 a2u2 82. arccos audu = u arccos au − − a Z 1 83. arctan audu = u arctan au ln(1 + a2u2) − 2a Z 1 84. arccot audu = u arccot au + ln(1 + a2u2) 2a Z 85. ln audu = u ln au u − Z ln au 1 86. un ln audu = un+1 n +1 − (n + 1)2 Z ∞ a2u2 √π 87. e− du = 2a Z0 8

WALLIS’ FORMULAS

(m 1)(m 3) (3)(1) π π − − ··· 2 (m)(m 2) (4)(2) 2 , m even; 88. sinm u du = − ··· 0  (m 1)(m 3) (4)(2) Z  (m−)(m −2) ···(5)(3) , m odd π π − ··· 2 2 89. cosm u du =  sinm u du 0 0 Z π Z 2 90. sinm u cosn u du = Z0 (m 1)(m 3) (3)(1)(n 1)(n 3) (4)(2) − (m−+n···)(m+n 2)− (3)(1)− ··· , m even, n odd; − ···  (m 1)(m 3) (4)(2)(n 1)(n 3) (3)(1)  − (m−+n···)(m+n 2)− (3)(1)− ··· , m odd, n even;  − ···   (m 1)(m 3) (4)(2)(n 1)(n 3) (4)(2)  − (m−+n···)(m+n 2)− (4)(2)− ··· , m odd, n odd; − ···  (m 1)(m 3) (3)(1)(n 1)(n 3) (3)(1) π  − (m−+n···)(m+n 2)− (4)(2)− ··· 2 , m even, n even  − ···   GAMMA FUNCTION

Deﬁnition :

∞ t x 1 Γ(x)= e− t − dt, x> 0 Z0 Shift Property : Γ(x +1)= xΓ(x), x> 0 Deﬁnition : Γ(x + 1) Γ(x)= , x< 0, x = 1, 2,... x 6 − − Special Values : 1 Γ( )= √π 2 Γ(n) = (n 1)!, n =1, 2,... − 9

LAPLACE TRANSFORMS

∞ st 1. f(t) F (s)= 0 e− f(t) dt 2. af(t)+ bg(t) aF (Rs)+ bG(s)

3. f ′(t) sF (s) f(0) − 2 4. f ′′(t) s F (s) sf(0) f ′(0) − − 5. tnf(t), n =1, 2, 3,... ( 1)nF (n)(s) − 6. eatf(t) F (s a) P − − e stf(t) dt 0 7. f(t + P ) f(t) 1 e−sP ≡ − Ras 8. f(t)H(t a) e− L f(t + a) − { } as 9. f(t a)H(t a) e− F (s) − − t F (s) 10. 0 f(u) du s t 11. 0 fR(u)g(t u) du F (s)G(s) f(t)− ∞ 12. R t s F (v) dv 1 13. 1, H(t)∗, u(t) R s n n! 14. t , n =1, 2, 3,... sn+1 a Γ(a+1) 15. t , a> 1 a − s +1 at 1 16. e s a − ω 17. sin ωt s2+ω2 s 18. cos ωt s2+ω2 1 at 1 19. a (e 1) s(s a) − − 1 at bt 1 20. a b (e e ) (s a)(s b) , a = b − − − − 6 at 1 21. te (s a)2 − n at n! 22. t e , n =1, 2, 3,... (s a)n+1 − a 23. sinh at s2 a2 − s 24. cosh at s2 a2 − 25. 1 (1 cos at) 1 a2 − s(s2+a2) 26. 1 (at sin at) 1 a3 − s2(s2+a2) * The Heaviside step function H is deﬁned by 0, if t< 0 H(t)= 1, if t 0. ≥ 10

LAPLACE TRANSFORMS (Continued)

27. 1 (sin at at cos at) 1 2a3 − (s2+a2)2 1 s 28. 2a t sin at (s2+a2)2 1 s2 29. 2a (sin at + at cos at) (s2+a2)2 s2 a2 30. t cos at (s2+−a2)2 b sin at a sin bt 1 31. ab(b2− a2) (s2+a2)(s2+b2) , a = b − 6 cos at cos bt s 32. (b2 −a2) (s2+a2)(s2+b2) , a = b − 6 a sin at b sin bt s2 33. (a2 −b2) (s2+a2)(s2+b2) , a = b − 6 bt ω 34. e− sin ωt (s+b)2+ω2 bt s+b 35. e− cos ωt (s+b)2+ω2 2 36. 1 cos ωt ω − s(s2+ω2) sa 37. δ(t a) e− − √(a b)2+ω2 − bt ω e− sin(ωt + Ψ) s+a 38. where (s+b)2+ω2 ω Ψ = arctan a b − √b2+ω2 bt 1+ ω e− sin(ωt Ψ) − b2+ω2 39. where s[(s+b)2+ω2] ω Ψ = arctan b − −act 1+ e sin(√1 c2at Ψ) √1 c2 − − − a2 40. where s[s2+2acs+a2] √1 c2 Ψ = arctan −c , 1

