ecture 6&7 Modulation Eytan Modiano AA Dept
Lectures 6&7 Modulation Eytan Modiano AA Dept. Eytan Modiano Slide 1
Modulation Representing digital signals as analog waveforms ° Baseband signals Signals whose frequency components are concentrated around zero ● Passband signals Signals whose frequency components are centered at some frequency fc away from zero Baseband signals can be converted to passband signals through modulation Multiplication by a sinusoid with frequency fc
Modulation • Representing digital signals as analog waveforms • Baseband signals – Signals whose frequency components are concentrated around zero • Passband signals – Signals whose frequency components are centered at some frequency fc away from zero • Baseband signals can be converted to passband signals through modulation – Multiplication by a sinusoid with frequency fc Eytan Modiano Slide 2
Baseband signals The simplest signaling scheme is pulse amplitude modulation(PAM) With binary pam a pulse of amplitude a is used to represent a"1" and a pulse with amplitude-a to represent a0 The simplest pulse is a rectangular pulse, but in practice other type of pulses are used For our discussion we will generally assume a rectangular pulse If we let g(t be the basic pulse shape than with pam we transmit gt) to represent a“1and-g(t) to represent a“0” 1→>S(t)=g(t) A 0=>S(t)=-g(t)
Baseband signals • The simplest signaling scheme is pulse amplitude modulation (PAM) – With binary PAM a pulse of amplitude A is used to represent a “1” and a pulse with amplitude -A to represent a “0” • The simplest pulse is a rectangular pulse, but in practice other type of pulses are used – For our discussion we will generally assume a rectangular pulse • If we let g(t) be the basic pulse shape, than with PAM we transmit g(t) to represent a “1” and -g(t) to represent a “0” g(t) 1 => S(t) = g(t) A 0 => S(t) = -g(t) Tb Eytan Modiano Slide 3
M-ary PAM Use M signal levels, Ay. A Each level can be used to represent Log2(M)bits °Eg,M=4=A1=3,A2=-1,A3=1A4=3 s()=A1g(t) Mapping of bits to signals si b1b2 SSs 011
M-ary PAM • Use M signal levels, A1…AM – Each level can be used to represent Log2(M) bits • E.g., M = 4 => A1 = -3, A2 = -1, A3 = 1, A4 = 3 – Si(t) = Ai g(t) • Mapping of bits to signals Si b1b2 S1 00 S2 01 S3 11 S4 10 Eytan Modiano Slide 4
Signal Energy Em=(Sm()dt=(Am)L(g )dt=(Am)E g The signal energy depends on the amplitude Eo is the energy of the signal pulse g(t) For rectangular pulse with energy ea=> A dt=TA=>A=E/2 g(t) 0<t≤T E。/2 g() otherwise
g t t T Signal Energy T T Em = ∫0 ( (t))2 dt = ( Am )2 ∫0 ( )2 dt = ( Am )2 Eg Sm gt • The signal energy depends on the amplitude • Eg is the energy of the signal pulse g(t) • For rectangular pulse with energy Eg => T Eg = ∫0 A2 dt = TA2 => A = Eg / 2 g(t) Eg / 2 0 ≤≤ () = 0 otherwise Eg /2 Eytan Modiano Slide 5 T
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
Energy content of modulated signals Cos(a) 1+cos(20 ∫ g()+」g(Co(4x)t e+ The cosine part is fast varying and integrates to O Modulated signal has 1 2 the energy as the baseband signal Slide 10
() Energy content of modulated signals ∞ ∞ Em = ∫−∞ U2 t dt = ∫−∞ Amg2 m () ( ) 2 t Cos2 (2πfct)dt Cos2 α = 1 + cos(2α) 2 A A2 ∞ 2 ∞ 2 () + 2 m ∫−∞g t Cos2 (4πfc Em = t)dt 2 m ∫−∞g t 2 () A2 m Em = E + ≈ 0 2 g • The cosine part is fast varying and integrates to 0 • Modulated signal has 1/2 the energy as the baseband signal Eytan Modiano Slide 10
