Chapter 3 · PAM is a 1D modulation method PSK is a 2D modulation method with vector space...

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