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
New Mexico State University
Las Cruces, New Mexico
May 1998
w f
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. w
ABSTRACT
i m ,
w
PULSE SHAPED 8-PSK
w 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
iii 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 w HP workstations which simulated the power spectral density of pulse shaped 8-PSK
U 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 w 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)
iv w
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.
w TABLE OF CONTENTS
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 w
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
w 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
vi E
w 3.1.3 Simulation Results of Particular 8-PSK Waveforms ...... 49
w 3.1.4 Analysis Results of Particular 8-PSK Waveforms ...... 53
3.2 Bit Error Rates Simulations ...... 55
4 CONCLUSIONS AND RECOMMENDATIONS ...... 61 i REFERENCE ...... 62
vii LIST OF TABLES
Table 2-1 Gray Codes and Phase ¢ ...... 6
Table 3-1 Traditional Filters Used in the Simulations ...... 38
Table 3-2 Utilization Ratio for Post-Modulation Pulse Shaping 8-PSK ...... 48
Table 3-3 Utilization Ratio for Pre-Modulation Pulse Shaping 8-PSK ...... 48
Table 3-4 PSD Line Components for Trapeziod Shaping ...... 53
Table 3-5 PSD Line Components for Weighted Raised Cosine Transit
Shaping ...... 53
Table 3-6 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
w w LIST OF FIGURES
Figure 2-1 8-PSK Signal Constellation ...... 6
Figure 2-2 Producing Phase Signals Block Diagram ...... 7
Figure 2-3 Modulation Schemes For Unfiltered And Filtered Phase Signals ...... 8
Figure 2-4 Amplitude Response of Various Orders of Butterworth Filters ...... 9
Figure 2-5 An Impulse Response of 5th-Order of Butterworth Filters ...... 9
Figure 2-6 Amplitude Response of Various Orders of Bessel Filters ...... 10
Figure 2-7 An Impulse Response of 3rd-Order of Bessel Filters ...... 1I
Figure 2-8 Frequency Response: Raised Cosine Characteristics with Different w rolloff factor a ...... 12
Figure 2-9 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 w Figure 2-12 The Waveforms of Real Part and Imaginary Part of Pre-Modulation
w Filtered 8-PSK ...... 15
Figure 2-13 Pulses with Different Overlapping Index K from [1 ] ...... 24 w 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
ix Figure 2-16 T-s Duration Raised Cosine Pulses Used as Phase Pulse and 8-PSK
i , Signaling ...... 27
Figure 2-17 2T-s Duration Raised Cosine Pulses Used as Phase Pulse and
8-PSK Signaling ...... 28
Figure 2-18 Weighted Raised Cosine Pulses Used as Phase Pulse and 8-PSK
= Signaling ...... 29
Figure 2-19 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
w
Figure 2-24 Block Diagram for the Receiver Error Rate Estimator ...... 34
2 S
Figure 2-25 Block Diagram of Synchronizer ...... 35
Figure 2-26 Simple Error Rate Estimator Block Diagram ...... 36
Figure 3-1 Block Diagram of Spectrum analysis ...... 37
Figure 3-2 Unfiltered 8-PSK Power spectral Density ...... 40
Figure 3-3 Power spectra of 8-PSK Pulse Shaped with 5th-Order Butterworth
Filter (BT=I) ...... 41
Figure 3-4 Power spectra of 8-PSK Pulse Shaped with 5th-Order Butterworth
w
X 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
_ I w 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
w
Figure 3-15 PSD of 8-PSK Used 2T-s Duration Trapezoid Pulse Shaping ...... 5o
xi
w 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 w 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 3-21 BER simulation System Block Diagram ...... 55 _,i w Figure 3-22 BER Plot for Pre-Modulation Pulse Shaping Filters with BT--1 ...... 56 t --
w Figure 3-23 BER Plot for Pre-Modulation Pulse Shaping Filters with BT=2 ...... 57
Figure 3-24 BER Plot for Pre-Modulation Pulse Shaping Filters with BT--3 ...... 57
Figure 3-25 BER Plot for Pre-Modulation Pulse Shaping With 3T Trapezoid ... 58
m Figure 3-25 Plot for ISI Loss as Utilization Ratios for Pre-Modulation Pulse
Shaping ...... 6O
r
xii
u k.., LIST OF ABBREVIATIONS
(X Rolloff Factor for SRRC filters
P Band Utilization Ratio w AWGN Additive White Gaussian Noise
dB Decibels
BER Bit Error Rate
L_. BPSK Binary Phase Shift Keying
BW Bandwidth - ==- BT Bandwidth-Time (symbol) product
CCSDS Consultative Committee for Space Data Systems
Eb/No Energy (bit) to Noise ratio
ESA European Space Agency
•r rmmB fs Sampling Frequency fc Carrier Frequency
w FFT Fast Fourier Transform
Hz Hertz
HP Hewlett Packard
IIR Infinite Impulse Response (Filter) m ISI Intersymbol Interference
JPL Jet Propulsion Laboratory
LSB Least Significant Bit
MSB w Most Significant Bit
MSK Minimum Shift Keying
NASA National Aeronautics and Space Administration
NMSU New Mexico State University
NRZ-L Non-Return-to-Zero Logic
QPSK Quaternary Phase Shift Keying
xiii PA Power Amplifier
PCM Pulse Coded Modulation
PM Phase Modulation
PSD Power Spectral Density
PSK Phase Shift Keying
QPSK Quartenary Phase Shift Keying i
Rb Bit Rate
Rs Symbol or Baud Rate
RC Raised Cosine
sec Seconds
.__ SFCG Space Frequency Coordination Group
SNR Signal-to-Noise Ratio
SPW Signal Processing Worksystem
SSPA Solid State Power Amplifier
SRRC Square Root Raised Cosine
i Tb Bit Period
TWT Traveling Wave Tube
W
= --
I
W
xiv
-- I L Chapter 1
INTRODUCTION
w
With the continuing development of communications and increasing users,
w 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
L--
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
r shaping which performs a low pass filtering before modulation to reduce bandwidth.
