Final Report May 23, 2011 Deepa Chaparala Tara Phillips Rob Slowik Samuel Sun.
Final Exam Review - University of Toronto · Final Exam Review Professor Deepa ... Professor Deepa...
Transcript of Final Exam Review - University of Toronto · Final Exam Review Professor Deepa ... Professor Deepa...
Final Exam Review
Professor Deepa Kundur
University of Toronto
Professor Deepa Kundur (University of Toronto) Final Exam Review 1 / 67
Final Exam Review
Reference:
Sections:2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.73.1, 3.2, 3.3, 3.4, 3.5, 3.64.1, 4.2, 4.3, 4.4, 4.6, 4.7, 4.85.1, 5.2, 5.3, 5.4, 5.56.1, 6.2, 6.3, 6.4, 6.5, 6.67.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7, 7.810.1, 10.2 of
S. Haykin and M. Moher, Introduction to Analog & Digital Communications, 2nded., John Wiley & Sons, Inc., 2007. ISBN-13 978-0-471-43222-7.
Professor Deepa Kundur (University of Toronto) Final Exam Review 2 / 67
Chapter 3: Amplitude Modulation
Professor Deepa Kundur (University of Toronto) Final Exam Review 3 / 67
Amplitude Modulation
I In modulation need two things:
1. a modulated signal: carrier signal: c(t)2. a modulating signal: message signal: m(t)
I carrier:I c(t) = Ac cos(2πfct); phase φc = 0 is assumed.
I message:I m(t) (information-bearing signal)I assume bandwidth/max freq of m(t) is W
Professor Deepa Kundur (University of Toronto) Final Exam Review 4 / 67
Amplitude Modulation
Three types studied:
1. Amplitude Modulation (AM)(yes, it has the same name as the class of modulation techniques)
2. Double Sideband-Suppressed Carrier (DSB-SC)
3. Single Sideband (SSB)
Professor Deepa Kundur (University of Toronto) Final Exam Review 5 / 67
Amplitude Modulation TechniquesAM:
sAM(t) = Ac [1 + kam(t)] cos(2πfct)
SAM(f ) =Ac
2[δ(f − fc) + δ(f + fc)] +
kaAc
2[M(f − fc) + M(f + fc)]
f
S (f )
f
S (f )
f
S (f )
f
S (f )
2W 2W
2W 2W
W W
W W
AM
USSB
LSSB
DSB
I highest power
I BT = 2W
I lowest complexity
Professor Deepa Kundur (University of Toronto) Final Exam Review 6 / 67
Amplitude Modulation TechniquesAM: For:
1. 1 + kam(t) > 0 (envelope is always positive); and
2. fc �W (message moves slowly compared to carrier)
m(t) can be recovered with an envelope detector.
AMwave
Output+
-+
-
Professor Deepa Kundur (University of Toronto) Final Exam Review 7 / 67
Amplitude Modulation TechniquesDSB-SC:
sDSB(t) = Ac cos(2πfct)m(t)
SDSB(f ) =Ac
2[M(f − fc) + M(f + fc)] f
S (f )
f
S (f )
f
S (f )
f
S (f )
2W 2W
2W 2W
W W
W W
AM
USSB
LSSB
DSB
I lower power
I BT = 2W
I higher complexity
Professor Deepa Kundur (University of Toronto) Final Exam Review 8 / 67
Amplitude Modulation Techniques
DSB-SC:
I An envelope detector will not be able to recover m(t); it willinstead recover |m(t)|.
I Coherent demodulation is required.
ProductModulator
Low-pass�lter
Local Oscillaor
DemodulatedSignal
v (t)0s(t)
Professor Deepa Kundur (University of Toronto) Final Exam Review 9 / 67
Amplitude Modulation TechniquesSSB:
sUSSB(t) =Ac
2m(t) cos(2πfct)− Ac
2m̂(t) sin(2πfct)
SUSSB(f ) =
{Ac
2 [M(f − fc) + M(f + fc)] |f | ≥ fc0 |f | < fc
sLSSB(t) =Ac
2m(t) cos(2πfct) +
Ac
2m̂(t) sin(2πfct)
SLSSB(f ) =
{0 |f | > fcAc
2 [M(f − fc) + M(f + fc)] |f | ≤ fc
H(f ) = -j sgn(f )M(f ) M(f )
f
H(f )j
-j
h(t) = 1/( t)m(t) m(t)
Professor Deepa Kundur (University of Toronto) Final Exam Review 10 / 67
Amplitude Modulation TechniquesSSB:
upper SSB
lower SSB
f
S (f )
f
S (f )
f
S (f )
f
S (f )
2W 2W
2W 2W
W W
W W
AM
USSB
LSSB
DSB
I lowest power
I BT = W
I highest complexity
Professor Deepa Kundur (University of Toronto) Final Exam Review 11 / 67
Amplitude Modulation Techniques
SSB:
I Coherent demodulation works here as well.