PROBABILITY AND STATISTICS

Formulas for counting permutations and combinations: n! n n! (a) P = (b) C = = n k (n k)! n k k k!(n k)! − − Let A and B be subsets (events) of the same sample space. (a) P (A B)= P (A)+ P (B) P (A B). ∪ − ∩ (b) The conditional probability of B given A, P (B A), is deﬁned by | P (A B) P (B A)= ∩ , P (A) =0. | P (A) 6 (c) A and B are statistically independent if and only if P (A B)= P (A)P (B). ∩ Let X be a discrete random variable with probability function f(x). (a) P (X = x)= f(x), where x S range of X ∈ ≡ (b) E(X)= xf(x)= µ = mean of X XS (c) var(X)= E((X µ)2)= E(X2) µ2 = x2f(x) µ2 = σ2 − − − XS = variance of X (d) σ = var(X) = standard deviation of X. Let X be a continuousp random variable with and probability density f(x). b (a) P (a X b)= f(x) dx. ≤ ≤ Za (b) E(X)= xf(x) dx = µ = mean of X ZS (c) var(X)= E((X µ)2)= E(X2) µ2 = x2f(x) dx µ2 = σ2 − − − ZS = variance of X (d) σ = var(X) = standard deviation of X. p Let X1,X2,...,Xn be n statistically independent random variables, each hav- ing the same expected value, µ, and the same variance, σ2. Let X + X + + X Y = 1 2 · · · n . n σ2 Then Y is a random variable for which E(Y )= µ and var(Y )= n . 12

PROBABILITY FUNCTIONS (Discrete Random Variables)

Binomial

n x n x f(x; n,p)= p (1 p) − , (x =0, 1, 2,...,n), x −

µ = np, σ2 = np(1 p). − Poisson e aax f(x; a)= − , (x =0, 1, 2,...), x!

µ = σ2 = a.

STANDARD NORMAL CUMULATIVE DISTRIBUTION FUNCTION

N(0,1)

z 1 t2 F (z)= e− 2 dt √2π Z−∞

0 z t

TABLE OF NORMAL AREAS

z F (z) z F (z) z F (z) z F (z) 0.0 0.500 0.8 0.788 1.6 0.945 2.4 0.992 0.1 0.540 0.9 0.816 1.7 0.955 2.5 0.994 0.2 0.579 1.0 0.841 1.8 0.964 2.6 0.995 0.3 0.618 1.1 0.864 1.9 0.971 2.7 0.997 0.4 0.655 1.2 0.885 2.0 0.977 2.8 0.997 0.5 0.691 1.3 0.903 2.1 0.982 2.9 0.998 0.6 0.726 1.4 0.919 2.2 0.986 3.0 0.999 0.7 0.758 1.5 0.933 2.3 0.989 13

PROBABILITY FUNCTIONS (Continuous Random Variables)

Uniform 1 f(x; a,b)= (a x b). b a ≤ ≤ −

a + b (b a)2 µ = ; σ2 = − . 2 12 Exponential

bx f(x; b)= be− (x 0, b> 0). ≥

1 1 µ = ; σ2 = . b b2 Normal

1 (x−a)2 f(x; a,b)= e− 2b2 (

µ = a; σ2 = b2.

FOURIER SERIES

The Fourier series expansion of a function f(t) is deﬁned by:

∞ nπt nπt f(t)= a + (a cos + b sin ), 0 n L n L n=1 X where 1 L a0 = f(t) dt 2L L Z−

1 L nπt an = f(t)cos dt, n 1 L L L ≥ Z−

1 L nπt bn = f(t) sin dt, n 1 L L L ≥ Z−

(Note: for the complex Fourier Series, see page 29) 14

FOURIER SERIES (Continued)

Examples of Fourier Series

f(t) 1 1, 0

4 πt 1 3πt 1 5πt sin + sin + sin + ) π L 3 L 5 L · · · f(t) 1 0, L < t

1 2 πt 1 3πt 1 5πt + cos cos + cos + ) 2 π L − 3 L 5 L − · · · L f(t) f(t)= t, L

2L πt 1 2πt 1 3πt sin sin + sin + ) π L − 2 L 3 L − · · · f(t) L f(t)= t , L

L t L 4L πt 1 3πt 1 5πt cos + cos + cos + ) 2 − π2 L 32 L 52 L · · · 15

SEPARATION OF VARIABLES

Eigenvalues and Eigenfunctions for the diﬀerential equation

ϕ′′ + λϕ =0.

1. ϕ(0) = ϕ(L)=0

nπ 2 nπ λn = ( L ) , ϕn = sin L x, n =1, 2, ...

2. ϕ′(0) = ϕ(L)=0

2 λ = (n 1 ) π , ϕ = cos(n 1 ) π x, n =1, 2, ... n − 2 L n − 2 L 3. ϕ(0) = ϕ′(L)=0 2 λ = (n 1 ) π , ϕ = sin(n 1 ) π x, n =1, 2, ... n − 2 L n − 2 L 4. ϕ′(0) = ϕ′(L)=0

nπ 2 nπ λn = ( L ) , ϕn = cos L x, n =0, 1, 2, ...

5. ϕ( L)= ϕ(L), ϕ′( L)= ϕ′(L) − − λ0 =0, ϕ0 =1, nπ 2 nπ nπ λn = ( L ) , ϕn = sin L x and cos L x, n =1, 2, ...