Pre-modulation pulse shaping with different modulation schemes, such Pulse Code
m_ Modulation (PCM), Binary Phase Shift Keying (BPSK), Quaternary Phase Shift
Keying (QPSK) and Gaussian filtered Minimum Shift Keying (GMSK) have been
_m u studied in [13]-[16]. The other pulse shaping method is post-modulation pulse
E--
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 w
and Square Root Raised Cosine (SRRC) filters, with different Bandwidth-Time (BT)
w 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 m spectrum when the signals go through a non-linear Solid State Power Amplifier w (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 lWW 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.
_I
N 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-
W modulation pulse shaping and post-modulation pulse shaping, Martin and Nguyen m.
I indicated that pre-modulation pulse shaping can overcome disadvantages produced by
post modulation pulse shaping, such as larger, heavy filters which are not compatible
U
with some mission operations requirements.
2
W m In contrast to using traditional filters by Martin and Nguyen, Greenstein [4] m_
has examined PSK with different overlapping baseband pulses. Greenstein
investigated triangular, cosine, raised cosine and Nyquest pulse shapes, yielding the
__i results for continuous and discrete line components of PSD. w Although many studies have investigated pre-modulation pulse shaping and
post-modulation pulse shaping with various filters, post-modulation pulse shaping can w
_i induce a non-constant envelope signaling which can spread the reduced bandwidth.
F Pre-modulation pulse shaping can induce power spectral spikes which are not good F__
for power efficiency and may produce adjacent channel interference. A study is
needed for pre-modulation pulse shaping which produces a constant modules signal m m with no spikes
=__ w 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 w
Bessel and SRRC, were used as pulse shaping filter applied to 8-PSK phase signals.
m 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
m wL 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
m 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.
m w
I Chapter 2 m
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 m
will be studied. Finally, the topic will be an Additive White Gaussian Noise (AWGN)
w channel and a matched filter used as receiver.
m_ 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 take one of 8 equally-
spaced values, such as in Table 2-1 and shown in Figure 2-1. A plot of 8PSK signal
constellation is shown in Figure 2-1. An 8-PSK signal, S(t), can be expressed as following.
S(t) = Acexp[j(coc t +_o (t))] = A¢cos[ co_t + (,o(t)l+jAcsin[o_ t + cp(t)] (2-1)
m
w
r IMAGINARY (QUADRATURE) A i 011 ! 001
]
Yi I...... :6 ooo OlO I
REAL ('1_ PHASE)
Xi
iio • • I00
111 I01
Figure 2-1 8PSK Signal Constellation
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) m
corresponding to the 3 bit Gray code. Table 2-I shows the relationship between Gray
w codes and _o.
w Table 2-1 Gray Codes and Phase _p
m 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 ° r 100 337.5 °
m In SPW, the Gray codes and phase q_ can be produced by the design shown in Figure 2-2.
N 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 u
= 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.
w
RandomData ][ "lNumericJBinary to Number to
control Symbol _p(t) OUTPUT I RandomData[ _._.__BinaryINumeric to t
xix2 control w Random JBinary to T Data "]Numeric
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.
2
V 8PSK v(t) Phase ]modulator
_1 Ace j(p
(a) Unfiltered 8PSK With Constant Envelope
= ,,
8-PSK vl (t) Phase (a Modulator - Shaping A c ejfO I Pulse
iv
Co) Post-Modulation Pulse Shaping Method
8-PSK v2(t) Phase Pulse [Modulator
IJ Shaping _[ Ac ej(p
(c) Pre-Modulation Pulse Shaping Method
Figure 2-3. Modulation Schemes for Unfiltered and Filtered Phase Signals.
w 2.2 Filters Used for 8-PSK Pulse Shaping
liii_ 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 u
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-
U Symbol Interference (ISI)[4]. The distortions due to the pulse shaping filters make the
implementation of an optimal receiver difficult.
b.
m 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 w frequency fc corresponds to -3dB amplitude point.
Buttorvvorth Filter fc(3dS) = 1 Hz & ordor=2.3.4.5,6.7,8 o. 3 : : _ ,_. :.<_'--.. _---..._ .._.... _ i'm : -2O ...... -t! i_:----_ i ...... i--:...... -'_ .__z.Z._._..:,-.z _,_ _,_ i_:,.-._ ...... __._Q_._[_._..q-._Q ...... _ .... : • . : f : \ \ _ _'_. _ "__ --.._ : _-.__ -40 ...... !-t!.....-.---,.... - i _ ill "q.."-,,_'-., '--.. i _ _ " Cut-Olaf _ i I i n = S "'--='="-""_:.'_ "_. _ _ "'-_._ -6o ..... E_,_,,._.,t_y..4.... -/--"...... "--":-_ .... _."--:"_ .... :_-_- u -80 ...... _ ..... ["" :.'.... _'"i ...... r ...... :r--'K .... " ....
-100
Frequency (H:z)
Figure 2-4 Amplitude Response of Various Orders of Butterworth Filters
m pulse R esponse of 5th-O rder B utterwo rth Filte
...... I ...... _...... i...... _...... r ...... i......
-0 iii 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
w 9
m 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).
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
w simulation. Figure 2-6 shows the amplitude response of various orders of Bessel Filters.
=w Note that the amplitude is not as fiat in the passband region as for the Butterworth filter.
w
Bessel Filter fc(3dB) = 1 HZ & n=2.3.4.5.8.7.8 -_ ! ; ,-----L-_ : n=2i i
-20
-40 m
-60 ...... _.3_¢__-____-___.:...... :._.__.__,_____: ! .=_ '-i----/i
-8O
-100 10 ° Frequency (I-12:)
w
Figure 2-6 Amplitude Response of Various Orders of Bessel Filters
10
u 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.
Impulse Response of 3rd-Order Bessel Filter 1 | ! i
0.8 ...... • ...... ,D ...... ,_ ......
