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Chapter Four:
Bandpass Modulation
What is bandpass modulation? Baseband modulation
The form of shaped pulses
Bandpass modulation The shaped pulses modulate a sinusoid, called a carrier
wave or a carrier Types
Phase shift keying Frequency shift keying Amplitude shift keying Continuous phase modulation Hybrid
Why using a carrier? The carrier converts to an electromagnetic field for
propagation through antennas The size of the antenna depends on the wavelength λ
Ex: Cellular telephone, antenna size λ/4
cm
MHzf
milesm
m
Hzf
smcf
c
84
10900/100.3 bemight size antenna The
,900carrier aWith
15105.24
bemight size antenna The
103000
100.3
,3000 signal Baseband
/100.3, Wavelength
68
4
58
8
Sinusoidal Waveforms The general form of the carrier wave
)(cos)()( ttAts
Time-varying amplitude Time-varying angle
)(cos)()( 0 tttAts
Radian frequency 2πf Phase
Coherent vs. non-coherent With/without the knowledge of the carrier’s phase to detect the signals Complexity vs. performance
Phase Shift Keying (PSK)
M
it
MiTtttT
Ets
i
ii
2)(
,...2,1,0,)(cos2
)( 0
duration symbol :energy symbol : TE
Waveform Vector
s2s1
M=2
T T T
Frequency Shift Keying (FSK)
MiTttT
Ets ii ,...2,1,0,cos
2)(
duration symbol :energy symbol : TE
Waveform Vector
s2
s1
M=3
s3T T T
Frequency Shift Keying (FSK) Orthogonal signals in FSK
Not all FSK signaling is orthogonal Example: f1=10000Hz, f2=11000Hz, are they
orthogonal?? To meeting the criterion the spacing between
the tones [on page 202, example 4.3]
A frequency separation is a multiple of 1/T Hz The minimum requirement spacing for noncoherent
detection is 1/T The minimum requirement spacing for coherent
detection is 1/(2T)
Amplitude Shift Keying (ASK)
MiTttT
tEts i
i ,...2,1,0,cos)(2
)( 0
Waveform Vector
Binary ASK (On-Off Keying)
s2 s1
M=2
T T T
Hybrid -- Amplitude Phase Keying
MiTtttT
tEts i
ii ,...2,1,0,)(cos
)(2)( 0
Waveform Vector
T
T
T M=8
Where are we?!
Format Pulse modulation Bandpass modulation
Format Demodulation, sampling and detection
Correlation Receiver
MiTttntstr i ...2,1,0)()()(
Receiver:
Step 1: reduce r(t) to a single random variable z(T) or a set of RVs zi(T)
Step 2: a symbol decision is made on the basis of comparing z(T) to a threshold or choosing the max zi(T)
Frequency Down-conversion
Receiving filter
Sampling at TEqualizing
filter Detection
Coherent Detection – Binary Case Use a single correlator
Use two correlators
Coherent Detection – Decision
021
2
0
2
0
2
2
0
1
0
1
2)(
2
1exp
2
1)|(
2
1exp
2
1)|(
aaTz
azszp
azszp
Conditional probability functions
02: Noise variance
0: standard deviation
Coherent Detection – PSK Review
The information is contained in the instantaneous phase of the modulated carrier
For a binary PSK: 0 and 180 are used In fact, PSK can be views as ASK signal with the
bipolar carrier amplitudes The ideal detector requires perfect knowledge of
the un-modulated carrier phase at the receiver
Coherent Detection Binary PSK (BPSK)
TttT
Et
T
Ets
TttT
Ets
0,cos2
cos2
)(
0,cos2
)(
002
01
)()()(&)()()(
cos2
)(:function basis a Assuming
1121211111
01
tEtatstEtats
tT
t
EdtttntEEszE
EdtttntEEszE
tT
t
T
T
0 12
112
0 12
111
01
)()()(|
)()()(|
cos2
)( reference with thez(T) of valueexpected The
Take notes
Coherent Detection Sampled Matched Filter
Nyquist rate
Sampling time needs to be equal to or less than the symbol time
Shift to the register
Pay attention!
)3()1()( nsnNsnc iii
1
0
1
0
)()()(
)()()(
K
niii
N
nii
ncnkskzE
ncnkrkz
Coherent Detection Difference between MF and Correlators
If the timing of the MF and correlators are aligned, their outputs at the end of symbol time are identical
MF: A new output value is available in response to each new input sample; Equated to several correlators operating at different starting points of the input time series
Correlators: Computes an output once per symbol time
Timing errors in the correlator badly degraded performance
Example: Consider the waveform set s1(t)=At and s2(t)=-At,
t=[0,T] where k=0,1,2,3. Illustrate how a sampled matched filter can be used to detect a received signal from this sawtooth waveform set in the absence of noise.