Solution of inhomogeneous ordinary diﬀerential equations

1. un′ + k λnun = hn(t)

kλnt t kλn (t τ) un(t)= ane− + 0 hn(τ)e− − dτ 2 2. un′′ + µn un = hn(t) R 1 t un(t)= an cos µnt + bn sin µnt + hn(τ) sin µn(t τ) dτ µn 0 − R 16

BESSEL FUNCTIONS

1. The differential equation 2 2 2 2 x y′′ + xy′ + (λ x n )y =0 − has solutions y = yn(λx)= c1Jn(λx)+ c2Yn(λx), where ∞ ( 1)mtn+2m J (t)= − . n 2n+2mm!Γ(n + m + 1) m=0 X 2. General properties: n J n(t) = ( 1) Jn(t); J0(0) = 1; Jn(0) = 0, n =1, 2, 3,... ; − − for fixed n, Jn(t) = 0 has infinitely many solutions.

3. Identities: d n n (a) [t Jn(t)] = t Jn 1(t). dt − d n n (b) [t− J (t)] = t− J (t); J ′ (t)= J (t) dt n − n+1 0 − 1 1 (c) Jn′ (t)= [Jn 1(t) Jn+1(t)] 2 − − n n = Jn 1(t) Jn(t)= Jn(t) Jn+1(t) − − t t − 2n (d) Jn+1(t)= Jn(t) Jn 1(t) (recurrence). t − − 4. Orthogonality:

Solutions yn(λ0x),yn(λ1x),...,yn(λix),... of the diﬀerential eigensystem 2 2 2 2 x y′′ + xy′ + (λ x n )y =0 − A y(x ) B y′(x )=0, A y(x )+ B y′(x )=0 1 1 − 1 1 2 2 2 2 have the property

x2 xyn(λix)yn(λj x) dx =0 Zx1 for i = j, and 6 2 x2 x2 2 2 2 2 2 d (λ x n )y (λix)+ x yn(λix) xy2 (λ x) dx = i − n dx n i 2λ2 x1 i Z x1

for i = j.

5. Integration properties:

ν ν (a) t Jν 1(t) dt = t Jν (t)+ C − Z ν ν (b) t− J (t) dt = t− J (t)+ C ν+1 − ν Z (c) tJ0(t) dt = tJ1(t)+ C Z (d) t3J (t) dt = (t3 4t)J (t)+2t2J (t)+ C 0 − 1 0 Z (e) t2J (t) dt =2tJ (t) t2J (t)+ C 1 1 − 0 Z (f) t4J (t) dt = (4t3 16t)J (t) (t4 8t2)J (t)+ C 1 − 1 − − 0 Z LEGENDRE POLYNOMIALS

The diﬀerential equation 2 (1 x )y′′ 2xy′ + n(n + 1)y =0 − − has solution y = Pn(x), where

[n/2] 1 m n 2n 2m n 2m P (x)= ( 1) − x − n 2n − m n m=0 X on the interval [ 1, 1]. The Legendre polynomials can also be obtained itera- tively from the ﬁrst− two,

P0(x) = 1 and P1(x)= x, and the recurrence relation,

(n + 1)Pn+1(x)=(2n + 1)xPn(x) nPn 1(x). − − The Legendre polynomials are also given by Rodrigues’ formula: ( 1)n dn P (x)= − [(1 x2)n]. n 2nn! dxn − Orthogonality: 1 0, n = m; Pn(x)Pm(x) dx = 2 , n =6 m. 1 2n+1 Z− Standardization: Pn(1) = 1. 18

TRIGONOMETRIC FUNCTIONS IN A RIGHT TRIANGLE

If A, B, and C are the vertices (C the right angle), and a, b, and r the sides opposite, respectively, then a b sine A = sin A = , cosine A = cos A = , r r a b tangent A = tan A = , cotangent A = cot A = , b a r r secant A = sec A = , cosecant A = csc A = , b a exsecant A = exsec A = sec A 1 B − versine A = vers A =1 cos A r − a coversine A = covers A =1 sin A 1 − haversine A = hav A = vers A A C 2 b

TRIGONOMETRIC FUNCTIONS OF SPECIAL ANGLES

π π π π 3π 0 6 4 3 2 π 2 sin 0 1 √2 √3 1 0 1 2 2 2 − cos 1 √3 √2 1 0 1 0 2 2 2 − tan 0 √3 1 √3 0 3 ∞ ∞ cot √3 1 √3 0 0 ∞ 3 ∞ sec 1 2√3 √2 2 1 3 ∞ − ∞ csc 2 √2 2√3 1 1 ∞ 3 ∞ −

COMPLEX EXPONENTIAL FORMS (i2 = 1) −

1 ix ix 2i sin x = (e e− ) csc x = 2i − eix e ix − − 1 ix ix 2 cos x = (e + e− ) sec x = ix ix 2 e + e− ix ix ix ix 1 e e− e + e− tan x = − cot x = i i eix + e ix eix e ix − − − 19

RELATIONS AMONG THE FUNCTIONS

1 1 sin x = csc x csc x = sin x 1 1 cos x = sec x sec x = cos x 1 sin x 1 cos x tan x = cot x = cos x cot x = tan x = sin x sin2 x + cos2 x =1 1+tan2 x = sec2 x 1+cot2 x = csc2 x 2 2 ∗ sin x = √1 cos x ∗ cos x = 1 sin x ± − ± − 2 2 ∗ tan x = √sec x 1 ∗ sec x = √1 + tan x ± − ±p 2 2 ∗ cot x = √csc x 1 ∗ csc x = √1+cot x ± π − ± π sin x = cos( 2 x) = sin(π x) cos x = sin( 2 x)= cos(π x) tan x = cot( π − x)= tan(−π x) cot x = tan( π − x)=− cot(π− x) 2 − − − 2 − − −

