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