Fund. of Digital Communications Chapter 2: Signals and Systems

Transcription

1 Fund. of Digital Communications Chapter 2: Signals and Systems Klaus Witrisal Signal Processing and Speech Communication Laboratory Graz University of Technology October 6, 2016 Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 1/54

2 2-1 Signal Spaces Outline 2-2 Linear Operators, Linear Systems, and a Little Linear Algebra 2-3 Frequency Domain Representation of Signals 2-4 Matrix diagonalizations 2-5 Bandpass signals Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 2/54

3 2-1 Signal Spaces References: (Figures taken from these books.) Barry, Lee, Messerschmitt: Digital Communications, 3rd Ed., Kluwer Academic Publishers, 2004 J. G. Proakis and M. Salehi, Communication System Engineering, 2nd Ed., Prentice Hall, 2002 M. Vetterli, et al., Foundations of Signal Processing, Cambridge, 2014 Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 3/54

4 2-1 Signal Spaces Idea: represent signals as vectors (in linear vector spaces) allows for geometric interpretations linear (vector-) algebra can be used for signal processing algoithms applies for continuous-time and discrete-time signals Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 4/54

5 2-1 Signal Spaces (cont d) Def: linear vector space set of vectors X, and scalars (in R or C) for which the following operations are defined: vector addition and scalar multiplication and the following properties must hold: additive identity (zero vector), additive inverse, multiplicative identity associative, commutative, and distributive laws results are vectors in vector space linearity follows in this case: x,y X; a,b R(or C) ax+by X Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 5/54

6 2-1 Signal Spaces (cont d) Elementary operations (in a 2D linear space) a) sum of two vectors b) multiplication of a vector by a scalar [Barry 2004] Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 6/54

7 2-1 Signal Spaces (cont d) Properties of the elementary operations in a vector space: Given vectors x,y,z X and scalars a,b R(or C): a) Commutativity: x+y = y +x b) Associativity: (x+y)+z = x+(y +z) and abx = a(bx) c) Distributivity: a(x+y) = ax+ay and (a+b)x = ax+bx d) Additive identity: There exists an element 0 X s.t. x+0 = 0+x = x for every x X e) Additive inverse: There exists a unique element x X s.t. x+( x) = ( x)+x = 0 for every x X f) Multiplicative identity: For every x X, 1x = x Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 7/54

8 2-1 Signal Spaces (cont d) Subspaces in 3D Euclidean space X = R 3 a) line (1D) ax X 1 X 1 X = R 3 b) plane (2D) ax+by X 2 X 2 X = R 3 [Barry 2004] Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 8/54

9 2-1 Signal Spaces (cont d) In digital communications: mapping of information onto sets of M waveforms signal space of M waveforms S = span{s 1 (t),s 2 (t),...,s M (t)} set of all signals, that can be represented as linear combinations of these M waveforms a subspace of continuous-time signals elements of this subspace; s(t) S s(t) =α 1 s 1 (t)+α 2 s 2 (t)+ +α M s M (t) = M i=1 α i s i (t) S Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 9/54

10 2-1 Signal Spaces (cont d) E.g.: Signal space spanned by two signals s 1 (t), s 2 (t) a) (basis) signals s 1 (t), s 2 (t) b) examples of signals in S = span{s 1 (t),s 2 (t)} Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 10/54 [Barry 2004]

11 2-1 Signal Spaces (cont d) inner product (scalar product) inner product space needed f. geom. interpretations of: distance, angle, length (of/between vectors) E.g.: inner product on X = C N x,y = N i=1 x i y i = y H x x = [x 1,x 2,...,x N ] T y = [y 1,y 2,...,y N ] T y H = [y 1,y 2,...,y N ] Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 11/54

12 2-1 Signal Spaces (cont d) Properties of the inner product: Def.: The inner product maps two vectors to a scalar, i.e. a = x,y, where x,y X, a C(or R) The following holds: Distributivity: x+y,z = x,z + y,z Linearity in the first argument: ax,y = a x,y (Conjugate linearity in the second: x,ay = a x,y ) Hermitean symmetry: x,y = y,x Positive definiteness: x, x > 0 for x 0 Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 12/54

