Outline Digital Communications. Lecture 02 Signal-Space Representation. Energy Signals. MMSE Approximation. Pierluigi SALVO ROSSI
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1 Outline igital Communications Lecture 2 Signal-Space Representation Pierluigi SALVO ROSSI epartment of Industrial and Information Engineering Second University of Naples Via Roma 29, 8131 Aversa (CE, Italy homepage: pierluigisalvorossi@unina2it 1 Energy Signals 2 Signal Constellation 3 Examples 4 Noise Representation P Salvo Rossi (SUNIII igital Communications - Lecture 2 1 / 21 P Salvo Rossi (SUNIII igital Communications - Lecture 2 2 / 21 Energy Signals L 2 ( is the set of the energy signals over R, ie s(t L 2 ( E s s(t 2 dt < + L 2 ( is a Hilbert space with (scalar inner product < s 1, s 2 > s 1 (ts 2(tdt s 1 (t, s 2 (t L 2 ( {ψ n (t} N is said an orthonormal set of signals in L2 ( if < ψ n, ψ m >= ψ n (tψm(tdt = δ n,m MMSE Approximation Given an orthonormal set of signals {ψ n (t} N in L2 (, which one is the best linear combination to represent a generic signal s(t L 2 (? efine the generic approximation ŝ(t c n ψ n (t the usual criterion to find the best vector of coefficients c = (c 1,, c n is the Minimum Mean Square Error (MMSE criterion efine the error signal and the Mean Square Error (MSE, ie its energy e(t s(t ŝ(t e(t 2 dt ɛ 2 rms P Salvo Rossi (SUNIII igital Communications - Lecture 2 3 / 21 P Salvo Rossi (SUNIII igital Communications - Lecture 2 4 / 21
2 MSE he MSE can be expressed as ɛ 2 rms = s(t 2 dt + = E s + = E s c n 2 2R { } ŝ(t 2 dt 2R s(tŝ (tdt { N } s(tψn(tdt s(tψn(tdt 2 c n + c n s(tψ 2 n(tdt }{{} he last term is the only depending on c thus the MMSE is achieved when such a term is null Constellation Point enote s = (s 1,, s N the coefficient vector achieving the MMSE s = arg min c C N ɛ2 rms Such vector is also called the constellation point in the signal space and its component are computed as s n =< s, ψ n >= s(tψn(tdt he corresponding MMSE is ɛ 2 rms,opt = min c C N ɛ2 rms = E s s n 2 P Salvo Rossi (SUNIII igital Communications - Lecture 2 5 / 21 P Salvo Rossi (SUNIII igital Communications - Lecture 2 6 / 21 Orthogonality Principle Gram-Schmidt Process It is a graphical interpretation of the MMSE result < e, ψ n > = < s ŝ, ψ n > = < s, ψ n > < ŝ, ψ n > = s n s n = Error (e(t and data ({ψ n (t} N are orthogonal It takes a set of signals {s m (t} M m=1 spanning a subset of L2 ( and provides an orthogonal set {ψ n (t} N, with N M, spanning the same (N-dimensional subset Iterate the following: n 1 ψ n (t = s n (t ψ n (t = l=1 ψ n (t < ψ n, ψ n > < s n, ψ l > ψ l (t P Salvo Rossi (SUNIII igital Communications - Lecture 2 7 / 21 P Salvo Rossi (SUNIII igital Communications - Lecture 2 8 / 21
