Signals and systems Fall 2003 Lecture#16 30 October 2003 AM with an arbitrary periodic carrier 2. Pulse Train Carrier and Time-Division Multiplexing 3. Sinusoidal Frequency Modulation 4. Dt Sinusoidal am 5. DT Sampling, Decimation and Interpolat
Signals and Systems Fall 2003 Lecture #16 30 October 2003 1. AM with an Arbitrary Periodic Carrier 2. Pulse Train Carrier and Time-Division Multiplexing 3. Sinusoidal Frequency Modulation 4. DT Sinusoidal AM 5. DT Sampling, Decimation, and Interpolation
AM with an Arbitrary Periodic carrier x(t y(t) c(t)-periodic with period T, carrier frequency wc Remember: periodic in t zx∑a(-M)(a=7 impulse train) k Y(w) X()*C0)=X)*∑ ∑akX(d-ku)
AM with an Arbitrary Periodic Carrier
Modulating a(periodic) rectangular Pulse train c(t) △ T 0 ()=m(+):c(+) y(t) 0
Modulating a (Periodic) Rectangular Pulse Train
Modulating a Rectangular Pulse Train Carrier, contd C()=27∑akb(u-k k △/T d and sin(kuc△/2) T k for rectangular pulse Y(o) yw)=xu)*clw (wc+aM Drawn assuming nyquist rate is met
Modulating a Rectangular Pulse Train Carrier, cont’d for rectangular pulse
Observations 1)We get a similar picture with any c(t that is periodic with period T' 2) As long as oc=2 /T>20M, there is no overlap in the shifted and scaled replicas of XGo. Consequently, assuming ao*0 x(t) 0/2 c(t)=2aejkot x(t can be recovered by passing y(t)through a LPF 3)Pulse Train Modulation is the basis for Time-Division Multiplexing Assign time slots instead of frequency slots to different channels, e. g. AT&t wireless phones 4)Really only need samples (x(nT))when oc>2 o*adulation Pulse amplitude
Observations 1) We get a similar picture with any c(t) that is periodic with period T 4) Really only need samples {x(nT)} when ωc > 2 ωM ⇒ Pulse Amplitude Modulation x(t) can be recovered by passing y(t) through a LPF 2) As long as ωc = 2π/T > 2ωM, there is no overlap in the shifted and scaled replicas of X(jω). Consequently, assuming ao ≠ 0: 3) Pulse Train Modulation is the basis for Time-Division Multiplexing — Assign time slots instead of frequency slots to different channels, e.g. AT&T wireless phones