ECE 4790 ELECTRICAL COMMUNICATIONS Spring 2000

1
ECE 4790ELECTRICAL COMMUNICATIONSSpring 2000

• Dr. Bijan Mobasseri
• ECE Dept.
• Villanova University

2
Policies and procedures

• 3 hours of lecture per week(MW)
• 2 hours of lab per week
• Homework assigned but not collected
• Lab is due weekly
• 2 tests and one final
• Grade break down
• 25 each test(2)
• 25 final
• 25 labs

3
Lab work

• Hands-on lab work is an integral part of the
  course
• There will be about 10 experiments done using
  MATLAB with signal processing and COMM toolboxes
• Experiments, to the extent possible, parallel
  theoretical material
• Professional MATALB code is expected

4
Going online

• Send a blank message to
• ece4790_s00-subscribe_at_egroups.com
• You will then have access to all class notes,
  labs etc.
• Notes/labs are in MS Office format
• Class material are for your own use only

5
Ethical standards

• This course will be online and make full use of
  internet. This convenience brings with it many
  responsibilities
• Keep electronic class notes/material private
• Keep all passwords/accounts to yourself
• Do not exchange MATLAB code

6
Necessary Background

• This course requires, at a minimum, the following
  body of knowledge
• Signal processing
• Probability
• MATLAB

7
Course Outlook

• Introduction
• signals, channels
• bandwidth
• signal represent.
• Analog Modulation
• AM and FM
• Source coding
• Sampling, PAM
• PCM, DM, DPCM

• Pulse Shaping
• best pulse shape
• interference
• equalization
• Digital Modulations
• Modem standards
• Spread Spectrum
• High speed data

8
SIGANLS AND CHANNELS IN COMMUNICATIONS

• AN INTRODUCTION

9
A Block Diagram
Information source
user
source decoder
source encoder
channel decoder
channel encoder
demodulator
modulator
channel
10
Information source

• The source can be analog, or digital to begin
  with
• Voice
• Audio
• Video
• Data

11
Source encoder

• Source encoder converts analog information to a
  binary stream of 1s and 0s

1 0 0 1 1 0 0 1 ...
Source encoder PCM, DM, DPCM, LPC
12
Channel encoder

• The binary stream must be converted to real pulses

polar
1 0 1 1 0
channel encoder
on-off
13
Modulator

• Signals need to be modulated for effective
  transmission

1 0 1 1 0
Modulator
14
Channel

• Channel is the medium through which signals
  propagate. Examples are
• Copper
• Coax
• Optical fiber
• wireless

15
Signals and Systems Review
16
Periodic vs. Nonperiodic

• A periodic signal satisfies the condition
• The smallest value of To for which this condition
  is met is called a period of g(t)

period
17
Deterministic vs. random

• A deterministic signal is a signal about which
  there is no uncertainty with respect to its value
  at any given time
• exp(-t)
• cos(100t)

18
Energy and Power

• Consider the following
• Instantaneous power is given by

i(t)

V(t)
R
-
19
Energy

• Working with normalized load, R1?

20
Average Power

• The instantaneous power is a function of time. An
  overall measure of signal power is its average
  power

21
Energy and Power of a Sinusoid

• Take
• Find the energy

22
Instantaneous Power

• Instantaneous power

23
Average Power

• The average power of a sinusoid is

24
Average Transmitted Power

• What is the peak signal amplitude in order to
  transmit 50KW? Assume antenna impedance of 75?.
• Note the change in Pavg for non-unit ohm load

25
Energy Signals vs. Power Signals

• Do all signals have valid energy and power
  levels?
• What is the energy of a sinusoid?
• What is the power of a square pulse?
• In the first case, the answer is inf.
• In the second case the answer is 0.

26
Energy Signals

• A signal is classified as an energy signal if it
  meets the following
• 0ltEltinfinity
• Time-limited signals, such as a square pulse, are
  examples of energy signals

27
Power Signals

• A power signal must satisfy
• 0ltPltinfinity
• Examples of power signals are sinusoidal functions

28
Exampleenergy signal

• Square pulse has finite energy but zero average
  power

A
T
29
Examplepower signal

• A sinusoid has infinite energy but finite power

A
30
SUMUP

• Energy and power signals are mutually exclusive
• Energy signals have zero avg. power
• Power signals have infinite energy
• There are signals that are neither energy or
  power?. Can you think of one?

31
DEFINING BANDWIDTH
32
WHAT IS BANDWIDTH?

• In a nutshell, bandwidth is the highest
  frequency contained in a signal.
• We can identify at least 5 definitions for
  bandwidth
• absolute
• 3-dB
• zero crossing
• equivalent noise
• RMS

33
ABSOLUTE BANDWIDTH

• The highest frequency

Spectrum
W
-W
f
34
3-dB BANDWIDTH

• The frequency where frequency response drops to
  .707 of its peak

Spectrum
W
f
35
FIRST ZERO CROSSING BANDWIDTH

• The frequency where spectrum first goes to zero
  is called zero crossing bandwidth.