ProductModulator
Low-pass�lter
Local Oscillaor
DemodulatedSignal
v (t)0s(t)
Professor Deepa Kundur (University of Toronto) Final Exam Review 12 / 67
Costas Receiver
-90 degreePhase Shifter
Voltage-controlledOscillator
ProductModulator
Low-passFilter
PhaseDiscriminator
ProductModulator
Low-pass�lter
DSB-SC wave
DemodulatedSignal
Coherent Demodulation
Circuit for Phase Locking
v (t)0local oscillator output
Professor Deepa Kundur (University of Toronto) Final Exam Review 13 / 67
Costas Receiver
-90 degreePhase Shifter
Voltage-controlledOscillator
ProductModulator
Low-passFilter
PhaseDiscriminator
ProductModulator
Low-pass�lter
DSB-SC wave
DemodulatedSignal
Q-Channel (quadrature-phase coherent detector)
I-Channel (in-phase coherent detector)
v (t)I
v (t)Q
Professor Deepa Kundur (University of Toronto) Final Exam Review 14 / 67
Quadrature Amplitude Modulation
-90 degreePhase Shifter
ProductModulator
ProductModulator
MultiplexedSignal
Messagesignal
Messagesignal
s(t) = Acm1(t) cos(2πfct) + Acm2(t) sin(2πfct)
Professor Deepa Kundur (University of Toronto) Final Exam Review 15 / 67
Quadrature Amplitude Modulation
s(t) = Acm1(t) cos(2πfct) + Acm2(t) sin(2πfct)
ProductModulator
Low-passFilter
ProductModulator
Low-pass�lter
MultiplexedSignal
-90 degreePhase Shifter
Professor Deepa Kundur (University of Toronto) Final Exam Review 16 / 67
Quadrature Amplitude Modulation
s(t) = Acm1(t) cos(2πfct) + Acm2(t) sin(2πfct)
upper SSB
lower SSB
f
S (f )DSB
f
S (f )
f
S (f )
f
S (f )
2W 2W
2W 2W
W W
W W
USSB
LSSB
QAM
Professor Deepa Kundur (University of Toronto) Final Exam Review 17 / 67
Chapter 4: Angle Modulation
Professor Deepa Kundur (University of Toronto) Final Exam Review 18 / 67
Angle ModulationI Phase Modulation (PM):
θi (t) = 2πfct + kpm(t)
fi (t) =1
2pi
dθi (t)
dt= 2πfckp
dm(t)
dt
sPM(t) = Ac cos[2πfct + kpm(t)]
I Frequency Modulation (FM):
θi (t) = 2πfct + 2πkf
∫ t
0
m(τ)dτ
fi (t) =1
2pi
dθi (t)
dt= fc + kfm(t)
sFM(t) = Ac cos
[2πfct + 2πkf
∫ t
0
m(τ)dτ
]
Professor Deepa Kundur (University of Toronto) Final Exam Review 19 / 67
carrier
message
amplitudemodulation
phasemodulation
frequencymodulation
Professor Deepa Kundur (University of Toronto) Final Exam Review 20 / 67
Angle Modulation
PM and FM:
Integrator PhaseModulator
m(t) s (t)FM
Di�erentiator FrequencyModulator
m(t) s (t)PM
Professor Deepa Kundur (University of Toronto) Final Exam Review 21 / 67
Properties of Angle Modulation
1. Constancy of transmitted power
2. Nonlinearity of angle modulation
3. Irregularity of zero-crossings
4. Difficulty in visualizing message
5. Bandwidth versus noise trade-off
Professor Deepa Kundur (University of Toronto) Final Exam Review 22 / 67
Narrowband FM
I Suppose m(t) = Amcos(2πfmt).
fi (t) = fc + kf Amcos(2πfmt) = fc + ∆f cos(2πfmt)
∆f = kf Am ≡ frequency deviation
θi (t) = 2π
∫ t
0
fi (τ)dτ
= 2πfct +∆f
fmsin(2πfmt) = 2πfct + βsin(2πfmt)
β =∆f
fmsFM(t) = Ac cos [2πfct + βsin(2πfmt)]
For narrow band FM, β � 1.