0.6
0.4 w !!!iiiiii!iiiii!!!!i!::::::i:::::i:::ii::::::ii:i::::::i 0.2 = ,
0 1 1 Time (Symbol _me "F) 3
Figure 2-7 An Impulse Response of 3rd-Order of Bessel Filter
L_ 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 m, (1 - a) "T, 0 _ RC(f)= (2-2) 2[,. L 2a ' _-_)-<1-'1-<2T (1 + a) o, I/F> 2T where the zeros will occur at t = nT (T is the sampling interval) and ct is the rolloff factor L_ 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 W I1 U frequency response of the Raised Cosine Function with different rolloff factors (t_ = 0, 0.25, 0.5 and 1) and T =1 seconds. Transfer Function of Raised Cosine Filter with r = 0. 0.25.0.5 and 1 1 0t = 025 0.9 m 0.8 •..... _V;--_-" _!_ _.-0...... i...... 0.7 ...... _...... ; ...... :" - IVT- W'h-_ t'e- _ aq- g_ ...... for_ -_ I 0.6 ..... 0.5 .... F...... -rrh_TT ...... _...... O4 ..... -1./(2a'-)-w_h._r,_-1:_-_1-s-e-e.:-..... fo!r ct = 0 0.3 .....!...... 0.2 0.1 0 i i 0 0.2 0.4 0.6 0.8 1 Normalized Frequency (I-_) Figure 2-8 Frequency Response: Raised Cosine Characteristics with Different Time I_esponse of F_sised Cosine _'lltef _th r = O. O :_. O _; and 1 r It z_ 08 ...... _...... i...... 7------_ ...... i...... 06 ...... --_i-;--;---_-_-,-i:----//-----i.....\_...... Y:.-_,;-_:-,_--- m, ...... _...... ,,___i....,...... _....._ ....i...... :;;...... -O2 -3 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 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 t 2 (2-4) rcVrT[(4a -_) -1] In this equation, cr is the roll off parameter and T is one symbol period (1/symbol rate). l 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 4 ...... _:...... i ...... i ...... i...... i-...... J 0 500 1000 1500 2000 filte re d p h a s e s a m p le s 4 2 0 0 500 1000 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 1 of v(t) 0.5 0 -0 .5 -1 0 .01 0 .02 0 .03 0 .04 0 .05 0 .06 Im ag Tim e s of vCt) 1 0.5 -0 .5 -1 0 0.01 0.02 0.03 0.04 0 05 0.06 Times 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 IiR 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 w dispersed which increases error rates and affects power spectrum. m 14 m X-. real part of v2 {t) 1 -0o.5o -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 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 i 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 w 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 15 _I _i 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. For convenience, define the equivalent complex baseband modulation v(t)=exp(j _p(t)), (2-6) According to Prabhu and Rowe, the spectrum of s(t) is Ps(f)= _ Pv(f-f¢)+ ¼ Pv(-f-f_), (2-7) where Pv is the spectrum of v(t). m For pre-modulation pulse shaped 8-PSK, go(t) takes on one of eight level pulses. g_(t)= 8 [8-(2i-1)]p(t), (i=1,2 .... 8) (2-8) Where p(t) is the pulse shaped waveform which may be the result of a rectangular pulse 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 . (2 -9) 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 (2-1o) Using vector notation, ¢ (t) becomes _0(t) =[0 1 0 0 0 0 0 0]g(t) (2-11) 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 = .. 16 m et=[10000000] T e2=[0 1 0 0 0 0 00 ]T m r e8=[0 0 0 0 0 0 0 1 IT (2-12) and ak takes on only one of with the same probability 1 then the values el, e2, ... es g, _o(t) may be expressed as oO (p (t)= Z ak T g(t) ;-m < t _< m (2-13) ],(_ --oo 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 ;0< t < T (2-16) 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 (2-17) Then v(t) can be written as 17 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 IT Mb=_ [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 i E(f) = _ e'J2_fnR_ (n). (2-22) w /'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[£ b£ r r(t+ r -kT) btn r*(t-lT)] k=--co ]=-_ --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) (2-24) k=-oo I=-_e 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 1i R_(t+ r ,t) dt (2-26) R_(r) =T- r 2 Substituting (2-24) in (2-26), T rV(t+ r -kT) R0 (k-l)r (t-lT) dt. I=--_ T 2 Let k-l--n and t-IT --x, T ---IT 1 £ £ 21 rT(t+l--nT) _ (n)r*(x)dx R_(r)=y n=-=o I=--_ _T_I T 2 19 rT(t+ r -nT) R, (n)r'(x)dx Let t--x, then Rv( )=7 1 _, _ rV(t+r-nT)_(n)r*(t)dt (2-27) n =-_0 --00 Power spectral density of v(t) is defined as Pv(f) = S Rv( r )e -j2rc r r d r (2-28) = = -_ e"J2 rT(t+ r -nT) _ (n)r*(t)dt d 2- --00 n= --00 --o0 w 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 .-_ R(f) = 5 e -j2rc ft r(t) dt, is the Fourier transform of r(t), and it is also a column vector. -00 Thus Pv(f) can be expressed as L..a RT(f) e-J2z _T Ro (n) R*(f) ! - 1 e "j2rc fnT _(n) }R*(f) (2-29) T 20 By using (2-22), Pv(f') becomes 1 Pv(f') =_ RT(f) _(fT)R*(f) (2-30) since _(f)= £ e-J2_f_R_ (n). Introducing covariance matrix Kb (n) = Rb (n)- M b Mb n, then w (f)= £ e-J2lr _n(_ (n)+ Mb Mb n) w /'1 = --¢0 L...d_ = £ e-J2 _ fnZ (n) + M b Mb tl _ e "j2n"fn (2-31) n=---_ n=--_ Using Fourier series, we have e-J2 _ fn = £ _ (f-n) (2-32) n= --oC} n=-_O | J (2-33) .