ee104: lecture 25 outline announcements review of last lecture probability of bit error in ask/psk...
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EE104: Lecture 25 Outline
Announcements
Review of Last Lecture
Probability of Bit Error in ASK/PSK
Course Summary
Hot Topics in Communications
Next-Generation Systems
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Announcements HW 7 due Monday at 9 pm (no late HWs)
Solutions will be posted then.
Final Exam is Th., 3/20, 8:30am in Gates B03 (basement) Exam is open book/notes, covers through today’s lecture. Emphasis is on material after midterm, similar to practice
finals SCPD student must make remote arrangements w/me by this
Friday
Practice finals posted on class website 10 bonus points if turned in by 3/20 at 8:30am (Joice has
solutions)
Final Review: Monday 7-8pm (room 380-380Y) Extra Office Hours
My extra hours: M 4:30-6:30, TW 12-2, and by appointment Jaron: W 4-6 (Bytes), Nikola: after review and T 5-7pm (110
Packard)
will be done at end of class (10 bonus points)
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Review of Last Lecture
Passband Digital ModulationASK/PSK special cases of DSBSCFSK special case of FM
ASK/PSK Demodulator:
Decision devices finds if r(iTb) is closer to r0 or r1
Noise immunity N is half the distance between r0 and r1
Bit errors occur when noise exceeds this immunity
s(t)
cos(2fct)
bT
dt0
)(
nTb
Decision Device
“1” or “0” r(nTb)
r0
r1
N
r0+N
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Noise in ASK/PSK
Probability of bit error: Pb=p(|N(nTb)|>N=.5|r1-r0|)
N(nTb) is a Gaussian RV: N~N(=0,2=.25NoTb)
For x~N(0,1), Define Q(z)=p(x>z)
ASK:
PSK:
s(t)
cos(2fct)
bT
0
nTb
r(nTb)+N(nTb) “1” or “0” +
N(t)
ChannelN
r1
r0
00
225.)25.( NE
NTA
bcbbbc QQTANpP
00
2 2)5.( NE
NTA
bcbbbc QQTANpP
Eb is averageenergy per bit
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Course Summary
Communication System Block Diagram Fourier Series and Transforms Sampling Power Spectral Density and
Autocorrelation Random Signals and White Noise AM Modulation FM Modulation Digital Modulation
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Communication System Block Diagram
Source encoder converts message into a message signal or bits. Source decoder converts back to original format.
Transmitter converts message signal or bits into a transmitted signal at some carrier frequency. Modulation, may also include SS, OFDM, precoding.
Channel introduces distortion, noise, and interference.
Receiver converts back to message signal or bits. Demodulation (for SS and OFDM too), may also include equalization.
SourceDecoderChannel ReceiverTransmitter
TextImagesVideo
)(ts )(ˆ ts)(ˆ...ˆˆ
21
tmbb
)(...21
tmbb
SourceEncoder
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Main Focus of This Class
Modulation encodes message or bits into the amplitude, phase, or frequency of carrier signal.
Channel filters signal and introduces noise
Demodulator recovers information from carrier
Analog or DigitalModulator
)(ts )()(ˆ tnts )(ˆ...ˆˆ
21
tmbb
)(...21
tmbb
Transmitter
Channel
h(t)
+Analog or Digital
Demodulator
Receiver
n(t)
Need tools for manipulating and filtering signals and noise
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Fourier Series
Exponentials are basis functions for periodic signals Can represent periodic signal in terms of FS coefficients Complex coefficients are frequency components of signal
tnfj
nnp ectx 02)(
dtetxT
c tnfj
T
pn0
0
2
0
)(1
0 T0 2T0-T0
.5T-.5T
c1 c2
c0
c3
c4
xp(t)
t f1/T0 2/T00-1/T0
c-4
c-3
c-2
c-1
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Fourier Transform Represents spectral components of a signal Signal uniquely represented in time or frequency
domain These coefficients are frequency components of signal
dtetxfX nftj 2)()(
dfefXtx ftj 2)()(
.5T-.5T
A
tf
Timelimited signals have infinite frequency contentBandlimited signals are infinite duration
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Key Properties of FTs
Frequency shifting (modulation)Multiplying signal by cosine shifts it by fc in
frequency.