SUMS AND MULTIPLES OF ANGLES

sin(x y) = sin x cos y cos x sin y cos(x± y) = cos x cos y± sin x sin y ± tan x tan∓ y tan(x y)= ± ± 1 tan x tan y sin2x = 2 sin x cos∓ x cos2x = cos2 x sin2 x = 2cos2 x 1=1 2 sin2 x sin3x = 3 sin x − 4 sin3 x − − cos3x = 4cos3 x− 3cos x sin4x = 8 sin x cos−3 x 4 sin x cos x cos4x = 8cos4 x 8cos− 2 x +1 2 tan x− cot2 x 1 tan2x = cot2x = − 1 tan2 x 2cot x − 3 tan x tan3 x tan3x = − 1 3 tan2 x − 1 1 cos x 1 1+cos x ∗ sin x = − ∗ cos x = 2 ±r 2 2 ±r 2 1 1 cos x 1 cos x sin x ∗ tan x = − = − = 2 1+cos x sin x 1+cos x ±r * The choice of the sign in front of the radical depends on the quadrant in which x, regarded as an angle, falls. 20

MISCELLANEOUS RELATIONS

sin x sin y = 2 sin 1 (x y)cos 1 (x y) ± 2 ± 2 ∓ cos x + cos y = 2cos 1 (x + y)cos 1 (x y) 2 2 − cos x cos y = 2 sin 1 (x + y) sin 1 (x y) − − 2 2 − sin x cos y = 1 [sin(x + y) + sin(x y)] 2 − cos x sin y = 1 [sin(x + y) sin(x y)] 2 − − cos x cos y = 1 [cos(x + y) + cos(x y)] 2 − sin x sin y = 1 [cos(x y) cos(x + y)] 2 − − sin(x y) sin(x y) tan x tan y = ± cot x cot y = ± ± ± cos x cos y ± sin x sin y 1+tan x π cot x+1 π 1 tan x = tan( 4 + x) cot x 1 = cot( 4 x) − − − sin x sin y 1 sin x sin y 1 cos x±+cos y = tan 2 (x y) cos x±cos y = cot 2 (x y) ± − − ∓ 1 sin x+sin y tan 2 (x+y) sin x sin y = tan 1 (x y) − 2 − sin2 x sin2 y = sin(x + y) sin(x y) − − cos2 x cos2 y = sin(x + y) sin(x y) − − − cos2 x sin2 y = cos(x + y) cos(x y) − − INVERSE TRIGONOMETRIC FUNCTIONS

The following table gives the domains of the inverse trigonometric functions.

Interval containing principal value Function x positive or zero x negative y = arcsin x and y = arctan x 0 y π π y 0 ≤ ≤ 2 − 2 ≤ ≤ y = arccos x and y = arccotx 0 y π π y π ≤ ≤ 2 2 ≤ ≤ y = arcsecx and y = arccscx 0 y π π y π ≤ ≤ 2 − ≤ ≤− 2

Two notation systems are in common use. For example, the inverse sine is 1 often denoted by arcsin or sin− . In many references, however, “principal 1 value”is used for the function and is denoted by Arcsin or Sin− . Thus, in one book, 1 1 π π Arc sin = Sin− x = and arcsin x = +2nπ, 2 6 6 while in another, only lower case is used: 1 π arcsin = . 2 6 Special care is advisable in the use of reference materials. 21

SIDE-ANGLE RELATIONS IN PLANE TRIANGLES

Letters A, B, and C denote the angles of a plane triangle with opposite sides a, b, and c, respectively. The letter s denotes the semi-perimeter of the triangle, that is, 1 s = (a + b + c). 2 a b c = = = diameter of the circumscribed circle sin A sin B sin C a2 = b2 + c2 2 bc cos A − a = b cos C + c cos B A B a b C tan − = − cot 2 a + b 2 2 sin A = s(s a)(s b)(s c) bc − − − area = sp(s a)(s b)(s c) − − − A p(s b)(s c) sin = − − 2 r bc A s(s a) cos = − 2 r bc A (s b)(s c) tan = − − 2 s(s a) s − a + b sin A + sin B tan 1 (A + B) cot 1 C = = 2 = 2 a b sin A sin B tan 1 (A B) tan 1 (A B) − − 2 − 2 −

sin α sin β d h = d = sin(α + β) cot α + cot β h α β d

sin α sin β d h = d ′ = sin(β α) cot α cot β h ′ − − ′ α β′ d 22

HYPERBOLIC FUNCTIONS

1 x x 1 x x sinh x = (e e− ), cosh x = (e + e− ) 2 − 2 x x e e− sinh x 1 tanh x = x − x = = e + e− ±cosh x coth x tanh x 1 sinh x = cosh2 x 1= = ± − 1 tanh2 x ± coth2 x 1 p − − 1 coth x cosh x = 1 + sinh2 x = p = p 1 tanh2 x ± coth2 x 1 p − − sinh x p cosh2 x 1 p 1 tanh x = = − = 1 + sinh2 x ±p cosh x coth x p1 + sinh2 x cosh x 1 coth x = = = p sinh x ± cosh2 x 1 tanh x − sinh(x y) = sinh x cosh y cosh x sinh y ± ± p cosh(x y) = cosh x cosh y sinh x sinh y ± ± tanh x tanh y coth x coth y 1 tanh(x y)= ± coth(x y)= ± ± 1 tanh x tanh y ± coth y coth x ± ± sinh 2x = 2cosh x sinh x cosh2x = cosh2 x + sinh2 x 1 1 sinh− x = ln(x + x2 +1) cosh− x = ln(x + x2 1) − 1 1 1+px 1 1 xp+1 tanh− x = ln coth− x = ln 2 1 x 2 x 1

