Lecture 6 Modulation, Demodulation (Detection): Part...

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EE4900/EE6420 Digital Communications Suketu Naik

EE4900/EE6420: Digital Communications

Lecture 6

Modulation,

Demodulation

(Detection): Part 1

2


Block Diagrams of Communication System

Digital Communication System

Informatio

n (sound,

video, text,

data, …)

Transducer &

A/D ConverterModulator

Source

Encoder

Channel

Encoder

Tx RF

System

Output

Signal

D/A Converter

and/or output

transducer

DemodulatorSource

Decoder

Channel

Decoder

Rx RF

System

Channel

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Introduction

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Digital Baseband Modulation Schemes

Baseband Modulation is done before the information

(bits) is sent out using a carrier signal

Popular methods for low data-rate are,

ASK: Amplitude Shift Keying

OOK: On-Off Keying

FSK: Frequency Shift Keying

ASK Signal

OOK Signal

FSK Signal

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Digital Baseband Modulation Schemes

Baseband Modulation is done before the information

(bits) is sent out using a carrier signal

Popular methods for high data-rate are,

PCM: Pulse Code Modulation

PAM: Pulse Amplitude Modulation

PWM: Pulse Width Modulation

PPM: Pulse Position Modulation

BPSK: Binary Phase Shift Keying

QPSK: Quadrature Phase Shift Keying

O-QPSK: Offset Quadrature Phase Shift Keying

QAM: Quadrature Amplitude Modulation

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Pulse Code Modulation (PCM)

PCM is commonly used for digital audio (Windows PCs,

CDs, digital telephony)

PCM: Analog signal (such as voice or music) is

converted to discrete samples and then quantized to bits

After quantization, companding (compressing and

expanding) takes place: μ-law in North America and

Japan and A-law in Europe and rest of the world

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Pulse Code Modulation (PCM)

PCM Example

PCM Signal

Original Signal

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Pulse Amplitude Modulation (PAM)

PAM is used in Digital TV, 10GBase-T Ethernet, LED

driver, Controls

PAM: Width is fixed, Amplitude varies

Original Signal

PAM Signal

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Pulse Width or Duration Modulation (PWM or PDM)

PWM is used in Controls (motors, servos for robots,

power, audio, telecomm.),

PWM: Position is fixed, Width varies

Original Signal

PWM Signal

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Pulse Position Modulation (PPM)

PPM is used in non-coherent RF comm., contactless

smart card, RFID tags, radio control, optical comm.

PPM: Width is fixed, Position varies

Original Signal

PPM Signal

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Binary Phase Shift Keying (BPSK)

BPSK is commonly used in communications

BPSK: Phase of the baseband signal changes from 0 deg

to 180 deg (2 transitions)=1 bit per symbol

Original Signal

BPSK Symbol

Q (out of phase)

I (in phase)Phase Diagram 01

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Quadrature Phase Shift Keying (QPSK)

QPSK (and O-QPSK) is commonly used in

communications

QPSK: Phase of the baseband signal changes from 45

deg, 135 deg, 225 deg, 315 deg (4 transitions)=2 bits per

symbol

Original Signal

QPSK Symbol

Q (out of phase)

I (in phase)Phase Diagram

11

10

01

00

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8-PSK

8PSK = 8 phase transitions= 3 bits per symbol

Q (out of phase)

I (in phase)8-PSK Phase

Diagram

000

001

011

010

101

111

110

100

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Quadrature Amplitude Modulation (QAM)

QAM (rhymes with Guam and VietNAM) is commonly

used in communications

QAM: Amplitude and Phase of the Baseband signal

changes, 16-QAM (16 transitions, 4-bits/symbol), 32-QAM

(5-bits/symbol), 64-QAM (64 transitions, 6-bits/symbol)

16-QAM

Phase Diagram

Q (out of phase)

I (in phase)

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Quadrature Amplitude Modulation (QAM)