36
EQUIVALENT NOISE BANDWIDTH

• Bandwidth which contains the same power as an
  equivalent bandlimited white noise

37
RMS BANDWIDTH

• RMS bandwidth is related to the second moment of
  the amplitude spectrum
• This measures the tightness of the spectrum
  around its mean

38
RMS BANDWIDTH OF A SQUARE PULSE

• Take a square pulse of duration 0.01 sec. Its
  spectrum is a sinc

39
RMS BANDWIDTH

• The RMS bandwidth can be numerically computed
  using the following MATLAB code

W rms 35.34 Hz
40
Bandwidth of Real Signals

• This is the spectrum a 3 sec. clip sampled at 8KHz

41
Gate Function

• Gate function is one of the most versatile pulse
  shapes in comm.
• It is pulse of amplitude A and width T

A
T/2
-T/2
42
rect function

• Gate function is defined based on a function
  called rect

43
Expression for gate

• Based on rect we can write
• Note that for -T/2lttltT/2, the argument of rect is
  inside -1/2,1/2 therefore rect is 1 and g(t)A

44
Generalizing gate

• Let say we want a pulse with amplitude A
  centered at tto and width T

T
A
tto
45
Arriving at an Expression

• What we have a is a rect function shifted to the
  right by to
• Shift to the right of f(t) by to is written by
  f(t-to)
• Therefore

46
gate function in the Fourier Domain

• The Fourier transform of a gate function is a
  sinc as follows

Zero crossing
47
Zero Crossing

• Zero crossings of a sinc is very significant.
• ZC occurs at integer values of sinc argument

48
Some Numbers

• What is the frequency content of a 1 msec. square
  pulse of amplitude .5v?
• We have A0.5 and T1ms
• First zero crossing at f1000Hz obtained by
  setting 10-3f1

49
RF Pulse

• RF(radio frequency) pulse is at the heart of all
  digital communication systems.
• RF pulse is a short burst of energy, expressed by
  a sinusoidal function

50
Modeling RF Pulse

• An RF pulse is a cosine wave that is truncated on
  both sides
• This effect can be modeled by gatingthe cosine
  wave

51
Mathematically Speaking

• Call the RF pulse g(t), then
• This is in effect the modulated version of the
  original gate function

52
Spectrum of the RF Pulsebasic rule

• We resort to the following
• Meaning, the Fourier transform of the product is
  the convolution of individual transforms

53
RF Pulse Spectrum Result

• We now have to identify each term
• Then, the RF pulse spectrum, G(f)

54
Interpretation

• The spectrum of the RF pulse are two sincs, one
  at f- fc and the other at ffc

55
Spectrum of 5 msec pulse
56
Code for RF pulse spectrum
57
Baseband and Bandpass Signals and Channels
58
DefinitionsBaseband

• The raw message signal is referred to as
  baseband, or low freq. signal

59
DefinitionsBandpass

• When a baseband signal m(t) is modulated, we get
  a bandpass signal
• The bandpass signal is formed by the following
  operation(modulation)

60
Bandpass ExampleAM
61
Digital Bandpass

• Baseband and bandpass concepts apply equally well
  to digital signals

1
1
1
1
0
RF pulses
62
Baseband vs. Bandpass Spectrum

• Creating a bandpass signal is the same as
  modulation process. We have the following

Interpretation
63
Showing the Contrast
baseband
frequency
Bandwidth doubles
bandpass
bandwidth
64
SIGNAL REPRESENTATION

• How to write an expression for signals

65
Introduction

• We need a formalism to follow a signal as it
  propagates through a channel.
• To this end, we have to learn a few concepts
  including Hilbert Transform, analytic signals and
  complex envelope

66
Hilbert Transform

• Hilbert transform is an operation that affects
  the phase of a signal

90
H(f)
f
-90
H(f)1
Phase response
67
More precisely
68
HT notation

• The HT of g(t) is denoted by
• In the frequency domain,

69
Find HT of a Sinusoid

• Q what is the HT of cosine?Anssine

70
HT Properties

• Property 1
• g and HT(g) have the same amplitude spectrum
• Property 2
• Property 3
• g and HT(g) are orthogonal, i.e.

71
Using HT Pre-envelope

• From a real-valued signal, we can extract a
  complex-valued signal by adding its HT as follows
• g(t) is called the pre-envelope of g(t)

72
Question is Why?

• It turns out that it is easier to work with g(t)
  than g(t) in many comm. situations
• We can always go back to g(t)

73
Pre-envelope Example

• Find the pre-envelope of the RF pulse
• We can re-write g(t) as follows

74
Pre-envelope is...

• Compare the following two
• Pre-envelope of
• is

75
Pre-envelope in the Frequency Domain

• How does pre-envelope look in the frequency
  domain?

