PULSE SHAPED 8-PSK BANDWIDTH EFFICIENCY AND … · narrowing bandwidth and increasing band...
Transcript of PULSE SHAPED 8-PSK BANDWIDTH EFFICIENCY AND … · narrowing bandwidth and increasing band...
PULSE SHAPED 8-PSK
BANDWIDTH EFFICIENCY
AND SPECTRAL SPIKE ELIMINATION
JIANPING TAO, B.EBG.
NMSU-ECE-98-003 MAY 1998
Presented to Dr. James LeBlanc
in partial fulfillment of the requirements
for the Degree
Master of Science in Electrical Engineering
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New Mexico State University
Las Cruces, New Mexico
May 1998
https://ntrs.nasa.gov/search.jsp?R=19980055189 2018-07-09T23:06:39+00:00Z
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ACKNOWLEDGMENTS
I would like to thank Dr. James P. LeBlanc, my advisor, for his help and
direction in the development and preparation of this report. I am truly indebted to Dr.
Sheila B. Horan who brought me to the communication field, for her advice, help and
direction in my research. I also wish to thank Dr. Phillip De Leon for his direction on
the sampling theorem and advice in my report. I thank Dr. Tonghui Wang for his
valuable advice in my report. I am grateful to Dr. Marvin K. Simon at the Jet
Propulsion Laboratory, for providing me his report on discrete spectrum.
I would like to thank the National Aeronautics and Space Administration
(NASA) for their support under Grant NASA# NAG 5-1491 and for giving me the
chance to work on this project. Also a special thank you to the people in the Center
for Space Telemetry and Telecommunications Systems at New Mexico State
University for their assistance.
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ABSTRACT
PULSE SHAPED 8-PSK
BANDWIDTH EFFICIENCY
AND SPECTRAL SPIKE ELIMINATION
BY
JIANPING TAO, B.EBG.
Master of Science in Electrical Engineering
New Mexico State University
Las Cruces, New Mexico, 1998
Dr. James P. LeBlanc, Chair
The most bandwidth-efficient communication methods are imperative to cope
with the congested frequency bands. Pulse shaping methods have excellent effects on
narrowing bandwidth and increasing band utilization. The position of the baseband
filters for the pulse shaping is crucial. Post-modulation pulse shaping (a low pass
filter is located after the modulator) can change signals from constant envelope to
non-constant envelope, and non-constant envelope signals through non-linear device
(a SSPA or TWT) can further spread the power spectra [6]. Pre-modulation pulse
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shaping (a filter is located before the modulator) will have constant envelope. These
two pulse shaping methods have different effects on narrowing the bandwidth and
producing bit errors. This report studied the effect of various pre-modulation pulse
shaping filters with respect to bandwidth, spectral spikes and bit error rate. A pre-
modulation pulse shaped 8-ary Phase Shift Keying (8PSK) modulation was used
throughout the simulations.
In addition to traditional pulse shaping filters, such as Bessel, Butterworth
and Square Root Raised Cosine (SRRC), other kinds of filters or pulse waveforms
were also studied in the pre-modulation pulse shaping method. Simulations were
conducted by using the Signal Processing Worksystem (SPW) software package on
HP workstations which simulated the power spectral density of pulse shaped 8-PSK
signals, end to end system performance and bit error rates (BERs) as a function of
Eb/No using pulse shaping in an AWGN channel. These results are compared with
the post-modulation pulse shaped 8-PSK results.
The simulations indicate traditional pulse shaping filters used in pre-
modulation pulse shaping may produce narrower bandwidth, but with worse BER
than those in post-modulation pulse shaping. Theory and simulations show pre-
modulation pulse shaping could also produce discrete line power spectra (spikes) at
regular frequency intervals. These spikes may cause interference with adjacent
channel and reduce power efficiency. Some particular pulses (filters), such as
trapezoid and pulses with different transits (such as weighted raised cosine transit)
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were found to reduce bandwidth and not generate spectral spikes. Although a solid
state power amplifier (SSPA) was simulated in the non-linear (saturation) region,
output power spectra did not spread due to the constant envelope 8-PSK signals.
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TABLE OF CONTENTS
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LIST OF TABLES .................................................................................................................. viii
LIST OF FIGURES ................................................................................................................ ix
LIST OF ABBREVIATIONS .............................................................................................. xm
Chapter
1 INTRODUCTION ................................................................................................................
2 THEORY ................................................................................................................................ 5
2.1 8-PSK Pulse Shaping Methods ......................................................................... 5
2.2 Filters Used for 8-PSK Pulse Shaping ............................................................... 8
2.3 Power Spectral Components and Pulse Shaping Waveforms ...................... 15
2.3.1 Baseband Pulse Shaped 8-PSK Vector-Matrix Notation ..................... 15
2.3.2 Power Spectral Components of Pulse Shaped 8-PSK ........................ 18
2.3.3 Pulse Shaping Waveforms ................................................................. 23
2.4 Solid State Power Amplifier (SSPA) .......................................................... 32
2.5 Additive White Gaussian Noise Channel ........................................................... 33
2.6 Receiver And Error Rate Estimation System ........................................................ 34
3 PULSE SHAPED 8-PSK SIMULATIONS ................................................................. 37
3.1 Simulations on Power Spectral Density ............................................................... 37
3.1.1 PSD Simulation Results of Using Traditional Filters ......................... 39
3.1.2 PSD Comparison Between Two Pulse Shaping Methods ............... 47
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3.1.3 Simulation Results of Particular 8-PSK Waveforms ......................... 49
3.1.4 Analysis Results of Particular 8-PSK Waveforms ............................ 53
3.2 Bit Error Rates Simulations ................................................................................. 55
4 CONCLUSIONS AND RECOMMENDATIONS ....................................................... 61
REFERENCE ........................................................................................................ 62
vii
LIST OF TABLES
= =
Table 2-1
Table 3-1
Table 3-2
Table 3-3
Table 3-4
Table 3-5
Table 3-6
Gray Codes and Phase ¢ ................................................................................... 6
Traditional Filters Used in the Simulations ................................................. 38
Utilization Ratio for Post-Modulation Pulse Shaping 8-PSK .............. 48
Utilization Ratio for Pre-Modulation Pulse Shaping 8-PSK ................ 48
PSD Line Components for Trapeziod Shaping .................................. 53
PSD Line Components for Weighted Raised Cosine Transit
Shaping ............................................................................................... 53
PSD Line Components for Raised Cosine Shaping ........................... 53
Table 3-7 PSD Line Components for Cosine Transit Shaping .......................... 54
Table 3-8 Utilization Ratio of Pre-Modulation Pulse Shaping with Different
Waveforms ........................................................................................... 54
Table 3-9 ISI Loss Measurements at 10 -3 BER (Compared to Ideal 8-PSK) ....... 59
Table 3-10 ISI Loss Measurements at 10 _ BER (Compared to Ideal BPSK) ....... 59
VII1
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LIST OF FIGURES
Figure 2-1
Figure 2-2
Figure 2-3
Figure 2-4
Figure 2-5
Figure 2-6
Figure 2-7
Figure 2-8
Figure 2-9
8-PSK Signal Constellation ......................................................................... 6
Producing Phase Signals Block Diagram ........................................... 7
Modulation Schemes For Unfiltered And Filtered Phase Signals ....... 8
Amplitude Response of Various Orders of Butterworth Filters ......... 9
An Impulse Response of 5th-Order of Butterworth Filters .............. 9
Amplitude Response of Various Orders of Bessel Filters .................. 10
An Impulse Response of 3rd-Order of Bessel Filters ........................ 1I
Frequency Response: Raised Cosine Characteristics with Different
rolloff factor a .................................................................................. 12
Impulse Responses of Raised Cosine Filters for Different a ............. 12
Figure 2-10 The Waveforms of Phase _ and Filtered Phase (p' .......................... 13
Figure 2-11 The Waveforms of Real Part and Imaginary Part of 8-PSK .............. 14
Figure 2-12 The Waveforms of Real Part and Imaginary Part of Pre-Modulation
Filtered 8-PSK .................................................................................. 15
Figure 2-13 Pulses with Different Overlapping Index K from [1 ] ....................... 24
Figure 2-14 T-s Duration Cosine Pulses Used as Phase Pulse and 8-PSK
Signaling ........................................................................................... 25
Figure 2-15 2T-s Duration Cosine Pulses Used as Phase Pulse and 8-PSK
Signaling ........................................................................................... 26
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Figure 2-16
Figure 2-17
Figure 2-18
Figure 2-19
T-s Duration Raised Cosine Pulses Used as Phase Pulse and 8-PSK
Signaling ........................................................................................... 27
2T-s Duration Raised Cosine Pulses Used as Phase Pulse and
8-PSK Signaling ............................................................................... 28
Weighted Raised Cosine Pulses Used as Phase Pulse and 8-PSK
Signaling ........................................................................................... 29
2T-s Duration Trapezoid Pulses Used as Phase Pulse and 8-PSK
Signaling ........................................................................................... 30
Figure 2-20 Weighted Raised Cosine Transit Pulses Used as Phase Pulse and
8-PSK Signaling ............................................................................... 31
Figure 2-21 The Output Characteristic of 10 Watts SSPA ................................... 32
Figure 2-22 The Structure of SSPA ...................................................................... 33
Figure 2-23 Power Spectral Density of White Gaussian Noise ............................. 33
Figure 2-24 Block Diagram for the Receiver Error Rate Estimator ..................... 34
Figure 2-25 Block Diagram of Synchronizer ....................................................... 35
Figure 2-26
Figure 3-1
Figure 3-2
Figure 3-3
Figure 3-4
Simple Error Rate Estimator Block Diagram ................................... 36
Block Diagram of Spectrum analysis ................................................ 37
Unfiltered 8-PSK Power spectral Density .......................................... 40
Power spectra of 8-PSK Pulse Shaped with 5th-Order Butterworth
Filter (BT=I) ...................................................................................... 41
Power spectra of 8-PSK Pulse Shaped with 5th-Order Butterworth
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Filter (BT=2) ......................................................................................41
Figure 3-5 Power spectra of 8-PSK Pulse Shaped with 5th-Order Butterworth
Filter (BT=2.8) ................................................................................... 42
Figure 3-6 Power spectra of 8-PSK Pulse Shaped with 5th-Order Butterworth
Filter (BT=3) ....................................................................................... 43
Figure 3-7 Power spectra of 8-PSK Pulse Shaped with 5th-Order Bessel
Filter (BT=I) ...................................................................................... 43
Figure 3-8 Power spectra of 8-PSK Pulse Shaped with 5th-Order Bessel
Filter (BT= 1.2) ................................................................................... 44
Figure 3-9 Power spectra of 8-PSK Pulse Shaped with 5th-Order Bessel
Filter (BT=2) ...................................................................................... 45
Figure 3-10 Power spectra of 8-PSK Pulse Shaped with 5th-Order Bessel
Filter (BT=3) ...................................................................................... 45
Figure 3-11 Power spectra of 8-PSK Pulse Shaped with SRRC Filter (a =1) ..... 46
Figure 3-12 Utilization Ratio of Using 5th-Order Butterworth Filters in
Pre-Modulation Pulse Shaping ........................................................ 48
Figure 3-13 Utilization Ratio of Using 3rd-Order Bessel Filters in
Pre-Modulation Pulse Shaping ........................................................ 49
Figure 3-14 PSD Comparison of 8-PSK Between Two Pulse Shaping Methods
(5th-Order Butterworth Filter, BT =1 ) ............................................ 49
Figure 3-15 PSD of 8-PSK Used 2T-s Duration Trapezoid Pulse Shaping ......... 5o
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Figure 3-16 PSD of 8-PSK Used 3T-s Duration Trapezoid Pulse Shaping ......... 51
Figure 3-17 PSD of 8-PSK Used 2T-s Duration Cosine Transit Signaling Pulse
Shaping ............................................................................................. 51
Figure 3-18 PSD of 8-PSK Used 2T-s Raised Cosine Transit Signaling Pulse
Shaping ............................................................................................. 52
Figure 3-19 PSD of 8-PSK Used 2T-s Duration Weighted Raised Cosine
Transit Signaling Pulse Shaping ........................................................ 52
Figure 3-20 Utilization Ratio of Different Pulses in Pre-Modulation Pulse
Shaping .............................................................................................. 54
Figure
Figure
Figure
Figure
Figure
Figure
3-21 BER simulation System Block Diagram .......................................... 55
3-22 BER Plot for Pre-Modulation Pulse Shaping Filters with BT--1 ...... 56
3-23
3-24
3-25
3-25
BER Plot for Pre-Modulation Pulse Shaping Filters with BT=2 ...... 57
BER Plot for Pre-Modulation Pulse Shaping Filters with BT--3 ...... 57
BER Plot for Pre-Modulation Pulse Shaping With 3T Trapezoid ... 58
Plot for ISI Loss as Utilization Ratios for Pre-Modulation Pulse
Shaping .............................................................................................. 6O
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k..,LIST OF ABBREVIATIONS
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•r rmmB
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(X
P
AWGN
dB
BER
BPSK
BW
BT
CCSDS
Eb/No
ESA
fs
fc
FFT
Hz
HP
IIR
ISI
JPL
LSB
MSB
MSK
NASA
NMSU
NRZ-L
QPSK
Rolloff Factor for SRRC filters
Band Utilization Ratio
Additive White Gaussian Noise
Decibels
Bit Error Rate
Binary Phase Shift Keying
Bandwidth
Bandwidth-Time (symbol) product
Consultative Committee for Space Data Systems
Energy (bit) to Noise ratio
European Space Agency
Sampling Frequency
Carrier Frequency
Fast Fourier Transform
Hertz
Hewlett Packard
Infinite Impulse Response (Filter)
Intersymbol Interference
Jet Propulsion Laboratory
Least Significant Bit
Most Significant Bit
Minimum Shift Keying
National Aeronautics and Space Administration
New Mexico State University
Non-Return-to-Zero Logic
Quaternary Phase Shift Keying
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i
W
= --
I
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-- I
PA
PCM
PM
PSD
PSK
QPSK
Rb
Rs
RC
sec
SFCG
SNR
SPW
SSPA
SRRC
Tb
TWT
Power Amplifier
Pulse Coded Modulation
Phase Modulation
Power Spectral Density
Phase Shift Keying
Quartenary Phase Shift Keying
Bit Rate
Symbol or Baud Rate
Raised Cosine
Seconds
Space Frequency Coordination Group
Signal-to-Noise Ratio
Signal Processing Worksystem
Solid State Power Amplifier
Square Root Raised Cosine
Bit Period
Traveling Wave Tube
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Chapter 1
INTRODUCTION
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With the continuing development of communications and increasing users,
frequency bands are becoming more and more congested. In order to cope with this
frequency congestion, methods to increase bandwidth utilization have be studied in
[1]-[6] and [13]-[16]. These methods consider efficient modulation which is
implemented using pulse shaping methods.
