ANALYSIS OF FREQUENCY HOPPING SYSTEM WITH 2-ARY
FSK AND BPSK MODULATION AND AN IMPLEMENTATION
OF A COHERENT 2-ARY FSK/FH MODEM :
A Thesis Presented to
The Faculty of the College of Engineering and Technology
Ohio University
In Partial Fulfillment
of the Requirements for the Degree of
Master of Science
b Y
Yaim B. Zaxawi
March 1983
OHIO UNIVERSITY LIBRARY ATHENS, OHIO Acknowledgment
I am indebted to my advisor, Dr. Joseph E. Essman, for his encouragement, valuable suggestions and guidance during the course of this work. Also the author wishes to express his thanks to
Dr. M. Jameel for his help, particularly in the real time simulation.
Special appreciation and affection is recorded to my parents for their support and understanding during the course of this work. TABLE OF CONTENTS
Page ACKNOWLEDGMENT ...... i LISTOFTABLES ...... iv LISTOFFIGURES ...... v
Chapter
INTRODUCTION ...... 2 . SPECTRAL ANALYSIS ...... 4
Introduction
2.2 Analysis of Frequency Shift Keying/Frequency Hopping (FSK/FH) ...... 4
2.2.a Analysis of Frequency Hopping Signal Using a PN-Code Sequence ...... 4
2.2.b Analysis of Frequency Shift Keying (FSK) Signal ...... 8 2.2.c Spectrum Analysis ...... 11
2.3 Spectrum Analysis of Binary Phase Shift Keying/ Frequency Hopping (BPSK/FH) ...... 27 2.4 Summaryand Conclusions ...... 34 3 . PERFORMANCE OF THE FREQUENCY HOPPING SYSTEE.1 ...... 39 3.1 Introduction ...... 39
3.2.a Partial-Band Noise Jan?ming Model ...... 39 3.2.b Partial-Band kltitone Jaming Model .... 40
3.3 Probability of Error Calculations in the Presence of Partial-Band Noise Jamming ...... 40
3.3.a Detection of Yon-Coherent Binary Frequency Shift Keying in Frequency Hopping Environment (2-ary FSK/FH) ...... 40 i i 3.3.b Worst Case Jamming Strategy Against Non-Cherent FSK/FH ...... 43
3.3 .c Detection of Coherent Frequency Shift Keying in Frequency Hopping Environment (Coherent FSK/FH) ...... 46
3.3.d Binary Differential Phase Shift Keying in Frequency Hopping Environment (BDPSK/FH) ...... 49
3.4 Calculations of the Probability of Error in the Presence of Partial-Band Multitone Jamming . . 52
3.4.a 2-ary Non-Coherent FSK/FH with Jam Tone Spacing Equal to the Bit Rate . . . . . 55
3.4.b 2-ary Non-Coherent FSK/FH with Jam Tone Spacing Equal to Twice Bit Rate . . . . 57
3.4.c Probability of Error of the BDPSK/FH with Jam Tone Spacing Equal to the Bit Rate ...... 57 3.5 Conclusions...... 60 4. REALTIMESIMULATION...... 62 4.1 Introduction ...... 62
4.2 Design and Description of the Coherent FSK/FH Modem 62
4.3 Results andcomments ...... 72 4.3 Summary and Conclusions ...... 54
BIBLIOGRAPHY...... 91 APPENDIX
A. COMPUTER LISTINGS LIST OF FIGURES
Figure Page
2.1 Spread Spectrum Frequency Hopping Model ...... 5
2.2 Carrier Frequency Versus Time for the FH Signal Using a PN-Code Sequence ...... 7
2.3 The Periodic Rectangular Function Used in the Expression of the FH Signal ...... 7
2 -4 Frequency Synthesizer's Waveforms with B = 2 ...... 9 a) Code Sequence b) Carrier Frequency Versus Time c) The Periodic Rectangular Function d) Output Waveform
2.5 The FSK Modulator Waveforms ...... 10 a) Baseband Signal (Input) b) Output Waveform c) Frequency Versus Time
2.6 Frequency Versus Time Relationship for the FSK/FH Signal Assuming T~ = 2~~ for B = 2 ...... 13
2.7 The General Periodic Gate Function Used in the FSK/FH Analysis ...... 13
2.3 Frequency Versus Time for the FSK/FH Signal with B = 3 and T~ = ~,/3 ...... 14 a) Code Sequence b) Rectangular Functions for the FSK and FH Signal c) Carrier Frequency Versus Time
2.9 Magnitude Line Spectra for the Plain FSK Signal Considering the Second Zero Crossing (Positive Frequencies Only) ...... 20
2.10 Magnitude Line Spectra for the Plain FH Signal with B = 2 Considering the Second Zero Crossing (Positive Frequencies Only) ...... 21
2.11 Magnitude Line Spectra for the Plain FH Sigcal with B = 3 Considering the First Zero Crossing (Positive Frequencies Only! ...... 22 Figure Page
Magnitude Line Spectra for the Plain FH Signal with B = 4 Considering the First Zero Crossing ...... 23
Magnitude Line Spectra for the FSK/FH Signal with B = 2 and rh = 2rm (urn = 4%) After Th Seconds, Considering the First Zero Crossing ...... 24
Magnitude Line Spectrum for the FSK/FH Signal with B = 2 and -rh = 2r for the Special Case Where Awl m
Magnitude Line Spectra for the FSK/FH Signal with B = 2 and rh = r,/4 (b+, = 2um) After 2Th Seconds, Considering the First Zero Crossing (Positive Frequencies Only) ...... 26 BPSKWaveforms ...... 28 a) Baseband Signal b) Output Signal
Carrier Frequency Versus Time for the BPSK/FH Signal with B = 2 and rh = 2rm ...... 30 Magnitude Line Spectra for the BPSK Signal ...... 32
Magnitude Line Spectra for the BPSK/FH Signal with B = 2 and r = 2rm 33 h ......
Partial-Band Noise Jamming Model ...... 41
Partial-Band Multitone Jamming Model ...... 41
Bit Probability of Error Versus Bit Energy to Jam Noise for the Non-Coherent FSK/FH in the Presence of Partial-Band Noise with a = 1.0 ...... 44
The Product XPB(x) Versus E/NJ for the Coherent and Non-Coherent FSK/FH in the Presence of Partial- BandNoiseJa~ming ...... 35
Bit Probability of Error Versus the Bit Energy of Jam Noise Density in the Presence of Partial-Sand Noise with a = 1.0 for the Coherent and Non- Coherent FSK/FH ...... 47 Figure Page
3.6 The Jammed Fraction of the RF Bandwidth vs the Bit Energy to Jam Noise Density Ratio in the Presence of Partial Band Noise (Worst Case) for the FSK/FH (Coherent), FSK-FH (Non-Coherent) , and BDPSK/FH Waveforms ...... 50
3.7 Maximum Bit Probability of Error vs Bit Energy to Jam Noise Density in the Presence of Partial-Band Noise for the Coherent and Non-Coherent FSK/FH Waveforms (Worst Case) . . . . 51
3.8 The Product XPB(x) Versus (E/NJ) for the BPSK/FH Waveform in the Presence of Partial-Band Noise Jamming...... 53
3.9 Bit Probability of Error vs the Bit Energy to Jam Noise in the Presence of Partial-Band Noise Jamming for the BFSK/FH and BDPSK/FH (Worst Case) . 54
3.10 Bit Probability of Error Versus the Bit Energy to Jam Noise Density in the Presence of Partial- Band Multitone Jamming (Worst Case) for the Non-Coherent FSK/FH and for the BPSK-FH Waveforms ....58
3.11 Bit Probability of Error Versus the Bit Energy to Jam Noise Density (Worst Case) for All the Different Waveforms Considered ...... 59
4.1 Block Diagram of the Implemented Coherent FSK/FH Modem...... 63
4.2 Circuit Diagram Connection for the Coherent FSK/FH Modem...... 64 a) Transmitter's Side b) Receiver's Side
4.3 Input Waveforms to the FSK/FH Modem ...... 74 a) Input Data (A Stream of Square Wave Pulses with r = 15 msec and duty cycle = 50%) m b) The Spreading Frequency Hopping Signal with a Center Frequency = 150 KHz, Number of Output Frequencies = 11, and Hopping Step Size = 5 KHz
4.4 Output Waveforms from the FSK Modulator ...... 75 a) The Mark Frequency f M = 2 KHz b) The Space Frequency fS = 1 KHz c) The FSK Signal Figure Page
Amplitude Response of the Receiver's 4-Pole Butterworth Band Pass Filter with Center Frequency f = 1.5 KHz and Quality Factor = C QBp=2...... 76 The Complete Transmitted FSK/FH Signal Waveform (Output of the Transmitter's Mixer) ...... 77 Output Waveform of the Receiver's BPF ...... 77 Output Data (Output of the FSK Demodulator) ...... 77
Frequency Spectrum for the FH Signal Coming from the Synthesizer with a Center Frequency = 150 KHz, Number of Output Frequencies = 11, Hopping Step Size = 5 KHz and Time/Hopping Step -rh = 30 msec ...... 78
Frequency Spectrum for the FSK Signal with the Mark Frequency f = 2 KHz, Space Frequency fS = 1 KHz M and Pulse Width rm = 15 msec ...... 78
Frequency Spectrum for the Frequency Hopping Signal of Fig. 4.9 ...... 79 a) When Mixed with the Mark Frequency Only b) When Mixed with the Space Frequency Only
Frequency Spectrum for the FSK/FH Signal with a Center Frequency = 150 KHz, Number of Output Frequencies = 11, Hopping Step Size (Awi) = 5 KHz, ;h = 30 msec and -rm = 15 msec (Slow Frequency Hopping) ...... 80
Frequency Spectrum for the De-spread Signal After Being Passed Through the BPF ...... 80
Frequency Spectrum for the FH Signal with a Center Frequency = 500 KHz, Number of Output Frequencies = 101, Hopping Step Size = 5 KHz and Time/Step = 30 msec...... S1
Frequency Spectrum for the FH Signal (Whose Spectrum is Shown in Fig. 4.14) When Mixed with the FSK Signal with -rm= 15 msec ...... 81
Portion of the Spectrum of the Signal Whose Spectrum is Shown in Fig. 4.15. [Horizontal Scale is Set to 10 KHz/cm.] ...... 82 Figure Page
4.17 Frequency Spectrum for the FSK/FH Signal with a Center Frequency = 150 KHz, Number of Output Frequencies = 11, Hopping Step Size = 0.5 KHz, T~ = 30 msec and T~ = 15 msec ...... 82
4.18 Frequency Spectrum for the FSK/FH Signal with a Center Frequency = 150 KHz, Number of Output Frequencies = 11, Hopping Step Size = 5 KHz and T~ = ~,/2 = 30 msec (Fast Frequency Hopping) ...... 83 a) After t < 3Th b) After t > 3Th Chapter 1
INTRODUCTION
Shannon's original work in 1948 in the field of statistical communication theory showed that the capacity of a channel to transfer error-free information is enhanced by increasing the bandwidth of the transmitted signal, and this was the basis of spread spectrum develop- ment. At the beginning, this new concept of communication did not draw much attention because it was very difficult to implement any such
system with the circuit technology that was available at that time.