QAM Signal in Time-domain

8-QAM=8PSK

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Basics of

Modulation

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Waveforms=Bits

M-ary (“emery”) digital communication system uses M

symbols to transmit log2(M) bits

Examples

BPSK: M=2, log2(M)=1 bit per symbol

Two symbols, one for each bit (0 and 1)

QPSK: M=4, log2(M)=2 bits per symbol

Four symbols, one for two bit pairs

(00, 01, 11, 10)

Binary PAM: M=2, 1 bit per symbol

8-PAM: M=8, 3 bits per symbol

16-QAM, 32-QAM, 64-QAM, 128-QAM: ?

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Waveforms, Symbols, Basis Functions

M-ary communication uses M symbols

Symbols are created by changing amplitudes of basis

functions

Thus a waveform is a linear combination or weighted

sum of basis functions

This waveform can be called the baseband signal or the

modulating signal

The modulating signal modulates (multiplied with) the

carrier signal so that the carrier signal can now carry

“bits”

Basis functions > Symbols > Waveform

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Signal Space and Basis Functions

Let’s represent the M waveforms as a signal (symbol) set

𝑺 = {𝒔𝟎 𝒕 , 𝒔𝟏 𝒕 , 𝒔𝟐 𝒕 , … , 𝒔𝑴−𝟏 𝒕 }with symbol time Ts

B=set of basis functions

Signal Space=Span{B}= linear

combination of basis functions

Baseband signal s(t) (with K=M)

𝒔(𝒕) =

𝒌=𝟎

𝑲−𝟏

𝒂𝒌∅𝒌(𝒕)

Energy 𝑬𝒌= 𝑻𝟏𝑻𝟐 ∅𝒌

𝟐 𝒕 𝒅𝒕

s0(t) s1(t) s2(t) s3(t) s4(t) s5(t) s6(t) s7(t)

Ts

Example

1) M=8 waveforms=8 signals (symbols)

2) Signal (symbol) set S contains 8

possible signals (symbols)

3) Each symbol represents 3-bits

e.g. bit patterns such as 101 or 100 are

mapped to specific symbols

4) s(t), the baseband signal, is the

linear combination of the basis

functions, constructed from symbols,

s0(t), …sM-1(t)

Example Baseband Signal s(t)

Basis FunctionAmplitude

2Ts 3Ts 4Ts …

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Properties of Basis Functions

Orthogonality

න𝑻𝟏

𝑻𝟐

∅𝒊 𝒕 ∅𝒋 𝒕 𝒅𝒕 = 𝟎

Interpretation:

Multiply two different signals and integrate, the result is 0

Orthonormal Set

න𝑻𝟏

𝑻𝟐

∅𝒊 𝒕 ∅𝒋 𝒕 𝒅𝒕 = 𝜹 𝒊 − 𝒋 = ቊ𝟏 𝒊𝒇 𝒊 = 𝒋𝟎 𝒊𝒇 𝒊 ≠ 𝒋

Interpretation:

Multiply two different signals and integrate, the result is 0

Multiply two identical signals and integrate, the result is 1

Example: cos(2πt) sin(2πt)

1) Two orthogonal signals are transmitted together to reduce noise (e.g.

QPSK, QAM)

2) The detector (receiver) is designed based on orthonormality

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Energy

Recall that baseband signal s(t) can be created from K

orthonormal functions as,

𝒔(𝒕) =

𝒊=𝟎

𝑲−𝟏


Interpretation:

Constants ak are the components of K-dimensional vector so that s(t)

can be represented as K-dim vector S,

𝑺 = [𝒂𝟎 𝒂𝟏 𝒂𝟐…𝒂𝑲−𝟏]

Energy

𝑬 = න𝑻𝟏

𝑻𝟐

𝒔𝟐 𝒕 𝒅𝒕 =

𝒌=𝟎

𝑲−𝟏

𝒂𝒌𝟐

1) Energy in the signal s(t) is equal to the square of the Euclidean

distance between the K-dimensional point and the origin

2) This result is important for designing the detector (receiver)

Analog signal

representing

the bit stream

An example

[𝒂𝟎 𝒂𝟏 𝒂𝟐]

0

1

2

An example

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Modulation

And Detection

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Approximation of Received Signal r(t)

Transmitted signal=s(t), Received signal=r(t), Noise=w(t)

r(t)=s(t) + w(t)

Goal: find the best approximation ො𝒓(t) to r(t) using basis functions

Why?