Two kinds of pulse shaping methods have been investigated to realize
bandwidth efficiency. One kind of pulse shaping methods is pre-modulation pulse
shaping which performs a low pass filtering before modulation to reduce bandwidth.
Pre-modulation pulse shaping with different modulation schemes, such Pulse Code
Modulation (PCM), Binary Phase Shift Keying (BPSK), Quaternary Phase Shift
Keying (QPSK) and Gaussian filtered Minimum Shift Keying (GMSK) have been
studied in [13]-[16]. The other pulse shaping method is post-modulation pulse
shaping which conducts low pass filtering after modulation. Caballero [6] has used
this method to simulate 8 level Phase Shift Keying (8-PSK) signaling over non-linear
satellite channels.
In [6] post-modulation pulse shaping method is investigated with various
filters. Three kind of filters, consisting of a fifth order Butterworth, third order Bessel
and Square Root Raised Cosine (SRRC) filters, with different Bandwidth-Time (BT)
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product or roll off factor, have been used to reduce bandwidth in the simulations of 8-
PSK over non-linear satellite channels. Because of post-modulation pulse shaping, it
produced a non-constant envelope pulse shaped 8-PSK which can spread the power
spectrum when the signals go through a non-linear Solid State Power Amplifier
(SSPA). The simulations also included measures of symbol error rate (SER) for post-
modulation pulse shaped 8-PSK signaling. Simulation results indicated that there is
trade-off between bandwidth efficiency and SER.
Prabhu and Rowe [1] presented a matrix method for computing the power
spectrum of digital phase modulation in 1974, and studied pre-modulation pulse
shaped pulse train with or without overlapping. Pre-pulse shaping may produce not
only a continuous component but also a discrete line component at some particular
frequencies in power spectral density (PSD). Based on Prabhu and Rowe, Simon
recently obtained explicit expressions for power spectrum when signaling is m-ary
PSK in the presence of intersymbol interference (ISI) produced by pulse shaping.
Martin and Nguyen in [13],[15] and [16], Otter in [14] have studied different
traditional filters (Butterworth, Bessel and SRRC filters) used as pre-pulse shaping
for different modulation types (PCM, BPSK, QPSK and GMSK). By comparing pre-
modulation pulse shaping and post-modulation pulse shaping, Martin and Nguyen
indicated that pre-modulation pulse shaping can overcome disadvantages produced by
post modulation pulse shaping, such as larger, heavy filters which are not compatible
with some mission operations requirements.
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In contrast to using traditional filters by Martin and Nguyen, Greenstein [4]
has examined PSK with different overlapping baseband pulses. Greenstein
investigated triangular, cosine, raised cosine and Nyquest pulse shapes, yielding the
results for continuous and discrete line components of PSD.
Although many studies have investigated pre-modulation pulse shaping and
post-modulation pulse shaping with various filters, post-modulation pulse shaping can
induce a non-constant envelope signaling which can spread the reduced bandwidth.
Pre-modulation pulse shaping can induce power spectral spikes which are not good
for power efficiency and may produce adjacent channel interference. A study is
needed for pre-modulation pulse shaping which produces a constant modules signal
with no spikes
This project conducted simulations on pre-pulse shaped 8-PSK over non-
linear channel. First, three kinds of traditional filters, 5th order Butterworth, 3rd order
Bessel and SRRC, were used as pulse shaping filter applied to 8-PSK phase signals.
Filtered phase signals were then modulated to become pulse shaped 8-PSK signals.
Pulse shaped 8-PSK signals were amplified by a non-linear SSPA. In order to
simulate channel noise, an additive white Gaussian noise (AWGN) was added. At the
receiver, an integrate and dump circuit was used. Power spectra produced by pre-
modulation pulse shaping were compared with those using post-modulation pulse
shaping. In addition, received bit error rate 03ER) was also compared between two
pulse shaping methods. Second, theoretical power spectrum for pulse shaped 8-PSK
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was studied based on [1] and [2]. Third, many pulse waveforms were found for 8-
PSK to reduce bandwidth and reduce or eliminate power spectral spikes. Simulations
and numerical results indicated that trapezoid and weighted raised cosine transit
pulses with more than two symbol periods can get rid of spectral spikes. These results
are in agreement with Simon's recent report [3]. Finally, According to simulation
results, recommendations were given for searching for optimal pulses to reduce
bandwidth without spikes.
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Chapter 2
THEORY
This section will introduce some of the concepts which were used for this project.
First, 8-PSK signals and pulse shaping methods will be described. Butterworth, Bessel and
Raised Cosine Filters are discussed as traditional pulse shaping filters. Second, a detailed
discussion of pulse shaping waveforms and power spectral components of pre-modulation
pulse shaped PSK signals will be presented. Pre-modulation pulse shaped 8-PSK power
spectra consist of continuous and discrete line (spikes) components. Various pulse
shaping waveforms, such as trapezoid and raised cosine are used. Third, since a Solid
State Power Amplifier (SSPA) was used in the simulations, the characteristics of the SSPA
will be studied. Finally, the topic will be an Additive White Gaussian Noise (AWGN)
channel and a matched filter used as receiver.
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2.1 8-PSK Pulse Shaping Methods
In 8 level Phase-Shift Keying (8-PSK), information is contained in the phase of the
signal, and each symbol has 3 bits. The phase of the carrier can
spaced values, such as in Table 2-1 and shown in Figure 2-1.
constellation is shown in Figure 2-1. An 8-PSK signal, S(t), can be expressed as following.
take one of 8 equally-
A plot of 8PSK signal
S(t) = Acexp[j(coc t +_o (t))] = A¢cos[ co_t + (,o(t)l+jAcsin[o_ t + cp(t)] (2-1)
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IMAGINARY
(QUADRATURE)
Ai
011 ! 001
]
Yi I............ :6 oooI
Xi
• I00
111 I01
REAL
('1_ PHASE)
Figure 2-1 8PSK Signal Constellation
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Where the envelope A c is constant in this project, _ =2 zc f_, and fc is the carrier frequency
in hertz, _o is the phase which takes on one of the 8 discrete values (in degrees)
corresponding to the 3 bit Gray code. Table 2-I shows the relationship between Gray
codes and _o.
Table 2-1 Gray Codes and Phase _p
Binary Gray Codes (p (Degree)
000 22.5 °
001 67.5 °,I
011 112-5 °
010 157.5 °
110 147.50
111 247.5 °
101 292.5 °
100 337.5 °
In SPW, the Gray codes and phase q_ can be produced by the design shown in Figure 2-2.
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The phase signals q_(t) can be modulated to 8-PSK signals v(t), where
v(t)=Ace j _p (t). To realize the pre-modulation pulse shaping, the phase signal, (# (t), goes
through a pulse shaping filter before it is modulated. In this project, various filters are
used. In addition, the change of the position of the pulse shaping filters can produce a
change in results. If the pulse shaping filter is placed after the modulator [5], this method is
called as post-modulation pulse shaping. Figure 2-3 shows the modulation for unfiltered
and filtered 8-PSK signals.
Random [ JBinary toData ] "lNumeric
I Random _._.__Binary to tData[ INumeric
Random
Data
JBinary to
"]Numeric
control
xix2 control
T
Number
to
Symbol
_p(t)OUTPUT
Figure 2-2 Producing Phase Signals Block Diagram
Where CI=I, C2=2, and C3=4. The Number to Symbol block functions as Table 2-1, and
completes the conversion from Gray codes to phase in degrees.
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= ,,
Phase ]modulator
_1 Ace j(p
8PSK v(t)
(a) Unfiltered 8PSK With Constant Envelope
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Figure 2-3.
Phase (a Modulator
A cejfO I Pulse- Shaping
8-PSK vl (t)
Co) Post-Modulation Pulse Shaping Method
Phase Pulse [Modulator
Shaping _[ Acej(p
8-PSK v2(t)
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(c) Pre-Modulation Pulse Shaping Method
Modulation Schemes for Unfiltered and Filtered Phase Signals.
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2.2 Filters Used for 8-PSK Pulse Shaping
Pulse Shaping Filters are used to narrow bandwidth and improve bandwidth
utilization. In this report several traditional types of filters such as the 5th order
Butterworth, 3rd-order Bessel and Square Root Raised Cosine (SRRC)( a =1) are utilized
in the simulations. Other kinds of filters with overlapping or non-overlapping pulses, will
be introduced in section 2.3.2.
Pulse shaping can induce distortions of 8-PSK signals and increase the risk of Inter-
Symbol Interference (ISI)[4]. The distortions due to the pulse shaping filters make the
implementation of an optimal receiver difficult.
b.
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Butterworth filters: Butterworth filters provide the maximally fiat response in the
passband. Response varies monotonically within the stopband, decreasing smoothly from
f2 =0 and £2 = oo. In the simulations, the 5th order Butterworth filter was used. The Figure
2-4 shows the amplitude response for different orders of Butterworth filters. The cut off
frequency fc corresponds to -3dB amplitude point.