− − Complex hyperbolic functions involve the Euler relations

iz iz iz iz e + e− e e− eiz = cos z + i sin z; cos z = , sin z = − . 2 2i cos z = cosh iz cosh z = cos iz sin z = i sinh iz sinh z = i sin iz − − tan z = i tanh iz tanh z = i tan iz − − cot z = i coth iz coth z = i cot iz sinh(x iy) = sinh x cos y i cosh x sin y ± ± cosh(x iy) = cosh x cos y i sinh x sin y ± ± 1 ln z = ln z + i arg z; so ln ix = ln x + (2n + )πi | | | | 2 and ln( x) = ln x + (2n + 1)πi (n =0, 1, 2,...) − | | ± ± 23

FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS

I. Separable equations : N(y) dy = M(x) dx. To solve, integrate both sides:

N(y) dy = M(x) dx + c. Z Z ∂M ∂N II. Exact equations : M(x, y) dx + N(x, y) dy = 0, where = . To ∂y ∂x solve, ∂f ∂f integrate = M(x, y) and = N(x, y) : ∂x ∂y

f(x, y)= M(x, y) dx + φ(y)= N(x, y) dy + ψ(x)= c. Z Z III. Linear equations : dy + p(x) y = q(x). dx The solution is given by 1 c y = P (x)q(x) dx + , where P (x)= e p(x) dx. P (x) P (x) Z R IV. Homogeneous equations : dy y dy x = f or = f . dx x dx y dy du dx dx Substitute y = ux, = x + u or x = vy, = y + v, dx dx dy dy integrate the resulting expression, then resubstitute for u or v.

V. Equations in the rational form dy ax + by + c = , dx dx + ey + f where a, b, c, d, e, f are constants such that ae = bd. Substitute 6 bf ce cd af x = X h, y = Y k, where h = − , k = − , − − ae bd ae bd − − to obtain the homogeneous equation dY aX + bY = . dX dX + eY This equation may be solved as in item IV above. 24

LINEAR SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS

rx I. Homogeneous equation ay′′ + by′ + cy =0. Substitute y = e to ﬁnd the characteristic equation, and write its roots r1 and r2 : b √b2 4ac ar2 + br + c = 0; r , r = − ± − . 1 2 2a

The solution, yc, is given by the following table.

2 r1x r2x Case 1. b 4ac > 0, yc = c1e + c2e real and distinct− roots.

2 rx Case 2. b 4ac =0, yc = (c1 + c2x)e two real equal− roots.

2 αx Case 3. b 4ac < 0, yc = e (c1 cos βx + c2 sin βx), two complex− roots. where b √4ac b2 α = − , β = − 2a 2a

II. Undetermined Coeﬃcients : ay′′ + by′ + cy = F (x).

1. Solve the homogeneous equation ay′′ + by′ + cy = 0 for yc.

2. Assume a particular solution yp for any term in F (x) that is one of three special types, as follows.

A. For a polynomial of degree N, yp is a polynomial of degree N. p(x) p(x) B. For an exponential ke , yp = Ae .

C. For j cos ωx + k sin ωx, yp = A cos ωx + B sin ωx.

3. If any term in yp found above appears in yc, multiply the corresponding trial term by the power of x just high enough that the resulting product contains no term of yc. 4. For any term in F (x) that is a product of terms of the three types listed above, let yp be the product of the corresponding trial solutions.

5. Substitute yp into the given O.D.E. and evaluate the unknown coeﬃcients by equating like terms. 6. The general solution of the given equation is

y = yc + yp.

Evaluate the two arbitrary constants c1 and c2 in the general solution y by applying boundary/initial conditions, as appropriate. 25

III. Variation of Parameters : ay′′ + by′ + cy = F (x)∗.

1. Write the solution of the homogeneous equation ay′′ + by′ + cy = 0 in the form yc = c1y1 + c2y2.

2. Assume a particular solution yp for any term in the form

yp = u1y1 + u2y2,

where u1 and u2 are functions determined by the conditions

u1′ y1 + u2′ y2 =0 F (x) u′ y′ + u′ y′ = G(x) where G(x)= . 1 1 2 2 a 3. The solution of this system of equations is given by

y2G y1G u′ = and u′ = , 1 − W 2 W where W denotes the Wronskian determinant: y y W = W (x)= 1 2 =0. y′ y′ 6 1 2

4. Integrate to ﬁnd the unknown functions: y (x)G(x) y (x)G(x) u (x)= 2 dx and u (x)= 1 dx. 1 − W (x) 2 W (x) Z Z

5. Form the particular solution yp = u1y1 + u2y2. 6. The general solution of the given equation is

y = yc + yp.

Evaluate the two arbitrary constants c1 and c2 in the general solution y by applying boundary/initial conditions, as appropriate.

IV. Laplace Transforms : ay′′ + by′ + cy = F (x), y(0) = y0 and y′(0) = y0′ . 1. Take the Laplace transform (see pages 9-10) of both sides of the given equation, using the given initial conditions. The result is an algebraic equation for F (s), the Laplace transform of the solution. 2. Solve the algebraic equation obtained in Step 1 to ﬁnd F (s) explicitly. 3. The inverse Laplace transform of F (s) is the solution of the given initial value problem.