13 2-1 Signal Spaces (cont d) Def: (induced) norm (squared length, energy ) x 2 = x,x = N i=1 x i 2 = x H x Def: angle (for real-valued vectors) 1 x,y x y = cos(θ) 1 distance: x y Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 13/54

14 2-1 Signal Spaces (cont d) Standard inner product spaces: DT signals (square-summable sequ.; X = l 2 (Z)) x[n],y[n] = x[n]y [n] x[n] 2 = n= n= x[n] 2 <, y[n] 2 < CT signals (square-integrable functions; X = L 2 (R)) x(t),y(t) = x(t) 2 = x(t)y (t)dt x(t) 2 dt <, y(t) 2 < Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 14/54

15 2-1 Signal Spaces (cont d) Projection onto a subspace Def: subspace S X for x X: projection of x onto S is unique element ˆx S, that is closest to x x ˆx = min y S x y ˆx = argmin y S x y Projection Theorem: projection error x ˆx must be orthogonal to the subspace S Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 15/54

16 2-1 Signal Spaces (cont d) Projection of x onto subspace S Projection of signal r(t) onto s 1 (t), s 2 (t) Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 16/54

17 2-1 Signal Spaces (cont d) In digital communications: (review) mapping of information onto sets of M waveforms signal space of M waveforms S = span{s 1 (t),s 2 (t),...,s M (t)} set of all signals s(t) S, that can be represented as linear combinations of these M waveforms, i.e. s(t) = M i=1 α i s i (t) Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 17/54

18 2-1 Signal Spaces (cont d) Orthonormal basis (of a signal space) minimum set of N orthonormal functions (N M) that can be used to represent the elements s(t) S (as linear combinations): N s(t) = s i ψ i (t) i=1 orthonormal (basis) functions {ψ i (t) i = 1,...,N}: { ψ i (t)ψk (t)dt = δ[i k] = 1 i = k 0 i k ψ i(t) = 1 Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 18/54

19 2-1 Signal Spaces (cont d) Different bases for the signals s 1 (t), s 2 (t) Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 19/54

20 2-1 Signal Spaces (cont d) Geometric representation of six waveforms in three different bases geometric relations remain the same! Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 20/54

21 2-1 Signal Spaces (cont d) arbitrary signals projection onto S, given an orthonormal basis for S r(t) = ˆr(t)+e(t) ˆr(t) = N i=1 r i ψ i (t) e(t) = r(t) ˆr(t) r j = r(t),ψ j (t) = r(t)ψ j (t)dt... projection error... coefficients Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 21/54

22 2-1 Signal Spaces (cont d) Gram-Schmidt orthogonalization find the N M orthonormal basis functions 1. ψ 1 (t) = s 1(t) s 1 (t), S 1 = span{ψ 1 (t)} 2. ψ 2 (t) = s 2(t) ŝ 2 (t) s 2 (t) ŝ 2 (t), k. ψ k (t) = s k(t) ŝ k (t) s k (t) ŝ k (t), projection of s 2 (t) onto S 1 ŝ 2(t) = c 21 ψ 1 (t) ŝ k(t) = k 1 i=1 c kiψ i (t) no basis function, if s k (t) ŝ k (t) = 0 Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 22/54

23 2-1 Signal Spaces (cont d) Representation of M waveforms {s k (t)} through an orthonormal basis vectors wheres k,i = s k = [s k,1,s k,2,...,s k,n ] T s k (t)ψ i(t)dt = s k (t),ψ i (t) operations on signals are equivalent to operations on vectors (preservation of inner products; unitarity) s j (t),s k (t) = s j,s k = s H k s j s k (t) 2 = s k 2 = s H k s k s j (t) s k (t) 2 = s j s k 2 Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 23/54

24 2-2 Linear Operators, Linear Systems, and a Little Linear Algebra Hilbert spaces: inner product spaces with properties such as completeness: space contains all convergence points of sequences separability: space contains a countable basis standard spaces C N,l 2 (Z),L 2 (R) are all separable Hilbert spaces Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 24/54