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-40
-6o
-80
-100
Buttorvvorth Filter fc(3dS) = 1 Hz & ordor=2.3.4.5,6.7,8
o. 3: : _ ,_. :.<_'--.. _---..._ .._.... _ i'm :
...... -t! i i i--_:----_ ........ :........... .... -'_ :,-.z _,_ _,_.__z.Z._._.. _._Q_._[_._i_:,.-._ ........ _ ..q-._Q ....... _ ....
: • . : f : \ \ _ _'_. _ "__ --.._ : _-.__
........... ....!-t!.....-.---,.... -i _ ill "q.."-,,_'-., '--.. i _ _ "
Cut-Olaf _ i I i n = S "'--='="-""_:.'_ "_. _ _ "'-_._
..... E_,_,,._.,t_y..4.... -/--"........................ "--":-_ .... _."- -:"_ .... :_-_-
...... _ ..... ["" :.'.... _'"i ........................ r ......... :r--'K .... " ....
Frequency (H:z)
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Figure 2-4 Amplitude Response of Various Orders of Butterworth Filters
-0
m pulse R esponse of 5th-O rder B utterwo rth Filte
...... I ......... _.......... i.......... _.......... r .......... i.......
0 1 2 3 4 5 6
Tim e (S ym bol tim e T)
Figure 2-5 An Impulse Response of 5th-Order of Butterworth Filter
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All orders of this filter pass through the 3 dB cut-off point. An impulse response of 5-
order Butterwouth filter is shown in Figure 2-5. Since the duration of the impulse response
is more than one symbol period T, it may produce Intersymbol Interference (ISI).
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Bessel Filters: The Bessel type of filter has the slowest transition between
passband and stopband and the passband phase response is better than the Butterworth. It
has approximately linear phase in the passband so that it has a property of preventing
dispersion of the signal. The amplitude for Bessel filters is not as flat in the passband
region as for the Butterworth filters. This project uses a 3rd order Bessel filter for the
simulation. Figure 2-6 shows the amplitude response of various orders of Bessel Filters.
Note that the amplitude is not as fiat in the passband region as for the Butterworth filter.
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-20
-40
-60
-8O
-100
Bessel Filter fc(3dB) = 1 HZ & n=2.3.4.5.8.7.8
-_ ! ; ,-----L-_ : n=2i i
........ _.3_¢__-____-___.:........................ .:._.__.__,_____:! .=_ '-i----/i
10 °
Frequency (I-12:)
Figure 2-6 Amplitude Response of Various Orders of Bessel Filters
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Also the transition from the passband to the stopband region is not as rapid as for the
Butterworth Filters for the various orders of filters. An impulse response of the 3-order
Buessel filter is shown in Figure 2-7.
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1
0.8
0.6
0.4
0.2
0
Impulse Response of 3rd-Order Bessel Filter| ! i
.............. • ................. ,D ................. ,_ .................
!!!iiiiii!iiiii!!!!i!::::::i:::::i:::ii::::::ii:i::::::i1
1 Time (Symbol _me "F) 3
Figure 2-7 An Impulse Response of 3rd-Order of Bessel Filter
Square Root Raised Cosine Filter (a =1) The Raised Cosine filters are used to eliminate
ISI in a linear channel. The transfer function of the Raised Cosine Filter is
RC(f)=
(1 - a)"T, 0 _<lfl <
2T
2[,. L 2a ' _-_)-<1-'1-<2T(1 + a)o, I/F>
2T
(2-2)
where the zeros will occur at t = nT (T is the sampling interval) and ct is the rolloff factor
or the excess bandwidth over the Nyquist Band Filter which can be varied from 0 to 1. The
rolloff factor, o_, determines the bandwidth of a Raised Cosine Filter. Figure 2-8 shows the
I1
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frequency response of the Raised Cosine Function with different rolloff factors
0.25, 0.5 and 1) and T =1 seconds.
1
0.9
0.8 • ..... _V;--_-"
............. _.................0.7
0.6
0.5
O4
0.3
0.2
0.1
00
..... -1./(2a'-)-w_h._r,_-1:_-_1-s-e-e.:-.....fo!r ct = 0
i i0.2 0.4
Transfer Function of Raised Cosine Filter with r = 0. 0.25.0.5 and 1
= 0250t
_!_ _.-0......i........................; ...... :" - IVT- W'h-_ t'e- _ aq- g_ ..................
for_ -_ I
.... F........ .-rrh_TT ........ _.............
.....!........................
0.6 0.8 1
Normalized Frequency (I-_)
Figure 2-8 Frequency Response: Raised Cosine Characteristics with Different
(t_ = 0,
r It
z_
m,
08
06
-O2
-3
Time I_esponse of F_sised Cosine _'lltef _th r = O. O :_. O _; and 1
............. _............ i........ 7------_ ........ i.........................
.........--_i-;--;---_-_-,-i:----//-----i.....\_............Y:.-_,;-_:-,_---
............_........,,___i....,......._....._ ....i..........:;;.........Ti me (t)
Figure 2-9 Impulse Response of Raised Cosine Filters for Different ct
Figure 2-9 gives impulse response for different rolloff factors. From Figure 2-9, it can
be seen that if the symbols are sampled at the nulls of the sinc function then the symbol
interference from the previous and next symbols will not exist. As mentioned in [17], if
the Raised cosine filters are used the ISI will be eliminated in a linear channel. To
eliminate ISI in a linear channel, a Square Root Raised Cosine filter is used in both the
12
l
transmitter and the receiver. If RC(f') is the frequency response of the Raised Cosine
Filter, the frequency response of Square Root Raised Cosine filter is
SRRC(f)= _ (2-3)
According to [10], the coefficients of SRRC filter can be obtained from the formula (2-4):
[cos((l+a)rc 7)+ sin((1-a)rc )]h(t)=4 a (2-4)t 2
rcVrT[(4a -_) -1]
In this equation, cr is the roll off parameter and T is one symbol period (1/symbol rate).
The Real Raised cosine block in SPW is selected for the project simulation. Ture SRRC
filtering is noncausal and has infinite coefficients, but actually a finite number of
coefficients are needed to determine the FIR tap length. This truncation can cause some
frequency distortion. Since pre-modulation pulse shaping makes the channel no longer
linear, simulation will show that SRRC can not eliminate ISI.
u n lille re p h a s e
i ............... i................ i- .............. J4 ................. . ............... _:................. i ...............
0 500 1000 1500 2000
filte re d p h a s e s a m p le s
4
2
0
0 5001000 1500 2000
sam pies
Figure 2-10 The Waveforms of Phase _pand Filtered Phase _p'
(Butterworth 5th filter is used, BT=I)
The Observation of Pulse Shaped 8-PSK Waveforms. With the use of pulse shaping
filters, the 8-PSK signals change their shapes. The main character of the waveform change
13
is a transient (rise and drop) in the filtered _p' (as shown in Figure 2-10). In Figure 2-10, it
can be seen that the filtered phase (a' has a smooth transition from 0 to 2 n., in sample
range 0 to 500. The transient (rise or drop) makes the big difference between unfiltered and
the filtered 8-PSK shown in Figure 2-11 and Figure 2-12.
"x,,,
Real 1of v(t)
0.5
0
-0 .5
-1
Im ag
of vCt) 1
0 .01 0 .02 0 .03 0 .04 0 .05 0 .06
Tim e s
0.5
-0 .5
-10 0.01 0.02 0.03 0.04 0 05 0.06
Times
-±
IiR
w
Figure 2-11 The Waveforms of Real Part and Imaginary Part of 8-PSK
The filtering process elongates data symbols such that they overlap one another. This
shows the existence of Intersymbol Interference (ISI) introduced by the filtering process.
When the pulse waveforms pass through the bandlimited system, they will be spread or
dispersed which increases error rates and affects power spectrum.
m
m
14
X-.
real part of v2 {t)
1
oo-0 .5
-1 .....
0 500 1000 1500 2000
im a gary part of v2 (t) sam pies
1
-0 .5
-1
500 1 000 1 500 2000
sam pies
Figure 2-12 The Waveforms of Real Part and Imaginary Part of Pre-
Modulation Filtered 8-PSK (512 Samples/Symbol)
r
i
w
2. 3 Power Spectral Components and Pulse Shaping Waveforms
Pre-modulation pulse shaping for phase modulation could produce power spectra
with not only continuous components but also spikes [1]-[4] and [13]-[16]. Prabhu and
Rowe [1] have given a rigorous discussion for components of pulse shaped PSK power
spectra. Simon [2] & [3] have also derived explicit formulas for continuous and discrete
components of pulse shaped PSK based on [1]. Greenstein [4] studied numerous pulse
shapes and compared their efficiency of narrowing bandwidth. In this section, 8-PSK
power spectral components based on [1] and [2] with an assumption of cyclostationary will
be discussed.
2.3.1 Baseband Pulse Shaped 8-PSK Vector-Matrix Notation
The following developments on power spectra of pre-modulation pulse shaped 8-
PSK are based on [1]. A digital angle-modulated carrier of the form can be described as
m
w
_I
_i
m
15
m
s(t)= cos [2 _r fct+ (p (t)]
=Re {exp[j2 n"let+ (,o(t)] }, (2-5)
m
= ..
where (p (t) is a digital angle modulation, and f¢ is the carrier frequency.
define the equivalent complex baseband modulation
v(t)=exp(j _p(t)),
According to Prabhu and Rowe, the spectrum of s(t) is
Ps(f)= _ Pv(f-f¢)+ ¼ Pv(-f-f_),
where Pv is the spectrum of v(t).
For pre-modulation pulse shaped 8-PSK, go(t) takes on one of eight level pulses.
Where p(t) is the
passed through an ISI-producing filter. Denote a column vector g(t) such that
g(t) =[gl(t) g2(t) g3(t) g4(t) gs(t) g6(t) g7(t) gs(t)] x .
If g_(t) is strictly time-limited to a bit interval T, for example, let i=2, then
(o (t) =g2(t) = _ 5p(t). ; 0 < t < T
Using vector notation, ¢ (t) becomes
_0(t) =[0 1 0 0 0 0 0 0]g(t)
For convenience, a column ak vector is defined as:
a k =[ ak (1) ak (2) ak (3) ak (4) ak. (5) ak (6) ak (7) akt8)] T,
and define the basis column vector
For convenience,
(2-6)
(2-7)
g_(t)= 8 [8-(2i-1)]p(t), (i=1,2 .... 8) (2-8)
pulse shaped waveform which may be the result of a rectangular pulse
(2 -9)
(2-1o)
(2-11)
16
m
m
r
and ak takes on only one of
_o(t) may be expressed as
et=[10000000] T
e2=[0 1 0 0 0 0 00 ]T
e8=[0 0 0 0 0 0 0 1 IT
the values el, e2, ... es
oO
(p (t)= Z ak T g(t)],(_ --oo
(2-12)
with the same probability 1 theng,
;-m < t _< m (2-13)
Using the same idea, define two vectors b k and r(t), such that
b k =[ bk (I)bk (2)bk (3)bk (4) bk (5)bk (6)bk (7)bk(8)]T, (2-14)
1
bk also takes on only one of the values el, e2, ... e8 with the same probability _,
r (t) =exp[jg(t) 1= [ exp(jgl(t)) expfjg2(t)) ... exp(jgs(t)) ]g. (2-15)
According to (2-10), the complex baseband 8-PSK modulation may be described as
v(t)=exp[j _o(t)]=exp[jgz(t)]=exp[ j -_ 5p(t)]
--[0 1 0 0 0 0 0 0]r(t) =b2 T r(t)
;0< t < T (2-16)
;0_< t < T (2-17)
Then v(t) can be written as
17
i
w
..+.
v(t)= _ bkTr(t-kT) ;-oo < t < oo (2-18)k=---_o
2.3.2 Power Spectral Components of Pulse Shaped 8-PSK
In order to determine the spectral density of (2-18), the signal r(t) is deterministic,
and bk is a random vector sequence from 2-14. Assume that bk is wide-sense stationary.