*The coeﬃcients may also be functions, i.e., b = b(x) and c = c(x). 26

VECTOR ALGEBRA

1. Scalar and vector products. Given vectors

a = α1e1 + α2e2 + α3e3 = axi + ayj + azk

b = β1e1 + β2e2 + β3e3 = bxi + byj + bzk,

c = γ1e1 + γ2e2 + γ3e3 = cxi + cyj + czk, let φ be the angle from a to b. The scalar (dot) product of a and b is the scalar a b = a b cos φ = a b + a b + a b . · | || | x x y y z z The vector (cross) product of a and b is the vector of magnitude a b = a b sin φ, | × | | || | perpendicular to a and b and in the direction of the axial motion of a right-handed screw turning a into b. In coordinate form,

e2 e3 α1 β1 a b = e × e α β × 3 × 1 2 2 e e α β 1 × 2 3 3 i a b x x a a a a a a = j a b = y z i + z x j + x y k y y b b b b b b k a b y z z x x y z z

= (aybz azby )i + (azbx axbz )j + (ax by aybx)k. − − − The box product of a, b, and c is the scalar quantity given by a b c [abc] = [bca] = [cab]= [bac]= [cba]= [acb] · × ≡ − − − α1 β1 γ1 ax bx cx = α β γ [e e e ]= a b c . 2 2 2 1 2 3 y y y α β γ a b c 3 3 3 z z z

The product of two box products is given by

a d a e a f [abc][def]= b · d b · e b · f . · · · c d c e c f · · ·

A special case gives Gram′s determinant :

a a a b a c [abc]2 = b · a b · b b · c · · · c a c b c c · · ·

= [(a b)(b c)(c a)] × × × = (a a)(b b)(c c)+2(a b)(b c)(a c) (a a)(b c)2 · · · · · · − · · − (b b)(a c)2 (c c)(a b)2. · · − · · 27

VECTOR ALGEBRA (Continued)

The vector triple product of vectors a, b, and c is given by b c a (b c) = (a c)b (a b)c = . × × · − · a b a c · ·

Three additional formulas for scalar and vector products:

a c b c (a b) (c d) = (a c)(b d) (a d)(b c)= · · . × · × · · − · · a d b d · · 2 2 (a b) = (a a)(b b) (a b) × · · − · (a b) (c d) = [acd]b [bcd]a = [abd]c [abc]d. × × × − −

VECTOR ANALYSIS

Diﬀerential operators ∂Φ ∂Φ ∂Φ Φ(x,y,z) = gradΦ(x,y,z) i + j + k ∇ ≡ ∂x ∂y ∂z ∂F ∂F ∂F F(x,y,z) = div F(x,y,z) x + y + z ∇ · ≡ ∂x ∂y ∂z F = curl F(x,y,z) ∇× ∂F ∂F ∂F ∂F ∂F ∂F z y i + x z j + y x k ≡ ∂y − ∂z ∂z − ∂x ∂x − ∂y i j k ∂ ∂ ∂ = ∂x ∂y ∂z

Fx Fy Fz

∂F ∂F ∂F (G )F = Gx + Gy + Gz · ∇ ∂x ∂y ∂z = (G F )i + (G F )j + (G F )k · ∇ x · ∇ y · ∇ z ∂2 ∂2 ∂2 2 = ( ) + + (Laplace operator) ∇ ∇ · ∇ ≡ ∂x2 ∂y2 ∂z2 div grad Φ = ( Φ) = 2Φ ∇ · ∇ ∇ grad div F = ( F)= 2F + ( F) ∇ ∇ · ∇ ∇× ∇× curl curl F = ( F)= ( F) 2F ∇× ∇× ∇ ∇ · − ∇ curl grad Φ = ( Φ)=0 ∇× ∇ div curl F = ( F)=0 ∇ · ∇× (ΦΨ) = Ψ Φ+Φ Ψ ∇ ∇ ∇ 2(ΦΨ) = Ψ 2Φ+2( Φ) ( Ψ)+Φ 2Ψ ∇ ∇ ∇ · ∇ ∇ 28

VECTOR ANALYSIS (Continued)

(F G) = (F )G + (G )F + F ( G)+ G ( F) ∇ · · ∇ · ∇ × ∇× × ∇× (ΦF)=Φ F + ( Φ) F ∇ · ∇ · ∇ · (F G)= G F F G ∇ · × ·∇× − ·∇× (G )(ΦF)= F(G Φ)+Φ(G )F · ∇ · ∇ · ∇ (ΦF)=Φ F + ( Φ) F ∇× ∇× ∇ × (F G) = (G )F (F )G + F( G) G( F) ∇× × · ∇ − · ∇ ∇ · − ∇ ·

Cylindrical : x = r cos θ,y = r sin θ,z = z ∂Φ 1 ∂Φ ∂Φ grad Φ = Φ= ˆr + θˆ+ kˆ ∇ ∂r r ∂θ ∂z 1 ∂(rF ) 1 ∂F ∂F div F = F = r + θ + z ∇ · r ∂r r ∂θ ∂z 1 ˆ 1 ˆ r ˆr θ r k ∂ ∂ ∂ curl F = F = ∂r ∂θ ∂z ∇× Fr rFθ Fz