25 2-2 Linear Operators,... Linear operator: generalizes finite-dimensional matrices used to perform operations with vector in-/outputs Def. of a linear operator (from H 0 to H 1 ): a function A : H 0 H 1 for all vectors x,y H 0 and α C (or R), the following properties hold: (i) additivity: A(x+y) = Ax+Ay (ii) scalability: A(αx) = α(ax) H 0... domain; H 1... codomain Def.: linear operator on H 0 : a function A : H 0 H 0 Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 25/54

26 2-2 Linear Operators,... a linear operator from C N to C M is the same as an M N matrix thus, concepts from linear algebra can be borrowed: e.g. the range of a lin. operator A : H 0 H 1 is a subspace of H 1 : R(A) = {Ax H 1 x H 0 } e.g. the null space of a lin. operator A : H 0 H 1 is a subspace of H 0 that A maps to 0: N(A) = {x H 0 Ax = 0} Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 26/54

27 2-2 Linear Operators,... Def.: inverse an operator A : H 0 H 1 is invertible if a linear operator B : H 1 H 0 exists s.t.: (a) BAx = x for every x H 0 and (b) ABy = y for every y H 1 then B = A 1 is the inverse of A if only (a) holds, then B is the left inverse of A if only (b) holds, then B is the right inverse of A Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 27/54

28 2-2 Linear Operators,... Unitary operators preserve the geometry of vectors (lengths and angles): Def.: A linear operator A : H 0 H 1 that (i) is invertible (ii) preserves inner products Ax,Ay H1 = x,y H0 for every for every x,y H 0 Prop. (ii) implies preservation of norms: Ax 2 = x 2 (Parseval theorem) Props. (i) and (ii) imply A 1 = A H (A H : H 1 H 0 is the adjoint of A : H 0 H 1 ) Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 28/54

29 2-2 Linear Operators,... Orthogonal projection via pseudoinverse: projection onto R(A); least-squares approximation given A : H 0 H 1 ; R(A) is a subspace of H 1 for any vector y H 1, how to find best approxim. ŷ R(A) of y? find unique solution for ŷ = Ax that minimizes y ŷ 2 projection theorem yields y Ax,a i = 0 i; a i...i-th column ofa A H (y Ax) = 0 A H Ax = A H y... normal equations Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 29/54

30 2-2 Linear Operators,... Orthogonal projection via pseudoinverse (cont d): normal equations yield x = (A H A) 1 A H y = By B... pseudoinverse of A BA = I... i.e. B is a left inverse of A furthermore: ŷ = Ax = A(A H A) 1 A H y = Py P = AB... orthogonal projection onto R(A) Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 30/54

31 2-2 Linear Operators, Linear Systems,... Eigenvectors and eigenvalues Def.: an eigenvector of an operator A : H H is a non-zero vector v H s.t. for some λ C λ... eigenvalue (λ,v)... eigenpair Av = λv for H l 2 (Z), v is called eigensequence for H L 2 (R), v is called eigensignal Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 31/54

32 2-2 Linear Operators, Linear Systems,... continuous-time (CT), linear systems x T y T : L 2 (R) L 2 (R) is a linear operator on L 2 (R) i.e. y = Tx; x, y are CT functions x(t), y(t) in particular, we are interested in linear time-invariant (LTI) systems Def. (LTI System): y = Hx y = Hx, wherex (t) = x(t τ) andy (t) = y(t τ) Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 32/54

33 2-2 Linear Operators, Linear Systems,... Eigenfunctions of LTI systems (complex exponentials) v(t) = e j2πft f... frequency in Hz (cycles per second) system response becomes (Hv)(t) = λ f v(t) = H(f)e j2πft λ f... eigenvalue for v(t) at frequency f H(f) = λ f... frequency response of the LTI system this motivates frequency domain signal representations Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 33/54

34 2-3 Frequency Domain Representation of Signals Why? develop intuition for many processing steps in digital communications: sinusoidal signals are eigensignals of linear systems understanding the system response of LTI systems frequency occupation of communications signals separation of RF signals in frequency domain Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 34/54