Professor Deepa Kundur (University of Toronto) Final Exam Review 23 / 67
Narrowband FM
Modulation:
sFM(t) ≈ Ac cos(2πfct)︸ ︷︷ ︸carrier
−β Ac sin(2πfct)︸ ︷︷ ︸−90oshift of carrier
sin(2πfmt)︸ ︷︷ ︸2πfmAm
∫ t0 m(τ)dτ︸ ︷︷ ︸
DSB-SC signal
Modulatingwave
IntegratorNarrow-band
FM waveProduct
Modulator
-90 degreePhase Shifter carrier
+-
Professor Deepa Kundur (University of Toronto) Final Exam Review 24 / 67
Carson’s Rule
A significant component of the FM signal is within the followingbandwidth:
BT ≈ 2∆f + 2fm = 2∆f
(1 +
1
β
)
I For β � 1, BT ≈ 2∆f = 2kfAm
I For β � 1, BT ≈ 2∆f 1β
= 2∆f∆f /fm
= 2fm
Professor Deepa Kundur (University of Toronto) Final Exam Review 25 / 67
Generation of FM Waves: Armstrong Modulator
Narrow bandModulator
FrequencyMultiplier
m(t) s(t) s’(t)Integrator
CrystalControlledOscillator
frequency isvery stable
Narrowband FM modulator
widebandFM wave
Professor Deepa Kundur (University of Toronto) Final Exam Review 26 / 67
Demodulation of FM Waves
Ideal EnvelopeDetector
ddt
I Frequency Discriminator: uses positive and negative slopecircuits in place of a differentiator, which is hard to implementacross a wide bandwidth
I Phase Lock Loop: tracks the angle of the in-coming FM wavewhich allows tracking of the embedded message
Professor Deepa Kundur (University of Toronto) Final Exam Review 27 / 67
Chapter 5: Pulse Modulation
Professor Deepa Kundur (University of Toronto) Final Exam Review 28 / 67
Pulse Modulation
I the variation of a regularly spaced constant amplitude pulsestream to superimpose information contained in a message signal
tTTs
A
I Three types:
1. pulse amplitude modulation (PAM)2. pulse duration modulation (PDM)3. pulse position modulation (PPM)
Professor Deepa Kundur (University of Toronto) Final Exam Review 29 / 67
Pulse Amplitude Modulation (PAM)
tTTs
At
TTs
t
m(t)m(0)m(-T )s
m(T )s
m(2T ) s
t
m(t)
s(t)
T
Ts
m(0)m(-T )sm(T )s
m(2T ) s
tT
Ts
t
m(t)m(0)m(-T )s
m(T )s
m(2T ) s
t
m(t)
s(t)
T
Ts
m(0)m(-T )sm(T )s
m(2T ) s
Professor Deepa Kundur (University of Toronto) Final Exam Review 30 / 67
Pulse Duration Modulation (PDM)
tT
Ts
t
m(t)m(0)m(-T )s
m(T )s
m(2T ) s
t
m(t) s(t)m(0)m(-T )s
m(T )s
m(2T ) s
PDM
Professor Deepa Kundur (University of Toronto) Final Exam Review 31 / 67
tT
Ts
t
m(t)m(0)m(-T )s
m(T )s
m(2T ) s
t
m(t) s(t)m(0)m(-T )s
m(T )s
m(2T ) s
PDM
m(t) s(t)
PPM
Professor Deepa Kundur (University of Toronto) Final Exam Review 32 / 67
Summary of Pulse ModulationLet g(t) be the pulse shape.
I PAM:
sPAM(t) =∞∑
n=−∞kam(nTs)g(t − nTS)
where ka is an amplitude sensitivity factor; ka > 0.
I PDM:
sPDM(t) =∞∑
n=−∞g
(t − nTs
kdm(nTs) + Md
)where kd is a duration sensitivity factor; kd |m(t)|max < Md .
