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Slide 1
Digital ModulationIr. Muhamad Asvial, MSc., PhD
Center for Information and Communication Engineering Research (CICER)Electrical Engineering Department - University of Indonesia
E-mail: asvial@ee.ui.ac.idhttp://www.ee.ui.ac.id/~cicer
Slide 2 Digital Modulation
Why digital modulation?
• High robustness against noise
• Low band occupancy
• Low power consumption
• Integration
• Frequency regions
Slide 3 Channel model
Discrete
Source
Series to
ParallelMapper Waveform
Selector
Waveform DetectorDemapperParallel to
Series
Discrete
sink
+
channels
Slide 4
Memory and Memoryless Modulator
• The modulator is said to have Memory:When the mapping from the digital sequence {Dn} to waveforms is performed under the constraint that a waveform transmitted in any time interval depends on one or more previously transmitted waveform
• The modulator is said to have Memoryless:When the mapping from the sequence {Dn} to the waveform is performed without any constraint on previously transmitted waveforms.
Slide 5 Channel model
• Compose through different waveforms
• Introductory example for modulation with and without memory
Slide 6 Introductory example
1 1 0 1 0 0 0 1 0 1 1 0
1
0
10110100
111101100110
010011001000
Binary Modulation
Quaternary Modulation
Octonary Modulation
)(8 tx ASK
)()(2 txtx BPSKASK
)(4 tx ASK
)(8 tx ASK
Slide 7
Bandwidth occupancy can be reduced by increasing
Introductory example
Form the previous figure we can conclude:
If we assume that the transmission power is the same for binary and quaternary modulation, than which waveform detection scheme is more reliable.
where might change in the interval
Slide 8 BPSK transmitted signal
0 1 1 0 0 1
Slide 9 BPSK transmitted signal
A generalization of BPSK for arbitrary is called PSK,
Any signal with a time-varying envelope uses PSK modulation would be clearly distorted.
A main disadvantage of a PSK-signal is that the bandwidth occupancy might be quite high.
Let us now introduce a so called continous phase modulated signal (or CPM-signal for short)
Slide 10 CPM transmitted signal
0 1 1 0 0 1
Slide 11 Error probability
Now let us calculate the error probability in AWGN channel
Slide 12 Error probability
Let us define
Which of the following constellations has the minimum error probability?
Slide 13 Average energy
The average energy for transmission of a constellation with equally-likely signal vectors is given by
Slide 14 Average energy
Slide 15 Minimum average energy
To minimize we have to set the partial derivatives equal to zero
Hence we obtain
Slide 16 Error probability (M=2)
Slide 17 Error probability (M=2)
Slide 18 Error probability (M=2)
By introducing the so-called error function
we finally obtain
which can be written as
By using the complementary error function
Slide 19 Error probability (M=2)
Correlation coefficient
so we can also write
Slide 20 Error probability (M=2)
If we further assume equal energy signals
Due to SCHWARZ-inequality
takes values between
Hence,
Slide 21 Error probability (M=2)
Slide 22 4-PSK
Slide 23 4-PSK error probability
Slide 24
For high signal-to-noise ratio (SNR) and when GRAY code is applied, could be approximated to its lower limit
4-PSK error probability
Which is more relevant in practice, symbol error probability or bit error probability?
The relation between symbol error probability and bit error probability could be written as
Beside error probability, what else are important measures for the quality of a communication link ?
Slide 25
The so-called SHANNON-bandwidth of a signal set with dimensions with approximate duration is defined as
Fundamental remarks
Bandwidth and power efficiency
bit duration
symbol duration
symbol energy
symbol power
bit energy
bandwidth efficiency
power efficiency
Slide 26 Fundamental remarks
Now we are able to define the signal-to-noise (power) ratio
The higher the more efficient the use of the available bandwidth made by the modulation scheme.