Coherent Detection Multiple Phase-Shift Keying
M
it
T
Etsi
2cos
2)( 0
)(2
sin)(2
cos
)()()(
sin2
)(&cos2
)( :Choose
21
2211
0201
tM
iEt
M
iE
tatats
tT
ttT
t
iii
Coherent Detection – Multiple PSK
Inphase component
Quadrature component
Noise estimate of the transmitted
Probability of Bit Error for Coherently Detected BPSK
Detection rule:
otherwise)(
02
if)(
2
2101
ts
aa z(T)ts
Two types of errors:
0
0
)|()|(
)|()|(
22
11
r
r
dzszpseP
dzszpseP
Probability of Bit Error for Coherently Detected BPSK
)()|()()|( 2211 sPsePsPsePPB
A priori probability (equally likely)
dz
az
dzszpsePsePP
aar
aarB
2
0
2
2/0
2/ 221
2
1exp
2
1
)|()|()|(
210
210
00
212
2/
02
2
22exp
2
1
/)(
021 N
EQ
aaQdu
uP
azuLet
bu
aauB
Q function: complementary error function/co-error function
The standard deviation of the noise
2/ variancenoise the: 020 N
Probability of Bit Error for Binary Coherent Signals
In general, the BER becomes
vectorssignalbetween angle the:
signals obetween twt coefficienn correlatio-cross timeThe cos
)1(
2exp
2
1
0/)1(
2
0
N
EQdu
uP b
NEBb
Binary PSK: 1
Binary FSK: 02/
OOK: 02/
Example: Find the expected number of bit errors made in one day for a
BPSK system with a bit rate of 5000 bps. The received waveforms and are coherently detected with a matched filter. The value of A is 1mV. Assume that the single-sided noise power spectral density is
and that signal power and energy per bit are normalized relative to 1 ohm load.
tAts 01 cos)( tAts 02 cos)(
HzwN /10 110
Special Case – Detection
Case 1: signal energy is different
Case 2: the decision threshold is not at the middle point
0
0
)|(2
1)|(
2
121 r
r
B dzszpdzszpP
Example
NonCoherent Detection Differential PSK (DPSK)
The procedure of encoding the data differentially No attempt is made to determine the actual value of the
phase of the incoming signal
)()(cos2
)(
...2,1,0,)(cos2
)(
tnttT
Etr
MiTtttT
Ets
io
ioi
Typically assumed as a random variable uniformly distributed between zero and 2
NonCoherent Detection cont.
Detection of Differential PSK (DPSK) Matched filters are not possible – less efficient The carrier phase of the previous signaling interval can be
used as a phase reference The detector calculates the coordinate of the incoming
signal by correlating it with locally generated waveforms The detector then measures the angle between the currently
received signal vector and the previously received signal vector
NonCoherent Detection – Binary DPSK
addition 2-modulo :
)()1()(
)()1()(
kmkckc
kmkckc
Read your textbook
…0111011011…
… 01110110111
NonCoherent Detection – Binary DPSK
Optimal: requires a reference carrier in frequency but not necessarily in phase with the received carrier
Complex envelope: inphase component and quadrature component of the carrier wave
ttyttx
tjttjytxts
sin)(cos)(
)sincos)()(Re)(
Example The bit stream 1 0 1 0 1 0 1 1 1 1 is to be transmitted
using DPSK modulation. Show the encoded message (first bit is 1) and the detected message.
addition 2-modulo :
)()1()(
)()1()(
kmkckc
kmkckc1 0 1 0 1 0 1 1 1 1
1
1
0 0 1 1 0 0 1 0 1 0
1 0 0 1 1 0 0 0 0 0
Example When cables are installed in a building, it is not
unusual for the engineers to get the connections of the twisted pair reversed. How can a binary signaling scheme be designed to cope with this eventuality and maintain correct polarity data transfer?
Probability of Bit Error for Binary DPSK The decision is based on the phase difference
between successively received signals
TttT
Etx
TttT
Etx
0cos2
)(
0cos2
)(
02
01
Ttxxorxxts
Ttxxorxxts
20),(),()(
20),(),()(
12212
22111
Probability of Bit Error for Binary DPSK
0
exp2
1
N
EP b
B
Example A DPSK transmitter can generate an average power
of 1 nW at the input to a receiver which has a noise power density of 0.5 10-12 Watts/Hz. If the symbol rate is 100 symbols per second, what is the BER performance for a DPSK decoder in the receiver?