I PPM:
sPPM(t) =∞∑
n=−∞g(t − nTs − kpm(nTs))
where kp is a position sensitivity factor; kp |m(t)|max < (Ts/2).
Professor Deepa Kundur (University of Toronto) Final Exam Review 33 / 67
Pulse-Code Modulation
I Most basic form of digital pulse modulation
PCM Data Sequence
ChannelOutput
Transmitter ReceiverTranmissionPathSO
URC
E
DES
TIN
ATIO
N
Professor Deepa Kundur (University of Toronto) Final Exam Review 34 / 67
PCM Transmitter
PCM Data Sequence
Anti-aliasedCts-timeSignal
Discrete-timeSignal
Digitalsignal
Continuous-timeMessageSignal
Source Sampler Quantizer EncoderLow-passFilter { {
Anti-aliasingFilter
Sampling aboveNyquist with
Narrow RectangularPAM Pulses
Maps Numbersto Bit Sequences
{ {Using a
Non-uniformQuantizer
Professor Deepa Kundur (University of Toronto) Final Exam Review 35 / 67
PCM Transmitter: Sampler
PCM Data Sequence
Anti-aliasedCts-timeSignal
Discrete-timeSignal
Digitalsignal
Continuous-timeMessageSignal
Source Sampler Quantizer EncoderLow-passFilter { {
Anti-aliasingFilter
Sampling aboveNyquist with
Narrow RectangularPAM Pulses
Maps Numbersto Bit Sequences
{ {
Using aNon-uniform
Quantizer
ts(t)
T
Ts
m(0)m(-T )sm(T )s
m(2T ) sm(t)
t
m(0)m(-T )sm(T )s
m(2T ) s
Professor Deepa Kundur (University of Toronto) Final Exam Review 36 / 67
PCM Transmitter: Non-Uniform Quantizer
PCM Data Sequence
Anti-aliasedCts-timeSignal
Discrete-timeSignal
Digitalsignal
Continuous-timeMessageSignal
Source Sampler Quantizer EncoderLow-passFilter { {
Anti-aliasingFilter
Sampling aboveNyquist with
Narrow RectangularPAM Pulses
Maps Numbersto Bit Sequences
{ {
Using aNon-uniform
Quantizer
AmplitudeCompressor
v
m2 3-2-3 -
23
-2
-3
-
UniformQuantizer
-1 10-2-3 2 3n
1
v[n] m[n]
0 0.25 0.5 0.75 1.0
0.25
0.5
0.75
1.0
Normalized input |m|
Nor
mal
ized
out
put |
v| large mu
Professor Deepa Kundur (University of Toronto) Final Exam Review 37 / 67
PCM Transmitter: Encoder
PCM Data Sequence
Anti-aliasedCts-timeSignal
Discrete-timeSignal
Digitalsignal
Continuous-timeMessageSignal
Source Sampler Quantizer EncoderLow-passFilter { {
Anti-aliasingFilter
Sampling aboveNyquist with
Narrow RectangularPAM Pulses
Maps Numbersto Bit Sequences
{ {
Using aNon-uniform
Quantizer
Quantization-Level Index Binary Codeword (R = 3)
0 0001 0012 0103 0114 1005 1016 1107 111
Professor Deepa Kundur (University of Toronto) Final Exam Review 38 / 67
Encoder: Example
Professor Deepa Kundur (University of Toronto) Final Exam Review 39 / 67
PCM: Transmission Path
PCM Data Sequence
ChannelOutput
Transmitter ReceiverTranmissionPathSO
URC
E
DES
TIN
ATIO
N
ChannelOutput
TranmissionLine
RegenerativeRepeater
TranmissionLine
RegenerativeRepeater
TranmissionLine
...PCM Data Shaped for Transmission
Decision-making
DeviceAmpli�er-Equalizer
TimingCircuit
DistortedPCMWave
RegeneratedPCMWave
Professor Deepa Kundur (University of Toronto) Final Exam Review 40 / 67
PCM: Regenerative RepeaterDecision-making
DeviceAmpli�er-Equalizer
TimingCircuit