Then, define a mean vector M b of bk as follows
Mb=E[bk]. (2-19)
If all signal pulses are equally likely,
1 ITMb=_ [1 1 1 1 1 1 1 1 . (2-20)
The autocorrelation function matrix of b k is
(k-I)=E[bk bin], (2-21)
and the power spectral density matrix of bk is
E(f) = _ e'J2_fnR_ (n). (2-22)/'1=--_
The goal is to find the power spectral density of the baseband 8-PSK signal v(t). The
autocorrelation function of v(t) is given by
Rv(t+ r ,t) = E[v(t+ r )v*(t)] (2-23)
=E{ _ bkTr(t+r-kT)[ _ bkTr(t-IT)] * }k=-n_ I=--_
=E[ _ bkTr(t+r-kT) _ blnr*(t-lT)]k=-co /=-_o
18
--E[£k=--co ]=-_
b£ r r(t+ r -kT) btn r*(t-lT)]
--E[£ £ rT(t+r-kT)bkblnr*(t-lT)]k=-co 1=--.o
=£ £ rm(t+r-kT)E[bkbin]r*(t-lT)]k=--_ 1=--_
=£ £ rV(t+r-kT)_(k-1)r*(t-IT)k=-oo I=-_e
(2-24)
From (2-24), it is easy to show
Rv(t + nT + r ,t + nT)= Rv(t + r ,t), (2-25)
then, v(t) is cyclostationary. According to [9], the time-average autocorrelation function
over a signal period is defined as
T
1iR_(r) =T- r
2
Substituting (2-24) in (2-26),
R_(t+ r ,t) dt (2-26)
T
I=--_ T
2
rV(t+ r -kT) R0 (k-l)r (t-lT) dt.
Let k-l--n and t-IT --x,
1
R_(r)=y
T---IT
£ £ 21 rT(t+l--nT) _ (n)r*(x)dx
n=-=o I=--_ _T_I T2
19
Let t--x, then
rT(t+ r -nT) R, (n)r'(x)dx
=
w
= :
1 _, _ rV(t+r-nT)_(n)r*(t)dtRv( )=7n =-_0 --00
Power spectral density of v(t) is defined as
Pv(f) = S Rv( r )e -j2rc r r d r
= -_ e"J2 rT(t+ r -nT) _ (n)r*(t)dt d 2---00 n= --00 --o0
e-J2 _ f 2"rT(t+ r -nT) R--'_(n)r*(t) dt d r.
Let x = t+ r, then
1 _-'_ _ " 2Z" fx -- S " 7Ff'/"Pv(f) = _ 0=-_ e -J2 rT(x-nT) _ (n) dx e J2 r*(t) dt d r.-oo .-_
(2-27)
(2-28)
L..a
R(f) = 5 e -j2rc ft r(t) dt, is the-00
Thus Pv(f) can be expressed as
Fourier transform of r(t), and it is also a column vector.
! -
RT(f) e-J2z _T Ro (n) R*(f)
1
Te "j2rc fnT _(n) }R*(f) (2-29)
20
w
w
L...d_
| J
w
By using (2-22), Pv(f') becomes
1Pv(f') =_ RT(f) _(fT)R*(f)
since
_(f)= £ e-J2_f_R_ (n).
Introducing covariance matrix Kb (n) = Rb (n)- M b Mb n, then
(f)= £ e-J2 lr _n(_ (n)+ Mb Mb n)/'1 = --¢0
= £ e-J2 _ fnZ (n) + M b Mb tl _ e "j2n"fnn=---_ n=--_
Using Fourier series, we have
e-J2 _ fn = £ _ (f-n)n= --oC} n=-_O
£ e'J2_'frn 1 £.=-_ =-T ,=-_ g(f- ).
(2-30)
(2-31)
(2-32)
(2-33)
Therefore,
e-J2 7/"fn/_ (n)+ M b Mb H
n=--oo n=-oo
= Pbc(f) + Pb, (f) (2-34)
B
where Pbc (f) is continuous spectrum component, and Pbt (f) is the discrete line spectrum
component.
21
=
From (2-30) and (2-34),
m
p_(f) =_ RT(f)i_oc(fT)+ _ (flr)]R*(f)
=1 RT(0_(IT)R*(f)+ 1 RT(f) Po_ (£T)R*(f)T T
=Pv¢(t')+ P,,(t') (2-35)
Pvc(f) is the continuous component of the pulse shaped 8-PSK power spectral density. A
further study on Pvl(f) is given as follows.
I RT(f) Po_t (fT)R*(0e ,(0=
1 RT(f){ 1 _ ;=-_ -_-M b Mb H 8(f- )}R*(f)/3="-o0
1 n
T 2 RT(f)MbMbHR*(f') _ _'(f--_)/1=--o0
m
_1 nT 2 MbTR(f)MbHR*(f) _ _(f-_)
_ 1 Iu 6(f_T)T2 IMb T R(f)r/=---co
_ 1 s 12_ nT282 I,T__.,R_(0 6(f--_). (2-36)
i=1 n=-co
The Pvt(f) is the discrete line component. (2-36) is the result of non-overlapping pulses,
that is they are strictly time-limited to one symbol time. When signal pulses have
overlapping, [1] defined a overlapping index K which stands for maximum signal pulse
22
duration. According to [1] and [2], Pvl(f) for arbitrary index K may be expressed as
w
u
=
u
_1 .. £Pvl(f) T282K I_ Ri(f)[ 2 6(f- ). (2-37)i=1 n=--_
For 8-PSK with overlapping K=2, power spectral line component is
Pv,(f) T28 , I_ R_(t312 8(f- ). (2-38)i=1 n=--_
and
R(f)= j" e-J2 z rt r(t) dt-oo
TR(f) = e-i2.1,
"ei{ ot (t)+g_( t- r) }
e,i[ gl (t)+OM (t--T)}
ei{ oz (t)+ gt (t--T)}
ej[ g_ (t)+oM (t- T)I
eYl uz(t)+ot (t--T) }
ei[ ¢__, (t)+o_ (t--T)}
e;{ o,, .)+o, (t--T) }
e,Jl _M (t)+oH (t-T)]
dr.(2-39)
where M=8 and gi(t) can be obtained from (2-8).
2.3.3 Pulse Shaping Waveforms
Some particular signal pulses (filters) which overlap and do not overlap will be
studied. According to [1], waveforms for different overlapping index K are shown as
follows.
23
K- !
-i o 2
.. ,,"",,//_\, I'"- "'t', X\• X K-7
..' : ..: / \ \
= =
-T 0 f 2
- 0 1 2
, _ K_4
-T 0 T 71' ]'T 4T
Figure 2-13 Pulses with different overlapping index K from [1]
Numerous pulse shapes with different pulse duration are simulated to investigate the effect
on narrowing bandwidth and reducing spectral spikes. In the following expressions, T is
used for one symbol time in seconds. In order to obtain a better effect on reducing spikes,
some pulses will have T duration fiat with different transits.
Cosine with Single-Interval
expressed as
T duration cosine with a single interval can be
P(t) = {
Tcos(_ t/T) ltl - -
2
70 Itl >-
2(2-40)
=
;--d
7
24
T duration cosine pulse shaped and non-pulse shaped 8-PSK phase signals are shown in
Figure 2-14. In Figures2-14 to 2-20, assume one symbol duration T=1.2
w
i
= =
-7
i i
0.8
0.6
0.4
0.2
-0.25
Cosine Signaling with Pulse Duration T
I 1 I 1 I I I I t
....... _- ...... I- ........... t" ...... t" ...... f- ............ I" ...... t" ......
I I I I ] I I I I
....... ;- ...... _ ...... F ...... F ...... r ...... F ...... F ........... T ......I I I I 1 I I I i
I I ! i i I I
I I I I I I I I I__ i
1 I I t I I I I I
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25
4
2
00
Phase Signals
i i l _i f\
......... ,'-........ ',................. : ..... 7:7--i -I- --1-ql_ i ._ i i i, ,, /\ , I,/
......... ,......... ,-/- - -- -V - -,- f- - %-- - _....... t- ,-I......I I I I _ I' :/, k/ \_ti
0.5 1 1.5 2 2.5
tI I
I I
l- ........ I.......
i
-/-X -,!_ _,
3 3.5
RealPartofPulseShaped 8-PSK Signals
....... _l .... i_; ....
/i / 1
2 _
W
0.5
0
-0.5
......... :........ -,,r--
......... - ....... l_,,I__' i-"-_
......... :........ t-:-k-.......i ,it,
0 0.5 1 1.5 2 2.5 3
l .......
/,I
3.5
Figure 2-14 T Duration Cosine Pulse Used as Phase Pulse and 8-PSK Signaling
= _
. =
25
Cosine with Double-lnterval Cosine with double interval can be expressed as
cos(_r _T ) Jtl< T
h(t)={ 0 I_>T (2-41)
2T duration cosine pulse used as
2-15.
phase pulse and 8-PSK signaling are shown in Figure
w
L_
0.8
0.6
0.4
0.2
Cosine Signaling with Pulse Duration 2T
I I I ...... t- ............ P ...... P ....................................
, _ , .... L ......................... 7 .......................... 7 ...... r-- , ' '
I I I I I t _-
I II I,- I,,.. ...... I- ...... ;- ...... g- ...... i.- ...........
- , , , ,, : ', ' , __
/ I I a s , I
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
6
4
2
1
0.5
0
-0.5
Phase Signals
,.........,........,____-..__.,____!........!........_........... I
!
........ t I i
' .... _,;.... _'x ' ' -/- ....I __L ....... L__
/ ...... _...... , , ,I
II I I I 1
O0 0.5 1 1.5 2 2.5 3 3.5
RealPartofPulseShaped S-PSK Signals
-: i ....... j .............
.........'......." ......LL_,___u__--_- l_ ::-\
......... ," ....... ; X / ; , ' \ / ', 1', \ J, ; \ / , ' .___'____I__L ........ ;--+, .... --r
................... _---_ .................... ,--\ / , _, ± _' ':\/ ', ', i _ i ;-\7i I "_¢ t I
0 0.5 1 1.5 2 2.5 3 3.5
Figure 2-15 2T Duration Cosine Pulse Used as Phase Pulse and 8-PSK Signaling
= =
26
Raised Cosine with Single-Interval
expressed as
Raised cosine with single interval can be
W
w
w
w
w
w
_=_
w
T
Tcos2(Kt_ Itl -< -
2h(t)= { T
0 It[ >-- (2-42)2
duration raisedcosine pulse used as phase pulse and 8-PSK signalingare shown in
Figure 2-16Raised-Cosine Signaling with Pulse Duration T
I I I I $ I I I I
0.8 ....... ,- ...... '- ...... _ .... " ...... " ...... " ..... _ ...... _ ...... _ ......I I I I I I I i l
= I = ' ___L ...... L ......
0.6 ....... r ...... r .... r ...... _"...... r ...... r ...... r -- ,I I I I I I i I I
0.4 ................................................ _ ...................i t I I I I I
I ; i I i 1 1 I i....... .....I I I I I I I
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25
Phase Signals
t i I I _, /\ i......... ........ ........ ........,....... ..........4 ..... , , , , , r, /\ l,'
I I l ! 1 _ _ ,,d i
, , ' ' '_ _ ' /_ I'' ' ' ' I' / \ ' /_ '
2 - -- _,--'_,---.-,- ....... "_........ _ ....... T r-] - - _- -- r"/----_--L'-- --I, / \ ",-- _ ..... / \ '/\ I', t / \ , / \ _, z I ' / _ II_ I
J,/ \ , / \ -,-_-_--,---_-_- ,/ \ !1 \i i l l i
o _"/ \ / \ _" _ ;/ \ z \.r _.., ,0 0.5 1 1.5 2 2.5 3 3.5
1
0.5
0
-0.5
0
Real Part of Pulse Shaped 8-PSK Signals
1 i ih f ,_ {,.,b,____L__ ,_,_ I , \I, , , ____2 ....... L'_........ L_,____,___,__,___,-__,-,,, _____ ....