Let us work on bandwidth again… A binary PSK modem is designed to work within a
bandwidth of 8 kHz. What is the maximum data rate that can be delivered if a raised cosine filter with a = 1 is used?
Answer: 4kHz
FSK Generation FSK
The information contained in the frequency of the carrier
Insensitive to amplitude fluctuations in the channel
Generating FSK signals Switching between distinct frequency sources Voltage Controlled Oscillator Quadrature Modulator
MiTttT
Ets ii ,...2,1,0,cos
2)(
FSK Generation Switching between distinct frequency sources
FSK Generation Voltage Controlled Oscillator
FSK Generation Quadrature (Vector) Modulator
Symbol 1: c+
Symbol 1: c-
Summing appropriate amount of an in-phase and quadrature version of the carrier signal
tccos tcsin
Coherent FSK Detection
Decision: if the output from the mark filter is larger than that from the space filter, a decision is made that a mark signal was transmitted.
A matched filter demodulator is optimum because its filters are matched” to the transmitted signal so that their response to the desired signal is maximized with respect to their noise response.
NonCoherent Detection of FSK
Pass the signal through two bandpass filters turned to the two signal frequencies
Data can be recovered using an envelope detector [diode + smoothing filter]
Detect which has the larger output averaged over a symbol period
NonCoherent Detection of FSK
Phase-locked loop: A voltage controlled oscillator: output frequency is proportional to the input voltage A phase detector: produce a voltage output proportional to the phase difference A loop filter: control the dynamics of the feedback circuit
Advantages/Disadvantages of FSK Advantages
A constant envelope modulation insensitive to amplitude variation in the channel compatible with non-linear transceivers
Detection is based on relative frequency changes does not require absolute frequency accuracy in the channel
Disadvantages Less bandwidth efficiency BER performance is worth than of PSK
Probability of Error
Example What is the bit error probability for non-coherent
binary FSK for an Eb/N0 value of 10 dB? What approximate Eb/N0 is required to achieve the same BER performance of coherent FSK and PSK?
Example: page 240, 4.17 Consider that a BFSK domodulator/detector has s
synchronization error consisting of a time bias pT, where p is a fraction of the symbol time T. In other words, the detection of a symbol starts early (late) and concludes early (late) by an amount pT. Assume equally likely signaling and perfect frequency and phase synchronization. Find the general expression for bit-error probability as a function of p. If the received SNR is 9.6 dB and p=0.2, compute the value of
degraded BER due to the timing bias. If one did not compensate for the timing bias in this example, how
much additional SNR must be provided in order to restore the BER that exists when p=0.
Amplitude Shift Keying [not from the textbook]
ASK The information contained in the amplitude of the carrier
On-off Keying: the simplest form of bandpass data modulation
MiTttT
Ets i
i ,...2,1,0,cos2
)(
ASK Generation Linear modulator: an ASK signal can be realized
using a mixer to multiply the carrier with the baseband symbol stream
ASK Generation Switch
Binary ASK: switch to gate the carrier on and off, driven by the data signal.
M-ary signals: with differing amplitudes to represent the required number of symbol states
Non-linear process
ASK Generation Bandpass filtering method
Filter is needed after modulation A high frequency modulated data signal can be eliminated
ASK Generation Baseband filtering method
Using the mixer-based approach the baseband data stream can be pre-filtered using a low pass (root raised cosine) filter
Non-Coherent Detection Envelope detection: the information is conveyed in
the amplitude or envelope of the modulated carrier signal
A diode rectifier and smoothing filter
Coherent Detection By mixing the incoming data signal with a locally
generated carrier reference and selecting the difference component from the mixer output.
Coherent vs. Non-Coherent Phase or vector representation diagram
Case study: off state with noise Non-coherent: envelope detection N Coherent: N/2
Carrier Recovery Method 1: send a reference signal along with the
data signal Method 2: recover the carrier from the modulated
data signal Phase-locked loop (PLL)
By locking an oscillator to the phase of the incoming carrier when a carrier-on symbol is sent, and holding this oscillator phase when the carrier is off, it is possible to produce the required coherent reference.