Bandwidth efficiency
Slide 27 Fundamental remarks
Power efficiency
Hence, power efficiency is defined as
For example for antipodal binary modulation
Symbol error probability depends on
Slide 28 Amplitude Shift Keying
Amplitude Shift Keying (ASK) (or PAM)
The transmitted signal is given by
The following values are usually chosen for
Slide 29 Amplitude Shift Keying
Possible values of
Possible values of
M=4, d=2
M=8, d=2
Slide 30 Probability of error in ASK
Slide 31 Probability of error in ASK
Slide 32 Probability of error in ASK
Slide 33 Expected average energy
Slide 34 Expected average energy
Slide 35 Error probability (M-ASK)
Slide 36 Phase Shift Keying(PSK)
only can take the values
Phase Shift Keying (PSK)
Slide 37 Analytical signal concept
Applying the concept of the analytical signal ...
Slide 38 Analytical signal concept
Slide 39 QPSK-(4-PSK)
Symbol duration for PSK is given by
Let us assume GRAY-coding as follows
Slide 40 QPSK-(4-PSK)
0 1 1 0 0 1
Slide 41 QPSK-(4-PSK)
The symbol error probability has been already calculated for 2-PSK ....
... and for 4-PSK
Slide 42 QPSK-(4-PSK)
For general M-ary PSK no closed form expression for the symbol error probability is available.
Instead, an upper and lower power limit can be given
Slide 43 QPSK-(4-PSK)
Power efficiency for M-PSK
Note that for both M=2 and M=4:
Bandwidth efficiency for M-PSK
which increases with M.
Slide 44 Offset QPSK (OQPSK)
Possible phase jumpsin 4-PSK
Example for phase jumpsin 4-PSK
Slide 45 Offset QPSK (OQPSK)
What does this delay mean for the vector represenation?
What will be the effect on the bandwidth?
OQPSK occupies less bandwidth than QPSK. Do you agree?
Slide 46 Offset QPSK (OQPSK)
0 1 1 0 0 1
Slide 47 Offset QPSK (OQPSK)
0 1 1 0 0 1
QPSK
OQPSK
Slide 48 Differential PSK (DPSK)
In DPSK only the phase difference are transmitted.
Note that if then only the phase values
are possible. K evenK odd
Slide 49 Quadrature Amplitude Modulation (QAM)
Constellation set for 4-, 16-, 64-QAM
Slide 50 Quadrature Amplitude Modulation (QAM)
The symbol error probability of M-QAM is bounded by
For large M this ratio approaches
Slide 51 QAM performance
Slide 52
Frequency Shift Keying (FSK)
where now takes values in the set
Suppose that
then the transmitted signal consists of the waveforms
Slide 53 Frequency Shift Keying
A closer look to these waveforms yields ...
Slide 54 Frequency Shift Keying
For nearly orthogonal waveforms, the minimum difference
frequency must fullfill
Error probability of M-FSK:
Slide 55 Performance of M-FSK
Slide 56 CPFSK
Continous Phase Frequency Shift Keying (CPFSK)
where
Slide 57 CPFSK
Integrating the data sequence
and use this expression as
the argument of the complex exponential. Then, we obtain
Slide 58 CPFSK
where denotes the largest integer less than or equal to
If we interprete
and an initial phase we finally obtain
Slide 59 MSK
MSK
CPFSK is known as a modulation scheme with memory and with a constant envelope.
Minimum Shift Keying (MSK)
Modulation index:
For binary transmission (M = 2) can only take thevalues
Slide 60 MSK
Suppose that the two equivalent lowpass waveforms are given by
so that the correlation yields
Observe that the signals are uncorrelated or also orthogonal if
which results in frequency differences
Slide 61 GMSK
So, the minimum modulation index for orthogonal signals is:
Gaussian Minimum Shift Keying (GMSK)
Why filtering of the rectangular signal is recommended?
Let us introduce a so-called Gaussian low pass filter
where is the 3dB cut-off frequency.
Slide 62 GMSK
With we obtain
with
and the constant
Slide 63 GMSK
Slide 64 CPM
Continuous Phase Modulation (CPM)
CPFSK-signal has been already defined as
Now let us draw MSK and GMSK signals ...
Let us define so-called the time dependent argument
Slide 65 CPM
0 1 1 0 0 1
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