Due to recent advances in statistical theory, coding theory, the advent of reliable high speed and inexpensive digital components and the
demand for increased message traffic from a larger number of users, all of which have created a need for improved communications, the field
of spread spectrum has drawn an increasing number of researchers in
the last few years.
In spread spectrum techniques,the transmitted signal spectrum
is spread over a bandwidth much larger than that required to transmit
the basic signal. This is done in order to provide systems anti-jam
characteristics and to provide multiple access capabilities. The
spreading is accomplished by some function other than the information
being sent. Generally, there are three types of spread spectrum techniques,
namely:
1. Direct spreading (pseudorandom sequences), in which a carrier
is modulated by a digital code sequence having a bit rate higher than
the information signal bandwidth. 1 2. Frequency-modulation pulse compression or "chirp," in which a carrier is swept over a wide band of frequencies during a given pulse.
3. Frequency hopping, "FH," in which the carrier is frequency shifted in discrete increments in a pattern determined by a digital code-sequence .
These basic spread spectrum systems are differentiated by their modulation formats. Other formats that are combinations of the above are also possible. A quick look at a typical frequency hopping system block diagram shows that it has a frequency synthesizer at both the transmitter and the receiver. It is the presence of this subsystem which has blocked the use of frequency hopping in the past, and much effort has been spent on the other spread spectrum techniques. But the development of improved small frequency synthesizers has open the way for widespread use of frequency hopping.
In this work, only the third technique mentioned above (frequency hopping) is used as a means of spectrum spreading. In general, there is no restriction on the choice of information modulation; however, in this report 2-ary frequency shift keying (FSK) and binary phase shift keying (BPSK) were considered. Chapter two is devoted to the spectrum analysis of the generalized transmitted FSK/FH and BPSK/FH signals which are represented by (2.9) and (2.34) respectively. Two general formulas for finding the frequency spectrum are derived using the Fourier analysis.
Examples treated include the following: FSK, PSK, FSK/FH and BPSK/FH.
In each case the transmitted data considered was a periodic square wave.
The third chapter of this thesis shows how the frequency hopping spread spectrum system combats the intentional noise introduced to the systen 3 by the jammer. Two common models were considered: partial-band noise jamming and partial-band multitone jamming. For each model the per- formance characteristics of the BFSK/FH and BDPSK/FH are presented. The maximum probability of error corresponding to the worst case jamming strategy, which consists of specifying the worst partial-band fraction, is determined. Finally a coherent FSK/FH modem was implemented and
Fig. 4.2a,b shows the circuit diagram of this system. It is to be noted that in this system the frequency synthesizer,which is controlled by a computer, is used to generate the frequency hopping signal. This signal is used for spectrum spreading at the transmitter's side and spectrum de-spreading at the receiver's side to assure frequency and phase coherence.
The transmitted data which is a stream of square wave pulses is finally recovered using a coherent FSK demodulator. A frequency spectrum analyzer was used to obtain the frequency spectrum for the FSK/FH signal. The design and details of the modem are discussed in chapter four followed by summary and conclusions. Chapter 2
SPECTRAL ANALYSIS
2.1 Introduction
In communication systems,it is important to know the spectral occupancy of the transmitted signal in order to efficiently design the modem. This chapter is devoted to the bandwidth BW analysis of a simplified frequency hopping model shown in Fig. 2.1 in which the central feature of the system is the code generator at both the trans- mitter and receiver. The generated code sequence is used to give com- mands to the frequency synthesizer to switch its output frequency in a pattern determined by the used code sequence. The baseband signal is modulated then mixed with the synthesizer output signal, to form the complete transmitted frequency hopping signal. At the receiver, the received signal is mixed with a synchronized replica of the transmitter's frequency hopping signal to remove the frequency hops. Assuming no timing'and other errors, the result will be the original modulated
signal which can be demodulated in a conventional manner. In this analysis, two types of modulations are considered, namely:
1. 2-ary frequency shift keying (FSK), and
2. binary phase shift keying (BPSK).
Analysis of Frequency Shift Keying/Frequency Hopping (FSKIFH)
2.2.a Analysis of Frequency Hopping Signal Using a PN-Code Sequence
"Frequency Hopping," or more accurately termed "multipie
frequency code-selected, frequency shift keying," is nothing more than a
6 plain FSK except that the set of frequency choices is greatly expanded [2].
Let the RF bandwidth W, which starts at we (lower limit of the
spread spectrum) and extends to w (upper limit of the spread spectrum), u be divided into M frequencies. The output waveform from the frequency
synthesizer is then given by:
(t)= f [COSwKt + yK(t)] [u(t-(~-l)\~- ~(t-Krh)] (2.1) K= 1 where sK is selected from M possible values according to the code sequence
= phase of the frequency synthesizer's output in the K t h ' K interval.
If a periodic PN-code sequence is used to give the frequency
commands to the synthesizer, then the synthesizer would change its
output frequency pseudorandomly every rh seconds to one of the ?4 fre-
quencies as suggested by Fig. 2.2 and the sequence of freq.uencies
repeats every Th seconds. The frequency increment is
Let the synthesizer output when using the PN-code be expressed as Fig. 2.2 Carrier frequency versus time for the FH s-ignal using a PN-code
sequence.
Fig. 2.3 The periodic rectangular function used in the ex~ressionof
the FH signal 8 B = number of bits for the code word specifying the frequency hop.
id = synthesizer's center frequency around which the M frequencies 0 are to be supplied.
Owl = hopping step size/2.
The [u{t-(~-l)r~)- u(t-KT )] rectangular function is illustrated in h Fig. 2.3. Each specific output frequency is given by [2] as
and the code word specifying the hop is given by
The ctfs in (2.3) and (2.4) are given by
Fig. 2.4 is an illustration of the synthesizer's waveforms when a code sequence with B = 2 is used and the code is such that the frequency stepped one increment at a time.
2.2 .b Analysis of Frequency Shift Keying (FSK) Signal
Let the baseband signal which is denoted by a(t) be the periodic function shown in Fig. 2.5a. If this signal is frequency modulated, then the instantaneous frequency of the 2-ary FSK modulator output would take on only two possible values as shown in Fig. 2.5b.
The FSK modulator output may be expressed as: Frequency
tine
Fig. 2.4 Frequency synthesizer's waveforms with 3 = 2. a) code sequence used. b) frequency versus zime. c) the qeriodic rectangular function. d) outrut waveform. tcoso t 70s%t 4 I
Frequency I 'm (c) - 94 w- us' 7 P. time Tm 1 Fig. 2.5 The FSK modulator waveforms. a) baseband si~nal(input). b) output waveform. c) frequency versus time relationship. L Vm(t) = [cos wLt] [u{t-(L-l)rmi - ~(t-LT,)] L= 1 where
in which
w = carrier frequency C
Gw = frequency deviation
us = space frequency
% = mark frequency
Fig. 2.5~illustratesthe time versus frequency for the FSK signal.
2.2.c Spectrum Analysis
The frequency synthesizer signal Vs(t) is mixed with the
FSK modulator output signal Vm(t) to form the complete transmitted frequency hopped signal VT(t). For simplicity let us assume that 'J T (t) is selected as the upper side band, then
2B 2 vT(t) = ~[cos(w~+u~)t+y(t)l [uit-(~-l)~~i K=l L=l
where
y(t) = time delay and other phase effects. 12
With no loss of generality, we assume that y(t) = 0, then V,(t) becomes
Fig. 2.6 shows the carrier frequency versus time for the above FH/FSK signal assuming that rh = 2rM (slow frequency hopping) and B = 2. Fig. 2.8 shows the same relationship assuming r = rM/3 (fast frequency h hopping) and B = 3. Yow, our goal is to find the frequency spectrum of V (t) and to do that we will use the frequency convolution property T of the Fourier transform which state that [5]
then
Let VT(t) be expressed as
where
f KL (t) = [cos(wK+wL)t] [u{~-(K-I)T~~- u(~-KT~)] and
gL(t) = [~{t-(L-I)?,} - ~(t-Lr,)] time Th 1 7 - I I
Fig. 2.6 Frequency versus time relationship for the FSK/FH signal assuming T~ = 2 T~ for B = 2.
Fig. 2.7 The periodic gate function ased in the FSK/FH analysis. Fig. 2.8 Frequency versus time for the FSK/FH si.pal with B = 3 and - - 'I, - ?,/3. aj Code sequence. b) Rectanwiar function for the FSK and FR slgnal.
To use the convolution property,we first find the Fourier transform of fKL(t) and gL(t). For this analysis, consider the general periodic gate function f(t) shown in Fig. 2.7. This function may be expressed
where
and
2 = Fourier coefficient of f (t) P
which is found to be
Taking the Fourier transform
Omitting the details, F(w) is found to be
sin pn~/T (u-pwp) p=-co
Waking use of the above result and of the time shifting property of the
Fourier transform which states that
-jut0 F{f(t-to)} = e F(o) (where F(o) =GjS {f(t)}) then, the Fourier transform of gL(t) and f (t) can be written as LK
where
= pulse width during which either the mark or space frequency 'm occurs = Tm/ 2.
