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Chapter 3
Fredrik Rusek
Proakis-Salehi book
Section 3.1
Memoryless modulation: Tuples of k bits are encoded into one waveform by a waveform modulator in an independent fashion
One waveform transmitted every Ts second
Symbol rate Bit interval
Bit-rate
Av. Energy per waveform
Av. Energy per bit
Av. Power
Section 3.2
Some methods to implement
Section 3.2-1 PAM
PAM: Real-valued signals. In baseband we have
Section 3.2-1 PAM
PAM: Real-valued signals. In passband
baseband
Complex baseband
How is energy of baseband and passpand related?
Section 3.2-1 PAM
PAM: Real-valued signals. In passband
baseband
Complex baseband
Section 3.2-1 PAM Make Gram-Schmidt orthogonalization process of
Section 3.2-1 PAM Make Gram-Schmidt orthogonalization process of
First basis function:
Second basis function:
General PAM Passband PAM
Section 3.2-1 PAM Make Gram-Schmidt orthogonalization process of
First basis function:
Second basis function: Does not exist. Signaling is 1-D.
General PAM Passband PAM
Section 3.2-1 PAM
Signal space representation of PAM
Transmitted signal can be written as
Vector representation (vector is a scalar since 1D)
Section 3.2-1 PAM
Euclidean distance of PAM
Minimum occurs for m-n=1/-1
It is illustrative to express this in terms of the average energy per bit
Bandpass PAM
Bandpass or baseband PAM
Section 3.2-1 PAM Baseband vs passband PAM
f
Baseband PAM
0
f
Passband PAM
fc >>1
Bandwidth is twice as large for the same data rate
Section 3.2-1 PAM Difficult Remedy: Single sideband transmission
f fc >>1
Bandwidth is the same with SSB bandpass as for baseband
Phase modulation: Always in passband. Defined as
Section 3.2-2 PSK
Phase modulation: Always in passband. Defined as
All these signals have the same energy. Maybe not so easy to see from last expression. But, recall that has half of the energy of
Section 3.2-2 PSK
Phase modulation: From the ”passband-baseband orthogonality” example of the last lecture, we know that
are orthogonal
Section 3.2-2 PSK
Phase modulation: From the ”passband-baseband orthogonality” example of the last lecture, we know that
are orthogonal
This is a PAM modulation of
Section 3.2-2 PSK
Phase modulation: From the ”passband-baseband orthogonality” example of the last lecture, we know that
are orthogonal
This is a PAM modulation of and this is an orthogonal PAM modulation of
Section 3.2-2 PSK
G-S of Phase modulation: We get a 2D G-S basis for phase modulation.
signal
Vector space representation
G-S basis
Section 3.2-2 PSK
Section 3.2-2 PSK Signal space and Euclidean distance:
PAM
The ”hole” in the middle kills the distance of PSK with large M
PSK
Section 3.2-3 QAM PAM is a 1D modulation method
PSK is a 2D modulation method with vector space representation
However, the first and the second coordinates are not independent, as they are constrained to give constant energy
QAM relaxes this, and use arbitrary (but always structured) coordinates in the two dimensions
Section 3.2-3 QAM
QAM. General definition of QAM:
There is no structure of and in the most general case
We can of course go into polar coordinates
Section 3.2-3 QAM
QAM. We can use the same basis as for PSK (or any other 2D basis)
Section 3.2-3 QAM
QAM. Different structures of the inphase and the quadrature
APSK (used in satellite communications)
Section 3.2-3 QAM
QAM. Different structures of the inphase and the quadrature
Rectangular QAM, or simply QAM. Used everywhere
Section 3.2-3 QAM
QAM. Different structures of the inphase and the quadrature
PSK is nothing but a special case…….
Section 3.2-3 QAM
QAM. Different structures of the inphase and the quadrature
But so is bandpass PAM PAM is QAM with one component set to 0
Section 3.2-3 QAM
Actually, QAM is just two PAMs sent on orthogonal dimensions.
These two dimensions can be (e.g.)
• sin/cos (the standard case),
• time (send one today, and the other one tomorrow),
• Frequency (use different bands to create two basis functions)
Section 3.2-3 QAM
Euclidean distance of QAM.
