EE373 : Digial Communicaions Week 6-7: Deecion Error Probabiliy Signal Space Orhogonal Signal Space MAJU-Digial Comm.-Week-6-7
Deecion Mached filer reduces he received signal o a single variable zt, afer which he deecion of symbol is carried ou The concep of maximum likelihood deecor is based on Saisical Decision Theory I allows us o formulae he decision rule ha operaes on he daa opimize he deecion crierion z T MAJU-Digial Comm.-Week-6-7 H > < H γ
Deecion of Binary Signal in Gaussian Noise The oupu of he filered sampled a T is a Gaussian random process MAJU-Digial Comm.-Week-6-7 3
Hence Baye s Decision Crierion and Maximum Likelihood Deecor z H > a + a < H γ where z is he minimum error crierion and γ is opimum hreshold For anipodal signal, s - s a - a z H > < H MAJU-Digial Comm.-Week-6-7 4
Error will occur if s is sen s is received P H s P e s s is sen s is received P H s P e s Probabiliy of Error P e s p z s dz γ P e s p z s dz γ The oal probabiliy of error is he sum of he errors P P e, s P e s P s + P e s P s B i i P H s P s + P H s P s MAJU-Digial Comm.-Week-6-7 5
If signals are equally probable P P H s P s + P H s P s B + [ P H s P H s ] by Symmery PB [ P H s + P H s ] P H s Numerically, P B is he area under he ail of eiher of he condiional disribuions pz s or pz s and is given by: P P H s dz p z s d z B γ γ z a ex p γ σ π σ dz MAJU-Digial Comm.-Week-6-7 6
P B z a exp γ σ π σ u z a σ u a a exp du σ π The above equaion canno be evaluaed in closed form Qfuncion Hence, a a dz PB Q equaion B σ Q z exp z π z.8 MAJU-Digial Comm.-Week-6-7 7
Recall: Error probabiliy for binary signals P B a a Q σ equaion B Where we have replaced a by a..8 To minimize P B, we need o maximize: a a σ We have Therefore, or a a σ a a Ed Ed σ N / N a a a a E E d d σ σ N N MAJU-Digial Comm.-Week-6-7 8
[ ] [ ] [ ] [ ] + T T T T d s s d s d s d s s E 3.63 N E Q P d B The probabiliy of bi error is given by: MAJU-Digial Comm.-Week-6-7 9 [ ] [ ] [ ] + s s d s d s
The probabiliy of bi error for anipodal signals: The probabiliy of bi error for orhogonal signals: N E Q P b B E MAJU-Digial Comm.-Week-6-7 The probabiliy of bi error for unipolar signals: N E Q P b B N E Q P b B
Error probabiliy for binary signals Table for compuing of Q-Funcions MAJU-Digial Comm.-Week-6-7
Relaion Beween SNR S/N and E b /N In analog communicaion he figure of meri used is he average signal power o average noise power raion or SNR. In he previous few slides we have used he erm E b /N in he bi error calculaions. How are he wo relaed? E b can be wrien as ST b and N is N/W. So we have: Eb STb S W N where σ N N / W N Rb Thus E b /N can be hough of as normalized SNR. Makes more sense when we have muli-level signaling. Reading: Page 7 and 8. MAJU-Digial Comm.-Week-6-7
Bipolar signals require a facor of increase in energy compared o orhogonal signals Since log 3 db, we say ha bipolar signaling offers a 3 db beer performance han orhogonal MAJU-Digial Comm.-Week-6-7 3
Comparing BER Performance For P P E b / B, orhogonal B, anipodal N db 9.x 7.8x 4 For he same received signal o noise raio, anipodal provides lower bi error rae han orhogonal MAJU-Digial Comm.-Week-6-7 4
Problem: Evaluaing Error Performance Consider a Binary Communicaion Sysem ha receives equally likely signals s and s plus AWGN see he following figure. Assume ha he receiving filer is a Mached Filer MF, and ha he noise Power Specral Densiy N is equal o - Wa/Hz. Use he values of received signal volage and ime shown on figure o compue he Bi Error Probabiliy. s millivols 3 µs 3 µs s millivols MAJU-Digial Comm.-Week-6-7 5