= 27/Tm = 'm baseband frequency
(2K- 1) -j (w-w -w ) K L 2 Th 6 (w-wK-uL-non)] + e
r sin mn~,/T, m 27~ - , w =- Gm - 7 Inn 'm/Tm " Tm and
Now, we can apply the frequency convolution properly and V T (w) becomes
(2L-1) - j (w-x) 2 m e ' 6 (w -x -mm) dxl For simplicity of integration let
and w - mu = b "K + "L = b~~ m
Substituting (2.23) in (2.22), VT(w) becomes
- j (bLKaK + uaL) [ 8 (x+bLK - nun) 6 (x-b) dx
j (bLKaK - maL) Jam ej (a L -aK 1' + e 6(x-bLK-nun) 6(x-b) dx]
Completing the above integrations V becomes T (w)
Csnsidering positive frequencies only and substituting (2.23) back in (2.25) , IVT (w) 1 becomes 2B 2 a a 1 sin (mv T~/T~) IvT(~)I = 1 1 T~(~~/~~)(rh/Th) ~=l~=l m=-w n=-oo m.rr rm/Trn
sin (nn rh/Tk,) B(w-w -w -mw -nw ) nn rh/Th KL m n
The spectrum of this FSK/FH is shown in Figs. 2.13, 2.14 and 2.15 for different cases. Without frequency hops (i.e. M = 1). the output of the frequency synthesizer would be a CW, i.e. V (t) = cos w t. When S 0 this signal is mixed with V (t) of (2.7), then the transmitted signal m V (t) (considering the upper sideband) becomes T
where
W -Aw=w when L = 1 C S
14 c +AW= % whenL=2 and
r m = pulse width during which either us or /AM occurs.
The magnitude of the Fourier transform of this plain FSK signal (con- sidering the positive frequencies) is given by
The frequency spectrum of this FSK signal is shown in Fig. 2.9.
2 7 Similarly, if one frequency (space or mark) is considered, then the output of the modulator Vm(t) becomes:
Vm(t) = cos wst y w = space frequency S
When this signal is mixed with VS(t) of (Z.3), then VT(t) becomes
2 VT(t) = 1 A[cos(~~+w~)~][u{t-(K-l)rh} - u(t-Krh)] (2.29) K= 1 where
r = hop interval h
The magnitude of the Fourier transform of this plain FH signal (considering positive frequencies only) is given by
2B T~ sin (nrr T~/T~) ~~~(~)l= 1 6(w-w K -w S-nw,) (2.30) K=l n=-a1 "Aq / nrr \/Th
The spectrum is shown in Figs. 2.10, 2.11 and 2.12 for different cases.
2.3 Spectrum Analysis of Binary Phase Shift Keying/Frequency Hopping
(BPSK/ FH)
Let the periodic baseband signal shown in Fig. 2.16a be phase modulated, then the output of the modulator would change its phase every
T seconds to one of two possible values (0 or T) according to the binary m stream as shohn in Fig. 2.16b. This waveform may be expressed as:
where Fig. 2.16 BPSK waveforms. a) Baseband signal. b) Output signal. 0 when L = 1 - OL - TT when L = 2
V (t) may be also written as m
where we get the positive sign when L = 1 and the negative sign when
L = 2. The plain frequency hopping signal coming from the frequency
synthesizer was given before as
This signal is mixed with Vm(t) (output of the BPSK modulator) to
form the complete transmitted BPSK/FH signal which would be denoted as VT(t) . Assuming that VT(t) is selected as the upper sideband and any time delay is neglected, then
2B 2 VT(t) = 1 1 t A[cos(wK+wc)t] [uit-(~-l)?~}- U(~-KT~)] K=l L=l
The carrier frequency versus time for this BPSK/FH signal is shown in
Fig. 2.17 for B = 2 and T~ = 2~~.
To obtain the frequency spectrum of the above signal, we need to
take its Fourier transform. Using the same method as that used to find
the Fourier transform of the FSK/FH signal represented by (2.9) and
omitting the details, we get Frequency
Fig. 2.17 Carrier frequency versus time for the BPSK/FH signal with B = 2 and rh = Z;,. where
r sin mn rm/Tm m Tr - W =-2 Gm - < mTr T~/T~ 9 Tm
b = w-mu and bcK = w +w m c K
Considering the positive frequencies only, V (w) becomes T
zB 2 a3 a j (bcKaK-waL) VT(w)= 1 kirAGmFn [e K=l L=l m=-w n=-oo
Taking the magnitude of VT(w), we get
'I I sin (mn 'I~/T,)
K=l L=l m=-co n=-a mT './Tm 1 I
sin (nn rh/Th) 6 (u-oK-oc -mum-nun) (2.38) nn 'Ih/Th
The magnitude line spectra for this BPSK/FH signal is shown in Fig. 2.19
for the special case where B = 2 and r = 2rm. h As before, without frequency hops, the output of the frequency
syntheizer would be
VS(t) = cos w t 0
Mixing this signal with Vm(t) of (2.31) we get
2 vT(t) = 1 t 4[cos(uo+wc)t] [u{t-(~-l)r~j- u(t-Lrm)] (2.39) L= 1
The magnitude of the Fourier transform of this plain PSK signal (con-
sidering the positive frequencies only) is given by
The spectrum is shown in Fig. 2.18.
2.4 Summary and Conclusions
This chapter was devoted to the spectrum analysis of the FSK/FH
and BPSK/FH signals which are given by (2.9) and (2.34) respectively.
Using Fourier analysis and the above two signals, two formulas to find
the frequency spectrum were derived. These are (2.25) for the FSK/FH
and (2.38) for the BPSK/FH. The bandwidth (BW) required for trarlsmission for each of the different modulations considered could be found from the magnitude line spectra corresponding to that modulation as follows:
1) The bandwidth of the plain FSK signal whose magnitude line spectra
of (2.28) is shown in Fig. 2.9 for the special case Aw > -2 .rr , Tm is given by
where
2Lw = Frequency separation between the two tones
BW1 = bandbase bandwidth
-n., 8 wm (considering the second zero crossing)
Two extreme cases are of interest:
a) If Aw >> BW then the bandwidth approaches 2Aw. This is 1' commonly called wideband FSK.
b) If Aw << BW 1 ' then the bandwidth approaches 2BW1 and this is called narrowband FSK.
2) The bandwidth of the plain BPSK signal whose magnitude line spectra of (2.40) is shown in Fig. 2.18, is given by
-n., 8wm (considering the second zero crossing)
3) The bandwidth of the plain frequency hopping signal (whose
magnitude line spectra of (2.30) is shown in Figs. 2.10, 2.11 and
2.12 for B = 2,3 and 4 respectively), is given by: 36
BW = (number of output frequencies from the frequency
synthesizer - 1.0) x hopping step size + BW2 (2.43)
where
BW2 = bandwidth of the frequency hopping baseband signal
Equation (2.43) may be also written as
BW = (number of frequency hops) x hopping step size
+ BW, L.
where
B = number of bits for the code word specifying the frequency
hop.
For large values of B (which is usually the case), the number of
output frequencies from the synthesizer would be large causing the
second term in the R.H.S. of the above equation to be negligible
compared to the first one, then the bandwidth can be expressed as
4) The magnitude line spectra of the general FSK/FH signal represented
by (2.26) is considered with B = 2 for the following cases:
a) Slow frequency hopping which is defined in this analysis as
> T For T - 2~,, we have the case in which -ch m . h - and
from (2.47) and (2.48) we get
The spectrum for this case of slow hopping is shown in
i) Fig. 2.13 for the special case Awl > (Aw+BW1+BW2). The bandwidth of this FSK/FH is given by B BW = (2 -1) x 2Awl x 20w + 2BW1 + 2BW2
= (zB-1) x 2Awl + 2BW2 + FSK bandwidth (2.50)
For large values of B and Awl, the above equation would
reduce to
ii) Fig. 2.14 for the special case where Awl < (Aw+BWl+BW2) and Aw < 2. Here, the spectrum would look like noise, but 'm we still would be able to detect our transmitted data, because
the spectrum centered at each carrier frequency would occur
at a different time from the others. b) Fast frequency hopping in which T~ < For T~ = ~~/4,we have
and from (2.52) and (2.53) we get
The spectrum for this case of fast frequency hopping is shown
in Fig. 2.15 for the special case Awl > (Aw+BWl+BW2). For large
values of B and Awl, the bandwidth is also given by
It is to be noted that the complete spectrum for the above slow
hopping and fast hopping cases would appear after T and 2Th h seconds respectively.
5) The bandwidth of the BPSK/FH whose magnitude line spectrum of (2.38)
is shown in Fig. 2.19 for the special case T = 2% and Awl > (BW1+BW2), h is given by
Again for large values of B, (2.56) would reduce to Chapter 3
PERFORMANCE OF THE FH SYSTEM
3.1 Introduction
Noise reduction is probably the most important single consideration in transmission of signals in noisy and/or hostile channels. This can be a major factor in the system design and performance. In the case of digital transmission,the noise can result in mistaken digits, and the performance of the system is evaluated in terms of probability of bit error.
In this chapter we will show how the spread spectrum technique
(in our case frequency hopping) is used to protect the system from deliberate interference (jamming). The probability of error is evalu- ated for the BFSK/FH and BDPSK/FH in the presence of two very common jamming models, namely:
a) partial-band noise jamming model, and
b) partial-band multitone jamming model.
Then the worst case jamming strategy is determined for each case.
3.2 Jamming Models
While many possible jammer models can be proposed, only the two most common models are considered in this work, and these are:
3.2. a Partial -Band Noise Jamming Model
By definition,noise jamming consists of filtered white Gaussian noise with the filtering restricting the noise to some or all the RF band. It will be assumed that the jammer has a total power J which is
3 9 40 uniformly spread across a fraction a of the total RF band-width W, then the jammer appears as an additive Gaussian noise with one-sided power spectral density given by
The partial band jamming model is shown in Fig. 3.1.
3.2.b Partial-BandMultitoneJammingModel
blultitone jamming consists of a series of equal amplitude tones coincident with the center frequencies of the frequency hopping channels. If the jammer evenly divides his total power J among q tones, then the power in each tone would be J/q. If the spacing between any two successive tones is denoted by Rc, then the fraction of band which is jammed is given by:
The multitone jamming model is shown in Fig. 3.2.
In the following probability of error's analysis the white Gaussian noise background is neglected since its contribution to error is small compared with the deliberate interference, and in general, could be considered as part of that system.
Probability of Error Calculations in the Presence of Partial-
Band Noise Jamming
3.3. a Detection of Non-Coherent Binary Frequency Shift Keying in
Frequency Hopping Environment (2-ary FSK-FH)
It was mentioned before that the partial-band noise jamming Fig. 3.1 Partial band noise jamming model
Fig. 3.2 Partial band multitone jamming model appears as an additive Gaussian noise with one-sided power spectral density N = J/aW. Therefore the probability of error for the non- J coherent FSK/FH reception is obtained directly from the results obtained by Schwartz [3] for the ordinary non-coherent FSK, by replacing n 0 (one-sided power spectral density of Gaussian noise) by our N as done J by Sam Houston [g] and Marvin Simon [16] . Thus the bit probability of error for the non-coherent FSK/FH is given by:
where
E = bit energy
a = fraction of the total RF band which is jammed
NJ = one-sided power spectral density
= J/aW
E/N J can be expressed as follows:
where
X = E/J/W = bit energy to jam noise density ratio.
