Lecture 9 Outline: AM and FM Modulation Announcements: HW deadline changed to 5pm Fridays Grader position open; let us know if you are interested Review of Last Lecture Amplitude Modulation AM Radio Quadrature Modulation Frequency Modulation

Review of Last Lecture Discrete-Time Downsampling: Digital Downsampling Removes samples of x(nTs) for n≠MTs Used under storage/comm. constraints Repeats Xd(ejW) every 2p/M and scales W axis by M This results in a periodic signal Xc(ejW) every 2p Introduces aliasing if Xd(ejW) bandwidth exceeds p/M Can prefilter Xd(ejW) by LPF with bandwith p/M prior to downsampling to avoid downsample aliasing 1 2 3 … xd[n]=x(nTs) xc[n] Downsample By M 1 2 3 4 p/M -p/M W’ Xd(ejW’) p -p Xc(ejW) -2p 2p … p/M -p/M W’ Xd(ejW’) p -p Xc(ejW) -2p 2p … W=MW’ W=MW’

W=wTs W=wMTs Xs(jw) Xd(ejW) w W Xs(jw) Xc(ejW) W Xc(ejW) Xd(ejW) W W 2p -2p W Xd(ejW) W=wTs Ts w Xs(jw) X(jw) -W 𝑋𝑑 𝑒 𝑗Ω = 1 𝑇𝑠 𝑘=−∞ ∞ 𝑋 𝑗 Ω−2𝜋𝑘 𝑇𝑠 MTs 2p -2p W Xc(ejW) W=wMTs w X(jw) -W MT Xs(jw) 𝑋𝐶 𝑒 𝑗Ω = 1 𝑀𝑇𝑠 𝑙=−∞ ∞ 𝑋 𝑗 Ω−2𝜋𝑙 𝑀𝑇𝑠 𝑙=𝑘𝑀+𝑚: 𝑙=−∞ ∞ = 𝑚=0 𝑀−1 𝑘=−∞ ∞ 𝑋𝐶 𝑒 𝑗Ω = 1 𝑀𝑇𝑠 𝑙=−∞ ∞ 𝑋 𝑗 Ω−2𝜋𝑙 𝑀𝑇𝑠 = 1 𝑀 𝑚=0 𝑀−1 1 𝑇𝑠 𝑘=−∞ ∞ 𝑋 𝑗 Ω−2𝜋(𝑘𝑀+𝑚) 𝑀𝑇𝑠 = 1 𝑀 𝑚=0 𝑀−1 𝑋𝑑 𝑒 𝑗(Ω−2𝜋𝑚)/𝑀 (9) 𝑋𝑐 𝑒 𝑗Ω = 1 𝑀 𝑚=0 𝑀−1 𝑋𝑑 𝑒 𝑗(Ω−2𝜋𝑚)/𝑀 1 2 3 4 xc[n]=xd[nM] 2p -2p W Xc(ejW) xd[n] 1 2 3 … 2p -2p -p/M p/M W Xd(ejW)

Communication System Block Diagram Modulator (Transmitter) Demodulator (Receiver) Channel Modulator (Transmitter) converts message signal or bits into format appropriate for channel transmission (analog signal). Channel introduces distortion, noise, and interference. Demodulator (Receiver) decodes received signal back to message signal or bits. Focus on modulators with s(t) at a carrier frequency wc. Allows allocation of orthogonal frequency channels to different users

Amplitude Modulation DSBSC and SSB Double sideband suppressed carrier (DSBSC) Modulated signal is s(t)=m(t)cos(wct) Signal bandwidth (bandwidth occupied in positive frequencies) is 2W Redundant information: can either transmit upper sidebands (USB) only or lower sidebands (LSB) only and recover m(t) Single sideband modulation (SSB); uses 50% less bandwidth (less $$$) Demodulator for DSBSC/SSB: multiply by cos(wct) and LPF W 2W USB LSB w -W W -wc w wc 2wc -2wc X cos(wct) s(t) wc -wc

AM Radio + + cos(wct) s(t)=[A+kam(t)]coswct ka A m(t) 1 W -W wc -wc m(t) + + X Broadcast AM has s(t)=[1+kam(t)]cos(wct) with [1+kam(t)]>0 Constant carrier cos(wct) carriers no information; wasteful of power Can recover m(t) with envelope detector (diode, resistors, capacitor) Modulated signal has twice bandwidth W of m(t), same as DSBSC 1/(2pwc)<<RC<<1/(2pW)

Quadrature Modulation Sends two info. signals on the cosine and sine carriers DSBSC Demod m1(t) LPF m1(t)cos(wct)+ m2(t)sin(wct) cos(wct) -90o sin(wct) m2(t) DSBSC Demod LPF

FM Modulation (not covered in lecture) Message signal m(t) encoded in carrier frequency FM modulated signal: Instantaneous frequency: wi=wc+kfm(t) Signal robust to amplitude variations and reflections Frequency analysis nonlinear (hard, will skip) Frequency Deviation: Df=kf max|m(t)| Maximum deviation of wi from wc: wi=wc+kfm(t) Carson’s Rule for bandwidth of s(t): FM Demod: Differentiator + Envelope Detector s(t)=Acos(q(t))=Acos(wct+kfm(t)dt) Bs2Df+2Bm Depends on max deviation from wc and how fast wi changes

Main Points Modulation is the process of encoding an analog message signal (or bits) into a carrier signal DSBSC multiplies the message signal and the carrier together. Synchronous demodulation multiplies by the carrier and then uses a LPF. Requires learning carrier phase at receiver (hard!) Broadcast AM uses extra carrier term to simplify reception SSB is a spectrally efficient AM technique with half the BW requirements of standard AM and DSBSC. Quadrature modulation sends two different signals in the same bandwidth using sin and cosine carriers (which are orthogonal) FM modulation encodes information in signal frequency. More robust to amplitude errors and signal reflections than AM Bandwidth depends on info. signal bandwidth and freq. deviation