Matched Filter Matched filter
Baseband transmission For optimizing the signal to noise ratio at the output of a
data receiver was discussed. Assumption: if the coherent detection is used
A matched filter pair such as the root raised cosine filters can thus be used to shape the source and received baseband data symbols in ASK
Timing Recovery Early-late gate synchronizer
The optimized filters are used Matched filter for instance
One with a slightly advanced timing reference One with a slightly retarded timing referenceComparator: periodically compared to see which is the larger The optimum timing signal is passed to a third data detector.
Example A coherent ASK demodulator has a 5o error in its
locally generated carrier reference. What will be the degradation in noise power immunity compared with an ideal demodulator?
M-ary Bandpass Modulation* Common knowledge
In principle, we can use any number of symbols for converting digital information.
A practical limit on the number of states to be used: the ability of receiving equipment to accurately resolve the individual states
A practical limit on the number of states to be used: the levels of noise and distortion introduced by the cannel and by the Tx and Rx units
Example: telephone modem 1024 symbol states vs. cellular systems two or four states
M-ary Bandpass Modulation M-ary signaling review
The processor considers k bits at a time The modulator produce one of M=2k waveforms
Does M-ary signaling improve the system performance? Error performance Bandwidth performance
M-ary ASK Implementation of M-ary ASK
Extension of binary ASK
Mixer at TX: to multiply the carrier with the baseband signal Coherent detectionMixer at RX: to multiply the received signal with a locally generated carrier referenceFilter: to select the DC value
M-ary ASK Performance
No opportunity to exploit orthogonally BER performance
Sensitivity to amplitude change Need for reasonable linearity
Constellation diagram* A representation of a signal modulated by a digital
modulation scheme To display the signal as a two-dimensional scatter
diagram in the complex Represents the possible symbols that may be selected
by a given modulation
M-ary FSK Increasing the noise immunity to achieve reliable
date transmission Possibility of using both orthogonal symbols or non-
orthogonal symbols Example: an orthogonal 8-ary FSK set with a symbol rate
of 1200 symbols/sec for coherent detection 1000Hz, 1600Hz, 2200Hz, 2800Hz, 3400Hz, 4000Hz, 4600Hz, and 5200Hz (same staring phase)
M-ary FSK Performance
M BER performance [cost: bandwidth]
M-ary PSK: Quadrature PSK PSK modulation scheme with four phase states
0, 90, 180, and 270 Twice the speed of BPSK in the same bandwidth
Modulator
Half the rate of the input data
Shape the data pulses in each channel
Continue…
M-ary PSK: Quadrature PSK
M-ary PSK: Quadrature PSK Demodulator
M-ary PSK: Quadrature PSK BER performance
Theoretical identical to that for BPSK
Differential QPSK Detection
/4 QPSK Widely used in the majority of digital radio modems Two identical constellations which are rotated by 45°
(π / 4 radians) with respect to one another reduces the phase-shifts from a maximum of 180°, but only to a maximum of 135° the filtered QPSK signal never passes through zero
Offset QPSK Staggering the input data streams to the two
quadrature BPSK modulators by half of symbol period
Symbol Error Performance
How about bit error probability? [Less]
How about bandwidth efficiency?
Probability of Symbol Error Equally-likely coherently detected M-ary PSK
Differentially coherent detection of M-ary DPSK
symbolper energy :log
sin2
2)(
2
0
MEE
MN
EQMP
bs
sE
MN
EQMP s
E2
sin2
2)(0
Probability of Symbol Error Equally likely coherently detected M-ary orthogonal
FSK
Equally likely noncoherently detected M-ary orthogonal FSK
0
)1()(N
EQMMP s
E
0
020
2exp
2
1)(
)!(!
!
exp)1(exp1
)(
N
EMMP
jMj
M
j
M
jN
E
j
M
N
E
MMP
sE
sM
j
jsE
Bit Error Rate vs. Symbol Error Rate
For orthogonal signals
For multiple phase signals
1
2/
12
2 1
M
M
P
Pk
k
E
B
M
PP E
B2log
Example: page 240, 4.12
Consider a 16-ary PSK system with symbol error probability 10-5. A Gray code (binary) is used for the symbol to bit assignment. What is the approximate bit error probability?
For a 16-ary orthogonal FSK system.
654
31
65
103.51012
2
12
2
105.24
10
Ek
k
B
EB
PP
k
PP
Gray code A binary numbering system two successive values differ in only one digit
Example: 00 01 11 10 Example: 000 001 011 010 110 111 101 100
Applications Sensor: angle detection Digital communication: error detection
