Mathematical Preliminaries and Review

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1 Mathematical Preliminaries and Review Ruye Wang for E59 1 Complex Variables and Sinusoids AcomplexnumbercanberepresentedineitherCartesianorpolarcoordinatesystem: z = x + jy= z z = re jθ = r(cos θ + j sin θ) where Re[z] =x = r cos θ Im[z] =y = r sin θ, he complex conjugate of z = x + jy is defined as: r = z = x2 + y 2 θ = z =tan 1 y/x z =(x + jy) = x jy =(re jθ ) = re jθ Arithmetic operations of complex variables (vectors in complex space): z + z =(x + jy)+(x jy)=2x =2Re [z] z z =(x + jy) (x jy)=2jy =2jIm[z] zz =(x + jy)(x jy)=x 2 + y 2 = z 2 z 1 ± z 2 =(x 1 + jy 1 ) ± (x 2 + jy 2 )=(x 1 ± x 2 )+j(y 1 ± y 2 ) z 1 z 2 = r 1 e jθ 1 r 2 e jθ 2 = r 1 r 2 e j(θ 1+θ 2 ) z 1 z 2 = r 1 e jθ1 r 2 e jθ 2 = r 1 r 2 e j(θ 1 θ 2 ) z 1/n =(re j(θ+2kπ) ) 1/n = r 1/n e j(θ/n+2kπ/n), (k =0, 1,,n 1) Some important values on the complex plane: e j2kπ =1, e j(2kπ+π/2) = j, e j(2kπ±π) = 1, e j(2kπ π/2) = j, (k =0, ±1, ±2, ) Euler s Formula: e jθ =cosθ + j sin θ e jθ =cosθ j sin θ, cos θ =(e jθ + e jθ )/2 sin θ =(e jθ e jθ )/2j 1

2 Asinusoidaltimefunctioncanberepresentedineitherofthetwoforms: he real and imaginary parts (horizontal and vertical projections)of a complex exponential(a vector rotating in CCW direction) represent a cosine and sine function, respectively: Re[Ae j(ωt+ϕ) ]=Re[A cos(ωt + ϕ)+jasin(ωt + ϕ)] = A cos(ωt + ϕ) Im[Ae j(ωt+ϕ) ]=Im[A cos(ωt + ϕ)+jasin(ωt + ϕ)] = A sin(ωt + ϕ) he rate of the rotation is determined by the angular frequency ω he vector sum of two complex exponentials for two vectors rotating in CW and CCW directions, corresponding to a positive frequency ω and a negative one ω, respectively: A cos(ωt + ϕ) = A 2 [ej(ωt+ϕ) + e j(ωt+ϕ) ] Similarly, a sine function can also be obtained as a vector sum: jacos(ωt + ϕ) = A 2 [ej(ωt+ϕ) e j(ωt+ϕ) ], ie A cos(ωt + ϕ) = A 2j [ej(ωt+ϕ) e j(ωt+ϕ) ] 2 Linear Algebra Vectors and matrices: An n-d vector is a column vector and its transpose is a row vector: x = x 1 x n, x =[x 1,,x n ] An m by n matrix is an array of m rows and n columns, which can also be represented in terms of its column (or row) vectors: a 11 a 1n A m n = =[a 1,, a n ] a m1 a mn 2

3 where a j is the jth column (j =1,,n): a j = a 1j a mj he transpose of A is: a 11 a 1n A = a m1 a mn a 11 a m1 = a 1n a nm =[a 1,, a n ] = a 1 a n he inverse A 1 of A satisfies: where I is the identity or unit matrix Inner (dot) product: A 1 A = AA 1 = I he inner (dot) product of two vectors x and y is a scaler defined as: < x, y >= x y =[x 1,,x n ] he following concepts are defined based on inner product: he norm (or length) of a vector is defined as Avectorx is normalized if x =1 y 1 y n n = x j y j x =< x, x > 1/2 n = x j x n j = x j 2 wo vectors x and y are orthogonal if < x, y >= 0 wo orthogonal vectors are orthonormal if they are both normalized he angle between two vectors is defined as: θ =cos 1 ( < x, y > ), ie, < x, y >= x y cos θ x y he scaler projection of x on y is defined as: P y (x) = < x, y > y = x cos θ, if y =1,then P y (x) =< x, y > (he projection of x on y could also be defined as a vector in the direction of y, whichisnot considered here) 3

4 Vector and matrix multiplication: An n-d column vector x n can be pre-multiplied by an matrix A m n to result an m-d column vector y m : y m = A m n x n which can be represented in element form as: y 1 y m and the ith element is (i =1,,m): = a 11 a 1n x 1 a m1 a mn x n n y i = a ij x j Given y and A, this linear system can be solved for x (by pre-multiplying both sides of the equation by A 1 ): A 1 y = A 1 Ax = x, ie x = A 1 y he product of two matrices A m k and B k n is also a matrix: C m n =[c 1,, c n ]=A m k B k n = A[b 1,, b n ] where the jth column is: c j = Ab j, ie, c 1j = a 11 a 1k b 1j c mj a m1 a mk b kj and its ith element is: k c ij = a il b lj, l=1 (i =1,,m) 4