Multiplication in time Convolution in Frequency
Convolution in time Multiplication in
Frequency
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Filtering
Convolution defines output of LTI filters
Convolution (time) Multiplication (freq.)
Easier to analyze filters in frequency domainFilters characterized by time or freq.
responseExponentials are eigenfunctions of LTI
filters x(t) h(t) y(t)=h(t)*x(t)
X(f) H(f) Y(f)=H(f)X(f)
LTI Filter
ej2fct H(fc)ej2fct
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Sampling
Sampling (Time):
Sampling (Frequency)
xs(t)
0 0 0
x(t) =n(t-nTs)
Xs(f)
0 0 0
X(f) =n(t-n/Ts)*
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Nyquist Sampling Theorem
A bandlimited signal [-B,B] is completely described by samples every .5/B secs.Nyquist rate is 2B samples/sec
Recreate signal from its samples by using a low pass filter in the frequency domainSinc interpolation in time domainUndersampling creates aliasing
Xs(f)X(f)
B-B B-B
X(f)
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Power Spectral Density
Distribution of signal power over frequency
PSD/autocorrelation FT pairs:
Rx(Sx(f)
Useful for filter and modulation analysis
Sx(f) dffSdttxT
P xT )(|)(|2
1lim 2
f
Sx(f) |H(f)|2Sx(f)H(f)
Sx(f) .25[Sx(f-fc)+ Sx(f+fc)]X
cos(2fct)
Assumes Sx(f) bandlimited [-B,B], B << fc
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Random Signals Not deterministic (no Fourier transform)
Signal contained in some set of possible realizations
Characterize by average PSD Sn(f)
Autocorrelation Rn() Sn(f) is the correlation of the random signal after time .Measures how fast random signal changes
Experiment
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Filtering and Modulation
Same PSD effect as for deterministic signals
Filtering
Modulation (no bandwidth
constraint on Sn)
Sn(f) |H(f)|2Sn(f)H(f)
Sn(f) .25[Sn(f-fc)+ Sn(f+fc)]X
cos(2fct)
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White Noise
Signal changes very fastUncorrelated after infinitesimally
small delay Good approximation in practice Filtering white noise: introduces
correlation
.5N0() .5N0
f
Sn(f)Rn()
Sw(f)=.5N0 .5N0|H(f)|2
H(f)
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Amplitude Modulation
Constant added to signal m(t)Simplifies demodulation if 1>|kam(t)|Constant is wasteful of power
Modulated signal has twice the BW of m(t)Simple modulators use nonlinear
devices
m(t)X
ka
X
cos(2fct)
+
1 s(t)=Ac[1+kam(t)]cos2fct
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Detection of AM Waves
Entails tradeoff between performance and complexity (cost)
Square law detector squares signal and then passes it through a LPFResidual distortion proportional to m2(t)Noncoherent (carrier phase not needed in receiver)
Envelope detector detects envelope of s(t)Simple circuit (resistors, capacitor, diode)Only works when |kam(t)|<1 (poor SNR), no
distortion.Noncoherent
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Double Sideband Suppressed Carrier
(DSBSC)
Modulated signal is
s(t)=Accos(2fct)m(t)
Generated by a product or ring modulator
Requires coherent detection (21)Costas Loop
m(t)
Accos(2fct+
ProductModulator
s(t) ProductModulator LPF
m´(t)
Accos(2fct+
Channel
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Noise in DSBSC Receivers
Power in m´(t) is .25Ac2P
Sn´(f)=.25[Sn(f-fc)+Sn(f+fc)]|H(f)|2
For AWGN, Sn´(f)=.25[.5N0+.5N0], |f|<B.
SNR=Ac2P/(2N0B)
ProductModulator
m´(t)+ n´(t)
Accos(2fct+
s(t)=Accos(2fct+m(t)+
n(t)LPF
1
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Single Sideband
Transmits upper or lower sideband of DSBSC
Reduces bandwidth by factor of 2Uses same demodulator as DSBSC
Coherent detection required.