Problem: Error Performance based Designing Consider ha NRZ binary pulses are ransmied along a communicaion cable ha aenuaes he signal power by 3 db from ransmier o receiver. The pulses are coherenly deeced a he receiver, and he daa rae is 56 kbis/s. Assume Gaussian noise wih N -6 Was/Herz. Wha is he minimum amoun of Power needed a he ransmier in order o mainain a bi-error probabiliy of Pe -3? MAJU-Digial Comm.-Week-6-7 6
Signals vs vecors Represenaion of a vecor by basis vecors Orhogonaliy of vecors Orhogonaliy of signals MAJU-Digial Comm.-Week-6-7 7
Wha is a signal space? Signal space Vecor represenaions of signals in an N-dimensional orhogonal space Why do we need a signal space? I is a means o conver signals o vecors and vice versa. I is a means o calculae signals energy and Euclidean disances beween signals. Why are we ineresed in Euclidean disances beween signals? For deecion purposes: The received signal is ransformed o a received vecors. The signal which has he minimum disance o he received signal is esimaed as he ransmied signal. MAJU-Digial Comm.-Week-6-7 8
Orhogonal signal space N-dimensional orhogonal signal space is characerized by N linearly independen funcions { ψ j } N called basis funcions. j The basis funcions mus saisfy he orhogonaliy condiion. ρ ji T < ψ i, ψ j > ψ ψ * K i j K d T j, i,..., N Where ρ ij i i j j is he correlaion coefficien. K is he normalizing consan which makes he signal space Orhonormal. MAJU-Digial Comm.-Week-6-7 9
Example of an orhonormal basis Example: -dimensional orhonormal signal space, / sin / cos > < < < d T T T T T T T ψ ψ ψ ψ π ψ π ψ ψ ψ MAJU-Digial Comm.-Week-6-7 Example: -dimensional orhonornal signal space, > < d ψ ψ ψ ψ ψ ψ T ψ T ψ ψ
Signal space Any arbirary finie se of waveforms where each member of he se is of duraion T, can be expressed as a linear combinaion of N orhonogal waveforms where. s i { ψ } N j j N N M { } M i i a ijψ j i,..., M j N M s where a ij < s, ψ > s ψ i j T i * j d j,..., N i,..., M T s i ai, ai,..., a in Vecor represenaion of waveform E i N j a ij Waveform energy Parseval s heorem MAJU-Digial Comm.-Week-6-7
Signal space a T * ij si ψ j d Waveform o vecor conversion i N s a ψ j Vecor o waveform conversion ij j s i ψ ψ N T T a i a in a ai M a in s m s m a ai M a in a i a in ψ ψ N s i s m ai, ai,..., a in MAJU-Digial Comm.-Week-6-7
Basis Funcions: An example A se of 8 orhogonal of basis funcions Wha signals can we form wih his se of basis funcions? MAJU-Digial Comm.-Week-6-7 3
Basis Funcions: An example Linear combinaion of basis funcions φ [n] + φ [n] + /3 φ 3 [n] + φ 4 [n] + /5 φ 5 [n] + φ 6 [n] + /7 φ 7 [n] + φ 8 [n] Waveforms in he span of basis funcions φ [n] + φ [n] + /9 φ 3 [n] + φ 4 [n] + /5 φ 5 [n] + φ 6 [n] + /49 φ 7 [n] + φ 8 [n] φ [n] + / φ [n] + /3 φ 3 [n] + /4 φ 4 [n] + /5 φ 5 [n] + /6 φ 6 [n] + /7 φ 7 [n] + /8 φ 8 [n] MAJU-Digial Comm.-Week-6-7 4
Represenaion of a signal in signal space MAJU-Digial Comm.-Week-6-7 5
Example: Baseband Anipodal Signals MAJU-Digial Comm.-Week-6-7 6
Example: BPSK MAJU-Digial Comm.-Week-6-7 7
Example QPSK MAJU-Digial Comm.-Week-6-7 8
Synhesis Equaion Modulaion MAJU-Digial Comm.-Week-6-7 9
Example: Baseband Anipodal Signals MAJU-Digial Comm.-Week-6-7 3
Example: BPSK MAJU-Digial Comm.-Week-6-7 3
Correlaion Measure of similariy beween wo signals c n + g z d. E E g z Cross correlaion ψ gz Auocorrelaion + τ g z + τ d. + ψ g τ g g + τ d. MAJU-Digial Comm.-Week-6-7 3
Analysis Equaion Deecion MAJU-Digial Comm.-Week-6-7 33
Correlaion Deecor MAJU-Digial Comm.-Week-6-7 34
Correlaion Deecor: Examples MAJU-Digial Comm.-Week-6-7 35
Correlaion Deecor Example: QPSK MAJU-Digial Comm.-Week-6-7 36