Now (3.3) can be written as
To show the effect of changing the RF bandwidth (W) on the system
performance, let a = 1.0 for simplicity, then we get We can see that by increasing W, the effective bit energy to jam noise ratio would increase. Fig. 3.3 shows the bit probability of error for two different RF-bandwidths with a = 1.
3.3.b Worst Case Jamming Strategy Against Non-Coherent FSK/FH
The purpose of the jammer is to degrade the system performance by increasing the probability of bit error. Since we are assuming that the jammer has an available power J, the only parameter left to him to vary is a, which is restricted to the range 0 < o -< 1. The bit probability of error was given by (3.5) as
multiplying both sides by X and substituting for ax, E/NJ we get 1 1 - 7 (E/NJ) X P, (X) = (E/NJ) e
Fig. 3.4 is a plot of the product X PB(X) versus (E/NJ). This curve
(E/N ) = = has a maximum value of 0.3679 at J max 2 amax X
a = 2/X .. max
This value is optimum only if it is less than unity (i.e. X > 2).
Using (3.8) in (3.5) we get PB (X) for X > 2. For X -< 2 we set max ci = 1 in (3.5) to get P (X) . Now for the non-coherent FSK/FH, the ,max probability of bit error corresponding to the worst case partial-band jamming can be written as: X (db) Fig. 3.3 Bit probability of error versus bit energy to jam noise ratio for the non-coherent FSK/FH in the presence of partial-band noise with a = 1.0
This is shown in Fig. 3.5. The optimum (which causes maximum bit error) jammed fraction of the total RF bandwidth W is expressed as follows :
u (X) = max
This is shown in Fig. 3.6.
3.3.c Detection of Coherent Frequency Shift Keying in Frequency
Hopping Environment (Coherent FSK-FH)
At the receiving end of the FSK/FH system, a FH-signal identical to the spreading one applied at the transmitter, is used to de-spread the incoming signal, and a conventional coherent FSK detection is used to decode the transmitted data. The bit probability of error for the ordinary coherent FSK is given by Schwartz [3] as
1 PB = Z erfc where
E = bit energy
= one-sided power spectral density of Gaussian noise n0
Since the partial-band noise jamming is considered as an additive
Gaussian noise, then bit probability of error for the coherent FSK/FH may be expressed as: Fig. 3.5 Bit probability of error versus the bit energy to jam noise density in the presence of partial-band noise with a = 1.0 a PB(E/NJ) = 7 erfc where E, NJ and a are defined as before. Equation (3.12) can be also written as:
a pB(x) = ;r erfc @ (3.13)
The bit probability of error is shown in Fig. 3.5 for both the coherent and non-coherent cases, with a = 1.0 for the purpose of comparison.
As before, to find the worst case jamming strategy against coherent FSK/FH, we multiply both sides of (3.9) by X to obtain
= 1.425 as This product has a maximum value of 0.16574 at (E/N J ) rnax shown in Fig. 3.4 from which
=-1.425 a max x
This value is optimum (in the sense of giving maximum bit error) if it is less than unity (i.e. X > 1.425). Using (3.15) in (3.13), the result would be PB (X) for X > 1.425. For X -< 1.425 we set a = 1.0 max in (3.13) . Omitting the details, the final form of %ax (X) and PB (X) rnax (worst case) for the coherent FSK/FH case would be given by:
and PB (XI= rnax ( 0.5 erfc X -< 1.425
Fig. 3.6 shows a X rnax (X) versus for the coherent and non-coherent FSK/FH cases, while Fig. 3.7 shows PB (X) versus X for both cases. max
3.3.d Binary Differential Phase Shift Keying in Frequency Hopping
Environment (BDPSK/FH)
The probability of bit error for the ordinary BDPSK in which the change in the carrier phase carries the information (i.e. a one dictates a change in the phase of the transmitted signal, and a zero dictates no change), is given by Seymour Stein [6] as:
where
n = one-sided power spectral density of Gaussian noise. 0 Now to determine an expression for the probability of error for the
BDPSK/FH in the presence of partial-band noise jamming, the same agru- ment as that used in the case of FSK/FH is used and we get:
where 2, E, and N J are defined as before. Equation (3.19) can be also written as amax (XI
Fig. 3.6 The jammed fraction of the RF bandwidth VS the bit energy to jam noise density ratio in the presence of partial band noise (worst case) FSK-FH (Non-Coherent) /
FSK-FH (Coherent)
X [dB)
Fig. 3.7 Max bit probability of error VS the bit energy to jam noise density in the presence of partial-band noise (worst-case) The bit probability of error for the BDPSK/FH which corresponds to the worst case partial-band jamming strategy may be found as before using Fig. 3 and equation (3-20). Omitting the details ci&X) and
PB (X) can be finally written as: max
a (X) = rnax
and
0.18394/X X > 1.0
PB (XI = (3.22) max 1 -X 7 e X -< 1.0
Figs. 3- 9 and 3- 6 show P (X) versus X(dB) and gaxagainst X(dB) Bmax respectively for both cases FSK/FH and BDPSK/FH for the purpose of comparison.
3.4 Calculations of the Probability of Error in the Presence of
Partial-Band Multitone Jamming
In this sectioq the second type of interference which is the partial-band multitone jamming is considered. The multitone jammer is assumed to have perfect knowledge of the system operation except for the frequency hopping code sequence, and that he has an available power J. The best strategy for him is to distribute his power equally among q contiguous tones and vary this number to maximize the probability of error of the system [g]
Fig. 3.9 Bit probability of error versus the bit energy to jam noise in the presence of partial-band noise jamming (worst case) 3.4.a 2-ary Ncn-Coherent FSK/FH with Jam Tone Spacing Equal to
the Bit Rate
1 Let Rc = - denote the bit rate, then the total number of T- frequency bins in the spread spectrum bandwidth (W) is given by
The probability that the jammer will hit a keyed tone is
where q = number of jamming tones.
The bit probability of error for the M-ary non-coherent FSK/FH was given by Sam Houston [9] as
For the 2-ary case we set M = 2 in the above equation to get
This equation can be expressed as
ap* (XI To determine the value of q which maximizes PB(X) we set = 0, aq to obtain
=- W "ma, 2Rc It is usually more convenient to express P in terms of the ratio h of the signal power (S) to the jamming power per tone (J/q), and let this ratio be denoted by a, i.e.
Now (3.24) can be written as
Substituting (3 -30) in (3.26) we get
aPB(x) We now maximize PB(X) with respect to a, setting aa = 0, to obtain
- X amax - 7
This value of a is optimum (gives maximum probability of error) if it is less than unity (i.e. X < 2). Using (3.32) in (3.31) we get
(X) for X < 2. For X -> 2 we set a = 1 in (3.31). Omitting the P~max details amm (X) and PB (X) can be expressed as : max and
PB (XI = max
Fig. 3.10 shows the worst-case performance represented by (3.34).
3.4.b 2-ary FSK/FH with Jam Tone Spacing Equals to Twice the Bit
Rate
In this strategy,the multitone jammer seeks to hit only one of the two transmitted symbols. The bit probability of error in this case was given by Sam Houston [9] as
The bit probability of error (worst case) can be found to be:
1 - X > 2.0 X - P (X) = Bmax 0.5 X < 2.0
This is shown in Fig. 3.10.
3.4.c Probability of Error of the BDPSK/FH with Jam Tone Spacing
Eaual to the Bit Rate
The bit probability of error (worst-case) for the BDPSK/FH with tone spacing = Rc was given by Sam Houston [9] as
PB (XI = max a Bmax 1.0 - - ,FSK/FH (Jam tone spacing = 2Rc) 0 /' ,' - - -- I' . -r---,\ 4 /- \ \ FSK/FH (Jam tone spacing = Rc) - \ \ -
0.1 - - . -P - BDPSK/FH (Jam tone spacing = Rc)
0.01 .- - -0 - - \ - \ \
I a I I 0.001 x I 1 * - -4 - 0 4 8 12 16 20 X (dB)
Fig. 3.10 Bit probability of error versus the bit energy to jam noise density in the presence of partial-band multitone jamming (worst case). (XI
-
,BDPSK/FH (jam tone spacing = Rc)
Non-Coherent FSK/FH (Jam tone spacing = R ) C
Son-Coherent FSK/FH (partial-band nolse) .
Coherent FSK/FH (partial -band noise) *
- 4 0 3 8 12 16 2 0 X(dB)
Fig. 3.11 Bit probability of error versus the bit energy to jam noise density (worst case). 60 This is shown in Fig. 3.10.
Fig. 3.11 shows the performance (worst-case) of the different types of modulation considered, in the presence of the partial-band noise jamming and partial -band multitone jamming for the purpose of comparison.
3.5 Conclusions
In this chapter, we have seen how the FH technique could be used to protect the communication system against the intentional inter- ference introduced to the system.
Following the previous analysis, we reach to the following conclusions:
1) Fig. 3.3 tells us that the bit probability of error can be
minimized by increasing the total RF bandwidth of the trans-
mitted spread spectrum signal.
2) In the presence of partial-band noise jamming we notice: i a) From Figs. 3.5 and 3.7, we can see that the non-coherent FSK/FH system requires more signal power for the same prob-
ability of error (i.e. there is a penalty paid for not
maintaining phase coherence in the non-coherent case).
b) From Fig. 3.6, it is seen that toobtain optimum results (from the point of view of the jammer), the total RF-bandwidth
(W) should be jammed for small values of X (bit energy to jam
noise density ratio), while for larger values of X, jamming
a fraction of W would be more effective.
(c) Referring to Fig. 3.5, one observes that when the total RF band-
width is j ammed (i.e. a = 1) and when X is small (X < 3 dB) , the coherent FSK/FH is the most effective technique against
the partial-band noise jamming. For larger values of X
(X > 3 dB), the BDPSK/FH system is the most effective one.
d) Considering the worst-case jamming strategy (in which the
bit probability of error is maxirmun), the most effective
system against the partial-band noise jamming is the coherent
FSK/FH followed by BDPSK/FH. This is shown clearly in Fig. 3.9.