DistortedPCMWave
RegeneratedPCMWave
t
t
t
t
t
THRESHOLD
BIT ERROR
“0” “0” “0”“1” “1” “1”
Original PCM Wave
Professor Deepa Kundur (University of Toronto) Final Exam Review 41 / 67
PCM: Receiver
Two Stages:
1. Decoding and Expanding:
1.1 regenerate the pulse one last time1.2 group into code words1.3 interpret as quantization level1.4 pass through expander (opposite of compressor)
2. Reconstruction:
2.1 pass expander output through low-pass reconstruction filter(cutoff is equal to message bandwidth) to estimate originalmessage m(t)
Professor Deepa Kundur (University of Toronto) Final Exam Review 42 / 67
Chapter 6: Baseband Data Transmission
Professor Deepa Kundur (University of Toronto) Final Exam Review 43 / 67
Baseband Transmission of Digital Data
ThresholdBinary input sequence
LineEncoder
0011100010Source
Threshold
Decision-Making Device
Sampleat time
DestinationTransmit-Filter G(f )
ChannelH(f )
Receive-�lter Q(f )
Outputbinary data
Threshold
Decision-Making Device
Sampleat time
Pulse Spectrum P(f )
bk = {0, 1} and ak =
{+1 if bk is symbol 1−1 if bk is symbol 0
s(t) =∞∑
k=−∞
akg(t − kTb)
x(t) = s(t) ? h(t)
y(t) = x(t) ? q(t) = s(t) ? h(t) ? q(t)
=∞∑
k=−∞
akg(t − kTb) ? h(t) ? q(t) =∞∑
k=−∞
akp(t − kTb)
Professor Deepa Kundur (University of Toronto) Final Exam Review 44 / 67
Baseband Transmission of Digital Data
ThresholdBinary input sequence
LineEncoder
0011100010Source
Threshold
Decision-Making Device
Sampleat time
DestinationTransmit-Filter G(f )
ChannelH(f )
Receive-�lter Q(f )
Outputbinary data
Threshold
Decision-Making Device
Sampleat time
Pulse Spectrum P(f )
∴ y(t) =∞∑
k=−∞
akp(t − kTb)
where p(t) = g(t) ∗ h(t) ∗ q(t)
P(f ) = G (f ) · H(f ) · Q(f ).
Professor Deepa Kundur (University of Toronto) Final Exam Review 45 / 67
Baseband Transmission of Digital Data
Threshold
Decision-Making Device
Sampleat time
Transmit-Filter G(f )
ChannelH(f )
Receive-�lter Q(f )
ThresholdBinary input sequence
LineEncoder
0011100010Source Destination
Outputbinary data
Threshold
Decision-Making Device
Sampleat time
Pulse Spectrum P(f )
P(f ) = G(f )H(f )Q(f )
Professor Deepa Kundur (University of Toronto) Final Exam Review 46 / 67
Baseband Transmission of Digital Data
Threshold
Decision-Making Device
Sampleat time
Pulse Spectrum P(f )
P(f ) = G(f )H(f )Q(f )
yi = y(iTb) and pi = p(iTb)
yi =√Eai︸ ︷︷ ︸
signal to detect
+∞∑
k=−∞,k 6=i
akpi−k︸ ︷︷ ︸intersymbol interference
for i ∈ Z
To avoid intersymbol interference (ISI), we need pi = 0 for i 6= 0.
Professor Deepa Kundur (University of Toronto) Final Exam Review 47 / 67
The Nyquist Channel
I Minimum bandwidth channel
I Optimum pulse shape:
popt(t) =√E sinc(2B0t)
Popt(f ) =
{ √E
2B0−B0 < f < B0
0 otherwise, B0 =
1
2Tb
Note: No ISI.
pi = p(iTb) =√E sinc(2B0iTb)
√E sinc
(2 · 1
2TbiTb
)=√E sinc(i) = 0.
Disadvantages: (1) physically unrealizable (sharp transition in freq domain); (2)
slow rate of decay leaving no margin of error for sampling times.