Because s(t) is corrupted by noise and we need to find an equivalent

of the corrupted s(t) or r(t) in Span{B}

Why?

-Span{B} contains the basis functions

-So that we can determine the symbols and hence the basis functions

Why?

Because the basis functions were used to create symbols which

represented the bits (information)

Symbols representing bits Symbols corrupted by noise

s(t) r(t)

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Least-Mean-Square Error

Transmitted signal=s(t), Received signal=r(t), Noise=w(t)

r(t)=s(t) + w(t)

Goal: find the best approximation ො𝒓(t) to r(t) using basis functions

How?

Find the energy of the error between the received signal r(t) and

approximation ො𝒓(t)

Energy of the error

𝒔(𝒕) =

𝒌=𝟎

𝑲−𝟏


ො𝒓(𝒕) =

𝒌=𝟎

𝑲−𝟏

𝒙𝒌∅𝒌(𝒕)

Transmitted Vector

[𝒂𝟎 𝒂𝟏 𝒂𝟐…𝒂𝑲−𝟏]

Approximate Vector

[𝒙𝟎 𝒙𝟏 𝒙𝟐…𝒙𝑲−𝟏]

𝜺 = න𝑻𝟏

𝑻𝟐

|𝒓 𝒕 − ො𝒓(𝒕)|𝟐𝒅𝒕

Energy of the error is minimized by selecting the

approximate vector [𝒙𝟎 𝒙𝟏 𝒙𝟐…𝒙𝑲−𝟏]

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Signal Space Projections or Phase Diagram or Constellation

𝒔(𝒕) =

𝒌=𝟎

𝑲−𝟏

𝒂𝒌∅𝒌(𝒕)𝒙𝒌 = න

𝑻𝟏

𝑻𝟐

𝒓 𝒕 ∅𝒌 𝒕 𝒅𝒕

Goal: find the approximate vector xk

Analysis

Equation

Synthesis

Equation

Received Signal

Approximate

Received Signal

Basis Function 2

Basis Function 1

Approximate

Vector, xk

An example with K=M=2

Things to remember:

1) Phase diagram shows

Points

2) Points are denoted by

Vectors from origin

3) Points and Vectors

represent an actual analog

baseband signal

4) The amplitude of the

baseband signal = point in

the phase diagram

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Modulator based on Synthesis Equation

Data/Audio/Video

[101 100 001 000 010 011 110 111]

Serial to Parallel

Group of

3 bits

One look-up table

per symbol

Constructing one

symbol so(t)

signal s(t) is created

from K-symbols

Example

Same group of bits

go to each path

1) Group of bits (e.g. 101)= the

decimal index (e.g. 5)

2) Now the coefficient with that

index is selected

Pulse-shaping filter

can be used here

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Summary: Modulation

It’s simple!

M-ary Modulation:

1) Serial bit stream represents data, audio, video

2) Create group of bits (Serial to Parallel) according to the M

3) Use the group of bits as the index in a look-up-table of

coefficients (amplitudes) of the basis functions

4) Pass the coefficient samples through pulse-shaping filter (one of

the pulse shapes, next lecture). Pulse shape is either the basis

function or a part of basis function. The output is called a

symbol.

5) Add all the symbols together (number of symbols depends on M

which depends on group of bits). This is the baseband signal.

6) Baseband signal then can modulate the carrier signal.