=-_£ e'J2_'frn =-T1 ,=-_£ g(f- ). w 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. e ,(0= I RT(f) Po_t (fT)R*(0 =-_1 RT(f){ -_-M1 b Mb H _ 8(f- ; )}R*(f) /3="-o0 1 n T 2 RT(f)MbMbHR*(f') _ _'(f--_) /1=--o0 _1 n T 2 MbTR(f)MbHR*(f) _ _(f-_) _ T21 IMb T R(f) Iu 6(f_T) r/=---co _ 1 s 12_ n T282 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 m 22 duration. According to [1] and [2], Pvl(f) for arbitrary index K may be expressed as Pvl(f) _1T282K 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 "ei{ ot (t)+g_( t- r) } e,i[ gl (t)+OM (t--T)} ei{ oz (t)+ gt (t--T)} w ej[ g_ (t)+oM (t- T)I (2-39) R(f) = e-i2.1, dr. u T eYl uz(t)+ot (t--T) } ei[ ¢__, (t)+o_ (t--T)} e;{ o,, .)+o, (t--T) } e,Jl _M (t)+oH (t-T)] 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 = u studied. According to [1], waveforms for different overlapping index K are shown as follows. 23 K- ! -i o 2 ..- "'t',• ,,"",,//_\, X\I'" 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 T duration cosine with a single interval can be expressed as T cos(_ t/T) ltl - - 2 P(t) = { 7 0 Itl >- (2-40) 2 = ;--d 24 7 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 Cosine Signaling with Pulse Duration T I 1 I 1 I I I I t 0.8 ...... _- ...... I- ...... t" ...... t" ...... f- ...... I" ...... t" ...... I I I I ] I I I I i 0.6 ...... ;- ...... _ ...... F ...... F ...... r ...... F ...... F ...... T ...... I I I I 1 I I I i 0.4 I I ! i i I I 0.2 __I I I iI I I I I I 1 I I t 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 i i l _i f\ t = = I I I I 4 l- ...... I...... ,'-...... ',...... : ..... 7:7-- i ,i ,,-I- --1-ql_/\ i ._ ,i I,/i i -/-X -,!_ _, 2 ...... ,...... ,-/- - -- -V - -,- f- - %-- - _...... t-,-I...... I I I I _ I 0 i' :/, k/ \_t -7 0 0.5 1 1.5 2 2.5 3 3.5 RealPartofPulseShaped 8-PSK Signals i i 0.5 ...... _l .... i_; .... l ...... 2 _ ...... :...... -,,r-- 0 ...... - ...... l_,,I__ ' i-"-_ -0.5 ...... :...... t-:-k-...... /i / 1 /, i ,it, I 0 0.5 1 1.5 2 2.5 3 3.5 W 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 phase pulse and 8-PSK signaling are shown in Figure 2-15. Cosine Signaling with Pulse Duration 2T I I I ...... t- ...... P ...... P ...... 0.8 ...... , _ , .... L ...... 7 ...... 7 ...... r-- , ' ' 0.6 I I I I I t _- I II I,- I,,...... I- ...... ;- ...... g- ...... i.- ...... 0.4 - , , , ,, : ', ' , __ 0.2 / 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 Phase Signals w 6 ,...... ,...... ,____-..__.,____!...... !...... _...... I ! 4 ...... t I i 'I .... _,;.... _'x__L ' ...... L__' -/- .... 2 I / ...... _...... , , , I I I I I 1 O0 0.5 1 1.5 2 2.5 3 3.5 RealPartofPulseShaped S-PSK Signals 1 : i ...... - j ...... 0.5 ...... '...... " ...... LL_,___u__--_- l_ ::-\ 0 ...... ," ...... ; X / ; , ' \ / ', 1', \ J , ; \ / , ' .___'____I__L ...... ;--+, .... --r ...... _---_ ...... ,--\ / , _, ± _ -0.5 ' ':\/ ', ', i _ i ;-\7 i I "_¢ t I 0 0.5 1 1.5 2 2.5 3 3.5 L_ Figure 2-15 2T Duration Cosine Pulse Used as Phase Pulse and 8-PSK Signaling = = 26 Raised Cosine with Single-Interval Raised cosine with single interval can be expressed as T cos2(Kt_ Itl -< - 2 h(t)= { T 0 It[ >-- (2-42) 2 T duration raisedcosine pulse used as phase pulse and 8-PSK signalingare shown in Figure 2-16 Raised-Cosine Signaling with Pulse Duration T W 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 w ...... 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 w 4 ..... I, ,I l, !, 1, _ _ ,,dr,i /\ l,' , , ' ' '_ _ ' /_ I' ' ' ' ' I' / \ ' /_ ' 2 - -- _,--'_,---.-,- ...... "_...... _ ...... T r-] - - _- -- r"/----_--L'-- -- I, / \ ",-- _ ..... / \ '/\ I' w , 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 w Real Part of Pulse Shaped 8-PSK Signals 1 1 h f ,_ {,.,b,____L__i i ,_ ,_I, I , \ , ____2 ...... L'_ 0.5 ...... L,,_,____,______,__,___,-__,-,.... I' _, I _ \ I ' , _ / , \/ d' ' ' ___L_' I __ILl ...... J 0 ...... l-r- -_-- -/- - - --r- _ _ -1...... _ ...... _ , _ /t .... I' _ I , , _ I, -I--':qA/--,, -0.5 ...... A- _ - - -I ...... - 1--_:11-1/-- ', L:_ __ _ ,, ,, ', I'_, _jl I U V l I ! I I |1 w 0 0.5 1 1.5 2 2.5 3 3.5 _=_ Duration Raised Cosine Pulse Used as Phase Pulse and 8-PSK Signaling Figure 2-16 T w _m w 27 Raised Cosine with double interval can be Raised Cosine with Double-Interval expressed as W c°s2(# "_7:) Itl < T h(t)={ 0 Itl>T (2-43) 2T duration raised cosine pulse used as phase pulse and 8-PSK signaling in Figure 2-17. w Raised-Cosine Signaling with Pulse Duration 2T , , _ _ ,, ------_, , , ...... • ...... w I I ...... ____ ---t- ...... t" ...... _" ..... l- ..... ! I 0.8 L I I ...... ! I I 0.6 ...... F ...... F-- ' i I i ) I i I- .... I- ...... l- ...... L- .... ; 0.4 ...... _ ...... I- .... i I I I i ' .... i i r -- , /___L ...... ;-...... r ...... r ...... r .... , ' 0.2 ...... 