35 2-3 Frequency Domain Representation of Signals Fourier series (complex-valued form) x(t) is periodic with period T, i.e. x(t) = x(t+t) functions with one period in (e.g.) L 2 ([ 1 2 T, 1 2 T)) x(t) = k= c k e jk(2π/t)t, ω 0 = 2π T,f 0 = 1 T fundamental freq. x(t) is represented in an orthogonal basis of complex exponentials e jk(2π/t)t computation of the Fourier coefficients c k = x(t),e jk(2π/t)t = 1 T 1 2 T 1 2 T x(t)e jk(2π/t)t dt Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 35/54

36 2-3 Frequency Domain Representation of Signals Fourier series properties given: Fourier series pair of a periodic signal x(t) = x(t+t) F {c k }; c k C, k Z;(i.e.{c k } l 2 (Z)) (the Fourier series reconstruction converges to x(t)) Hermitean symmetry: for x(t) R: c k = c k Parseval theorem (signal power) P = 1 T T 0 x(t) 2 dt = k= c k 2 (normalized) power of a spectral component is c k 2 Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 36/54

37 2-3 Frequency Domain Representation of Signals (LTI) system response in frequency domain x(t) H(f) y(t) x(t) = c k e jk(2πf 0)t y(t) = c k H(kf 0 )e jk(2πf 0)t k= k= H(f)... frequency response of the LTI system x(t) = x(t+t) y(t) = y(t+t) F {c k } F {d k = c k H(kf 0 )} Exploits linearity: αx(t)+βy(t) F {αc k +βd k } Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 37/54

38 2-3 Frequency Domain Representation of Signals E.g. periodic pulse train represented as Fourier series passed through first-order lowpass filter Zeitsignal time t Koeffizienten der Fourier Reihe index k (frequency k/t 0 ) coefficients c k, d k signals x(t), y(t) Fourier series coefficients index k (frequency k f ) truncated reconstruction of the time-domain signals c k x(t) d k y(t) x(t) y(t) time t Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 38/54

39 2-3 Frequency Domain Representation of Signals nonperiodic signals: Fourier transform X(f) = x(t)e j2πft dt inverse Fourier transform x(t) = X(f)e j2πft df Fourier transform pair: (if both exist t, f R) x(t) F X(f); x(t) L 2 (R), X(f) L 2 (R) Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 39/54

40 2-3 Frequency Domain Representation of Signals Fourier transform properties given: Fourier transform pair x(t) F X(f) Hermitean symmetry: for x(t) R: X(f) = X ( f) Parseval theorem (signal energy) E = x(t) 2 dt = X(f) 2 df X(f) 2... energy density at frequency f Note: E < for x(t) L 2 (R) Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 40/54

41 2-3 Frequency Domain Representation of Signals Properties of the Fourier transform (cont d) linearity ax(t)+by(t) ax(f)+by(f) ( ) time scaling x(at) 1 a X f a convolution x(t) y(t) X(f)Y(f) x(t)y(t) X(f) Y(f) time shift x(t τ) X(f)e j2πfτ frequency shift e j2πf0t x(t) X(f f 0 ) real part Re{x(t)} = 2( 1 x(t)+x (t) ) 1 ( 2 X(f)+X ( f) ) d m x(t) dt m (j2πf) m X(f) ( ) jt mx(t) d 2π m X(f) df m t x(τ)dτ 1 j2πf X(f)+ 1 2 δ(f) x(τ)dτ x(t)cos(2πf 0 t) 1 2 (X(f f 0)+X(f +f 0 )) X(t) x( f) Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 41/54

42 2-3 Frequency Domain Representation of Signals Fourier transform pairs e j2πf 0t δ(f f 0 ) δ(t T) e j2πft cos(2πf 0 t) 1 2 [δ(f f 0)+δ(f +f 0 )] sin(2πf 0 t) 1 2j [δ(f f 0) δ(f +f 0 )] sinc(ft) 1 F rect(f,f/2) rect(t,t/2) Tsinc(fT) e αt 1 u(t) j2πf+α ; Re{α} > 0 u(t) δ(f) j2πf ( δ(t)+ j ) πt u(f) k= δ(t kt) 1 T 1 πsgn(f) jt + m= δ( f 1 T m) Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 42/54