Hence, the Euclidean distance study of QAM collapses into a study of PAM.
PAM
PSK
QAM
M – QAM is equivalent to M-PAM
2
QAM is fully equivalent to SSB-PAM in terms of • Spectral
efficiency • Euclidean
distance • BER
Short Interlude
We saw different examples of the design of 2D signals.
They had different minimum Euclidean distances
Natural question: Which signal design (2D) provides the best minimum distance ?
This depends on the number of points to transmit.
With large number of points, the optimal organization of points is the A2 lattice
These points generate more distance for the same transmit energy
We have 1 D signals (PAM) and 2D signals (QAM/PSK).
Of course we can also create high-D signals.
Assume M orthogonal dimemsions. These can come from space, time. Frequency etc
In analogy to the A2 lattice, the optimal (densest) packing of objects in M dimensions are known for M=3…9, and M=24.
Unknown in all other dimensions
Section 3.2-4 high dim
Solution in 3D
Some standard methods: Orthogonal signals.
Distances between all signals are equal
Section 3.2-4 high dim
Looks great! What is the catch? QAM
Some standard methods: FSK. Special case of orthogonal signals
This is a non-linear modulation
Section 3.2-4 high dim
How to prove that FSK is an instance of orthogonal signaling?
We need the signals to be orthogonal at passband, hence
Section 3.2-4 high dim
Not orthogonal at baseband!
So far, all modulation schemes were memory less
The meaning of this is that the first group of symbols does not matter when constructing the signal for the second group
Section 3.3 CPM
So far, all modulation schemes were memory less
The meaning of this is that the first group of symbols does not matter when constructing the signal for the second group
CPM is one the most well studied modulations with memory.
Much of the most prominent work on CPM was carried out in this building by Sundberg and Aulin
Section 3.3 CPM
Begin with a PAM signal
Where is M-PAM data symbols and g(t):
Now define the complex baseband signal
Section 3.3-1 CPFSK
Section 3.3-1 CPFSK
Discontinuous
Continuous
Continuous phase signal
Section 3.3-1 CPFSK
Accumulated phase Modulation index
x 2T Typo in book. Eq. 3.3-10
Section 3.3-2 general CPM
CPFSK is a member of a more general family of modulation techniques: CPM (continuous phase modulation)
Phase satisfies
We get different CPMs by changing: M, h, g(t)
Section 3.3-2 general CPM
GSM
Section 3.3-2 general CPM
Now plot different phases
What kind of CPM is this?
Section 3.3-2 general CPM
Now plot different phases
What kind of CPM is this? M=2 (binary inputs In)
Section 3.3-2 general CPM
Now plot different phases
What kind of CPM is this? M=2 (binary inputs In) q(t) behaves as t So g(t) must be flat
CPFSK
Section 3.3-2 general CPM
A quaternary example CPFSK again
What is the number of states?
Section 3.3-2 general CPM
A quaternary example CPFSK again
What is the number of states? When kh equals we get a ”wrap-around” Hence, for h=1/2, We get 4 states
Section 3.3-2 general CPM
A quaternary example CPFSK again
What is the number of states? When kh equals we get a ”wrap-around” Hence, for h=1/2, We get 4 states
Section 3.3-2 general CPM
A partial response with GMSK (L=3)
This phase goes into CPM has: (i) constant envelope (nice for power amplifiers) (ii) continuous phase (nice for PSDs) (iii) memory in its modulation (drawback of demodulation) (iv) tradeoff beetween Euclidean distance and spectral efficiency
Section 3.3-2 general CPM
An important Special case: MSK
Transmitted signal
Section 3.3-2 general CPM
An important Special case: MSK
Transmitted signal
With
From FSK modulation: 1/2T is minimum frequency separation to make signals orthogonal
Section 3.3-2 general CPM
An important Special case: MSK
We will get back to MSK soon and show that it is actually a linear modulation, closely related to QPSK
Section 3.3-2 general CPM
Offset QPSK
Normal QPSK
Section 3.3-2 general CPM
Offset QPSK
Here
Section 3.3-2 general CPM
Offset QPSK
Here We go from here to here 180 degree phase jump => bad Spectral properties. Try to avoid. => Offset QPSK
Section 3.3-2 general CPM
Offset QPSK
Let the in-phase have an offset of half a symbol interval No 180 degrees phase jumps are present FBMC (candidate for 5G is based on Offset QAM)
Section 3.3-2 general CPM
Offset QPSK
Let the in-phase have an offset of half a symbol interval No 180 degrees phase jumps are present FBMC (hot candidate for 5G is based on Offset QAM)
Looks like state transition graph of MSK
Section 3.3-2 general CPM
When the smoke clears:
MSK is linear modulation (although it also qualifies to be in the CPM class)
MSK has the same description as Offset QPSK
In fact, all CPM signals have a linear basis (just perform G-S or K-L)
But not all CPMs are linear (MSK is a very special case)
Section 3.4 Power spectrum
Skip part about CPM spectrum
CPM spectrum was unknown until the late 70s.