7) Carrier signal is transmitted…

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Detector based on Analysis Equation

Received

Signal

Basis

Function

Find the

area

Find the

approximate

vector

Subtract the transmitted symbols

from the approximate vector and

find the minimum

Decision

Rule (derived from

Least Mean Square

Error = the Energy of

the Error)

𝜺 = න𝑻𝟏

𝑻𝟐

|𝒓 𝒕 − ො𝒓(𝒕)|𝟐𝒅𝒕

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Practical Detector: Matched Filter

Matched filter (matched to the pulse-shaping filter) can be used

for detection Direct Application of

Analysis Equation

Matched Filter

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Comparison of Two Approaches

Impulse response of the matched filter=time-reversal of pulse-shaping filter*

*= this is generally true. For PAM, pulse shape=basis function. However, note that BPSK, QPSK, 16-QAM may not have identical pulse shape

and basis function. For example for QPSK, basis functions=p(t)*cos and -p(t)*sin; where p(t) is the pulse or the impulse response of the pulse-

shaping filter. We will cover pulse shapes, PAM, QAM in upcoming lectures.

Pulse or the impulse

response of the pulse-

shaping filter*

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Summary: Demodulation (Detection)

It’s simple!

M-ary Demodulation (Detection):

1) Carrier signal is filtered, down-converted, filtered to baseband

signal

2) Now we reverse the process of modulation: pass it through

matched filter whose impulse response is the time-reversal of the

pulse shaping filter.

3) Output samples (x) from the matched filter are passed through

decision block which implements Least-Mean-Square error logic

4) Least-Mean-Square error logic:

1) x = we hope that these are the symbol amplitudes

2) a = we transmitted these symbol amplitudes=constellation

points

3) Take the differences x-a (Euclidean distance) and find the

minimum. This is the output or our best guess for what was

transmitted.

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Examples

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Example 5.1.1: Synthesis (Modulator)

Signal Space Projections

or Phase Diagram or

Constellation

Note that the points So and S1

are actual signals (symbols)

in time

0.5 1 0.5 1 0.5 1

0.5 1 0.5 1 0.5 1

basis function 1

(note: it can have different

amplitudes than 1 and -1)

basis function 2

K=2=two symbols

Two Orthonormal

Basis Functions

Coefficient chosen according

to the group of bits(=index)

Index Coeff

Symbols are constructed

from sum of the two basis

functions above

Coefficient chosen according

to the group of bits(=index)

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Example 5.1.1: Analysis (Detector)How do we figure out what was transmitted given r(t)?

A: Find approximate vector [x0 x1]

The approximate vector

shown on the phase diagram

Received Signal r(t)

𝒙𝒌 = න𝑻𝟏

𝑻𝟐

𝒓 𝒕 ∅𝒌 𝒕 𝒅𝒕

Analysis Equation

Multiply r(t) with basis function,

then find the area (integral)

ො𝒓 𝒕 = 𝟐∅𝟎 𝒕 +𝟐

𝟑∅𝟏(𝒕)

ො𝒓(𝒕) =

𝒌=𝟎

𝑲−𝟏

𝒙𝒌∅𝒌(𝒕)

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Example 5.1.1: Analysis (Detector)Transmitted signal s(t) Received signal r(t)

+0.5 1 0.5 1

2

4

2

Shape is changed and

Amplitude is changed. Why?

Approximate signal ො𝒓(𝒕)

0.5 1 0.5 1

1 1

-1

8/3=2.67

4/3=1.33

0.5 1

=

0.5 1

6

2

s(t)

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Example 5.1.1: Analysis (Detector)Approximate vector

x=[2 2/3]

What was transmitted?

Sm=[6 2]

What did we determine as the transmitted signal?

Decision Rule:

|2-6|=4

|2/3-2|=1.33

Answer: Minimum of {4, 1.33}=1.33

What was transmitted? 0d (000b) or 1d (001b)or 2d (010b)?

0.5 1

6

2

s(t)

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