Baseband Digital Transmission Multidimensional Signals
Baseband Digital Transmission Multidimensional Signals
Multidimensional Vs.Multiamplitude Multiamplitude signal is One-dimensional 00 01 10 11 -3d -d 3d With One basis signal:g(t) Multidimensional signal Can be defined using multidimensional orthogonal signals
Multidimensional Vs. Multiamplitude ◼ Multiamplitude signal is One-dimensional ◼ With One basis signal: g(t) ◼ Multidimensional signal ◼ Can be defined using multidimensional orthogonal signals -3d -d d 3d 00 01 10 11
Multidimensional Orthogonal Signals Many ways to construct In this text,Construction of a set of M=2k waveforms s(t),i=1,2,.,M-1 Mutual Orthogonality ■ Equal Energy Using Mathematics ∫s,(0s0dh=56k,i,k=0,l,M-1 where Kronecker delta is 0,i≠k
Multidimensional Orthogonal Signals ◼ Many ways to construct ◼ In this text, Construction of a set of M=2k waveforms si (t), i=1,2,.,M-1 ◼ Mutual Orthogonality ◼ Equal Energy ◼ Using Mathematics ◼ where Kronecker delta is 0 ( ) ( ) , , 0,1,., 1 T i k ik s t s t dt i k M = = − 1, 0, ik i k i k = =
Example of M=22 Multiamplitude Vs.Multidimensional 3d S3 A1So S1 S2 S3 S2 -d S1 -3d 50 All signal has identical energy 3=0h-=01M- See figure 4.27,page 202,for signal constellation
Example of M=22 ◼ Multiamplitude Vs. Multidimensional ◼ All signal has identical energy ◼ ◼ See figure 4.27, page 202, for signal constellation s3 s2 s1 s0 T 3d d -d -3d s0 s1 s2 s3 T A 2 2 0 ( ) , , 0,1,., 1 T i A T s t dt i k M M = = = −
Receiver for AWGN Channe Received signal from AWGN channel ■r()=S,(t)+(t),m=0,l,M-l,0≤t≤T White Gaussian process With power spectrum No/2 Receiver decides which of M signal waveform was transmitted by observing r(t) Optimum receiver minimize Probability of Error
Receiver for AWGN Channel ◼ Received signal from AWGN channel ◼ ◼ Receiver decides which of M signal waveform was transmitted by observing r(t) ◼ Optimum receiver minimize Probability of Error ( ) ( ) ( ), 0,1,., 1, 0 i r t s t n t m M t T = + = − White Gaussian process With power spectrum N0 /2