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
Sinusoidal Frequency Modulation(FM) Oscillator y(t=A cos(e(t) Amplitude fixed Phase modulation: A(t)=wct+.Bo+ kp c(t) Frequency modulation: dt=;wc+ka(t) x(t) is signal to be …………… instantaneous transmitted Chirp FⅣ x(t=u(t) x(t=tou(t)
Sinusoidal Frequency Modulation (FM) FM x(t) is signal to be transmitted
Sinusoidal Fm (continued) ransmitted power does not depend on x(t): average power=A/2 Bandwidth ofy(t) can depend on amplitude of x(t) Demodulation a) Direct tracking of the phase at (by using phase-locked loop b) Use of an lti system that acts like a differentiator v(t y Envelope detector HG@- Tunable band-limited differentiator, over the bandwidth of y(t) H(j0)≈j 2()s looks like am (wc+ kf()Asin e(t) envelope detection de/ di
Sinusoidal FM (continued) • Transmitted power does not depend on x ( t): average power = A 2/2 • B a n d wid t h o f y ( t) can depend on amplitude of x ( t ) • Demodulation a) Direct tracking of the phase θ( t) (by using phase-locked loop ) b) Use of an LTI system that acts like a differentiator H(j ω) — Tunable band-limi ted differentiator, over the bandwidth of y ( t) … looks like AM envelope detection
DT Sinusoidalam Multiplication <> Periodic convolution yn]=ml·cml←Y(c)=2J2X(e)C(ea-)d x(ela) Example #1: cln )=2x∑6(-+27k) c(e) 2m+ 2;+ (e)=o x(e)oc(
DT Sinusoidal AM Multiplication ↔ Periodic convolution Example #1:
Example #2: Sinusoidal AM COS wn 丌∑6(0-+276)+0(a+4+27k) k X(e) Drawn assuming C 7-w I>WctWM 0.e UM<W<丌- →M<丌/2 Y(el) 2T (e4)=。X(e)C(c14) 人人 o overlap o -2-2a+uc 2T-ωc shifted spectra
Example #2: Sinusoidal AM No overlap of shifted s pectr a
Example #2(continued): Demodulation COS on Lowpass filter Possible as long as there is H(e/ no overlap of shifted replicas ofX(el/ 1e., and w+uM<2丌-u-uM →wM<We<丌-wM 2丌+ (e y W(e)=o y(e)c(e 人 W 2 Ⅹ(e) Misleading drawing-shown for a very special case of o=T/2 2T
Example #2 (continued): Demodulation Possible as long as there is no overlap of shifted replicas of X( ej ω): Misleading drawing – shown for a very special case of ωc = π/2