3) Fig. 3.10 shows that in the presence of partial -band
noise jamming (worst-case) and when the values of X are small
(X < 3 dB), the most effective system against the tone jamming
is the FSK/FH (jam tone spacing = Rc), while for larger values
of X (X > 3 dB), the most effective system is the BDPSK/FH (jam
tone spacing = Rc)
4) Fig. 3.11 shows that
a) The most effective interference (higher bit error) against
our frequency hopping system is the partial-band multitone
j amrning with the jam tone spacing = 2 Rc and the non-coherent
FSK is considered.
b) The least effective interference (lower bit error) against our
system is the partial-band noise jamming when the coherent
FSK is considered.
5) Although the worst case bit probability of error in our cases is - 1 relatively high, (approximately 10 - for X(0 - 20 dB),
it is considered good and reasonable, because without the spread
spectrum technique which we are using, the probability of error
would be much higher and we would certainly lose our transmitted
data. Chapter 4
REAL TIME SIMULATION
4.1 Introduction
Most of the concepts of spread spectrum have been known for many years, but the components and techniques for implementing reliable
systems have only been available recently. The primary reason for this
is that only recently has the technology in integrated circuitry area come to the point of making small, high speed and reliable electronic components available at a reasonable cost.
Fig. 4.1 shows the block diagram of the spread spectrum system
implemented in this investigation. The input data is a square wave and the modulation assumed is frequency shift keying (FSK). The spread
spectrum technique used is frequency hopping. It can be seen that the same signal is used for spectrum spreading at the transmitting
side, and for spectrum de-spreading at the receiving end to assure fre- quency and phase coherence. A coherent FSK demodulator is finally used
to recover the transmitted data. The design and details of this system
is discussed in the next section followed by conclusions.
4.2 Design and Description of the Coherent FSK/FH Modem
A FSK/FH Modemwas implemented and Fig. 4-2a,b shows the circuit
diagram of this system. The input data to the system (Fig. 4-3a) is a
stream of square wave pulses produced by the first I.C. (XR-2206). The
pulse width and the duty cycle can be adjusted by the choice of R1 and
R2, and are given by: -
Pulse Iloub 1e - Coherent FSK not~b1e - BP na t a Data u ';- Buffer a = Balanced =I Buffer Buffer -L * t:SK Gen . In ' Mod. PBalanced- Filter Out Mixer Yixer Den~od. u u u u i - u u u J, i i
1 "equency Synth. .
A
Code Gen.
i
Fig. 4.1 Block diagram of the implemerited coherent FSK/FIi Modem.
2 1 (-) = baseband frequency (4.1) % = R1+R2
L Duty Cycle = - R1+R2
In this work a 50% duty cycle was used. The output pulses of the first stage is input to I.C. (XR-2206), which is the FSK generator.
To this modulator, two timing resistors R3 and R4 are connected to pins 7 and 8 respectively. Depending on the voltage level of the input pulses at pin 9, either one or the other of these timing resistors is activated. A high level voltage selects the mark frequency and is given by:
where
and a low voltage level selects the space frequency and is given by:
where
Thus it can be seen that the mark and space frequencies can be indepen- dently adjusted by the choice of the timing resistors. Potentiometers
R6 and R should be adjusted is for 7 for minimum harmonic distortion (R 6 sineshaping while R7 is for symmetry adjustment). The output waveform at pin 2 is a sinusoidal FSK and it has a continuous phase during the
frequency transition between f and Its magnitude is controlled by S %. potentiometer R,. 3 A frequency counter can be used to measure the mark and space frequencies separately as follows:
(a) When pin 9 of the FSK modulator is disconnected or connected to a positive voltage, a sinusoidal waveform with frequency equal to
% is obtained at the output (pin 2) and may be measured.
(b) When pin 9 is grounded, the frequency of the output sinusoidal
waveform at pin 2, would be fM, and may be measured using the
frequency counter.
The first two 741 operational amplifiers shown in Fig. 4.2a,b acts as buffers, one between the FSK modulator and the transmitter's mixer, while the other one between the transmitter and receiver's mixer. Each mixer is a doubly balanced one, and employs SL-640C. Pin 2 of each mixer must be decoupled to earth via a capacitor which represents the lower possible impedance at both the carrier and signal frequencies.
At the transmitter's mixer the sinusoidal FSK signal (fed to pin 7) is mixed with the frequency hopping signal (fed to pin 3 of the mixer).
The output signal (obtained at pin 5) contains the sum and difference frequencies of the two mixed signals. At the receiver's mixer the incoming signal (fed to pin 3) is mixed with the same frequency hopping signal that was used at the transmitter's side to assure phase coherence.
Rg and Rg of the transmitter's mixer are the carrier and signal null potentiometers. It is necessary to adjust these controls alternately.
First with the carrier but no FSK signal potentiometer Rg is adjusted for minimum output. Conversely, with the FSK signal and no carrier, potentiometer Rg is set for minimum leakage to the output. The same is said about potentiometers R10 and RI1 of the receiver's mixer. 6 8
The frequency hopping signal, which is used at both the transmitter and receiver, is provided by the HP-3330B Automatic Synthesizer which could be programmed manually by using the front panel keyboards, or remotely by using a seven-bit parallel ASCII code. The remote control is accomplished by the HP-9835A computer system. An interactive BASIC program "SWEEP" was written and when executed, gives commands to the synthesizer to generate the required hopped frequencies. The de-spread signal which is taken from pin 5 of the receiver's mixer, is fed to the receiver's band pass filter (BPF). Two Burr-Brown UAF-31 1.C.s are used to implement a two stages, 4-pole Butterworth BPF. The center frequency of this filter is chosen to fall mid-way between the mark and space frequencies and a low quality factor is chosen to enable the filter to pass both frequencies. The following steps are considered in designing the two stages BPF [12] :
(1) Low pass filter parameters (quality factor Q and normalized natural
frequency fn ) are given by Burr-Brown for different number of poles. Xow, for the required number of poles, fn and Q are selected.
(2) A computer FORTRAN program "ZAFILTER" is used to transform the low
pass design to the equivalent band pass design and the required
component values to be connected to the UAF-31 are calculated.
The input data to this program are Q, fn (selected in step I),
ABP (band pass gain) and Q (quality factor for the BPF). The BP amplitude response for the implemented BPF with center frequency =
1.5 KHz and QBp = 2 is shown in Fig. 4.5. The filtered signal (Fig. 4.7) is fed to the last I.C. (XR-2211) at pin 2. This I.C. is an FSK demodulator which operates on the phase- locked loop (PLL) principle. The following steps are taken in designing the FSK demodulator [13]:
(1) The center frequency of the PLL should be calculated to fall midway
between the mark and space frequencies, i .e.
f is also given by C
where
A suitable value for R12 is chosen, then c4 is calculated using
(4.8).
(2) The tracking range (2 Aft) which is the range of frequencies over
which the PLL can retain lock with a swept input signal is calculated
using
It is also given by
Af, - -R12fc R13
Using (4.10) and (4.11), RI3 is determined.
(3) The PLL damping factor (p) determines the amount of overshoot,
undershoot orringing present in the phase-locked loop's response to a step change in frequency and is given by:
For most Modem application p is set equal to 0.5, then
Knowing c (for step 1) c5 can be determined. 4 (4) The FSK output filter time constant (T~)is given by
0.3 TF=R c = 146 BaudRate
Choosing a suitable value for RI4, and knowing the baud rate, c 6 can be calculated.
(5) c7 can be determined using the formula
16 --- 16 C7("Ff 2 Capture range in HZ 2*fc
where the capture range (+ Afc) is the range of frequencies over
which the phase locked loop can acquire lock. In most Modem appli-
cations
Afc = (80% - 99%) Aft (4.16)
Finally the output data are obtained at pin 7 of the FSK demodulator,
Table 4.1 shows the integrated circuits and the component values used for the design of the Modem. Table 4.1 7 1 The Integrated Circuits and Components Values Used for the Design of the FSK/FH Modem
I.C. .WFACTURER FUNCTION COMWNENTS VALUES COMMENTS
XR-2206 Exar Pulse Generator Rl = 25 kR (Poten.) A 1.2 kilobaud data rate with R2 = 25 kli (Poten.i 50% duty cycle is generated. C1 = 0.022 uF
XR-2206 Exar FSK R3 = 25 kQ Mark frequency = 2 KHz. Sinusoidal R4 = 58 kR Space frequency = 1 KHz. Modulator These frequencies may be measure1 R5 = 25 kR (Poten.) using a frequency counter. used to control the ampli- R6 = 25 kQ (Poten.) R 5 is tude of the FSK signal. R7 = 1 kR (Poten.) R used for sine shaping. I 6 is 1 i R7 is used for symmetry adjust. 74 1 Archer Buffer Between the FSK modulator and the transmitter's mixer.
SL-640C Plessey Transmitter's R8 = 10 kn (Poten.) R and R are the carrier and 8 9 Semi- doubly Rg = 10 kc (Poten.) signal null potentiometers. conductors balanced mixer
74 1 Archer Buffer Buffer between the transmitter's and receiver's mixer. I SL-640C Plessey Receiver's R10 = 25 kR (Poten.) R10 and Rll are the carrier and Semi- doub 1y Rll * 25 kj2 (Poten.) signal null potentiometers. conductors I mixer 1 741 i Archer Buffer between the receiver's
mixer and the BPF. j 1 I Buffer i UAF-31 / ~urr-~rown i BPF RF1 = 89 kn This filter is n two stages, j four pole Butterworth BPF 1 RF2 = 89 kn with center frequency = 1.5 KHz I R = 250 kc and quality factor = 2. i Q1 Computer program "ZAFILTER" is RG1 = 27 ki?. i used for the design of this RPl:. RF3 = 127 kc i RF4 = 127 kQ R = 370 kg Q2 I RG2 = 39 kc XR-221 Exar Coherent RI2 = 25 kR PLL center frequency = 1.5 KHz Track Bandwidth = 2 KHz C4 = 27 nF ~~~odulator Capture Bandwidth = 1.6 Mz 1 R13 = 39 kc Loop Damping Factor = 0.5 C5 = 7 nF R14 = 102 kc? C6 = 2.5 nF R15 = 470 kQ C7 = 10 nF I 72 4.3 Results and Comments
The input signals to our coherent FSK/FH modem, which are the spreading FH signal coming from the synthesizer and the transmitted data
(stream of binary pulses) are shown in Fig. 4.3. The binary pulses are frequency modulated and the output of the modulator which is shown in
Fig. 4.4, is mixed with the FH signal to form the FSK/FH signal. This signal is shown in Fig. 4.6. At the receiver's side, the received signal is multiplied with the same FH signal as that used at the trans- mitter's side. Fig. 4.7 shows the de-spread signal after being passed through the BPF whose amplitude response is shown in Fig. 4.5. The output of the BPF is FSK de-modulated and the output of the demodulator is shown in Fig. 4.8.