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 Phase Signals , i, I',- - -___t , I , ______"_-.._.._._ ...... -' _-- -/ ..... " - - -'_- - -;-'- - -2" - , 4 ...... ,-...... _---7 .... q.... _'.._ _',- - -"---J-;-_ ' , IL/', 7- _' ', i ] i_ ...... =...... 2 -- -- _',,...... _ ...... _ .... i ' I I i • ' ' _ I I w IJ I_ I I 00 0.5 1 1.5 2 2.5 3 3.5 =._ Real Part of Pulse Shaped 8-PSK Signals w 1 I I I I I 1I I'_ , , ._ " -,'_ , __ '\ _ ------Zr_ ' - .... Jr ..... -/-L',-N ...... ,..... -3 0.5 ...... L,___\.... r,...... /1-','_...... i-- ,, / i! \ : ,\I ,,I ___l_____\..... _I ...... _.---/--r_-\i -- -- ..... ,,t-" ...... 7 0 _---_ ,-: I _! \ ', ,', I _! \ .... L:..... /-- -0.5 ...... ,, ,, / 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 Weighted raised cosine [5] with double interval can be expressed as 2rot l[cos(x t)]_ 12_.__kk-cos-_ ] Itl _ where 0.5 are shown in Figure 2-18. Weighted Raised-Cosine Signaling with k=1.5 I I I " I I i .... 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 0.5 ...... :-..... ,'-...... L...... '- ...... L...... '-...... ,_..... ,_...... I I I I I I m 0 I I J -0.5 -0.4 -0.3 -0.2 -0. 0 O. 0.2 0.3 0.4 0.5 L = Phase Signals I I I 1 I I I t I I I 6 ...... I...... -_ol ...... -I ...... t ..... 1...... I- I 4 .... _ _ __2 .... 2 .... L__ , , -- 'rrI ._.I '1 "_- ,I "_'_- ,t ...... F--I 7_"-_ .... I =-- I l l I eft" -- I 2 .... t, I _:__ _ _x._;/ _ _X._Z_ __ ., -/- -'_: w rF7 ...... __ ..... I.... r: .... \-- _,-t ...... ,.... I' ' i 0 I I I I I I 0 0.5 1 1.5 2 2.5 3 3.5 =-- Real Part of Pulse Shaped 8-PSK Signals 1 I I I w 'h ',fh I I I __ __ 0.5 ---2__ ' rl f , !AA ...... L:__...... :_/_q ' ...... _ ...... "ir ..... -P I'1 '/_ I I i I / 0 f I I I ...... _:__...... :_/____----'q ...... "r ...... '4"1" ..... /- ...... TiI ,I A ,l A _,I / -0.5 ...... :/ __ / "1...... ::-/...... i-V- 'i VN:,) w 0 0.5 1 1'.5 2 2.5 3 3.5 Figure 2-18 Weighted Raised Cosine Pulse Used as Phase Pulse and 8-PSK Signaling w 29 =-- Trapezoid Signaling A trapezoid signal can be expressed as T 1 ltl < - 2 2 T< h(t)= T 2- -_ Itl -_ - Itl < 0 Itl >T (2-45) where 0.5 _ Trapezoid pulse used as phase and 8-PSK signals are shown in Figure 2-19. Trapezoid Signaling with Pulse Duration 2T i y I I It I ,I ",4.. : I I I I I I I I I I t I l I I I [ I I I I I I I I I I ...... --_ r ..... P ...... 0.5 ...... r...... r" ...... r- ...... I- ...... r'---- I I I I I i I I I l i 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 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 Phase Signals i i i i i/\ i i 6 .....-- -- -1 '...... 'I ....-_- i' ....-'--_-'1--- .... - - -- '-I ..... - --'I-- - - ..... I I / _ t _ I I I i 4 ---, I...... /I ..... "X,- -_-//-I -\-- l_ I / ...... __I ...... rI ...... ,I...... -_ I I I I _ _[I t === r, T----'? .... _ \_/ _:-- : J I r,_, ---q ------_ut,..... ,_...... __y------x-:--_- - - - - 1 0 1 i i I t _l 0 0.5 1 1.5 2 2.5 3 3.5 Real Part of Pulse Shaped 8-PSK Signals i i\ , , A, , / ,itI ,I I,/I ',:I \// I I, / , /_ , , / ', '_1 I , I' / == 0 ...... i__ ,I t- -_ ,,I 7_ /_,t r ..... _r.... l ,,I -,F;_'1 .... ' ',i\ ,, L,.,' ' :, I -1 0 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 --= I 3O Weighted Raised Cosine Transit Signaling A pulse with weighted raised cosine transit and flat within one symbol period can be expressed as T Itl< - 2 h(t)= Itl_ T 2 i 0 Itl >T where 0.5_ w 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 Phase Signals I I I /\ I I I I w 8 ...... I...... I ..... 7 ...... 3" ...... T ...... I- ...... I...... 'I :I :"-I '/:xI 'I 'I I' 6 ...... a...... £ - -..-"x- .... L ...... L ...... i ...... : _7----c---_-/ \ : : :/_ 4 w ...... 7:- 7__, ...... _...... _:.... T--r ...... _,- 7---Tr----X-; 2 ...... ,_,,_/ ...... :...... _...... __:=.... _____.-_/ ..... '__:_._:_ 0 1.1/ I i i _l _V i 0 0.5 1 1.5 2 2.5 3 3.5 Real Part of Pulse Shaped 8-PSK Signals .I ! 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 z 3] 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 w 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. SSPA, ESA 10WattsAmplifier:MagnitudeResponseCurve 0 nn i t w i i I i , , ,,! , i I -10 O EL > J -20...... 013...... r_ .= i "5 i © i -3[? i -30 -25 -20 -15 -10 -5 0 Input LevelPower (riB) SSPA, ESA 10Watts Amplifier:PhaseResponseCurve I 1 i I w ._--.- ...... , , m u._t _...... n , -6 ; ; l -30 -25 -20 -I5 -10 -5 0 InputLevelPower (dB) m Figure 2-21 The Output Characteristic of 10 Watts SSPA 32 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 i-- and input amplitude to output phase. Figure 2-22 shows the structure of the SSPA. TWT D in out = (TABLE) iave_pwr hold AVERAGE COMPLEX J, x POWER reset hold SSPA w I' 't w Figure 2-22 The Structure of SSPA. w 2.5 Additive White Gaussian Noise Channel m Additive White Gaussian Noise (AWGN) is added to the signals out of the SSPA to i PSD,(f) No/2 w w -fs/2 0 fJ2 w Figure 2-23 Power Spectral Density Of White Gaussian Noise. w 33 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 r_ the white noise, and Fs is the sampling frequency. 