5 Orthogonal and unitary matrices Some special matrices of interest are defined below: Ais symmetric if A = A Ais Hermitian if A = A, wherea is the conjugate transpose of A IfA = A is real, it is symmetric Ais orthogonal if A = A 1,ie,A A = AA = I Ais unitary if A = A 1 IfA = A is real, it is orthogonal All column (or row) vectors of a unitary matrix A =[a 1,, a n ]areorthonormal: 1 i = j < a i, a j >= a i a j = δ[i j] = 0 i j where δ[n] is the delta function defined as: δ[n] = 3 Inner Product (Hilbert) Space Euclidean space: 1 n =0 0 n 0 An n-d inner product vector space, called a Euclidean space, which is a set of all n-d vectors with inner product defined his space can be spanned by a set of n linearly independent basis vectors b 1,, b n } (none of them can be represented as a linear combination of the rest), so that any vector x in the space can be expressed as a linear combination of these basis vectors: c 1 b 11 b 1n c 1 n x = k=1 c k b k = c 1 b c n b n =[b 1,, b n ] c n = b n1 b nn c n = Bc where B = [b 1,, b n ]isannbynmatrixwiththenbasisvectorsasitscolumns,andc = [c 1,,c n ] is a column vector composed of n coefficients or weights for the basis vector hese coefficients can be obtained by solving this linear system (by pre-multiplying B 1 on both sides): c = B 1 x In particular, if the basis vectors are orthonormal: < b i, b j >= b i b j = δ[i j] then B becomes: and the kth element = B 1 is a unitary matrix (or orthogonal if B = B is real), and the equation above c = c 1 c n = B 1 x = B x = b 1 b n c k =< x, b k >= x b k = P bk (x) is the projection of vector x onto the kth basis vector b k 5 x

6 Any set of n linearly independent vectors can be used as a basis of an n-d vector space Any given basis of n vectors can be converted in to a set of orthogonal basis vectors by Gram- Schmidt process Any rotation of an orthogonal basis results another orthogonal basis, ie, any two different sets of orthogonal bases are related by a rotation corresponding to an orthogonal matrix All discussions above for n-d Euclidean space can be generalized to an infinite dimensional vector space by letting n Function space: he concept of n-d vector space can be generalized to n-d function space, which a set of all functions x(t) definedoveraparticularrange0 <t< with the inner product defined as: <x(t),y(t) >= x(t)y (t)dt, (compared to < x, y >= i x i yi ) he norm of a function is defined as: x(t) =< x(t),x(t) > 1/2 = x(t)x (t)dt = Afunctionx(t) isnormalizedifithasunitynorm x(t) =1 wo functions x(t) andy(t) areorthogonaliftheinnerproductiszero: <x(t),y(t) >= x(t)y (t)dt =0 x(t) 2 dt his function space can also be spanned by a set of basis functions b k (t) sothatanygivenfunction x(t) inthespacecanbeexpressedasalinearcombinationofthesebasisfunctions: x(t) = k c k b k (t) In particular, if the basis functions are orthogonal: <b k (t),b l (t) >= b k (t)b l (t)dt =0 (k l) then the coefficient c k can be obtained by taking an inner product with b l (t) onbothsidesofthe equation above: <x(t),b l (t) >=< k c k b k (t), b l (t) >= k c k <b k (t), b l (t) >= c l <b l (t),b l (t) >= c l b l (t) 2 he kth coefficient becomes: c k = <x(t),b k(t) > <b k (t),b k (t) > = x(t)b k (t)dt b k (t) 2 Moreover, if the basis functions are orthonormal (orthogonal and normalized) b k (t) =1,thenwe get c k =<x(t),b k (t) >= x(t)b k (t)dt ie, c k is the projection of x(t) ontothekthbasisfunctionb k (t) 6

7 4 Miscellaneous Integration of these expressions: sin 2 (at)dt, te t dt, t sin(at)dt Solution of 1st and 2nd-order ordinary differential equations: ẏ(t)+ 1 y(t) =af(t), τ ÿ(t)+2ζω nẏ(t)+ωn 2 y(t) =af(t) where τ, ζ, ω n and a are real constants, and the input f(t) canbe: f(t) =0[freeresponse]; f(t) =u(t) [unitstepresponse]; f(t) = sinωt [sinusoidal response]; f(t) =e jωt [harmonic (complex exponential) response]; Geometric series: Finite: Infinite ( x < 1): rigonometry identities: n 1 k=0 x k =1+x + x x n 1 = 1 xn 1 x k=0 x k =1+x + x 2 + = 1 1 x sin(α ± β) =sinα cos β ± cos α sin β cos(α ± β) =cosα cos β sin α sin β sin 2 α =(1 cos 2α)/2 cos 2 α =(1+cos2α)/2 sin 2α =2sinα cos α cos 2α =cos 2 α sin 2 α sin(α/2) = ± (1 cos α)/2 cos(α/2) = ± (1 + cos α)/2 7