I' _, I _ \ I ', _ / , \/ d' ' '
...... l-r- -_-- -/- - - --r- _ _ -1........ _ ....... _, _ /t ....
I' _ I , , _ I,......... A- _ -- -I.........................
L:_ __ _ ,, ,, ', I '_,I ! I I |1
0.5 1 1.5 2 2.5
Figure 2-16 T
___L_' I __ILl ....... J
-I--':qA/--,,- 1--_:11-1/-- ',_jl I U V l
3 3.5
Duration Raised Cosine Pulse Used as Phase Pulse and 8-PSK Signaling
_m
w
27
W
w
Raised Cosine with Double-Interval
expressed as
Raised Cosine with double interval can be
2T
c°s2(# "_7:) Itl < Th(t)={ 0 Itl>T (2-43)
duration raised cosine pulse used as phase pulse and 8-PSK signaling in Figure 2-17.
w
L
w
=._
w
0.8
0.6
0.4
0.2
Raised-Cosine Signaling with Pulse Duration 2T
, , _ _ ,, ------_, , , ...... • ......I I ...... ____ ---t- ...... t" ...... _" ..... l- ..... ! I
I I ..................... ! I I
....... F ...... F-- ' i I i ) I
i I- .... I- ...... l- ...... L- .... ;
....... _ ...... I- .... i I I I i ' ....
i i r --, /___L ...... ;-...... r ...... r ...... r .... , '
....... r_ _f---r .... i i L ' ! ',; ; i , ' ' ' ' 0.5
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
4
1
0.5
0
-0.5
Phase Signals
, i, I',- - -___t, I , _ _ _ _ ___"_-.._.._._ ....... -' _-- -/ ..... " - - -'_- - -;-'- - -2" - ,
......... ,- ....... _---7 .... q.... _'.._ _',- - -"---J-;-_ ', ', 7- _' ',IL/
i ] i_ ........ =........
-- ,................ _ .... i I2 -- _', _ ....... 'I i •
' ' _ I IIJ _ I II
00 0.5 1 1.5 2 2.5 3 3.5
Real Part of Pulse Shaped 8-PSK Signals
I I I I I 1II'_ , , ._ " -,'_ , __'\ _ -- -- --Zr_ ' - .... Jr ..... -/-L',-N ........ ,..... -3
........ L,___\.... r,...... /1-','_...... i-- ,, / i! \ :I I I i -- -- ..... t-" .......,\ ,, ___l_____\..... _....... _.---/--r_-\ ,, 7
_---_ ,-: I _! \ ', ,', I _! \ .... L:..... /--......... ,, ,, /
i,._ - - _.. / .--_ ,l / ,'._ 3.5
0 0.5 1 1.5 2 2.5 3
Figure 2-17 2T Duration Raised Cosine Pulse Used as Phase Pulse and 8-PSK Signaling
28
Weighted Raised Cosine with Double-Interval
double interval can be expressed as
Weighted raised cosine [5] with
where 0.5<k<1.5 2T
are shown in Figure 2-18.
l[cos(x t)]_ 12_.__kk
h(t)= { 2 0
2rot
-cos-_ ] Itl _<Tm
Itl> T (2-44)
duration raised cosine pulse used as phase and 8-PSK signals
I
m
L =
=--
w
=--
w
w
w
=--
0.5
0-0.5
Weighted Raised-Cosine Signaling with k=1.5
I I " I Ii .... r ..... r" ...... r" -
I I I l I I I
I I I I I
I I I I ;
I I I 1 I I 1 I
....... :-..... ,'-...... L...... '- ............. L...... '- ...... ,_..... ,_......I I I
I I I
I I J
-0.4 -0.3 -0.2 -0. 0 O. 0.2 0.3 0.4 0.5
6
4
2
00
Phase Signals
I I I 1 II I t I I I
......... I.........-_ol ...... -I ........ .t ..... 1. ...... I- I
I ._.I 1 I t I I.... _ _ __2 .... 2 .... L__ , , --'rr ' "_- , "_'_- , ...... F-- 7_"-_ ....
I l l I eft" -- I.... t, I _:__ _ _x._;/ _ _X._Z_ __ ., -/- -'_:rF7 ....... __ ..... I.... r: .... \-- _,-t ....... ,....I' ' i
I I I I I I
0.5 1 1.5 2 2.5 3 3.5
1
0.5
0
-0.5
Real Part of Pulse Shaped 8-PSK Signals
'h ',fh........ L:__....... :_/_q
I'1 '/_........ _:__....... :_/____........ :/ __
i-V-0 0.5 1
I I II I I __ __
---2__ ' rl f' ...... _....... "ir ..... -PI I i I /
f I I I /-----'q ........ "r ....... '4"1" .....
I l I /, A , A _, /
1'.5 2 2.5
, !AA............. TiI
"1...... ::-/......'i VN:,)3 3.5
Figure 2-18 Weighted Raised Cosine Pulse Used as Phase Pulse and 8-PSK Signaling
29
Trapezoid Signaling A trapezoid signal can be expressed as
1
h(t)=
Tltl < -
2
2 T<2- -_ Itl -_ -
0 Itl >T
where 0.5 _<k< 1.5
Itl < T
(2-45)
Trapezoid pulse used as phase and 8-PSK signals are shown in Figure 2-19.
i
===
I
i
==
I
--=
I
0.5
0-0.5
Trapezoid Signaling with Pulse Duration 2T
y I I I II I t I
I I I I I I
I t I l I I
I I I I I I
....... r. ..... r" ...... r- ............ I- ...... r'----
I I I I I
i I I [ I I
I I I i I i
I I I l I I
I I I I I I
I I I I
-0.4 -0.3 -0.2 -0.1 0 0.1
, ",4.. :I
I I I
I [ I
I I I
.......--_ r ..... P ......
i I I
I l i
I [ I
i I i
i
I I I
0.2 0.3 0.4 0.5
Phase Signals
i i i i i/\ i i
I I I I _ _[I t
6 ..... ' '-_- ' ....-'--_-' .... '- - --' .....-- -- -1 ...... I .... i 1--- - - -- I ..... I-- - -
I I / _ t _ I I I i
I I I I I I I
4 ---, ......... / ..... "X,- -_-//- -\-- l_ / ...... __........ r ....... ,....... -_r, T----'? .... _ \_/ _:-- : Jr,_, ---q -- - ----_ut,..... ,_....... __y------x-:--_- - - - - 1
0 1 i i I t _l0 0.5 1 1.5 2 2.5 3 3.5
0
-10
Real Part of Pulse Shaped 8-PSK Signals
i__ I I t I '1 '
i\ , , A, , /I I I I I I,it , I,/ ',: \// , /, /_ , , / ', '_1 I , I' /
....... , t- -_ ,, 7_ /_,r ..... _r.... l ,, -,F;_ ....
' ',, ., :,',i\ L,0.5 1 1.5 2 2.5 3 3.5
Figure 2-19 2T Duration Trapezoid Pulse Used as Phase Pulse and 8-PSK Signaling
3O
Weighted Raised Cosine Transit Signaling A pulse with weighted raised cosine
transit and flat within one symbol period can be expressed as
i
w
w
w
.I
!
z
TItl< -
2
h(t)=
2
0 Itl >T
Itl_ T
where 0.5_<k< 1.5
The pulse with weighted raised cosine transit used as phase and 8-PSK signals are shown
in Figure 2-20
Weighted Raised-Cosine Transit Signaling with Pulse Duration 2T
, , I , , , , , ,1 I I I I
I I I I I I I I I
I I I ! I I I I I
I I I I I I I I I
U._ ....... r- ..... i- ...... i- ...... r ...... it- ...... r = ...... i'- ----_- ..... 7" ......
I I I I I I I
, , , , , , , , ,I I I I I
I I I I I I I I I
' i ' ' I ' 'I_ I I I I
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
8
6
4
2
00
Phase Signals
I I I /\ I I I I
' : :"-'/:x' ' '...... I....... I ..... 7 ...... 3" ........ T ........ I- ........ I........
I I I I I I I
......... a.......... £ - -..-"x- .... L ........ L ........ i ........
: _7----c---_-/ \ : : :/_........ 7:- 7__, ........ _....... _:.... T--r ....... _,- 7---Tr----X-;........ ,_,,_/ ...... :........ ._....... __:=.... _____.-_/ ..... '__:_._:_
1.1/ I i i _l _V i0.5 1 1.5 2 2.5 3 3.5
Real Part of Pulse Shaped 8-PSK Signals
0 0.5 1 1.5 2 2.5 3 3.5
Figure 2-20 Weighted Raised Cosine Transit Pulse Used as Phase Pulse
and 8-PSK Signaling
3]
w
2.4 Solid State Power Amplifier (SSPA)
A Solid State Power Amplifier (SSPA) is used in the project. The magnitude and
phase plots of the SSPA were obtained from JPL. From the plot (Figure 8), the SSPA has
two main ranges which include a linear and non-linear region. If the input power is less
than -5 dB, the SSPA is worked in the linear region, and the phase offset is approximately
zero. If the input power is larger than -5 dB, the SSPA is working in the nonlinear region.
This project sets the SSPA to work at saturation level of 0 dB.
.=
w
m
m
0nn
-10O
EL
>
J -20
r_
"5©
SSPA, ESA 10WattsAmplifier:MagnitudeResponseCurve
, ,,i t w ii I i ,, ! ,i I
............ 013.............
iii
-3[? i-30 -25 -20 -15 -10 -5 0
Input LevelPower (riB)
SSPA, ESA 10Watts Amplifier:PhaseResponseCurveI 1 Ii
._--.- .............................. , ,
u._t _.............
n ,-6 ; ; l-30 -25 -20 -I5 -10 -5 0
InputLevelPower (dB)
Figure 2-21 The Output Characteristic of 10 Watts SSPA
32
i--
The main part of the SSPA simulation is an analytical Traveling Wave Tube (TWT)
block which can simulate the nonlinear channel. By looking up the SSPA table file
provided by JPL, the TWT realizes a conversion from input amplitude to output amplitude
and input amplitude to output phase. Figure 2-22 shows the structure of the SSPA.
=
w
w
DTWT
in out
(TABLE)
AVERAGE
COMPLEX
x POWER
reset hold
I' 't
iave_pwr hold
J,SSPA
Figure 2-22 The Structure of SSPA.
w
m
i
w
w
w
2.5 Additive White Gaussian Noise Channel
Additive White Gaussian Noise (AWGN) is added to the signals out of the SSPA to
PSD,(f)
No/2
-fs/2 0 fJ2
Figure 2-23 Power Spectral Density Of White Gaussian Noise.
w
33
r_
simulate channel noise. Figure 2-23 Shows the power spectral density for the AWGN used
in the simulation. Actual white noise has a flat spectrum from -oo to oo. The AWGN is
taken to have zero mean and with variance o- n2 =No/2. N0/2 is the power spectral density of
the white noise, and Fs is the sampling frequency.
W
w
w
m
m
2.6 Receiver and Error Rate Estimation System
There are three parts in the receiver and estimation system, receiver, synchronizer
and error rate estimator. Figure 2-24 shows the block diagram for this system.
.. ...................................... _ .................................
r(t):s(t)+n(t)MATCHED
FILTER' i:
Receiver
DELAY
&
PHASE
METER
Synchronizer
• ...... ? ........................ •
ERROR
"_ RATE
ESTIMATOR
Figure 2-24 Block Diagram for the Receiver Error Rate Estimator
Receiver In an ideal receiver, a matched filter is used. The matched filter is a
linear filter which maximizes the signal to noise ratio (SNR). In this project, an integrate
and dump block (which match only a rectangular pulse) is used as the receiver.
w
34
w
Synchronizer The synchronizer performs a correlation between the reference
signals and received signals (delayed and distorted), and gives an estimate of the number
of samples of delay and the phase between the two samples. The reference signal is then
delayed the same number of samples so that the two signals can be compared to get the
error rate. Figure 2-25 shows the block diagram for the synchronization. The DELAY &
PHASE METER can detect the number of sample delays of the received signals. The
COMPLEX VARIABLE DELAY lets the reference signals be delayed by the same
number of samples as the received signal so they can be compared in the error rate
estimator.
u
w
w
u
reference
8 PSK
I Complex I
Variable
x Delay y
delay
received
8PSK
DELAY & PHASE
Figure 2-25 Block Diagram of Synchronizer
Error Rate Estimator
same delay), are put into the
The reference signal and received signal (now having the
error rate estimator to count the number of bit errors. The
35
main block used to count bit errors is SIMPLE ERROR RATE ESTIMATOR(Figure 2-
26) which compares the reference and the received signals to find bit errors.