Professor Deepa Kundur (University of Toronto) Final Exam Review 48 / 67
Raised-Cosine Pulse Spectrum
I has a more graceful transition in the frequency domain
I more practical pulse shape:
p(t) =√E sinc(2B0t)
(cos(2παB0t)
1− 16α2B02t2
)
P(f ) =
√E
2B00 ≤ |f | < f1
√E
4B0
{1 + cos
[π(|f |−f1)(B0−f1)
]}f1 < f < 2B0 − f1
0 2B0 − f1 ≤ |f |
α = 1− f1B0
BT = B0(1 + α) where B0 =1
2Tband fv = αB0
Note: No ISI. ∵ pi = 0.Professor Deepa Kundur (University of Toronto) Final Exam Review 49 / 67
Raised-Cosine Pulse Spectrum
f (kHz)
A= sqrt(E)/2B
A/2
0
0B0-B 02B0-2B0B /2 0B /2 0B /2 0B /2
Raised-Cosine
NyquistPulse
Professor Deepa Kundur (University of Toronto) Final Exam Review 50 / 67
The Eye Pattern
Slope dictates sensitivityto timing error
Best samplingtime
Distortion at sampling time
NO
ISE
MA
RGIN
Time interval over which waveis best sampled.
ZERO-CROSSINGDISTORTION
Professor Deepa Kundur (University of Toronto) Final Exam Review 51 / 67
Chapter 7: Digital Band-Pass ModulationTechniques
Professor Deepa Kundur (University of Toronto) Final Exam Review 52 / 67
Binary Modulation Schemes
c(t) = Ac cos(2πfct + φc)
I Binary amplitude-shift keying (BASK): carrier amplitude is keyed between
two possible values (typically√Eb and 0 to represent 1 and 0, respectively);
carrier phase and frequency are held constant.
I Binary phase-shift keying (BPSK):carrier phase is keyed between twopossible values (typically 0 and π to represent 1 and 0, respectively); carrieramplitude and frequency are held constant.
I Binary frequency-shift keying (BFSK): carrier frequency is keyed betweentwo possible values (typically f1 and f2 to represent 1 and 0, respectively);carrier amplitude and phase are held constant.
Professor Deepa Kundur (University of Toronto) Final Exam Review 53 / 67
Preliminaries
c(t) = Ac cos(2πfct + φc)
I Tb represents the bit duration
I Eb represents the energy of the transmitted signal per bit
I In digital communications the carrier amplitude is normalized to have unitenergy in one bit duration; thus we set
Ac =
√2
Tb
I The carrier frequency fc = kTb
for k ∈ Z to ensure an integer number ofcarrier cycles in a bit duration.
Professor Deepa Kundur (University of Toronto) Final Exam Review 54 / 67
Carrier for Digital Communications
Therefore
c(t) =
√2
Tbcos(2πfct + φc).
For k = 4
0t
Professor Deepa Kundur (University of Toronto) Final Exam Review 55 / 67
Binary Amplitude-Shift Keying (BASK)
Let φc = 0 and the carrier frequency is fc .
b(t) =
{ √Eb for binary symbol 1
0 for binary symbol 0
c(t) =
√2
Tbcos(2πfct + φc) =
√2
Tbcos(2πfct)
s(t) = b(t) · c(t)
=
{ √2Eb
Tbcos(2πfct) for symbol 1
0 for symbol 0
Professor Deepa Kundur (University of Toronto) Final Exam Review 56 / 67
BASK Transmitter and Receiver
Transmitter
BASKTransmittedSignal
LevelEncoding
ProductModulator
carrier
EnvelopeDetector
Low-passFilterBASK
wave
Output+
-
Threshold
Decision-Making Device
Binary inputsequence
t0011100010
Sampleat time
Receiver
EnvelopeDetector
Low-passFilterBASK
wave
Output+
-
Threshold
Decision-Making Device
BASKTransmittedSignal
LevelEncoding
ProductModulator
carrier
Binary inputsequence
t0011100010
Sampleat time
Professor Deepa Kundur (University of Toronto) Final Exam Review 57 / 67
Binary Phase-Shift Keying (BPSK)
si(t) =
√
2Eb
Tbcos(2πfct) for symbol 1 (i = 1)√
2Eb
Tbcos(2πfct + π) for symbol 0 (i = 2)
=
√
2Eb
Tbcos(2πfct) for symbol 1 (i = 1)
−√
2Eb
Tbcos(2πfct) for symbol 0 (i = 2)
Professor Deepa Kundur (University of Toronto) Final Exam Review 58 / 67
BPSK Transmitter and Receiver
Transmitter
Low-passFilter
Threshold
Decision-Making Device
Sampleat time
BPSKTransmittedSignal
Binary inputsequence
LevelEncoding
ProductModulator
carrier
ProductModulator
local carrierfrom PLL
(Non-return-to zero)
t0011100010
Receiver
Low-passFilter
Threshold
Decision-Making Device
Sampleat time
BPSKTransmittedSignal
Binary inputsequence
LevelEncoding
ProductModulator
carrier
ProductModulator
local carrierfrom PLL
(Non-return-to zero)
t0011100010
Professor Deepa Kundur (University of Toronto) Final Exam Review 59 / 67
Binary Frequency-Shift Keying (BFSK)
Let φc = 0, |f1 − f2| = 1Tb
and fi = kiTb
(integer number of cycles in a
bit duration).