USB LSB
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FM ModulationInformation signal encoded in
carrier frequency (or phase)
Modulated signal is s(t)=Accos((t))(t)=f(m(t))
Standard FM: (t)=2fct+2kf m()dInstantaneous frequency: fi=fc+kfm(t)Signal robust to amplitude variationsRobust to signal reflections and
refractions
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FM Bandwidth and Carson’s Rule
Frequency Deviation: f=kf max|m(t)|Maximum deviation of fi from fc: fi=fc+kfm(t)
Carson’s Rule:
B depends on maximum deviation from fc AND how fast fi changes
Narrowband FM: f<<BmB2Bm
Wideband FM: f>>Bm B2f
B2f+2Bm
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Generating FM Signals
NBFMCircuit based on product modulator
WBFM
Direct Method: Modulate a VCO with m(t)
Indirect Method: Use a NBFM modulator, followed by a nonlinear device and BPF
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FM Generation and Detection
Differentiator/Discriminator and Env. Detector
Zero Crossing DetectorUses rate of zero crossings to estimate fi
Phase Lock Loop (PLL)Uses VCO and feedback to extract m(t)
t
fcfcc dmktftmkfAts0
])(22sin[)](22[)(
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ASK, PSK, and FSK
Amplitude Shift Keying (ASK)
Phase Shift Keying (PSK)
Frequency Shift Keying
0)(0
1)()2cos()2cos()()(
b
bcccc nTm
nTmtfAtfAtmts
1)()2cos(
1)()2cos()2cos()()(
bcc
bcccc nTmtfA
nTmtfAtftmAts
1)()2cos(
1)()2cos()(
2
1
bc
bc
nTmtfA
nTmtfAts
1 0 1 1
1 0 1 1
1 0 1 1
AM Modulation
AM Modulation
FM Modulation
m(t)
m(t)
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ASK/PSK Demodulation
Probability of bit error: Pb=p(|N(nTb)|>N=.5|r1-r0|)
N(nTb) is a Gaussian RV: N~N(=0,2=.25NoTb)
For x~N(0,1), Define Q(z)=p(x>z)=.5erfc(z/2 )
ASK:
PSK:
s(t)
cos(2fct)
bT
0
nTb
r(nTb)+N(nTb) “1” or “0” +
N(t)
ChannelN
r1
r0
00
225.)25.( NE
NTA
bcbbbc QQTANpP
00
2 2)5.( NE
NTA
bcbbbc QQTANpP
Eb is averageenergy per bit
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FSK Demodulation (HW 7)
Comparator outputs “1” if r1>r2, “0” if r2>r1
Pb=p(|N1-N2|>.5AcTb)=Q(Eb/N0) (same as PSK)
Minimum frequency separation required to differentiate |f1-f2|.5/Tb (MSK uses this minimum separation)
s(t)+n(t)
cos(2f1t)
bT
0
nTb r1(nTb)+N1
“1” or “0”
cos(2f2t)
bT
0
nTb
r2(nTb)+N2
Comparator
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Megathemes in EE104
Fourier analysis simplifies the study of communication systems
Modulation encodes information in phase, frequency, or amplitude of carrier
Noise and distortion introduced by the channel makes it difficult to recover signal
The communication system designer must design clever techniques to compensate for channel impairments or make signal robust to these impairments.
Ultimate goal is to get high data rates with good quality and low cost.
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Hot Topics in Communications
All-optical networksComponents (routers, switches) hard to
buildNeed very good lasersCommunication schemes very basicEvolving to more sophisticated techniques
Advanced RadiosAdaptive techniques for multimediaDirect conversion radiosSoftware radiosLow Power (last years on a battery)Ultra wideband
Wireless Communications
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Future Wireless Systems
Nth Generation CellularWireless Internet (802.11)Wireless Video/Music Wireless Ad Hoc NetworksSensor Networks Smart Homes/AppliancesAutomated Vehicle NetworksAll this and more…
Ubiquitous Communication Among People and Devices
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The End
Good luck on the final
Have a great spring break