1 ∂ ∂Φ 1 ∂2Φ ∂2Φ Laplacian Φ = 2Φ= r + + ∇ r ∂r ∂r r 2 ∂θ2 ∂z2 Spherical : x = ρ cos θ sin φ, y = ρ sin θ sin φ, z = ρ cos φ ∂Φ 1 ∂Φ 1 ∂Φ grad Φ = Φ= ρˆ+ φˆ + θˆ ∇ ∂ρ ρ ∂φ ρ sin φ ∂θ 1 ∂(ρ2F ) 1 ∂(sin φ F ) 1 ∂F div F = F = ρ + φ + θ ∇ · ρ2 ∂ρ ρ sin φ ∂φ ρ sin φ ∂θ 1 1 ˆ 1 ˆ ρ2 sin φ ρˆ ρ sin φ φ ρ θ ∂ ∂ ∂ curl F = F = ∂ρ ∂φ ∂θ ∇× Fρ ρFφ ρ sin φ Fθ

2 2 1 ∂ 2 ∂Φ 1 ∂ ∂Φ 1 ∂ Φ Φ= ρ + sin φ + ∇ ρ2 ∂ρ ∂ρ ρ2 sin φ ∂φ ∂φ ρ2 sin2 φ ∂θ2

Integral Theorems F(r) dV = F(r) dA (Divergence theorem) V ∇ · S · R Φ Ψ dV +R Ψ 2Φ dV = (Ψ Φ) dA (Green’s theorem) V ∇ · ∇ V ∇ S ∇ · R (Ψ 2Φ Φ 2Ψ)R dV = (Ψ ΦR Φ Ψ) dA (symmetric form) V ∇ − ∇ S ∇ − ∇ · R 2Φ dV = Φ dA R (Gauss’ theorem) V ∇ S ∇ · R [ F(r)] RdA = F(r) dr (Stokes’ theorem) S ∇× · C · R H 29

FUNDAMENTALS OF SIGNALS AND NOISE

Complex numbers (j2 = 1) − a + bj = Rejθ, where R = a2 + b2, b parctan a , a> 0; arctan b π, a< 0; θ =  a π ,± a =0,b> 0;  2  π , a =0,b< 0. − 2   Aejθ Bejφ = ABej(θ+φ) · ejθ = ej(θ+2nπ), n = 1, 2,... ± ±

Unit phasor j π j = e 2

jπ j 0 j2π e = 1 1= e · = e −

j π j 3π j = e− 2 = e 2 − Euler Identity

− ejx +e jx jx cos x = 2 ; − e± = cos x j sin x; ejx e jx ± ( sin x = −2j .

Fourier series (Periodic signals. For the real number form, see p. 13.)

∞ j2πnf1t x(t)= cne , where n= X−∞ T1 2 1 j2πnf1 t cn = x(t)e− dt, T1 T1 Z− 2 T1 = period, 1 f1 = = fundamental frequency, T1

(For real signals, c n = cn∗ (complex conjugate).) − 30

Fourier transform (aperiodic signals)

∞ j2πft ∞ j2πtf X(f)= x(t)e− dt; x(t)= X(f)e df. Z−∞ Z−∞ The correspondence between x(t) and X(f) is denoted by x(t) X(f). ↔ Transform of Fourier Series

∞ ∞ If x(t)= c ej2πnf1t, then X(f)= c δ(f nf ). n n − 1 n= n= X−∞ X−∞

Convolution ( ) and Correlation (⋆) ∗

∞ ∞ x y = x(τ)y(t τ) dτ y(τ)x(t τ) dτ = y x ∗ − ≡ − ∗ Z−∞ Z−∞ ∞ x ⋆y = x(τ)y(t + τ) dτ Z−∞

Properties of the delta function

∞ δ(t) dt =1 Z−∞ ∞ δ(t t )x(t) dt = x(t ) − 0 0 Z−∞ x(t) δ(t t )= x(t t ) ∗ − 0 − 0 x(t)δ(t t )= x(t )δ(t t ) − 0 0 − 0

Parseval theorem (Average Power in a periodic signal)

T 2 1 2 ∞ 2 P = x(t) dt = cn T T | | | | 2 n= Z− X−∞

Rayleigh theorem (Total Energy in an aperiodic signal)

∞ ∞ E = x(t) 2 dt = X(f) 2 df | | | | Z−∞ Z−∞ 31

Fourier theorems

If x(t) X(f)is a transform pair, then ↔ LINEARITY ax(t)+ by(t) aX(f)+ bY (f) ↔ j2πt f SHIFT/MODULATION x(t t ) X(f)e− 0 − 0 ↔ x(t)ej2πf0 t X(f f ) ↔ − 0 PRODUCT/CONVOLUTION x(t)y(t) X Y ↔ ∗ x y X(f)Y (f) ∗ ↔ 1 f SCALE x(at) X ↔ a a | | d DIFFERENTIATION x(t) j2πfX(f) dt ↔ t X(f) INTEGRATION x(τ) dτ . ↔ j2πf Z−∞ Fourier pairs