3 Signal Constellation Example 1 (1/6 A set of signals {s m (t} M m=1 can be described through an orthogonal set {ψ n (t} N, with N M, via a set of M (N-dimensional vectors {s 1,, s M } denoted signal constellation he mth vector (or constellation point associated to s m (t is s m,1 < s m, ψ 1 > s m = s m,n = < s m, ψ n > < s m, ψ N > s m,n Consider a binary modulation (M = 2 with signals ( ( t /2 t 3/4 s 1 (t = A 1 rect s 2 (t = A 2 rect /2 he signals have the following energies: E 1 = A 2 1 and E 2 = A 2 2 /2 P Salvo Rossi (SUNIII igital Communications - Lecture 2 9 / 21 P Salvo Rossi (SUNIII igital Communications - Lecture 2 1 / 21 Example 1 (2/6 Example 1 (3/6 A possible orthonormal set of signals is ψ 1 (t = 1 ( t /2 rect ψ 2 (t = 1 ( ( t /4 t 3/4 (rect rect /2 /2 he constellation is the following ( +A1 s 1 = s 2 = ( +A2 /2 A 2 /2 P Salvo Rossi (SUNIII igital Communications - Lecture 2 11 / 21 P Salvo Rossi (SUNIII igital Communications - Lecture 2 12 / 21
4 Example 1 (4/6 Another possible orthonormal set of signals is ( 2 t /4 ϕ 1 (t = rect /2 ( 2 t 3/4 ϕ 2 (t = rect /2 Example 1 (5/6 he constellation is the following ( +A1 /2 s 1 = +A 1 /2 ( s 2 = +A 2 /2 P Salvo Rossi (SUNIII igital Communications - Lecture 2 13 / 21 P Salvo Rossi (SUNIII igital Communications - Lecture 2 14 / 21 Example 1 (6/6 Example 2 - QPSK (1/3 Selecting a different orthonormal set of signals for representation corresponds to a rotation of the reference axis of the signal space Consider a quaternary modulation (M = 4 with signals ( ( 2π t /2 s m (t = A cos t + (m 1π rect 2 m = 1, 2, 3, 4 he signals have all the same energy E = A 2 /2 P Salvo Rossi (SUNIII igital Communications - Lecture 2 15 / 21 P Salvo Rossi (SUNIII igital Communications - Lecture 2 16 / 21
5 Example 2 - QPSK (2/3 A possible orthonormal set of signals is ( ( 2 2π t /2 ψ 1 (t = + cos t rect ( ( 2 2π t /2 ψ 2 (t = sin t rect Example 2 - QPSK (3/3 he constellation is the following ( +A /2 s 1 = ( A /2 s 3 = ( s 2 = ( s 4 = +A /2 A /2 P Salvo Rossi (SUNIII igital Communications - Lecture 2 17 / 21 P Salvo Rossi (SUNIII igital Communications - Lecture 2 18 / 21 Representation of Stochastic Processes (1/2 Consider a stochastic process x(t L 2 (, ie such that each realization is an energy signal over, and an orthonormal set of signals {ψ n (t} x(t = x n ψ n (t x n = < x, ψ n > he dimension is in order to represent each possible realization he stochastic process is represented by a random vector with infinite components x(t x = x 1 x n Representation of Stochastic Processes (2/2 Each component x n = x(tψ n(tdt is a random variable { } E {x n } = E x(tψn(tdt = µ x (tψn(tdt Cov {x n, x m } = E {(x n E{x n }(x m E{x m } } { } = E (x(t µ x (t(x(s µ x (s ψn(tψ m (sdtds 2 = K x (t, sψn(tψ m (sdtds 2 For a WSS process µ x (t = µ x and K x (t, s = R x (t s µ x 2 P Salvo Rossi (SUNIII igital Communications - Lecture 2 19 / 21 P Salvo Rossi (SUNIII igital Communications - Lecture 2 2 / 21
6 WGN Charachterization Consider a White Gaussian Noise (WGN w(t L 2 ( with zero-mean independent real and imaginary parts, each with η /2 flat PS Each component w n = w(tψ n(tdt is a complex Gaussian rv E {w n } = Cov {w n, w m } = η δ(t sψn(tψ m (sdtds 2 = η ψn(tψ m (tdt = η δ n,m w is a complex Gaussian random vector with uncorrelated thus independent components w n N C (, η R{w n }, I{w n } N (, η /2 P Salvo Rossi (SUNIII igital Communications - Lecture 2 21 / 21
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