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. W ...... _ ...... DELAY r(t):s(t)+n(t) & ERROR w MATCHED ' i: PHASE "_ RATE FILTER METER ESTIMATOR Receiver Synchronizer w • ...... ? ...... • Figure 2-24 Block Diagram for the Receiver Error Rate Estimator m 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. m 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 & w 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. Variable x Delay y I delayComplex I u reference DELAY & PHASE w 8 PSK w received u 8PSK Figure 2-25 Block Diagram of Synchronizer Error Rate Estimator The reference signal and received signal (now having the same delay), are put into 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. in SIMPLE rate ERROR = i'ef RATE errs w ESTIMATOR N reset T L-- Figure 2-26 Simple Error Rate Estimator Block Diagram m w w 36 w Chapter 3 PULSE SHAPED 8-PSK SIMULATIONS 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. w--- 3.1 Simulations on Power Spectral Density Figure 3-1 shows the procedure for obtaining the power spectra for pre-pulse shaped 8-PSK signal. Solid State Spectrum Phase .I Power Analyzer Random Modulator Data "]Amplifier I BasebandFilter (FF ] (SSPA) .... T Spectrum Analyzer w (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 filters are used. For Butterworth and Bessel filters, bandwidth (BT) is equal to the cutoff w 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 i -- w Table 3-1. Table 3-1 Traditional Filters Used in the Simulation Filters BTs or roll-off factor ( a ) 5th-Order Butterworth BT=I BT=2 BT=2.8 BT=3 3rd-Order Bessel BT=i BT=1.2 BT=2 BT=3 SRRC a=l 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 w 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. ,,,,,,t,- w 38 3.1.1 PSD Simulation Results of Using Traditional Filters The plots of the power spectra are given in Figures 3-2 to Figure 3-11. The following parameters were used to obtain the power spectra. w Data Source • a° Ideal data with p(0) ( probability of zero )=0.5; c° Sample Rate (fs) =512P_ Hz Traditional Filters for Pulse Shaping: a. None b. Butterworth 5th Order (BT=1,2,2.8,3) m c. Bessel 3rd Order (BT=1,1.2,2,3) i d. SRRC (a =1, Number of tap length=32*P,_=8192, Bartlett window Power Amplifier: m_ m European Space Agency (ESA) 10-watt SSPA SPW Spectrum Analyzer m a. FFT length=512R s E_ m b. Window type: Bartlett =__ c. Average: 100 FFTs d. Sampling frequency =512R_ Hz Spectrum Plots Ranges + 10 Rb; + 75 R b ! 39 °I , , , ^/_^ ' , =.-. !!T . tall= -80 -10 -8 -6 -4 -2 0 2 4 6 8 0 Frequency (Rb) (a) Unfiltered 8 PSK + 10 1_ = m -75 -60 -40 -20 0 20 40 60 75 Frequency (Rb) (b) Unfiltered g PSK +_.75 1_ I' Figure 3-2 Unfiltered 8-PSK Power Spectral Density i SSPA Output, +/-10Rb (5th Order Butterworth BT=I ,Ideal Data) 0 , , , i_t , , , ii==w .i nn "10 -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, m 40 L_ SSPA Output, +/-75Rb (5th Order Butterworth BT=I ,Ideal Data) 0 ! I I I I I -100 I 1 1 -75 -60 -40 -20 0 20 40 60 75 .+ , Frequency (Rb) With All Spikes (b) 5th Order Butterworth Filter (BT=I), +_751% Figure 3-3 Power Spectra of 8-PSK Pulse Shaped with 5th-order Butterworth filter (BT=I) SSPA Output, +/-10Rb (5th Order Butterworth BT=2,1deal Data) 0 I I I ' I I I -20 -40 13:1 -60 -80 -100 - iiiiiiii!i,ii,,, I i/ill'fill+iiii_ -10 -7.5 -5 -2.5 0 2.5 5 7.5 10 Frequency (Rb) With All Spikes i i (a) 5th Order Butterworth Filter (BT=2), __.1ORb t m_ SSPA Output, +/-75Rb (5th Order Butterworth BT=2,1deal Data) 0 ! ! ! I1[ I , i -20 -40 lll -60 -80 -100 L -75 -60 -40 -20 0 20 40 60 75 Frequency (Rb) With All Spikes (b) 5th Order Butterworth Filter (BT=2), :t:75R_ Figure 3-4 Power Spectra of $-PSK Pulse Shaped with 5th-order Butterworth filter ('BT=2) 41 SSPA Output, +/-10Rb (5th Order Butterworth BT=2.8,1deal Data) 0 -20 -40 -60 ID -80 -100 -10 Frequency (tbo) With All Spikes (a) 5th Order Butterworth Filter (BT=2.$), + 10R_ SSPA Output, +/-75Rb (5th Order Butterworth BT=2.8,1deal Data) 0 I I I =IL I I I w -20 -40 rn --60 "O -8O -100 I -75 -60 -40 -20 0 20 40 60 75 Frequency (Rb) With All Spikes (b)5th Order Butterworth Filter (BT=2.