Time Domain Frequency Domain

sin(πT0f) h(t)=1 u T0 ( t ) H(f)= − 2 | | ↔ πf h(t) H(f) T 1 0

-T0/2 T0/2 t 1/T0 2/T0 f

sin(2πf0t) 1 h(t)= H(f)= (1 uf0 ( f )) 2πf0t ↔ 2f0 − | | h(t) H(f) 1 1/2f0

-f0 f0 f 1/2f0 1/f0 t 32

Fourier pairs Continued Time Domain Frequency Domain h(t)= K H(f)= Kδ(f) h(t) ↔ H(f) K K

t f

h(t)= Kδ(t) H(f)= K h(t) ↔ H(f) K K

t f

A h(t)= A cos(2πf t) H(f)= [δ(f + f )+ δ(f f )] 0 ↔ 2 0 − 0 h(t) H(f) A A/2

-f0 f0 f 1/f0 t

A h(t)= A sin(2πf t) H(f)= j[δ(f + f ) δ(f f )] 0 ↔ 2 0 − − 0 h(t) Im[H(f)] A A/2 1/f0 f0 t -f0 f

t sin2(πT f) h(t)= 1 | | 1 u ( t ) H(f)= 0 − T − T0 | | ↔ T π2f 2 0 0 h(t) H(f) T 1 0

-T0 T0 t 1/T0 2/T0 f 33

Fourier pairs Continued Time Domain Frequency Domain

α t 2α h(t)= e− | | H(f)= ↔ α2 +4π2f 2 h(t) H(f) 1 2/α

t f

π π2f 2 h(t) = exp( αt2) H(f)= exp − − ↔ α α r h(t) H(f) 1 √π/α

t f

e αt, t> 0 1 h(t)= − H(f)= 0, t 0 ↔ α +2πjf ≤ h(t) |H(f)| 1 1/α

t f j h(t) = sgn(t) H(f)= − ↔ πf h(t) Im[H(f)] 1

t f -1 34

Linear time invariant systems − Basic Properties If x(t) y(t)

g(t) q(t)

then ax(t t )+ bg(t t ) ay(t t )+ bq(t t ) − 0 − 0 − 0 − 0

Transform properties

Input Output

Time Domain δ(t) h(t)

x(t) y(t)= x(t) h(t) ∗ system

Frequency Domain X(f) Y (f)= X(f) H(f) · h(t) H(f) h(t) impulse response; H(f) transfer function ↔ ≡ ≡ Random signals

∞ x = xp(x) dx Mean value Z−∞

∞ x2 = x2p(x) dx Second moment Z−∞

T 1 2 = x 2 = x(t) 2 dt Norm square k k T T | | Z− 2

T 1 2 = x(t) dt Time average T T Z− 2 For an ergodic process: Second moment = Norm square and Mean value = Time average. 35

Autocorrelation

T 1 2 Rxx(τ) = lim x(t)x(t + τ) dt T T T →∞ Z− 2 Power Spectral Density

The power spectral density Gxx is the Fourier transform of the autocorrelation function:

∞ j2πtf ∞ j2πft R (t)= G (f)e df G (f)= R (t)e− dt xx xx ↔ xx xx Z−∞ Z−∞

Noise power (Gxx denotes the power spectral density)

2 2 P Total = Gxx df = Rxx(0) = x = (x) Z

2 P ac = N = σx = noise power

P = R ( ) = (x)2 dc xx ∞

Noise bandwidth (Gxx denotes the power spectral density)

G = G H 2 (linear system) yy xx · | |

∞ N = G H 2 df = noise power output y xx · | | Z−∞

n N = n B = noise power, where 0 = white noise 0 N 2

∞ B = H 2 df = noise bandwidth N | | Z−∞ 36

Ideal ﬁlter (low pass) − sin(2πf t) 1 u ( f ) h(t)= 0 H(f)= − f0 | | 2πf0 ↔ 2f0 h(t) H(f) 1 1/2f0

-f0 f0 f 1/2f0 1/f0 t

RC ﬁlter (low pass) − 1 t/RC 1 V = h(t) E(t) h(t)= e− H(f)= C ∗ RC ↔ 1+2πjfRC V h(t) |H(f)| C 1 1 C R

E(t) t f

Ideal ﬁlter (high pass) − sin(2πf t) h(t)= δ(t) 0 H(f)= u ( f ) − πt ↔ f0 | | h(t) H(f) 1 1/2f0 1/f0

t -f0 f0 f

RC ﬁlter (high pass) − 1 t/RC 2πjfRC V = h(t) E(t) h(t)= δ(t) e− H(f)= R ∗ −RC ↔ 1+2πjfRC h(t) |H(f)| 1 C VR R

E(t) t f 37

Ideal ﬁlter (band pass) − sin(2πf t) sin(2πf t) h(t)= 2 1 H(f)= u ( f ) u ( f ) πt − πt ↔ f2 | | − f1 | | h(t) H(f) 1

t -f2 -f1 f1 f2 f

LRC ﬁlter (band pass) − 2πjfRC V = h(t) E(t) h(t) H(f)= R ∗ ↔ 1 (2πf)2LC +2πjfRC − h(t) |H(f)| 1 C VR RL t E(t) −1/(2π√LC ) 1/(2π√LC ) f

Ideal sampling

Sampled signal

∞ x (t)= x(t) s (t) where s = δ(t kT ) δ · δ δ − s k= X−∞

Spectrum after sampling

∞ X (f)= X(f) S (f) where S (f)= f δ(f nf ) δ ∗ δ δ s − s n= X−∞ Sampling theorem If 1 T < or f > 2W (“Nyquist rate”) s 2W s then the signal can be reconstructed with a ﬁlter of bandwidth B, given by: W