A frequency spectrum analyzer was used to obtain the frequency spectrum for each of the waveforms of interest. Fig. 4.9 shows the frequency spectrum for the FH signal with a center frequency fo =
150 KHz, number of output frequencies M = 11, frequency increment Afi =
5 KHz and hopping interval rh = 30 msec, while Fig. 4.10 shows the spectrum for the FSK signal with the mark frequency fM = 2 KHz, the space frequency f = 1 KHz and the binary pulse width rm = 15 msec. S When the above two signals are mixed together, we get the FSK/FH signal
(slow hopping case) and the spectrum of this signal is shown in Fig. 4.12 from which the bandwidth is measured and found to be BW = 54 KHz.
Recalling that the BW for FSK/FH signal was given as
BW = (number of output frequencies - 1.O) x hopping step
size + bandwidth of the FSK signal + 2 BW2 where
BW2 = bandwidth of the frequency hopping baseband signal
In our case BW2 is small and can be neglected because T is large. Then h the bandwidth is given by
BW -'L (M-1) x Afi + bandwidth of the FSK signal
= (11-1) x 5 KHz + 4 KHz = 54 KHz which agrees with the measured bandwidth.
Fig. 4.17 shows the frequency spectrum for the above FSK/FH signal with Afi = 0.5 KHz. The spectrum looks like noise, but our transmitted data still could be detected as expected before. Fig. 4.13 shows the spectrum for the de-spread signal after being passed through the BPF.
The mark frequency component (2 KHz) is shown to be smaller than the space frequency (1 KHz) because the BPF is not exactly symmetrical about the center frequency (1.5 KHz) as shown in Fig. 4.5. Fig. 4.15 shows the frequency spectrum for the FSK/FH signal when using a FH signal with fo = 500 KHz, M = 101 and Afi = 5 KHz. The bandwidth of this signal is measured and found to be
BW = 500 KHz
Again, recalling that for large number of frequencies from the synthesizer, the bandwidth for the FSK/FH signal was given by
BW -a (M-1) x Afi
Applying this to our case we get
BW = (101-1) x 5 KHz = 500 KHz Fig. 4.3 Input waveforms tothe FSK/FH modem.
a) Input data (a stream of square wave pulses with pulse width rm = 15 msec and duty cycle = 50%).
b) The spreading frequency hopping signal with a center frequency = 150 KHz, number of output frequencies = 11 and hopping stcp size = 5 KHz. Fig. 4.4 Output waveforms from the FSK modulator. \ [Vertical scale = 15 mV/cm] ' (a) The mark frequency (fM = 2 KHz).
i
(b) The space frequency (fS = 1 KHz).
i
(c) The FSK signal.
Fig. 4.6 The complete transmitted FSK/FH signal waveform (out- put of the transmitter's mixer).
Fig. 4.7 Output waveform of the receiver's BPF.
Fig. 4.8 Output data (output of the FSK demodulator1 . Fig. 4.9 Frequency spectrum for the F'H signal coming from the frequency synthesizer with a center frequency = 150 KHq, number of out- put frequencies = 11, hopping.step size = 5 KHz and time/ hopping step (r = 30 msec). [Horizontal scale= 10 KHz/cm. h vertical scale = 2 mV/cm.]
Fig. 4.10 Frequency spectrum for the FSK signal with the mark frequency fM = 2 KHz, space frequency fS = 1 KHz and pulse width T~ = 15 msec. [Horizontal scale = 500 Hz/cm, vertical scale = 1 mV/cm.] Fig. 4.11 Frequency spectrum for the frequency hopping signal of Fig. 4.9, a) when mixed with the mark frequency only (sum and difference freqs are shown), b) when mixed with the space frequency only. [Horizontal scale = 10 KHzjcm, vertical space = 2 mV/cm.] Fig. 4.12 Frequency spectrum for the FSK/FH signal (output from the transmitterfs mixer) with a center frequency = 150 KHz, number of output frequencies = 11, hopping step size (Awi) = 5 KHz, T~ = 30 msec and rm = 15 msec (slow frequency hopping) .
Fig. 4.13 Frequency spectrum for the de-spread signal after being passed through the receiver's BPF. Fig. 4.14 Frequency spectrum for the FH signal coming from the synthesizer with a center frequency = 500 KHz, number of output frequencies; 101, hopping step size = 5 KHz and time/step = 30 msec. [Vertical scale = 2 mV/cm, horizontal scale = 100 KHz/cm.]
Fig. 4.15 Frequency spectrum for FH signal (whose spectrum is shown in Fig. 4.14) when mixed with FSK signal with T~ = 15 msec. Fig. 4.16 Portion of the spectrum of the signal whose spectrum is shown in Fig. 4.15. [Horizontal scale is set to 10 KHz/cm.]
Fig. 4.17 Frequency spectrum for the FSK/FH signal with a center frequency = 150 KHz, number of output frequencies = 11, hopping step size = 0.5 KHz, rh = 30 msec and -rm = 15 msec. [Vertical scale = 2 mV/cm, horizontal scale = 10 KHz/cm.] Fig. 4.18 Frequency spectm for the FSK/FH signal with a center frequency = 150 KHz, number of output frequencies = 11, hopping step size = 5 KHz and rh = T*/Z = 30 msec (fast frequency hopping) . a) after t < 3 Th, b) after t > 3 Th . 84 which is the same as the measured bandwidth. Finally, Fig. 4. shows the spectrum for the FSK/FH signal with fo = 150 KHz, M = 11, Afi = 5 KHz and rh = r,/2 = 30 msec (fast hopping case).
4.4 Summary and Conclusions
A spread spectrum coherent FSK/FH modem was successfully implemented and Fig. 4.2a,b shows the circuit diagram of this system from which we notice:
1. The input data to the system is a stream of square wave pulses.
The pulse width and the duty cycle can be adjusted by the choice of R1 and R2. These pulses are modulated and the modulation assumed is fre- quency shift keying (FSK). The mark and space frequencies can be independently adjusted by the choice of the timing resistors R3 and R4.
2. The spectrum of the FSK signal is spread using the frequency hopping signal (FH) coming from the frequency synthesizer. The center
frequency, number of output frequencies, frequency increment and the hopping interval for the FH signal can be adjusted by using front panel or remote programming.
3. The FH signal is mixed with the FSK signal to form the FSK/FH
signal. The bandwidth of this signal is measured for the following
two cases:
a) small number of output frequencies (M = 11) and found to be
BW -Q (M=1) x Afi + 2 FSK bandwidth
b) large number of output frequencies (M = 101) and found to be
BW -^. (bl-1) x Afi
The above two results agree with our theoretical results obtained earlier
in chapter two. 4. At the receiving end of the modem, spectrum de-spreading is accomplished by correlating the received signal with the same frequency hopping signal as that used for spectrum spreading at the transmitter's side, to assure frequency and phase coherence. In the same process in which the desired signal is de-spread any undesired incoming signal is spread by being multiplied with the same FH signal. Therefore, by passing the resulting signals through the BPF, which is designed to pass the mark and space frequencies, any undesired signal would be rejected.
Finally, a coherent FSK demodulator is used to recover the transmitted data. Chapter 5
SUMMARY AND CONCLUSIONS
A new analytical technique is developed which provides the communications engineer a means for investigating various trade-offs in spread spectrum signals and their effects on performance. The techniques developed are general and can be used to investigate both slow and fast hopping. Performance characteristics are determined and verified.
Generally, there is no restriction on the choice of information modulation; however, in this work two types of modulations were con- sidered, namely: i) FSK and ii) BPSK. The generalized transmitted
FSK/FH and BPSK/FH signals were represented by (2.9) and (2.34) respectively. Starting with the above two equations and using Fourier analysis, two general formulas to find the frequency spectrum were derived. These are (2.26) for the FSK/FH and (2.38) for the BPSK/FH.
Using a code sequence such that the frequency stepped one increment at a time, the magnitude line spectra was shown for the following cases: a) slow frequency hopping (which is considered here as the case when rh > rm), b) fast frequency hopping (T~< T,), and c) different number of output frequencies. It was found that when using a large number of output frequencies from the frequency synthesizer, which is usually the case, the bandwidth (BW) required for transmission of the complete frequency hopping signal would be approximated by
BW -% (number of output frequencies - 1) x hopping step size 87
In Chapter Three, two common jamming models were introduced to the system and these are partial-band noise jamming and partial-band multitone jamming. For each model, the performance of the BFSK/FH and
DBPSK/FH is presented and the maximum probability of error corresponding to the worst case jamming strategy (from the point of view of the system designer) is determined. Fig. 3.11 shows the system performance
(worst case) for the different types of modulation considered from which one observes that the coherent FSK/FH system is the most effective (lower probability of error) against the partial-band noise jamming while the
BDPSK/FH is the most effective against the multitone jamming.
Finally, a coherent FSK/FH modem was implemented and Fig. 4.2a,b shows the circuit diagram of this system. The input data to the modem is a stream of square wave pulses and the modulation assumed is FSK.
It is to be noted that at the receiving end of the modem, spectrum de-spreading is accomplished by multiplying the received signal with the same frequency hopping signal that was used for spectrum spreading at the transmitter's side to assure frequency and phase coherence.
The transmitted data is finally recovered using a coherent FSK demodu- lator. A frequency spectrum analyzer was used to obtain the frequency
spectrum of the FSK/FH signal for different cases and it was found that the bandwidth required for transmission is a function of the number of frequency hops and the hopping step size which agrees with the results obtained in Chapter Two. REFERENCES REFERENCES
1. R.C. Dixon, "Spread Spectrum Systems," Wiley Interscience Publication, 1976.
2. Joseph E. Essman and Paul R. Blasche, "WPAFB O.U. Communication Simulator," Vol. 1, June, 1980.
3. Robert C. Dixon, !'Spread Spectrum Techni,ques," IEEE Press, 1976.
4. Mischa Schwartz, Information Transmission, Modulation and Noise, Third Edition, McGraw-Hill Book Company, 1980.
5. Ferrel G. Stremler, Introduction to Communication Systems, Second Printing, Addison-Wesley Publishing Company, 1979.
6. Mischa Schwartz, William R. Bennett and Seymour Stein, Communication Systems and Techniques, McGraw-Hill Book Company, 1966. 7. Heinz H. Schriber, "Self-Noise of Frequency Hopping SignalsYM-IEEE Transactions on Communication Technology, October 1969. 8. William F. Utlant, "Spectrum Principles and Possible Applications to Spectrum Utilization and Allocati~n,~IEEE Communication Society Magazine, Sept. 1978.