$), + 751_, w 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) 0 I I I -20 -40 rn -60 nO -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 SSPA Output, +/-75Rb (5th Order Butterworth BT--3,1deai Data) 0 I I I -20 -40 -6O -8O -100 -75 -60 -40 -20 0 20 40 60 75 Frequency (Rb) With All Spikes (b) 5th Order Butterworth Filter (BT=3), + 75Rb 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) 0 -20 -40 "U -60 -80 -100 -10 -8 -6 -4 -2 0 2 4 6 8 10 m Frequency (Rb) With All Spikes m (a) 3rd Order Bessel Filter ('BT= 1), _+10R4, SSPA Output, +/-75b (3rd Order Bessel BT=I ,Ideal Data) -2o0 / ...... !, ...... i,...... :,...... , , , -40 _-...... ' ...... :...... i...... n,1 U .Io -100 -75 -60 -40 -20 0 20 40 60 75 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) 43 SSPA Output, +/-10b (3rd Order Bessel BT=1.2,1deal Data) 0 -20 -40 '10m --60 -80 -100 -10 -8 --6 -4 -2 0 2 4 6 8 10 / Frequency (Rb) With All Spikes (a) 3rd Order Bessel Filter ('BT=1.2), + 10R_ SSPA Output, +/-75b (3rd Order Bessel BT=1.2,1deal Data) 1 ! ! i , t -20! ..40 ' en -60 "10 -80 -100 I 1 -7.: -60 -40 -20 0 20 40 60 75 Frequency (Rb) With All Spikes (b) 3rd Order Bessel Filter (BT=I.2), + 75P,_ .,=11 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) 0 1 I I I 1 I i I -20 -40 co -60 -80 -100 0 --8 -6 -4 -2 0 2 4 6 8 0 Frequency (Rb) With All Spikes (a) 3rd Order Bessel Filter (BT=2), + 10R_ t_ 44 blJ SSPA Output, +/-10Rb (3rd Order Bessel BT---3,1deai Data) 0 -2O m -40 "O -6O --8O -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) 0 I I I I I I -20 -40 rn -60 -80 -100 J .-.----Lm -75 -60 -40 -20 0 20 40 60 75 Frequency (Rb) With All Spikes m (b) 3rd Order Bess¢l Filter (BT=2), + 75P_ SSPA Output, +/-75Rb (3rd Order Bessel BT--3,1deal Data) 0 I : I t w ...... :...... ° ...... _...... :...... i ...... !...... ,L .L_ -75 -60 -40 -20 0 20 40 60 75 w Frequency (Rb) With All Spikes , = 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 SSPA Output, +I-10Rb (Raised Cosine,a=1 ,Ideal Data) 0 -20 -40 m "o -60 -8O -100 0 -7.5 -5 -2.5 0 2.5 5 7.5 10 Frequency (Rb) With All Spikes (a) Square-Root Raised Cosine (a = l), + I 0R_ SSPA Output, +/-75Rb (Raised Cosine,a=l,ldeal Data) 0 I I I I 0Q "to -75 --60 --40 -20 0 20 4O 6O 75 Frequency (Rb) With All Spikes Co)Square-Root Raised Cosine (a =l), + 751_ m .,.-,, Figure 3-11 Power Spectra of 8-PSK Pulse Shaped with SRRC filter (a =1) n 46 3.1.2 PSD Comparison Between Two Pulse Shaping Methods The PSD related to pre-modulation pulse shaping will be compared with m 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 Band P = 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 w 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_ =27.34 1.28Rb 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 : : w 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) uiiiization None 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 BT=3 Utilization Ratio BT=2.8" i , BT=2" =___ BT=I" 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' Utilization Ratio BT=1.2" BT=I' 0 5 10 15 20 25 30 Figure 3-13 Utilization Ratio of Using 3rd-Order Bessel filters in Pre-Modulation Pulse Shaping SSPA Output, Pulse Shaping filter: 5th Order Butterworth BT=I , t ! ! [,_l ! ! ! ! .... t_"i %..i 2Post-,PuseShaping ...... : ...... -40 ...... :...... :...... :.... _r .... i...... • ,. - i ] m -60 -o -80 -100 ...... ! ...... 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 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 m 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. Trapezoid Signaling with Pulse Duration 2T 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 F 0.5 I I I I I I I I I 1 I I 'I 'I 'I 'I 'I 'I 't '7I ';t I I I I I I I I I I I I I i l I I I I I I I 0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Time (T) 7 PSD of 8--PSK used 2T duration trapezoid pulse shaping w o -20 nn "o_40 -60 I I I I I I I I I -5 -4 -3 -2 -1 0 1 2 3 4 w. Frequency (Rb) Figure 3-15 PSD of 8-PSK Used 2T Duration Trapezoid Pulse Shaping 50 Trapezoid Signaling with Pulse Duration 3T - I I I I I ! I I I I I | I I ...... "r ...... ----V ...... I I 05 I I I I i 1 I i i I I 1 I I I I I I | I 2.5 00 0.5 1 1.5 2 Time (T) PSD of 8-PSK used 3T duration trapezoid pulse shaping I ] I...... ! I i i i z : 10 -10 -8 -6 -4 -2 0 2 4 6 8 Frequency (Rb) Figure 3-16 PSD of 8-PSK used 3T-s duration trapezoid pulse shaping Cosine Transit Signaling with Pulse Duration 2T I I j I I I I I III I 11 = ,, ___J , , , _ --_ ' , 0.8 1 I l l l I I .... I ...... I ...... 0.6 ...... 7 ...... F ...... F ...... T ...... i"- --7 ...... I ...... I .I ...... 1 .t ...... I ...... I I I 0.4 I I I I I ...... pl ...... pI ..... pI ...... pI ...... pl ...... | I _P ¢I ...... plI ...... 0.2 -- - I I --fI ...... rI ...... I"I ...... rI -- --r I / i iI I LI ,I _I I , , -0.8 -0.6 -0.4 ...0.2 0 0.2 0.4 0.6 08 1 Time (T) PSD 0 I I -20 -4O i...... i...... :...... :...... "...... "...... "...... :...... " ...... " " i i , J i I I I I 10 -8 -6 -4 -2 0 2 4 6 8 Frequency (Re) Figure 3-17 PSD of 8-PSK used 2T-s duration cosine transit signaling pulse shaping 5] Raised-Cosine Signaling with Pulse Duration 2T i t I i I i i l | i i 0.8 I [ 1 I I l I I E I I I I t I I I I ...... 0.6 ...... F ...... F ..... F ...... y ...... F ...... F ...... F ...... y ...... T I l I I I I I I I .... _ ...... _ ..... _ ...... _ ...... _ _ ...... _ .... _ _ ...... 