76
Pre-envelope in positive and negative frequencies

• Lets evaluate G(f) for fgt0

G(f)
f
G(f)
f
77
Interpretation

• Fourier transform of Pre-envelope exists only for
  positive frequencies
• As such per-envelope is not a real signal. It is
  complex as shown by its definition

78
Corollary

• To find the pre-envelope in the frequency domain,
  take the original spectrum and chop off the
  negative part

79
Example

• Find the pre-envelope of a modulated message

G(f)
G(f)
AM signal
80
Another Definition for Pre-envelope

• Pre-envelope is such a quantity that if you take
  its real part, it will give you back your
  original signal

original signal
81
Bringing Signals Down to Earth

• Communication signals of interest are mostly high
  in frequency
• Simulation and handling of such signals are very
  difficult and expensive
• Solution Work with their low-pass equivalent

82
Tale of Two Pulses

• Consider the following two pulses
• Which one carries more information?

83
Lowpass Equivalent Concept

• The RF pulse has no more information content than
  the square pulse. They are both sending one bit
  of information.
• Which one is easier to work with?

84
Implementation issues

• It takes far more samples to simulate a bandpass
  signal

Sampling rate200Hz
0.01 sec.
4cycles/0.01 sec --gtfc400Hz
Sampling rate800Hz
85
Complex Envelope

• Every bandpass signal has a lowpass equivalent or
  complex envelope
• Take and re write as

86
Complex Envelope The Quick Way

• Rewrite the signal per following model
• The term in front of is the complex envelope
  shown by

m and theta contain all the information
87
Signal Representation Summary

• Take a real-valued, baseband signal

G(f)
g(t)
88
Pre-envelope Summarybaseband
G(f)
G(f)
Nothing for flt0
89
Pre-envelope Summary bandpass

• Baseband signal

G(f)
G(f)
90
Complex Envelope Summary

• Complex/pre envelope are related

G(f)
91
RF Pulse Complex Envelope

• Find the complex envelope of a T second long RF
  pulse at frequency fc

92
Writing as Re

• Rewrite g(t) as follows

Comp.Envjust a square pulse
93
RF Pulse Pre-envelope

• Recall
• Then

94
Story in the Freq. Domain
Pre-env. Spectrum (only fgt0 portion)
Original RF pulse spectrum
95
Complex Envelope Spectrum

• Complex envelopelow pass portion

96
CHANNELS AND SIGNAL DISTORTION

• Some of the material not in the book

97
Signal Transmission Modeling

• One of the most common tasks in communications is
  transmission of RF pulses through bandpass
  channels
• Instead of working at high RF frequencies at
  great computational cost, it is best to work with
  complex envelope representations

98
Channel I/O

• To determine channel output, we can work with
  complex envelopes

99
Passing an RF Pulse through a Bandpass Channel

• Here is the problem what is the output of an
  ideal bandpass channel in response to an RF pulse?

H(f)
100
What is the Complex envelope of H(f)?

• It is the lowpass equivalent of H(f)

2
1
B
2B
H(f)
101
What is the Complex Envelope of the RF Pulse?

• We found this before

102
Channel Output

• Here is what we have
• Channel complex envelope
• Input complex envelope
• Output

Bbandwidth
103
Interpretation
For the pulse to get through unscathed, channel
bandwidth must be larger than pulse
bw Bgt1/Tbit rate
1/T
B
104
What Does Distortion Do?

• Channel Distortion creates pulse dispersion

Channel
interference
105
Case of No Distortion

• There are two distortions we can live with
• Scaling
• Delay

To
106
Modeling Distortion-free Channels

• The input-output relationship for a
  distortion-free channel is
• y(t)Ax(t-Td)
• x(t)input
• y(t)output
• Ascale factor
• Td delay

107
Response of a Distortion-free channel

• What is channels frequency response?
• Take FT of the I/O expression
• Then

108
Amplitude and Phase Response
H(f)
Const amplitude response
f
/_H(f)
Linear phase response
f
109
Complete Model

• The complete transfer function is
• Since this is a lowpass function, its complex
  envelope is the same as H(f)

110
Lowpass Channel

• Is a first order filter an appropriate model for
  a distortion-free channel?
• To answer this question we have to test the
  definition of the ideal channel

R
C
111
Amplitude and Phase Response
3-dB bandwidth a/2pi1/(2piRC)
112
Response for RC10-3
bandwidth159 Hz
113
An ideal Channel?

• We must have constant amplitude response and
  linear phase response.
• Do we?. Deviation of H(f) from the ideal is
  tolerated up to .707form the peak.
• The frequency at which this occurs is the 3dB
  bandwidth

No signal distortion if input frequencies are
kept below 3dB bandwidth or 159 Hz here
114
Linear Distortion

• If any of the ideal channel conditions are
  violated but we are still dealing with a linear
  channel, we have linear distortion

amplitude
f
phase
115
Pulse Dispersion

• Putting a pulse g(t) through this filter produces
  3 overlapping copies

channel with distortion
T
gtT
116
Why?

• Let g(t) and r(t) be the transmitted and received
  signals. Then

117
Nonlinear Distortion

• This is the most serious kind where input and
  output are related by a nonlinear equation

Nonlinear channel
r
g
r
rg2
g
118
Impact of Nonlinear Dist.

• Nonlinear channels generate new frequencies at
  the output that did not exist in the input
  signal. Why?

G(f)
f
W
R(f)
f
2W
119
Practice Problems

• For pre-envelope 2.23
• For filtering using complex envelope 2.32