9. Sam W. Houston, "Modulation Techniques for Communication, Part I: Tone and Noise Jaming Performance of Spread Spectrum M-ary FSK and 2,4-ary DPSK Waveforms," Proceedings of the IEEE National Aeorspace and Electronics Conference, Dayton, Ohio, June 10-12, 1975, pp. 51-58.
10. Marvin K. Simon, !?ThePerformance of M-ary, FH-DPSK in the Presence of Partial-Band Multitone Jamming,ll IEEE Transaction on Communications, May 1982, Special Issue on Spread Spectrum Communication.
11. Edward M. Noll, Linear IC Principles, Experiments and Projects, Howard W. Sams and Company, Inc., 1974. 12. Burr-Brown Research Coruoration General Catalog. 1981.
13. Exar Complete Data Books, 1981.
14. The Radio Amateur's Handbook, 1982, (fifty-ninth edition).
15. "Plessy semiconductor^,^^ Linear IC Handbook, May, 1982.
16. Marvin K. Simon, "Differential Coherent Detection of QASK for Frequency
Hopping Systems, Part 11. Performance in the Presence of Jamming,"
IEEE Transaction on Communications, January 1982. BIBLIOGRAPHY BIBLIOGRAPHY
Digital Methods Synthesize Frequency Electronic Design, May 23, 1968.
Golomb, Baumert, Easterling, Stiffler and Viterbi, Digital Communica- tions with Space Applications. Prentice-Hall, Inc.: Englewood Cliffs, N.J.
Hekimiam, N.C. Digital Frequency Synthesizers. Washington, D.C.: Page Communications Engineers, Inc.
Huth, G.K. Detailed Frequency-Hopper Analysis. Magnavox Technical Library.
IEEE Transaction on Communication. Special Issue on Spread Spectrum Communications. August, 1977.
Kendall Webster Sessions. I.C. Schematic Source Master. John Wiley and Sons.
Noordanus, J. Frequency Synthesizers: A Survey of Techniques. IEEE Transactions on Communication Technology, April, 1969.
Nossen, Edward J. Fast Frequency Hopping Synthesizer. Camden, New Jersey: RCA/Communications System Division.
Pawula, R.F. and R.F. Mathis. A Spread Spectrum System with Frequency Hopping and Sequentially Balanced Modulation: Parts I E 11. IEEE Transaction on Communication, May, 1980.
Powers, Thomas R. The Master Handbook of I.C. Circuits. Tab Books, Inc., 1982.
Renschler, Ed and Brent Welling. An Integrated Circuit Phase-Locked Loop Digital Frequency Synthesizer. Motorola Semiconductor Products, Inc.
Schmidt H. and P.L. McAdam. "Anti-Jam Performance of Spread Spectrum Coded System." Proceedings of IEEE National Aerospace and Electronics Conference, Dayton, Ohio, June 10-12, 1975.
Simon, Marvin K. and Gaylord K. Huth. Differentially Coherent Detection of QASK for Frequency Hopping System. Part I: Perfor- mance in the Presence of a Gaussian Noise Environment. Part 11: Performance in the Presence of Jamming. IEEE Transaction on Communication, Vol. Com. 30, No. 1, January, 1982.
Viterbi, Andrew J. Principles of Coherent Communications. EIcGravi-Hill Book Company. Appendix A
Computer Listings
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FHFOOOLi FHF00Q7 FHF0009 2ATA P1/3.1459/ FHFil00Q B IS THE RAT13 OF THE 2IJLSE ,41074 T3 TYE PERIOD SF THE FSK FhFCQLC C IS THE 3PTIi3 9F THE PULSE iiIOTH r13 TIJE PER133 :IF THE FP FHF 00 1 ! !3=1o3/2..3 FHFOC 12 WRITf(6rl3) FHF0013 0 FilRVAT(6Xr 'C03E 'JOQD LENGTY'rBXt'N'tlbX I'Y*~ZLX~*ZC') FHF0014 CO 300 N=lr5 FHFGO 15 C=1.'3/FLOsT( Z:=::N) FHF0016 03 200 I=lr3L FHFC017 Y=I-lS F HFQO 1F! k..V XP=FLO.AT(F) FHFOOL3 IF(KP~E3~5.0)GO T3 LO FHF002'Y Zp=C<:(Sfhd( tp::p[:::C))/ (XP%PI<:C) FHFOO2 1 GO T0 30 FHFOOZZ Z P=C FHFOC23 DO LC9 J=1~31 F HFC1024 K=J-16 FHF0025 XU=FLUAT(K) FHFOD25 IF(XM .EQ. 9.0) ~3 TCI 40 FHFOOZ? ZC=PI~ZP~~S(SIN(XY%PI*~))/ (x~"!*PI:FD) FYFOOZ!! GO TO 53 F!4FOQ2? :O z;=p 1::: zp3': FHF003Q 50 dRITE(br60) fitr>q*KrZG cHF0031 30 FO2YAT(4X~i6rL5Y~2(Ibrl2X)~F15*6) FHFil03Z .OO CONTINUE FHF0033 100 CCI?:Tl?.iUE FHF0034 300 C3XTThUE FHF0035 STOP FHF0035 E NC FHF3C37 FILE: ALI FORTRAN A OHIO UNIVERSITY DEPABTYENT OF ELECTBICAL ENC C *+***************$***t,*t***t*t,*t**#*****************************************A~Iooc C NAflEl ZBHAHI NAId ALXOO( C DATE! JULY 27T3, 1982 ALIOOC C PROGBAII FOR CALCULATIHG ALPA-HAX- FOR TEE PSK-FH CASE IN THE ALIOOC C PRESENCE OF PARTIAL-BAND YOISE JA8flING ALIOOC C *****t*****$*************~*******ALIOOC C BLIOO~ #RITE (6,10) ALIOOC 10 PORffRT(13X,rX*, 15X, 'X (DB) ', 19X,*ALPA-SAX, *) ALIOOC DO 200 K=1,150 ALI001 h=0,25+0,25*FLOAT fK- 1) BLIOO 7 B= IO,OO*ALOG 10 (A) ALIOO ! I? {A, GT, 1,425) GO TO 20 ALIOOJ ALFA=1.0 ALI 00 l GO TO 30 ALIOO 1 20 ALFA=1,425/A ALIOO 1 30 WaITE (6,40) A,B, ALFB ALIOO f 40 Fc)RHAT(5Xp2(F12-5,9X) ,SX,El5-5) ALIOOS 200 CONTINUE hLIOOl S T3 P ALIO02 END ALI 002 PILE: ALI POBTBB?I A OHIO dNI7EBSfTY DIEAST3EHT OP 2LECT2XCXL SYC C **********************************************************************~~~~~c C BAMEJ ZBBAiiI NAI3 ALIOOf C DATE1 JULY 27T3, 1982 ALIOOr C PROGBAY FOR CALZULATIYG BUA-5BX- FOR TEE FSK-PH CASE IN THE ALIOOf C PRESEYCE OF PAaTIBL-BAND YOISE JAdflI3G ALI 00 C C **********************$************************************** &LxgOC C hLf OOC BRITE (6.10) ALIOO C 10 PORffAT(13X,'Xg,15X,'X (OB)*,193,tALPA-3AX-8) ALIOOC DO 200 K=l,ISO BLI0O7 A=0.25+O125*PLOAT [K- 1) ALTO07 B=?O.O0*8LOG10 (A) BLXOO : IF(A,GT, 1,425) GO TO 20 ALIOO1 BLFA=I * O ALIOO 1 GO TO 30 ALIOO 7 20 ALFB=?,42f/B PLIOO 7 30 UilITE(6.40) A.3,ALFA BLIOO 1 40 P~BMAT(~X,~(P?~,~,~X).5S,E15-5) ALf 00 7 200 COITIYUE ALIOOf S TC3P ALI002 END ALI002 *****6****t*t*ftC***ktrPf*tIrb-L*,Z****f************4********* BIrf)qQ1q NAME] 34'iA;'T NAIM FEGRPD 31T0002C ??OGDA:: T(? CXLCUIAX ?T4? bIT T'R6B.131LTT'1' OF PARqS ?OR T.19 FSS/FH SIT??"3C CASE IN "BESENC? 017 PABTIIL-92NG ?6313? JIM'l.IlJi; (BC3ST CAST) 3IT3C0-1" ****ta~*t~**********#*t*9~*44~It****~~t*~**31~30'35~ 21roo3~ir URITF: (6,13) RIT?337C 0 FO~Y~lT{11X,'X',13X,yX {QB) ' ,13~~' ?aT) 2iTi333511 DO 200 I=1,1-50 STT30r)9P X=3.25*7T OAT (I) 13~~oolcr Y=IO*ALOG10 (4) 3IT0011" rn L.C (XrGTrZ.i2) GiJ Ti> 23 3ITCC 12: PI3=3.5*EY 2 (-Y/2,3) 31T00 133 GO TO 30 3IT0313f 3 P3=3. 3679,'Y 3;30 15( 3 WcITE (6,401 X,K, i?B 3TT00 16: (3 PCR,'IRT(5X,2 (F?2-5,4'X) ,515-7) 3,Tr)n77{1 1 COYTTITUE 3~~gf~la.' STOP STT*?O?';il E t: n l3TPO9'>Oi :,::;:c<.