Signal space representations of communication signals, optimal detection and error probability
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1 Presentation Signal space representations of communication signals, optimal etection an error proaility Jac Romme Feruari 5 Outline Representation of anpass signal Discrete Time equivalent moel Signal space representations Optimal etection on AWGN Channel Symol Error Proaility on AWGN Channel
2 anpass signal Banpass Signal Real value signal S(f S * (-f finite anwith B infinite time span f c enotes center frequency f c f c Negative Frequencies contain no Aitional Info Banpass to Basean Two step proceure: + S ( f U ( f S( f S ~ ( f S + ( f f c ~ s ( t s + ( texp( jπ Characteristics: Complex value signal No information loss, truely equivalent Reconstruction: [ ~ s ( texp( jπf ] s( t Re c + s ( t s( t + j f c s( τ sˆ( t τ π t τ sˆ( t
3 Graphical impression S( f U ( f S + ( f ~ S ( f Discrete Time equivalent moel Ieal Sampling process with perio T s [] ˆ s( T Reconstruction Process s ( t ˆ s[]( δ T s( t δ ( T ( S f ˆ S f results is spectral copies Use ieal filter to get ri of them s ( t s[] sinc( f t sin(π t sinc( t πt s f < fs H ( f otherwise T
4 Graphical Proof of Nyquist f s f s fs fs f s f s fs fs f s f s fs fs Aliasing Time Domain Sampling Proceure
5 Time Domain Sampling Proceure ( Time Domain Sampling Proceure (
6 Time Domain Sampling Proceure ( General Communication signal General Shape: is value of -th symol element of {,,..,N s } Waveform (t has: w Unit energy for simplicity Duration <T, Banwith <B for every possile s( t ˆ E w ( t T T
7 Signal Space Representation Signal/vector space is a set of vectors together with two operators, aition of vector an multiplication y a scalar Define a set of BT real-value orthonormal functions f (t,f (t,...,f BT (t spanning the BT-imensional space Signal Space Representation Each waveform can e escrie y a vector containing BT elements w w ( t f ( t t f ( f ( t f ( t f K BT [ ( t ] T t Some properties: w( t t w v( t w( t t v, w
8 Signal Space Representation Example w T t Optimal etection on AWGN Channel Receive signal vector r w + n Noise is not anwith limite, ut a proper receiver loos only in the waveform space Noise elements are i.i.. Gaussian RV Receiver must mae ecision on transmitte waveform ase on r Maximum lielihoo (ML receiver Given AWGN case an equal liely symols, Maximum Lielihoo Minimum Distance
9 Optimal etection on AWGN Channel Symol Error Pro. (SEP is minimize if: ˆ arg min r w Β ( Distance can e written out to: r w r r, w + w In case of equal energy symols ˆ arg max Β ( r, w The same for all Output of -th Matche Filter Optimal etection on AWGN Channel Signal Space Matche Filter receiver r, w r r, w argmax ˆ r,w Ns Note: for equal Energy Symols
10 Optimal etection on AWGN Channel -th orer Bi-orthogonal (BPSK BPSK is imensional Moulation 4-th orer Bi-orthogonal (QPSK QPSK is imensional r, w w r, r, w w r, Noise at the two MF outputs is inepenent The waveform space is suspace of BT space N s -th orer Bi-orthogonal moulation Biorthogonal moulation is N s / imensional Maximum orer is M < BT r, w w 3 r, w 4,, w3, 4 Symol Error Proaility on AWGN channel BPSK case P( e P( r, w > r, w + (, >, P r w r w P( r, w > P( e Q r, w N Q Q E N N pf E r
11 Symol Error Proaility on AWGN channel Close form erivation of higher orer moulation often impossile Solution: Union-oun B P e P e P ( ( ( B B N Q M e P, ( > B P e P,,, ( ( w r w r N Q M Symol Error Proaility on AWGN channel 6-th orer i-orthogonal moulation Each symol has the same error proaility Mirror Symmetry B N Q e P, (
12 Symol Error Proaility on AWGN channel 6-th orer i-orthogonal moulation Each symol has the same error proaility mirror Symmetry P( e P( e 4Q Q N B, E N s + Q 4E N s E s E s Question: 6-th orer Bi-orthogonal worse than BPSK??? Symol Error Proaility on AWGN channel Answer: yes an no E s E Yes: with respect to No: with respect to { E log (6 } s E P( e 4Q log(6 E N + Q 4log(6 E N Secon No: Symol Error is not same as Bit error Depens on its to symol mapping Gray Coing See Proais
13 SEP of Bi-orthogonal moulation Note: numerically otaine exact SEP Any Questions??
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