=
w
L--
in SIMPLE
ERROR
i'ef RATE
ESTIMATOR
rate
errs
N
reset
TFigure 2-26 Simple Error Rate Estimator Block Diagram
m
w
w
w
36
Chapter 3
PULSE SHAPED 8-PSK SIMULATIONS
w---
This chapter will provide simulation procedures and results of pre-pulse shaped 8-PSK.
Simulations included power spectral density (PSD) and bit error rates (BER). In addition
to simulating traditional filters, some particular 8-PSK waveforms will first be simulated to
eliminate power spectral spikes.
3.1 Simulations on Power Spectral Density
Figure 3-1 shows the procedure for
shaped 8-PSK signal.
obtaining the power spectra for pre-pulse
Random
Data I BasebandFilter
Phase
Modulator
Solid State.I Power
"]Amplifier
] (SSPA)
Spectrum
Analyzer
(FF
w
.... T
Spectrum
Analyzer
(FFT)
Figure 3-1 Block Diagram of Spectrum Analysis
The random data source provides ideal data with a symbol rate (Rs), and the
sample frequency, Fs, is 512Rs Hz. For the pulse shaping, traditional filters and particular
w
37
w
i --
w
filters are used. For Butterworth and Bessel filters, bandwidth (BT) is equal to the cutoff
frequency which corresponds to the -3dB amplitude point. The BT represents the product
of the bandwidth with the symbol time. Traditional filters used in the simulations are in
Table 3-1.
Table 3-1 Traditional Filters Used in the Simulation
Filters
5th-Order Butterworth
3rd-Order Bessel
SRRC
BT=I
BT=i
BTs or roll-off factor ( a )
BT=2
a=l
BT=1.2
BT=2.8 BT=3
BT=2 BT=3
w
,,,,,,t,-
Particular 8-PSK waveforms, such as trapezoid, weighted raised cosine transit signaling,
cosine transit and raised cosine transit signaling, are also investigated to test their ability to
get rid of or reduce spectral spikes. The constant envelope pulse shaped 8-PSK signal
which comes from modulator goes to the input of the Solid State Power Amplifier (SSPA).
The spectrum Analyzer can be put before and after the SSPA. Since the SSPA is
worked at the saturation level and a constant envelope 8-PSK signal is used, the spectrum
of the signals at the input of the SSPA is almost the same as that of the signal output from
the SSPA. The power spectrum is obtained by taking the Fourier Transform of the input
signals of length 131072, using a Bartlett window. The power spectrum is then estimated
by averaging 100 such FFTs.
w
38
w
3.1.1 PSD Simulation Results of Using Traditional Filters
The plots of the power spectra are given in Figures 3-2
following parameters were used to obtain the power spectra.
Data Source •
to Figure 3-11. The
a°
c°
Ideal data with p(0) ( probability of zero )=0.5;
Sample Rate (fs) =512P_ Hz
m
i
m_m
m
E_
m
=__
!
Traditional Filters for Pulse Shaping:
a. None
b. Butterworth 5th Order (BT=1,2,2.8,3)
c. Bessel 3rd Order (BT=1,1.2,2,3)
d. SRRC (a =1, Number of tap length=32*P,_=8192, Bartlett window
Power Amplifier:
European Space Agency (ESA) 10-watt SSPA
SPW Spectrum Analyzer
a. FFT length=512R s
b. Window type: Bartlett
c. Average: 100 FFTs
d. Sampling frequency =512R_ Hz
Spectrum Plots Ranges
+ 10 Rb; + 75 R b
39
=.-.
tall=
=
°I , , , ^/_^ ' ,
!!T .-80
-10 -8 -6 -4 -2 0 2 4 6 8
Frequency (Rb)
(a) Unfiltered 8 PSK + 10 1_
0
m
-75 -60 -40 -20 0 20 40 60 75
Frequency (Rb)
I'
i
ii==w
.i
m
L_
nn"10
(b) Unfiltered g PSK +_.75 1_
Figure 3-2 Unfiltered 8-PSK Power Spectral Density
SSPA Output, +/-10Rb (5th Order Butterworth BT=I ,Ideal Data)0 , , , i_t , , ,
-80 .
-100
* I
-10 -7.5 -5 -2.5 0 2.5 5 7.5 10
Frequency (Rb) With All Spikes
(a) 5th Order Butterworth Filter (BT=I), -+ 10 P,q,
40
.+ ,
0
-100
I 1 1
-75 -60 -40 -20 0 20 40 60
Frequency (Rb) With All Spikes
SSPA Output, +/-75Rb (5th Order Butterworth BT=I ,Ideal Data)! I I I I I
75
(b) 5th Order Butterworth Filter (BT=I), +_751%
i i
t
m_
13:1
lll
Figure 3-3 Power Spectra of 8-PSK Pulse Shaped with 5th-order Butterworth filter (BT=I)
0
-20
-40-60
-80
-100
SSPA Output, +/-10Rb (5th Order Butterworth BT=2,1deal Data)I I I ' I I I
- iiiiiiii!i,ii,,, i/ill'fill+iiii_-10
I
-7.5 -5 -2.5 0 2.5 5 7.5 10
Frequency (Rb) With All Spikes
(a) 5th Order Butterworth Filter (BT=2), __.1ORb
0
-20
-40
-60
-80
-100
-75 -60 -40 -20 0 20 40 60
SSPA Output, +/-75Rb (5th Order Butterworth BT=2,1deal Data)! ! ! I1[ I , i
L
75
Frequency (Rb) With All Spikes
(b) 5th Order Butterworth Filter (BT=2), :t:75R_
Figure 3-4 Power Spectra of $-PSK Pulse Shaped with 5th-order Butterworth filter ('BT=2)
41
0SSPA Output, +/-10Rb (5th Order Butterworth BT=2.8,1deal Data)
w
ID
rn"O
-20
-40
-60
-80
-100
-10
Frequency (tbo) With All Spikes
(a) 5th Order Butterworth Filter (BT=2.$), + 10R_
0
-20
-40
--60
-8O
-100
SSPA Output, +/-75Rb (5th Order Butterworth BT=2.8,1deal Data)I I I =IL I I I
-75I
-60 -40 -20 0 20 40 60 75
Frequency (Rb) With All Spikes
w
0
(b)5th Order Butterworth Filter (BT=2.$), + 751_,
Figure 3-5 Power Spectra of 8-PSK Pulse Shaped with 5th-order Butterworth filter
03T=2.8)
SSPA Output, +/-10Rb (5th Order Butterworth BT---3,1deal Data)I I I
rnnO
-20
-40
-60
-80
-100
-10 -7.5 -5 -2.5 0 2.5 5 7.5 10
Frequency (Rb) With All Spikes
(a) 5th Order Butterworth Filter (BT=3), + 10P_
42
m
"U
0
-20
-40
-6O
-8O
-100
SSPA Output, +/-75Rb (5th Order Butterworth BT--3,1deai Data)I I I
-75 -60 -40 -20 0 20 40 60 75
Frequency (Rb) With All Spikes
(b) 5th Order Butterworth Filter (BT=3), + 75Rb
0
-20
-40
-60
-80
-100
-10
Figure 3-6 Power Spectra of $-PSK Pulse Shaped with 5th-order Butterworth filter (BT=3)
SSPA Output, +/-10Rb (3rd Order Bossel BT=I ,Ideal Data)
-8 -6 -4 -2 0 2 4 6 8 10
Frequency (Rb) With All Spikes
m (a) 3rd Order Bessel Filter ('BT= 1), _+10R4,
SSPA Output, +/-75b (3rd Order Bessel BT=I ,Ideal Data)
0 / , , , , , ,-2o ........!............i.............:...........-40 _- ........ ' ............ : ............. i...........
U
n,1.Io
-100
-75 -60 -40 -20 0 20 40 60
Frequency (Rb) With All Spikes
Co) 3rd Order Bessel Filter (BT=!), +_.75Rb
Figure 3-7 Power Spectra of g-PSK Pulse Shaped with 3rd-order Bessel filter (BT=I)
75
43
0SSPA Output, +/-10b (3rd Order Bessel BT=1.2,1deal Data)
-8 --6 -4 -2 0 2 4 6 8 10
-20
-40
m --60'10
-80
-100
-10/
Frequency (Rb) With All Spikes
(a) 3rd Order Bessel Filter ('BT=1.2), + 10R_
.,=11
t_
blJ
en"10
co
1
-20!
..40 '
-60
-80
-100
-7.:
0
-20
-40
-60
-80
-100
SSPA Output, +/-75b (3rd Order Bessel BT=1.2,1deal Data)
! ! i , t
I 1
-60 -40 -20 0 20
Frequency (Rb) With All Spikes
40 60 75
(b) 3rd Order Bessel Filter (BT=I.2), + 75P,_
Figure 3-8 Power Spectra of 8-PSK Pulse Shaped with 3rd-order Bessel filter (BT=1.2)
SSPA Output, +/-10b (3rd Order Bessel BT=2,1deal Data)1 I I I 1 I i I
0 --8 -6 -4 -2 0 2 4 6 8
Frequency (Rb) With All Spikes
(a) 3rd Order Bessel Filter (BT=2), + 10R_
0
44
0SSPA Output, +/-10Rb (3rd Order Bessel BT---3,1deai Data)
m
rn
-2O
m -40"O
-6O
--8O
0
-20
-40
-60
-80
-100
-10 -7.5 -=; -2.5 0 2.5 5 7.5 10
Frequency (Rb) With All Spikes
(a) 3rd Order Bessel Filter (BT=3), _+]ORb
Figure 3-9 Power Spectra of 8-PSK Pulse Shaped with 3rd-order Bessel filter (BT=2)
SSPA Output, +/-75b (3rd Order Bessel BT=2,1deai Data)I I I I I I
-75
0
J .-.----Lm
-60 -40 -20 0 20
Frequency (Rb) With All Spikes
40 60 75
(b) 3rd Order Bess¢l Filter (BT=2), + 75P_
SSPA Output, +/-75Rb (3rd Order Bessel BT--3,1deal Data)I : I t
w
w
, =
........ :. ............ ° ............ _. ........
.... :............. i ............. ! ........
,L .L_
-75 -60 -40 -20 0 20 40 60
Frequency (Rb) With All Spikes
75
Co) 3rd Order Bessel Filter (BT=3), + 75Rb
Figure 3-10 Power Spectra of 8-PSK Pulse Shaped with 3rd-order Bessel filter (BT=3)
45
0
-20
-40m"o -60
-8O
-1000
SSPA Output, +I-10Rb (Raised Cosine,a=1 ,Ideal Data)
-7.5 -5 -2.5 0 2.5 5 7.5
Frequency (Rb) With All Spikes
(a) Square-Root Raised Cosine (a = l), + I 0R_
10
0SSPA Output, +/-75Rb (Raised Cosine,a=l,ldeal Data)
I I I I
m.,.-,,
0Q"to
-75 --60 --40 -20 0 20 4O 6O
Frequency (Rb) With All Spikes
Co)Square-Root Raised Cosine (a =l), + 751_
Figure 3-11 Power Spectra of 8-PSK Pulse Shaped with SRRC filter (a =1)
75
n
46
m
w
: :w
3.1.2 PSD Comparison Between Two Pulse Shaping Methods
The PSD related to pre-modulation pulse shaping will be compared with
simulations using post-modulation pulse shaping. In order to compare the effect of two
methods, a frequency band utilization ratio p [16] can be defined as:
Number of spacecraft with Filtering Accommodated in Frequency BandP = Number of spacecraft without Filtering Accommodated in Frequency Band
An important assumption in deriving this ratio was made: the "spectra from
spacecraft in adjacent channels will be permitted to overlap one another provided that, at
the frequency where the overlap occurs, the signals are at least 50 dB below that of the
main telemetry lobe (1 st data sideband)." [16].
