1. 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

2. 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’

3. W=wTs W=wMTs Xs(jw) Xd(ejW) w W Xs(jw) Xc(ejW) W Xc(ejW) Xd(ejW) W W2p
-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)

4. 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

5. 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

6. 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)

7. 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

8. 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

9. 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