si(t) =
√
2Eb
Tbcos(2πf1t) for symbol 1 (i = 1)√
2Eb
Tbcos(2πf2t) for symbol 0 (i = 2)
0t
0 1 0 1 1 0 0 0
4 cycles 4 cycles 4 cycles 4 cycles 4 cycles5 cycles 5 cycles 5 cycles
Professor Deepa Kundur (University of Toronto) Final Exam Review 60 / 67
BFSK Transmitter and Receiver
Transmitter
BFSKTransmittedSignal
Binary inputsequence 0011100010
controlsswitchposition
localoscillator
localoscillator
Receiver
EnvelopeDetector
EnvelopeDetector
Band-pass�lter, f1
Comparator
Band-pass�lter, f2
v1
v2
{ 1 if v1 > v20 if v2 > v1
Professor Deepa Kundur (University of Toronto) Final Exam Review 61 / 67
Summary
BASK s1(t) =√
2Eb
Tbcos(2πfct) for symbol 1
s2(t) = 0 for symbol 0
BPSK s1(t) =√
2Eb
Tbcos(2πfct + 0) for symbol 1
s2(t) =√
2Eb
Tbcos(2πfct + π) for symbol 0
BFSK s1(t) =√
2Eb
Tbcos(2πf1t) for symbol 1
s2(t) =√
2Eb
Tbcos(2πf2t) for symbol 0
Professor Deepa Kundur (University of Toronto) Final Exam Review 62 / 67
Summary: Phasor Diagrams
BASK
BPSK
BFSK
symbol1
symbol1
symbol1
symbol0
symbol0
symbol0
Professor Deepa Kundur (University of Toronto) Final Exam Review 63 / 67
Summary: Phasor Diagrams
BASK
BPSK (antipodal)
BFSK (orthogonal)
symbol1
symbol1
symbol1
symbol0
symbol0
symbol0
Professor Deepa Kundur (University of Toronto) Final Exam Review 64 / 67
M-ary Digital Modulation SchemesFor M = 2m and T = mTb,
I M-ary Phase-Shift Keying
si (t) =
√2E
Tcos
(2πfc +
2π
Mi
)i = 0, 1, . . . ,M − 1, 0 ≤ t ≤ T .
I M-ary Quadrature Amplitude Modulation
si (t) =
√2E0
Tai cos(2πfct)−
√2E0
Tbi sin(2πfct)
i = 0, 1, . . . ,M − 1, 0 ≤ t ≤ T .
I M-ary Frequency-Shift Keying
si (t) =
√2E
Tcos( πT
(n + i)t)
i = 0, 1, . . . ,M − 1, 0 ≤ t ≤ T .
Professor Deepa Kundur (University of Toronto) Final Exam Review 65 / 67
Signal Constellations
M-PSK M-QAM M-FSK
1. For a given basis set: φ1, φ2, . . . , φn. Compute correlation of si (t) witheach of the bases members.
2. Graph the result in signal space.
Professor Deepa Kundur (University of Toronto) Final Exam Review 66 / 67
Important Identities
cos(A + B) = cos(A) cos(B)− sin(A) sin(B)
cos(A) cos(B) =1
2cos(A + B) +
1
2cos(A− B)
cos(A) sin(B) =1
2sin(A + B)− 1
2sin(A− B)
cos(A) = sin(A +
π
2
)cos(A + π) = − cos(A)
cos(A) = cos(−A) sin(A) = − sin(−A)
cos2(A) =1
2+
1
2cos(2A)
cos2(A) + sin2(A) = 1
cos(A) ≈ 1 for |A| � 1
sin(A) ≈ A for |A| � 1
�Professor Deepa Kundur (University of Toronto) Final Exam Review 67 / 67