43 2-3 Frequency Domain Representation of Signals (LTI) system response in frequency domain x(t) H(f) y(t) x(t) F X(f) y(t) F Y(f) = X(f)H(f) H(f)... frequency response of the LTI system System response in time domain: make use of convolution property: y(t) = x(t) h(t) F Y(f) = X(f)H(f) h(t)... impulse response of the LTI system (see below) Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 43/54

44 2-3 Frequency Domain Representation of Signals convolution integral (Faltungsintegral) x(t) h(t) = = x(λ)h(t λ)dλ h(λ)x(t λ)dλ convolution property of Fourier transform y(t) = x(t) h(t) F Y(f) = X(f)H(f) Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 44/54

45 2-3 Frequency Domain Representation of Signals Dirac pulse, δ-pulse, Dirac distribution defined by integral (sampling, sifting property Ausblendeeigenschaft): x(t)δ(t)dt = x(0) δ(t)dt = 1 Fourier transform: δ(t) F 1 δ(t T) F e j2πft Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 45/54

46 2-3 Frequency Domain Representation of Signals impulse response (of an LTI system): x(t) = δ(t) y(t) = δ(t) h(t) = time shifting property: y(t) = δ(t T) x(t) = δ(λ)h(t λ)dλ = h(t) δ(λ T)x(t λ)dλ = x(t T) Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 46/54

47 2-4 Matrix factorizations LTI operator H on C N (circular convolution) H = F 1 HF F... DFT matrix F 1... inverse DFT F 1 = N 1 FH H... frequency response (diagonal matrix) arbitrary lin. operator A on C N (full rank!) A = VΛV 1... spectral theorem; EV decomposition V... Matrix of (N independent) eigenvectors Λ... eigenvalues (diagonal matrix) Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 47/54

48 2-4 Matrix factorizations (cont d) arbitrary lin. operator A : C N C M singular value decomposition (SVD) A = UΣV H V... (Unitary) matrix of N right singular vectors Λ... Singular values (diagonal matrix; sorted) U... (Unitary) matrix of M left singular vectors for A having rank r, there are r non-zero singular values Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 48/54

49 2-5 Bandpass signals Definition: A bandpass (or narrowband) signal x(t) has an f-domain representation X(f) that is nonzero in a (usually small) neighborhood of some (usually high) frequency f 0, i.e.: X(f) = 0 for f f 0 W, where W f o f 0... center frequency W... bandwidth Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 49/54

50 2-5 Bandpass signals Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 50/54

51 2-5 Bandpass signals (cont d) Monochromatic signal has bandwidth W = 0 x(t) = Acos(2πf 0 t+θ) represented by phasor X = Ae jθ Bandpass signal time-varying phasor V(t) θ(t) x l (t) = V(t)e jθ(t) envelope phase x l (t) lowpass representation of x(t) Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 51/54

52 2-5 Bandpass signals (cont d) Relation between x(t) and x l (t): x(t)... bandpass signal x(t) = Re {x } l (t)e j2πf 0t z(t)... analytic signal corresponding to x(t); preenvelope of x(t) z(t) = x l (t)e j2πf 0t x l (t)... lowpass representation of x(t) x l (t) = x c (t)+jx s (t) Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 52/54

53 2-5 Bandpass signals (cont d) Generation and demodulation of bandpass signals (in practice) x(t) = x c (t)cos(2πf 0 t) x s (t)sin(2πf 0 t) Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 53/54

54 2-5 Bandpass signals (cont d) Transmission of bandpass signals through LTI systems given: x(t), h(t), y(t)... input (bandpass), linear system, output we find: Y l (f) = 1 2 X l(f)h l (f) y l (t) = 1 2 x l(t) h l (t) system response can be computed using lowpass equivalent representations of x(t), h(t), y(t) Fund. of Digital CommunicationsChapter 2: Signals and Systems p. 54/54