The Power Spectrum plots were used to determine this 50 dB level. For example,
for 8-PSK signals without a pulse shaping, the -50dB point is situated at approximately 35
Rb and using the Butterworth Filter (BT=I), the -50 dB point is approximately at 1.28 Rb,
then the Utilization ratio, p, is equal to
P = 35R_ =27.341.28Rb
From the simulations, the utilization ratios of pre-modulation pulse shaping were
calculated and listed in Table 3-3 along with the utilization ratios for post-modulation
pulse shaping [6]. From the utilization ratio table and Figure 3-12 to 3-13, for the same BT
and baseband filters, pre-modulation pulse shaping has a higher band utilization ratio than
that of post-modulation pulse shaping. Although the pre-modulation pulse shaping method
has a narrower bandwidth, it produces spikes which are discrete line components of power
47
spectral density at particular frequencies. Figure 3-14 gives a PSD comparison of two
methods with 5th-order butterworth filter.
Table 3-2 Utilization Ratio for Post-Modulation Pulse Shaping 8-PSK
Filter Type -50 dB point (Rb) uiiiizationNone 35 I
Butterworth(5th order)BT= 1 2.5 14
Besssel (3rd order) BT=I 2.95 11.86
SRRC rolloff =I 2.5 14
Table 3-3 Utilization Ratio of Pre-Modulation Pulse Shaping 8-PSK
Filter Type -50 dB point (R,o) Utilization
None 35 1
Butterworth(5th order) BT=I 1.28 27
Butterworth(5th order) BT=2 2.43 14.4
Butterworth (5th order) BT=2.8 3.23 10.8
Butterworth (5th order) BT=3 3.45 10.1
Besssel (3rd order) BT=I 1.75 20
Besssel (3rd order) BT=i .2 2.04 17.1
Besssel (3rd order) BT=2 3.22 10.8
Besssel (3rd order) BT=3 4.53 7.7
SRRC rolloff= 1 1.23 28.5
i ,
=___
BT=3
BT=2.8"
BT=2"
BT=I"
Utilization Ratio
0 5 10 15 20 25 30
Figure 3-12 Utilization Ratio of Using 5th-Order Butterworth filters in Pre-Modulation Pulse Shaping
48
BT=3_
BT=2'
BT=1.2"
BT=I'
Utilization Ratio
0 5 10 15 20 25 30
Figure 3-13 Utilization Ratio of Using 3rd-Order Bessel filters in Pre-Modulation Pulse Shaping
-40
m -60-o
-80
-100
SSPA Output, Pulse Shaping filter: 5th Order Butterworth BT=I
, t ! ! [,_l ! ! ! !
.... t_"i %..i 2Post-,PuseShaping ....... : ................ ........... :......... :.......... :.... _r .... i...... • ,. - i ]
......... ! ....... i ...... !........ / ....... :......... _....Pre-,Pulse Shaping ...... ; .........
-8 -6 -4 -2 0 2 4 6 8 I0
Frequency (Rb)
Figure 3-14 PSD Comparison of 8-PSK Between Two Pulse Shaping Methods
(5th Order Butterworth Filter ,BY=l)
3.1.3 Simulation Results of Particular 8-PSK Waveforms
In order to reduce power spectral spikes produced by the pre-modulation pulse
shaping method, some particular filters (8-PSK waveforms) were simulated. Since
Greenstein [4] has already shown that triangular, cosine, raised cosine and Nyquist
waveforms could not eliminate spectral spikes, formula (2-37) indicated that new
49
u
m
waveforms have to be investigated to get rid of spikes. The simulation parameters are the
same as in section 3.1.1 except for different filters (waveforms) as follows.
8-PSK Particular Pulse Shaping Signaling
a. Trapezoid Signaling (2T and 3T duration)
b. Cosine Transit Signaling (2T duration)
c. Raised Cosine Transit Signaling (2T duration)
d. Weighted Raised Cosine Transit Signaling (2T duration)
The above waveforms share a common characteristic, that is to keep fiat within one symbol
time T. Figures 3-15 to Figure 3-19 show these particular waveforms not only can reduce
bandwidth but also can eliminate or reduce spectral spikes.
0.5
0
Trapezoid Signaling with Pulse Duration 2T
' ' ' ' ' ' ' '7 ';
I I I I ] I I I
I I I I I I I
I I I I I I I I I
I I i I t I I I I
t I I I I I I I i...... m ...... F---
r --F -F FI I I I I
I I I I 1 I I
I I I I I I t I t
I I I I I I I I I
I I I I
i l I I I I I I I
-0.8 -0.6 -0.4 0.4 0.6 0.8-0.2 0 0.2
Time (T)
7
w
w.
o
-20
nn
"o_40
-60
-5
PSD of 8--PSK used 2T duration trapezoid pulse shaping
I I I I I I I I I
-4 -3 -2 -1 0 1 2 3 4
Frequency (Rb)
Figure 3-15 PSD of 8-PSK Used 2T Duration Trapezoid Pulse Shaping
50
05
Trapezoid Signaling with Pulse Duration 3T
I
I I II
----V ....... I II
I I
1 I
I I 1
I I I
I I I
- I
I I
I !
I |
I I
................... "r .........
I i
i i
| I
00 0.5 1 1.5 2Time (T)
2.5
PSD of 8-PSK used 3T duration trapezoid pulse shaping
z :
.......................... i i iI ] I ! I
-10 -8 -6 -4 -2 0 2 4 6 8Frequency (Rb)
Figure 3-16 PSD of 8-PSK used 3T-s duration trapezoid pulse shaping
10
=
0
-20
-4O
0.8
0.6
0.4
0.2
Cosine Transit Signaling with Pulse Duration 2T
I I j I I I I I IIII 11
,, ___J , , , _ --_ ' ,
....... p ...... p ..... p ...... p ...... p ................ _P ...... p ......l I I I l | I I l
1 I l l l I I .... I ...... I ......
....... 7 ............ F ...... F ...... T ...... i"- --7I I I 1 t I I I I................................................
I I I I I
I I I I ...... I ...... I -- I
¢ I
-- - --f ...... r I" r --r
I I I I I/ i i L , _ I , ,-0.8 -0.6 -0.4 ...0.2 0 0.2 0.4 0.6 08 1
Time (T)
PSDI I
i..........i........:.........:.........."..........".........."..........:.........".......... " " i i , J i
I I I I
-8 -6 -4 -2 0 2 4 6 8Frequency (Re)
Figure 3-17 PSD of 8-PSK used 2T-s duration cosine transit signaling pulse shaping
10
5]
w
0.8
0.6
0.4
0.2
0
-20
Raised-Cosine Signaling with Pulse Duration 2T
i t I i I i i l | i i
I [ 1 I I l I I E
I I I I t I I I I ......
....... F ...... F ..... F ...... y ...... F ...... F ...... F ...... y ...... T
I l I I I I I I I
.... _ ...... _ ..... _ ...... _ ...... _ _ ...... _ .... _ _ ......
I I I l I I I
I I I I
I I I I I __l__ F ...... nil ill--...... -- I 1 [ ...... F ...... [
I I I I I
-" -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Time (T)PSD
"o
t I I I I I I I
-8 -6 -4 -2 0 2 4 6 8
Frequency (Rb)
Figure 3-18 PSD of 8-PSK Used 2T Raised Cosine Transit Signaling Pulse Shaping
10
- 5
;l=d
= ;:
m
0.5
0
0
-20
-40
-60
180
-10
Weighted Raised-Cosine
I I_1 i I I I
II I I I I II l I I
I I I I l l
I I l I I I
I I I I I I
l I I I I
] I I I t I
I I I l
II I I I I
I t
43.8 -0.6 -0.4 -0.2 0Time (T)
Transit Signaling with Pulse Duration 2T
I t I
I
I I I
1 I I
____ ...... _ ..... ,_......
I
I I I I
I I
I I I I
0.2 0.4 0.6 0.8 1
PSD
I I I I I I I I
-8 -6 -4 -2 0 2 4 6 8
Frequency (Rb)
Figure 3-19 PSD of 8-PSK Used 2T Duration
Weighted Raised Cosine Transit Signaling Pulse Shaping
10
52
= .
m
3.1.4 Analysis Results of Particular 8-PSK Waveforms
According to formula (2-38) and (2-39), numerical analysis results of PSD for
trapezoid and weighted raised cosine signaling at some particular frequencies have been
calculated and are shown in Table 3-4 and Table 3-5. The discrete line components are
virtually zero, and they can eliminate all spikes. Simon's recent report [3] also indicated
3T duration trapezoid can eliminate power spectral line components. Tables 3-6 and Table
3-7 show that the raised cosine and cosine transit signaling can reduce line components,
but could not get rid of them. Analysis results are consist with simulation results.
r
w
Table 3-4 PSD Line Components for Trapezoid Shaping
iT 0
PSD 0.0819xl 0 "'_
1 2
0.0048xl 0-"': 0.0048x10 -j'_ 0.0098x 10-'_
4
O.O091xlO -j':
Table 3-5 PSD Line Components for Weighted Raised Cosine Shaping
wfT
PSD
0
O. 1736x 10 '_
1 2
0.0217x10 -_ 0.0087xl 0-'_ 0.0164xl 0 -j_ 0.0052x 10 -_z
i
Table 3-6 PSD Line Components for Raised Cosine Transit Shaping
fT
PSD 4.213x10 _ 4.213x10 _
2
4.213x10 _ 4.213x10 _ 4.213x10 _
= =
w
53
Table3-7PSDLine Componentsfor CosineTransitShaping
fff
PSD
0
4.333x10 -_ 4.333x10 -_
2
4.333x10 -_
3
4.333x10 -_
4
4.333x10 -_
From the simulations, the utilization ratios of pre-modulation pulse shaping with these
waveforms were calculated and listed in Table 3-8 and Figure 3-20
Table 3-8 Utilization Ratio of Pre-Modulation
Pulse Shaping 8-PSK with Different Waveforms
i .
Pulse Types -50 dB point Utilization
0 b)None 35 1
2T-s Trapezoid Pulses 3 11.6
3T-s Trapezoid Pulses 1 35
2T-s Pulses with Cosine Transit 7.5 4.6
2T-s Pulses with Raise-Cosine Transit 10 3.5
2T-s Pulses with Weighted Raised-Cosine Transit (k=1.5) 1.8 19.4
w
Weighted Raised Cosine'
Raised -Cosine Transit'
Cosine Transit'
3T-s Trapezoid'
2T-s Trapezoid'
0 1"0 t"5 2"0 2'5 3'0 35
Utilization Ratio
Figure 3-20 Utilization Ratio of Different Pulses in Pre-Modulation Pulse Shaping
54
• J
w
3.2 Bit Errors Rates Simulations
Figure 3-21 shows the system block diagram for pulse shaped 8-PSK end to end
system simulations
w
L.d
r =
Random
Phase
Baseband
Filter
Phase
Modulator
iDATA SOURCE! !PULSE SHAPING! ! MODULATION AMPLIFIER• . ........................ , ........................... • .................................................................
ERROR
RATE
ESTIMATOR
ERROR RATE...................................
AWGN CHANNEL
& INTEGRATEPHASE
& DUMPMETER
SYNCHRONIZER RECEIVER
Figure 3-21 BER Simulation System Block Diagram
m
w
AWGN is added to the signals out of the SSPA to simulate channel noise. The bit errors
are produced due to the ISI and AWGN. This report provides the measure of bit errors
according to bit energy to noise ratio (Eb/No). Figure 3-21 shows the system block
diagram for measuring bit error rates. For different pre-modulation pulse shaping filters
with the same BT, the plots of the BER are in Figure 3-22 to Figure 3-24.