Typically people measured the spectrum, but did not know how to compute it.
Tor Aulin and Carl-Erik Sundberg derived the spectrum (and won two best-of-the-best paper awards for it - Two out of the best 39 papers ever published within IEEE-comsoc/it/sig.proc)
Section 3.4 Power spectrum
Skip part about CPM spectrum
CPM spectrum was unknown until the late 70s.
Typically people measured the spectrum, but did not know how to compute it.
Tor Aulin and Carl-Erik Sundberg derived the spectrum (and won two best-of-the-best paper awards for it - Two out of the best 39 papers ever published within IEEE-comsoc/it/sig.proc)
It was easy according to Tor
Section 3.4 Power spectrum
M possible signals to transmit At time n, determines which of the M to use Hence, we catch modulation with memory in the derivations.
Signal in complex baseband
Section 3.4 Power spectrum
Steps in computing the power spectrum of a modulation: 1. Compute the spectrum of the complex baseband signal. Bandpass is
then given by 2.9-14
Section 3.4 Power spectrum
Steps in computing the power spectrum of a modulation: 1. Compute the spectrum of the complex baseband signal. Bandpass is
then given by 2.9-14
2. Compute the auto-correlation of the signal
Section 3.4 Power spectrum
Steps in computing the power spectrum of a modulation: 1. Compute the spectrum of the complex baseband signal. Bandpass is
then given by 2.9-14
2. Compute the auto-correlation of the signal 3. Check if cyclo-stationary process!! If so, average the auto-correlation ttttover one period!
Section 3.4 Power spectrum
Steps in computing the power spectrum of a modulation: 1. Compute the spectrum of the complex baseband signal. Bandpass is
then given by 2.9-14
2. Compute the auto-correlation of the signal 3. Check if cyclo-stationary process!! If so, average the auto-correlation ttttover one period! 4. Take Fourier transform of auto-correlation (or its average)
Section 3.4 Power spectrum
Compute the auto-correlation of the signal This was step 2. We cannot continue since we cannot solve the Expectation
Section 3.4 Power spectrum
Compute the auto-correlation of the signal This was step 2. We cannot continue since we cannot solve the Expectation Step 3: Average over one period
Section 3.4 Power spectrum
Compute the auto-correlation of the signal This was step 2. We cannot continue since we cannot solve the Expectation Step 3: Average over one period
Section 3.4 Power spectrum
Compute the auto-correlation of the signal This was step 2. We cannot continue since we cannot solve the Expectation Step 3: Average over one period
Section 3.4 Power spectrum
Step 3: Average over one period Define We get
Section 3.4 Power spectrum
Step 3: Average over one period Define We get Step 3 is now done
Section 3.4 Power spectrum
Step 4: Take Fourier transform
Section 3.4 Power spectrum
Step 4: Take Fourier transform
Section 3.4 Power spectrum
Step 4: Take Fourier transform
Section 3.4 Power spectrum
Recall
Section 3.4 Power spectrum
Much simpler example
What do we need to compute spectrum?
Section 3.4 Power spectrum
Much simpler example
What do we need to compute spectrum?
Section 3.4 Power spectrum
Much simpler example
What do we need to compute spectrum?
Section 3.4 Power spectrum
Much simpler example
What do we need to compute spectrum?
Spectrum can be shaped via pulse or inputs
Problems
3.2 3.5 3.15 3.22 3.26