Symmetric PAM Signal amplitudes are equally distant and symmetric about zero -7-5-3-101357 Am=(2m-1M,m=1.M Em=∑(2m-1-M03=E(M2-1/3
Symmetric PAM • Signal amplitudes are equally distant and symmetric about zero -7 -5 -3 -1 0 1 3 5 7 Am = (2m-1-M), m=1…M E M Eave = g ∑(2m − 1 − M)2 =Eg (M2 − 1)/ 3 M m=1 Eytan Modiano Slide 6
Gray Coding Mechanism for assigning bits to symbols so that the number of bit errors is minimized Most likely symbol errors are between adjacent levels Want to maP bits to symbols so that the number of bits that differ between adjacent levels is mimimized Gray coding achieves 1 bit difference between adjacent levels Example M=8 (can be generalized) 000 001 011 010 110 101 A8100
Gray Coding • Mechanism for assigning bits to symbols so that the number of bit errors is minimized – Most likely symbol errors are between adjacent levels – Want to MAP bits to symbols so that the number of bits that differ between adjacent levels is mimimized • Gray coding achieves 1 bit difference between adjacent levels • Example M= 8 (can be generalized) A1 000 A2 001 A3 011 A4 010 A5 110 A6 111 A7 101 Eytan Modiano A8 100 Slide 7
Bandpass signals To transmit a baseband signal s(t through a bandpass channel at some center frequency f, we multiply s(t by a sinusoid with that frequency m(t)=Sm(t)Cos(2ft) Amg(t) Cos(lift) Cos(2Tft F[COS(2+)]=((ff)+(f+f)2
Bandpass signals • To transmit a baseband signal S(t) through a bandpass channel at some center frequency fc, we multiply S(t) by a sinusoid with that frequency Sm(t) Um(t)= Sm(t)Cos(2πfct) = Amg(t) Cos(2πfct) Cos(2πfct) F[Cos(2πfct)] = (δ(f-fc)+δ(f+fc))/2 Eytan Modiano -fc f Slide 8 c
Passband signals, cont FlAmgt]=depends on gO [Amg(t) Cos(2 ft) A/2 A/ diano Recall: Multiplication in time convolution in frequency
Passband signals, cont. F[Amg(t)] = depends on g() Am -w w F[Amg(t) Cos(2πfct)] Am /2 Am /2 -fc fc Eytan Modiano Slide 9 Recall: Multiplication in time = convolution in frequency