55
m
= <
L.2
The following parameters were used to obtain the bit error rates:
Data source:
a. Ideal data with p(0) (probability of zero)=0.5;
c. Symbol rate Rs=l
b. Sample Rate (fs) = 16Rs
Pulse Shaping Filters:
a. Butterworth 5th Order (BT=1,2,2.8,3)
b. Bessel 3rd Order (BT=I, 1.2,2,3)
c. SRRC( a = 1,Number of tap length=256R s , rectangular window)
Power Amplifier:
European Space Agency(ESA) 10-watt SSPA, Back off--0dB as obtained
from JPL
BER for Pre-Modulation Pulse Shaping Filters with BT=I
1.00E+00
oILl:= 1.00E-04m
1.00E-05 j 1
1.00E-06
-1 1 3 5 7 9 11 13
Eb/No(dB)
•--_fr-Theoretical 8PSK
Butterworth
--B--Bessel
SRRC(alpha=l)
--0.-- Bessel BT=1.2
Figure 3-22. BER Plot for Pre-Modulation Pulse Shaping Filters with BT=I
56
BER for Pre-Modulation Pulse Shaping Filters with BT=2
L ;
w
1.00E+O0
I .OOE-01
i .00E-02
o 1.00E-03
m__ 1.00E-04
1.00E-05
i ' I '
i i
! i
t II : Ii =
I, E i ] \
' ! I I
, II
t I iI _ I , 1 i
L I • ,1.00E-06
-1 1 3 5 7 9 11 13 -
Eb/No(dB)
Figure 3-23. BER Plot for Pre-Modulation Pulse Shaping Filters with BT=2
--:J[--- Theoretical 8PSK
Butterworth
--I!-- Bessel
--_k-- SRRC(alpha=l)
= ,
m
JDm
.on
LLI
in
1.00E+00
1.00E-01
1.00E-02
1.00E-03
1.00E-04
1.00E-05
1.00E-06
BER for Pre-Modulation Pulse Shaping Filters with BT=3
r i
m
k I
-1 1 3
I=
i
f
5 7
Eb/No (dB)
9 11 13
---X---Theoretical 8PSK
--(k-- Butterworth
--I1--- Bessel
--A--- SRRC(alpha=l)
--'0-- Butterworth BT=2.8
Figure 3-24. BER Plot for Pre-Modulation Pulse Shaping Filters with BT=3
57
Although the Utilization ratio (Figure 3-20) for 3T Trapezoid is very high and the power
spectra are no spikes ( Figure 3-6) , this kind of waveform has overlapping within 3
symbol periods. This overlapping introduce a very serious ISI such that the integrate and
dump circuit can not work. Figure 3-25 shows the BER of pre-modulation pulse shaping
by using 3T Trapezoid. Since this BER is not accepted, other kind of receiver should be
used to correct ISI. An equalizer or sequential detector may be used to improve BER
performance.
m
w
m
etl.
tU
m
1.00E+00
1.00E-O1
1.00E-02
1.00E-03
1.00E-04
1.00E-05
1.00E-06 i
-1 1 3
: ,L ,L, r T
:_J, ! i !
i i
T
i i T5 7 9 11 13
Eb/No (dB)
-qY:-- Theoretical 8PSK
_(k--3T-s Trapezoid Pulse
Figure 3-25. BER Plot for Pre-Modulation Pulse Shaping With 3T Trapezoid Signaling
w
ISI Losses Due to Filtering
The ISI losses due to the pulse shaping filters can be determined by the BER
simulations. One needs to note that the receiver here was not an optimal receiver. Since the
2
-.-
. =
m
58
wintegrate and dump was used as receiver, it could not matched the pulse shaped signals.
Furthermore, there was no equalization for correcting ISI. The following tables give the
results of baseband filter ISI losses at 10 -3 BER for theoretical 8-PSK(or BPSK) [7] and the
filtered 8-PSK.
Table 3-9 ISI Loss Measurements at 10 -3 BER (Compared to ideal 8PSK)
Filter Type Loss Loss Loss Loss [,oss
BT=I BT=1.2 BT=2 BT=2.8 BT=3
(dB) (dB) (dB) (dB) (dB)
Butterworth ...........
5th Order 12.5 2.32 1.48 1.48
Bessel .....................
3rd Order 10 2.85 1.20
SRRC(a =1) ...................
4.22
Table 3-10 ISI Loss Measurements at 10 -3 BER (Compared to ideal BPSK)
=z
=
Filter Type Loss
BT=I
(dB)
Loss
BT=I.2
(dB)
Loss
BT=2
(dB)
Loss
BT=2.8
(dB)
Loss
BT=3
(dB)
13.2
Butterworth ...........
5th Order 15.7 5.52 4.68 4.68
Bessel ...........
3rd Order 6.05
SRRC(a =1)
7.42
4.88
-A
As stipulated in [16], the optimum filter should have the smallest BT value and provide
losses<0.4 dB. Thus an ideal matched filter and equalizer should be applied to meet the
= =
59
w
requirement of ISI loss <0.4. The Figure 3-25 shows the
utilization ratio for different filters.
ISI Loss (dB)
relationship of ISI loss and
m
20
15
10
5
iiiiiii iiiiiiiiiii!!!!!!!ii.: : i
5 10 15 2O 25 30
Utilization Ratios
Figure 3-25. Plots for ISI Loss as Utilization Ratios for Pre-Modulation Pulse Shaping
( ...... for 5th Order Butterworth Filter ; ...... for 3rd Order Bessel Filter; * for SRRC filter)
= -
w
w
60
W
w
w
Chapter 4
CONCLUSIONS AND RECOMMENDATIONS
This simulation based investigation on the pulse shaped 8-PSK modulations lead to
the following conclusions.
Pulse shaping is an effective way to reduce bandwidth and cope with the increasing
congestion of frequency band. Pre-modulation pulse shaping can keep constant envelope
and is more efficient on narrowing bandwidth than post-modulation pulse shaping. On the
other hand, pre-modulation pulse shaping using Butterworth, Bessel and SRRC filters can
produce discrete spectral components, and using integrate and dump receiver could not
ideally match pulse shaped signals. Thus the BER simulations by the pre-modulation pulse
shaping method are a best-case scenario. In order to reduce power spectral spikes and
improve BER, specialized 8-PSK pulse waveforms have been studied.
Pre-modulation pulse shaping and post-modulation pulse shaping have different
effects on narrowing bandwidth or frequency band utilization ratio. In general, pre-
modulation pulse shaping had better results for narrowing the bandwidth than post-
modulation pulse shaping did for the same traditional filters and BT values due to the
spectrum spreading effect of a non-constant modules through the non-linear amplifier.
Pre-modulation pulse shaping and post-modulation pulse shaping also have
different effects on producing bit error rates. In general, post-modulation pulse shaping had
lower bit error rates than pre-modulation pulse shaping for the non-ideal integrate & dump
receiver. That means that the narrower the bandwidth, the larger the bit error rate. For the
3rd Order Bessel filter, the BER produced by post-modulation pulse shaping (BT=2) is
61
i
w
w
2_
m
almost the same as that produced by pre-modulation pulse shaping (BT=3). For the 5th
Order Butterworth filters, the BER produced by pre-modulation pulse shaping (BT=2.8 or
3) is between those produced by post-modulation pulse shaping BT=I and BT =2.
Several pulse waveforms have been simulated, the pulse with cosine and raised
cosine can reduce power spectral line components. Furthermore, simulation and analysis
have shown that trapezoid and the pulse with weighted raised cosine transit can get rid of
discrete spectrum. Simon[3] and simulations also indicated that keeping at least one
symbol period flat and changing pulse transit can generate different effect on narrowing
bandwidth with no discrete spectrum. For further study, an improved method may be
found to change the pulse transit such that an optimal bandwidth with no spikes can be
achieved. On the other hand, any such method may also introduce potentially severe ISI.
A matched filter is crucial for receiving the signal to lower the BER. Matched
filters which can match pre- modulation pulse shaped signals are needed so that the BER
can be reduced. Since trapezoid and the pulse with weighted raised cosine transit have been
found to eliminate power spectral spikes, and they have deterministic waveforms which
have at least two symbol period duration, matched filters can be obtained according to
these waveforms.
Pulse shaping methods can introduce ISI and increasing BER. In order to obtain
better performance on BER, in addition to using an ideal matched filter, an equalizer may
be needed to reduce ISI.
i iw
62
REFERENCE
[1] V. K. Prabhu and H. E. Rowe, "Spectra of Digital Phase Modulation by Matrix
Methods," Bell Syst. Tech. J., vol. 53, pp. 899-935, May/June 1974.
[2] M.K. Simon, "On the Power Spectrum of Angle-Modulated PSK Signals Corrupted
by ISI," Private Communication, pp. 1-7, March 1996.
[3] M. K. Simon, "On the Power Spectrum of Angle-Modulated PSK Signals Corrupted
by Intersymbol Interference," TDA Progress Report 42-131, pp. I-10, November
1997.
[4] L. J. Greenstein, "Spectra of PSK Signals with Overlapping Baseband Pulses," IEEE
Trans. Commun., vol. COM-25, pp. 523-530, May 1977.
r_.
i[5] D. Subaainghe-Dias and K. Feher, "Baseband Pulse Shaping for rc/4 FQPSK in
Nonlinearly Amplified Mobile Channels," IEEE Trans. Commun., vol. 42, pp. 2843-
2852, October 1994.
[6] R. Caballero, "8-PSK Signaling over Non-Linear Satellite Channels," Masters Thesis,
New Mexico State University, pp. 1-200, May 1996.
[7]
[8]
P. J. Lee, "Computation of the Bit Error Rate of Coherent M-ary PSK with Gray code
Bit Mapping," 1EEE Trans. Commun., vol. COM-34, pp. 488-491, May 1986.
A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing. Englewood
Cliffs, NJ: Prentice-Hall, 1989.
u
[9] J. G. Proakis, Digital Communications. New York: McGraw-Hill, 1995.
i
63
[lO] SPW- The DSP Framework Communications Library Reference. Comdisco Systems
Product Number: SPW8015, March 1994.
[ 11] Signal Processing WorkSystem DSP Library Reference, Alta Group of Cadence
Design Systems, Inc. Product Number: SPW8015, March 1995.
[12] Signal Processing WorkSystem Interactive Simulation Library Reference. Alta Group
of Cadence Design Systems, Inc. Product Number: SPW8015, March 1995.
m
[13] W. L. Martin and T. M. Nguyen, "CCSDC - SFCG Efficient Modulation Methods
Study, A Comparison of Modulation Schemes, Phase 1: Bandwidth Utilization
(Response to SFCG Action Item 12/32), " SFCG- 13, Ottawa Canada, 13-21 October
1993.
[14] M. Otter, "CCSDC - SFCG Efficient Modulation Methods Study, A Comparison of
Modulation schemes, Phase lb: A Comparison of QPSK, OQPSK, BPSK, and
GMSK Modulation Systems (Response to SFCG Action Item 12/32), "European
Space Agency/ESOC, Member CCSDS Subpanel IE, (RF and Modulation), June
1994.
w
[15] W. L. Martin and T. M. Nguyen, "CCSDC - SFCG Efficient Modulation Methods
Study, A Comparison of Modulation Schemes, Phase 2: Spectrum Shaping (Response
to SFCG Action Item 12/32), " SFCG Meeting, Rothenberg, Germany 14-23,
September 1994.
[16] W. L. Martin, T. M. Nguyen, A. Anabtawi, S. M. Hinedi, L. V. Lain and M. M.
Shihabi, "CCSDC - SFCG Efficient Modulation Methods Study Phase 3." End-to-End
System Performance," Jet Propulsion Laboratory, May 1995.
64
[17] S. Haykin, Communication Systems. New York: John Wiley & Sons, 1994.
65