\:<.z:A.: ,:A 8- ' .x,;xq~'?~i:g.6,,:6:K-i.,:z: <..-:.,<.; x*,,:k.x,*z:F* *****.&*++;$+*** **.,! ***=.:;:+**** .:{, j >,3 , ? . '.. ~l'.-l,. ..1:.1. ' , ,. '"I:'\ <) - A ril L J?. .' -: 1 7 j:.Y 1 .I ' i .:.I '.!\):j ,) j ,J ... 1.I ' : . 4-2, 1- >.--.\,,,-. 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FILE: ILIAN FORTRAN A OHIO UNIVEBSITY DBPABPUEIT OP ELECTRICAL Ell C **+****+*+****~**~~********~****~~**~~**IHAQO C HbHE ZAYAUI Ma18 r nroG C DATE AUG. 16,1982 IHAOC C PROGRAB FOBCALCULATING THE PROBABILITY OF BEBOB FOB THE IPlAOC C BDPSK/PH VITX ALPB=l,O IN THE PRESEICB OF PABTXU-BAHD IHAOC C HOISE JAHMIW- IMAOC ~**************+**t***+r****.++*t***~r***l)i**t************-~rt+ rnaoc c znsot RPITE (6.10) IRAQI 10 PORHAT(t6X.'X',16X,*X (DB) ',13X,'PB(X) ') IHAOI DO 300 K=I,IOO rnaoc X=0,25+0,25*PLOAT (K- 1) IMOC PB=O. 5*EXP (-XI XMAOt Y= 1O,O*ALOG 10 (X) IMAOt VBLTE(6.20) X,P.PB I~AO~ 20 PORHAT (lOX.2 (P12.5,51) ,B15.5) IUllO( 300 COBTIIBE IflAOt STOP z naoc EBD IHAOt FILE: BDPSY FORTRAN R OHIO UNIVERSITY DEPASTUENT OF ELECTRICAL ENG C *****t***************~******************BDPOf)O C NAHEJ ZAUABI MAX?!! EEGRAD BDPOOO C DATE1 APRIL 27TH, 1982 BDPOOO C PROGRAR FOR CALCULATION OF BIT PROBABILITY OF ERROR FOB THE BDP00.3 C BDPSK/FH CASE IW THE PRESENCE OF PARTIAL-BAND JAflMING, BDPOOO C ***********************~*********+**t******BDPOOo n t BDPOOO C CALCULATION OP THE PRODUCT X*PB(X) BDPOOOt BRITE (6,10) BDPOOOi 10 PO%!lAT(13X,'E/NJt ,12X,'X*PBt) BDPOO 11 DO 300 5=1,50 BDP001- Z=0,5*FLOAT (J) BDPOO1; XP=Z/2- O*EXP (-2) BDP00 1 ' URIT5(6,20) Z ,XP B DPOO 1 20 FORYAT (5X,F12,5,5X,El5,7) BDP00 1' 300- COHTfNUE BDP00 16 4 BDPOOl* : CALCULATION OF' PS-MAX (UOST CASE) BDPOOl WRITE (6,30) BDP001 30 FORBAT(IlX,'X'.13X,'X (DB) ' ,14X,'PB (YAX) ') BDP002 DO 200 T=1,160 BDP002 X=0.25*FLOAT (I) BDPOOZ Y=10,0*BLOGlO(X) BDP002 IF (X. GT. 1.0) GO TO 40 BDP002 PB=O. 5*EXP (-X) BDP002 GO 'PO 50 BDP002 4 0 PB=O. 18394/X BDP002 50 URfTX(6,60) X ,Y,PB BDPO02 60 FORPIAT (5X,2 (Pl2.5,4X), 315.7) BDP002 200 CONTINUS BDP003 1- 9DP003 : CALCULATTOR OF ALFA-MAX BDP003 P RITE (6,7 0) BDP003 70 FOEMAT(13X,'Xt, 15X, 'X (DB) '.19X,'ALFA-HhXt) BDPO03 DO 100 K=1,50 BDP003 A=OW5*FLOAT(K) 3DP003 B=lO,O*ALOG10 (A) BDP003 IP(A,GT,1,0) GO TO 90 BDP003 ALFA=1,0 BDP003 GO TO 90 BDPOOY 8 0 ALFA= 1.O/A BDPO34 '30 Si4IfE(6,91) A,B,ALPA BDPOO4 91 . FORflAT(6X,F12,5,6X,F?3~5,7X,F15,7) BDP004 100 CONTINUE BDPOOY STOP BDP004 END BDPOOY * PILE: HABED FORTRAN A OHIO UNIVERSITY DEPARTAENT OF ZLECTQIC4L ENG C ********************************************************************* !J91700(1 C YAME] ZA2AUI NAIH EEGBAD HAHOOD C DATE] MAY 22ND,1982 NAAOOO 2 PBOGRAPl FOR CALCULATION OF BIT PROBABILITY OF ERROR FOR TYE NAHOOO C 2-ABP FSK/FH CASE I8 THE PRESENCE OF P83TIAL-BAND MULTI-TONE NAHOOO C JAHMING (VORST CAS E) , YAHOO3 c ********************************************************************* ??AH000 C NAHOOO c A) JAB TONE SPACING= arr RATE NAHOOO WRITE (6,lO) HAHOO 1 10 ?OREIAT(lIX,'Xr ,l3X,'X (DB) l,14X,'PB(flAX) ') YAHOO? DO 200 I=I,160 YAHOO? X=0,25*FLOAT (I) NAHOO 1 Y=10,O*ALOG10 (X) YAHOO1 IF(X ,GT. 2-01 GO TO 20 NAB001 PB=0,25 YAHOO1 GD TO 30 NAHOOl 20 PB= (1,O/X) *(I. 0-I,O/X) NBHOO 1 -3 0 URITE(6.40) X,Y,PB Y AH09 1 40 FORHAT (5X.2 (F12.6,4X) ,E15.7) NAH002 200 COHTINDE NA3OOZ ?. ?. - YAHOO2 C 3) JAM TONE SPACING=2* (BIT !?ATE) NAH00 2 %RITE (6,50) YAHOO2 50 FOB3AT(13Xf1X',15X,'X (DB) ' ,128,'PS (YAX) ') '?AH002 DO 300 J=1,163 NAHO32 ?=0. 25*f LOAT (J) NXHOO2 3= 10*O*BLOG to (2) V AH 002 IP(Z .GT. 2.0) GO TO 60 NAH002 Pil=O- 5 YhR303 GO TO 70 WAAOD7 60 P~=I,O/Z NBH003 70 URTTE (6,801 Z,W,PH %A8033 80 POR1IAT(5X,2(F12-6.4X) ,E:l5-6) N AH00 7 300 COHTTYUE NAB00 3 STQP NAH003 E ND NAB00 3 ?ILE: WABEEL FOZTRAM X OHIO UTIVSRSITY DEPARTYENT OF ELECTSICAL EYG ****9*****~*t*~***9**~****~**~**~***********3A9393 3 NA?lE1 ZBWAYT YBTZ EEGRAD NAB000 I DATE1 YAP 23RD,1982 NA9000 PROGRAY FOR CALCULATINZU D 9 D DON OF PROBABILITY OF ERROR f09 THE LJABOOF) J i3DPSK/F;S CASE IY THE PRESENCE OF PABTTAL-BAWi) HULTT- YABOOO : TOPE JAfiFIING (V9RST CASS) NAB000 ..- YAB00~ VRITE(6,10) NAB000 10 FOBiYAT(IlX,' DgX- D',13X,,X (DB) 't1YX,,P!3(MAX) ') HABOOO DO 100 1=1,160 NAB001 X=0,25*FLOAT (I) NAB001 Y=lO.O*ALOGlO (X) YBBOOl IF(X ,GT, 2.0) GO TO 20 W ABOO 1 PB=G. 5 Y ABOO 1 GO TO 30 YABOO? 20 P9= 1, O/X XAB00 1 3 0 PRIT3(6,40) X,P,PB 'TABOO 7 40 POH3AT (5X,2(Fl2,6,r)X), E15-7) NAB001 1GO CONTINUX VAB001 STOP HABOO2 END VAB002 z *************++*+*****************************************************~~~~ C NAM9:ZAUBWI Y913 ,EE SZAD ZAFC C DATE:3ETZ'.2OT8,13P2 ZAFC Z PSOGRbY TQ 7ESTGN 1 PGrJR ?OL? 3!rTTER73BTY BAY? PASS FILTER,USINS 'ZAP9 C 'UAP-30*, 4IT!i 2~2.0, CENTER FREQU?ML'Y=?. 5 KBZ,AND A (BP)=10.0 Z A FC Z THIS Fa3GBAY TSXNSFORT3S Log PASS ??I,?POSTTIOY 13TO THE EgOIVALZRT ZAPC C BaND ?RSS P?SIT'ION,THEY CALCULATES THT COflPOHENTS VALUES WIIIZE! ABE ZAPG C C3N"IECTED T3 TYE IJAF-30, ZAfO ~**********t****t********************************************ZIPS C ZAf9 2 BI-1JIJAD COYFfGiJRATTON 15 USED fit THESE: CALCULATrONS ZAFO CCYPLEX P,S, U ZAFO W3ITE(6,4) ZAP0 9 FO9?IAT(SX,"FiF?',13X,*BF2',14X,9RQ ',16Y,*RGq) ZbFO DATA FN, 2, aBP/1.0,0. 7071 1,2.13/ Z AFO Y=F?J*SQ2T(l.3- (1 .3/(Q*2.01 f **2) ZAP0 Y=-FN/ {2*2.3) 2AFO P=CYPLX fX,Y) Z R PD IJ=CONJG (P) ZAP0 ABP=10,0 ZAP0 DO 30 I=1,3 ZAP0 S=?/f?.O*QB?) ZAFO P=S**2-1.0 Z3FO T-AT~N~ AXH HA^ (P) ,~~ALIP)) ZAFO IF(T,CS.O.J 33 TO 10 ZAP0 T=2,3*3. 1459+f Z AFO T=T/2.0 ZAP0 A=SQRT (CABS (2)f *COS (T) ZA Fg3 B=SQRT (CABS (?) ) *SIU (T) ZBfQ S=S+C?!PLX {A,3) ZAPO! F?i=L'Ai3S (S) Z AFQ Q=-FN/ (2. *RE44L (3) ) ZAFf?' FO=1.5 E03+PY ZAFO' FA=Q*PO ZBFOI IF (FA .LT, 10309) GO TO 13 ZXFO( U9ITE (6,12) 1,.9FO( FI~EEIAT(RX, 'SZE dURR-i3ROHY CATALOS FOB Y3RE DETAILS9) ZAFO; G3 TO 63 ZAFO' RPI- t, 592 ooa/ro z AFo L ?F2=R21 ZAPt); ~P=Q ZAROi BQ=QP*RF1 Z\PDT I7 G =l?Q/A ?3 P ZA?Of BRITE(6,2O) RFJ,BF2vRQ,RG ZAFO C F?FYAT(~'X,~[E15.5,1X) ) ZAPOC I? (XIYAG {U) .XQ.3.) GO TO 6? TAFOC P=fJ ZAFQC STOP LAP0 2 9" D ZAFOC ZAF'3C