0.4 I I I l I I I I I I I I I I I I __l__ F ...... nil ill-- 0.2 ...... -- 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 0 -20 "o t I I I I I I I w 10 -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 Weighted Raised-Cosine Transit Signaling with Pulse Duration 2T II I I lI I I I I I I I l l I I I I I l I I I 1 I I I I I I I I 0.5 ____ ...... _..... ,_...... l I I I I ] I I I t I I t I I I_1 iI I I Il I I I I I II I I I I I I 0 I t I I I I 43.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Time (T) PSD 0 -20 - 5 -40 ;l=d m -60 = ;: 180 I I I I I I I I -10 -8 -6 -4 -2 0 2 4 6 8 10 Frequency (Rb) Figure 3-19 PSD of 8-PSK Used 2T Duration Weighted Raised Cosine Transit Signaling Pulse Shaping 52 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. m r w Table 3-4 PSD Line Components for Trapezoid Shaping iT 0 1 2 4 PSD 0.0819xl 0 "'_ 0.0048xl 0-"': 0.0048x10 -j'_ 0.0098x 10-'_ O.O091xlO -j': Table 3-5 PSD Line Components for Weighted Raised Cosine Shaping fT 0 1 2 w PSD O. 1736x 10 '_ 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 2 PSD 4.213x10 _ 4.213x10 _ 4.213x10 _ 4.213x10 _ 4.213x10 _ 53 = = w Table3-7 PSDLine Componentsfor CosineTransitShaping fff 0 2 3 4 PSD 4.333x10 -_ 4.333x10 -_ 4.333x10 -_ 4.333x10 -_ 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 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 i . 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 3.2 Bit Errors Rates Simulations w Figure 3-21 shows the system block diagram for pulse shaped 8-PSK end to end system simulations Baseband Phase Random Modulator Phase Filter iDATA SOURCE! !PULSE SHAPING! ! MODULATION AMPLIFIER • ...... , ...... • ...... AWGN CHANNEL w ERROR & INTEGRATE L.d PHASE RATE & DUMP METER r = ESTIMATOR ERROR RATE SYNCHRONIZER RECEIVER ...... Figure 3-21 BER Simulation System Block Diagram AWGN is added to the signals out of the SSPA to simulate channel noise. The bit errors m 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 w 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 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 •--_fr-Theoretical 8PSK Butterworth --B--Bessel SRRC(alpha=l) o --0.-- Bessel BT=1.2 m ILl := 1.00E-04 m 1.00E-05 j 1 1.00E-06 -1 1 3 5 7 9 11 13 = < Eb/No(dB) Figure 3-22. BER Plot for Pre-Modulation Pulse Shaping Filters with BT=I L.2 56 BER for Pre-Modulation Pulse Shaping Filters with BT=2 1.00E+O0 i ' I ' i i --:J[--- Theoretical 8PSK I .OOE-01 Butterworth --I!-- Bessel --_k-- SRRC(alpha=l) i .00E-02 ! i o 1.00E-03 t I I : I i = I m , E i ] \ __ 1.00E-04 ' ! I I 1.00E-05 I , I t I i I _ I , 1 i 1.00E-06 L I • , -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 L ; w BER for Pre-Modulation Pulse Shaping Filters with BT=3 1.00E+00 r i ---X---Theoretical 8PSK I --(k-- Butterworth 1.00E-01 = m --I1--- Bessel 1.00E-02 k I --A--- SRRC(alpha=l) JD m f --'0-- Butterworth BT=2.8 .o n 1.00E-03 LLI 1.00E-04 in 1.00E-05 i 1.00E-06 -1 1 3 5 7 9 11 13 = , Eb/No (dB) Figure 3-24. BER Plot for Pre-Modulation Pulse Shaping Filters with BT=3 57 m 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. 1.00E+00 m : ,L ,L -qY:-- Theoretical 8PSK 1.00E-O1 , r T :_J, ! i ! _(k--3T-s Trapezoid Pulse w T 1.00E-02 e i i tl. 1.00E-03 tU 1.00E-04 m 1.00E-05 i i 1.00E-06 i T -1 1 3 5 7 9 11 13 Eb/No (dB) Figure 3-25. BER Plot for Pre-Modulation Pulse Shaping With 3T Trapezoid Signaling m ISI Losses Due to Filtering w 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 -.- 58 . = m integrate and dump was used as receiver, it could not matched the pulse shaped signals. w 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) Filter Type Loss Loss Loss Loss Loss BT=I BT=I.2 BT=2 BT=2.8 BT=3 (dB) (dB) (dB) (dB) (dB) Butterworth ...... 5th Order 15.7 5.52 4.68 4.68 Bessel ...... =z 3rd Order 13.2 6.05 4.88 SRRC(a =1) 7.42 = -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 = = requirement of ISI loss <0.4. The Figure 3-25 shows the relationship of ISI loss and utilization ratio for different filters. w ISI Loss (dB) 20 15 m 10 iiiiiii iiiiiiiiiii!!!!!!!ii. 5 : : 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 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 W 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- w 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 w 3rd Order Bessel filter, the BER produced by post-modulation pulse shaping (BT=2) is 61 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 i 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 w 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 w 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. 2_ m 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. 62 i i w 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] 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. [8] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing. 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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. [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. w [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