國 立 交 通 大 學
電信工程學系
碩 士 論 文
應用於無線干擾環境與多波單音干擾下結合時
空編碼技術及跳頻展頻系統之研究
Combined Space-Time Coding with
Frequency-Hopping Spread Spectrum for
Wireless Channels with Multitone Jammers
研究生：沈晏麟
指導教授：王忠炫
應用於無線干擾環境與多波單音干擾下結合時
空編碼技術及跳頻展頻系統之研究
Combined Space-Time Coding with
Frequency-Hopping Spread Spectrum for
Wireless Channels with Multitone Jammers
研究生：沈晏麟 Student: Yan Lin Shen
指導教授：王忠炫 Advisor: Chung-Hsuan Wang
國立交通大學
電信工程學系碩士班
碩士論文
A Thesis
Submitted to Department of Communication Engineering
College of Electrical and Computer Engineering
National Chiao Tung University
in Partial Fulfillment of the Requirements
for the Degree of
Master of Science
in
Communication Engineering
October, 2008
應用於無線干擾環境與多波單音干擾下結合時
空編碼技術及跳頻展頻系統之研究
學生：沈晏麟 指導教授：王忠炫
國立交通大學電信工程學系碩士班
摘要
在無線傳輸的環境中，傳輸的信號常遭受到惡意的干擾源及通道衰減效應， 導致接收訊號產生嚴重的失真。跳頻展頻系統是一般最常用來抑制干擾效應的技 術，而具有分集增益及編碼增益的時空編碼技術可有效的降低通道衰減效應。因 此，在本篇論文裡吾人便結合了兩者之優點，提出了時空編碼結合跳頻展頻技術 以提升傳輸系統在無線干擾環境中之整體效能。 為了能夠專注於分析時空碼的解碼設計，吾人考慮了兩種較簡單的跳頻方 式。第一種定義為所有傳送天線的訊號都跳至相同的頻帶上，稱此為最差跳頻。 第二種情形為所有傳送信號皆設計為避免互相發生碰撞，稱此為最佳跳頻。其中 最差及最佳跳頻方式分別代表為此系統效能分析的上界與下界。針對上述跳頻方 式，吾人推導出在路徑增益已知或未知情況下此系統的最大可能性解碼。吾人亦 針對兩種不同的跳頻系統推導出建立適合的空時編碼的準則。此外，吾人亦針對 此系統提出了在無線干擾環境下好的時空碼準則。最後經由模擬結果驗證出，在 相同的訊雜比及頻寬效益的考量之下，此系統比傳統單進單出編碼效能來的更 佳。Combined Space-Time Coding with
Frequency-Hopping Spread Spectrum for Wireless
Jamming Channels with Multitone Jammers
Student: Yan Lin Shen Advisor: Chung-Hsuan Wang Department of Communication Engineering
National Chiao Tung University
Abstract
In wireless jamming environments, the transmitted signals usually suffer from hos-tile jammers and undesired channel impairments, e.g., multipath fading. Conventionally, frequency-hopping spread spectrum (FHSS) systems are most effective anti-jamming tech-niques, and space-time coding (STC), which introduces temporal and spatial correlation into the transmitted signals to achieve transmitter diversity without sacrificing the bandwidth, has been shown to provide excellent performance against multipath fading. Therefore, in this thesis, we combine STC with the FHSS to construct a powerful high-rate transmission scheme for wireless jamming channels.
Two cases of FH are considered here to simplify the design of STC. One is the worst-case frequency hopping which hops the symbols from all transmitter antennas into the same frequency band, and the other is the perfect frequency hopping which avoids any possible collision of the transmitted symbols. The actual performance of the combined STC/FHSS system with arbitrary hopping patterns can then be upper and lower bounded by the evalu-ated performance of the worst case and perfect case, respectively. The maximum likelihood decoding of space-time codes is derived with respect to different reception conditions, and the design criteria for constructing good space-time codes with respect to two kinds of FH are also derived. Verified by the simulation results, the proposed system can provide bet-ter performance than the conventional schemes in bet-terms of both bandwidth efficiency and signal-to-noise ratio.
誌謝
時間過的非常的快，回顧這兩年的研究生活，真是讓我人生多了很多歷練， 首先要感謝指導老師王忠炫教授，這段時間在研究上給予細心的指導與耐心教 誨，令我在研究上及做事的態度上成長許多、獲益匪淺，非常感謝您！也要感謝 口試委員：翁詠祿教授與翁芳標教授在口試期間給予我許多指導與建議。除此之 外，還要感謝實驗室的同學們：大師兄、力仁學長，感謝你們在我研究遇到難題 時，總是很熱心的給予我意見與幫助。同屆的一哥、老菜和小白，因為有你們的 陪伴，彼此支持鼓勵，兩年來讓我在遇到困難的時候不會覺得孤獨。學弟郭胖、 阿標、和白兔，很開心能夠和你們相處，一同聊天打屁，真的是一群活潑的學弟 妹！還有謝老師實驗室的強哥、宏益、小湯、duck、施施、振偉及冠亨，常常一 起去吃飯聊天，紓解壓力，很謝謝你們的陪伴。最後，要感謝姿靜和我的家人們， 總是默默的陪伴，在我心情低落時給予最大的鼓勵與支持。因為有你們的陪伴與 支持，我的研究所生活多了很多色彩，感謝你們！
Contents
Chinese Abstract I English Abstract II Appreciation III Contents IV List of figures VList of tables VII
1
Introduction
1
2
Overview of Frequency-Hopping Spread Spectrum Systems
and Jamming Environments
3
2.1
FHSS System
. . . 32.2
Jamming Environments
. . . 42.2.1
Broaband Noise Jammer
. . . 42.2.2
Partial-Band Noise Jammer
. . . 52.2.3
Multitone Noise Jammer
. . . 83
Review of Space-Time Coding
11
3.1STC System Model
. . . 113.2
Encoder Structure and Maximum Likelihood Decoding for STTC
12 3.3Design Criteria for Constructing Good Space-Time Codes
. . . . 134
Design of Space-Time Coding with FHSS Technique in
4.1
STC/FHSS System Model
. . . 184.2
STC Combined with the Worst Case Frequency Hopping
. . . 214.2.1
Decoing with CSI and JSI Available
. . . 214.2.2
Decoing with CSI but without JSI
. . . 224.2.3
Decoing with JSI but without CSI
. . . 234.2.4
Decoing without JSI and CSI
. . . 254.2.5
Design Criteria for Constructing Good Space-Time Codes
26 4.3STC Combined with the Optimum Case Frequency Hopping
. . 284.3.1
Decoing with CSI and JSI Available
. . . 284.3.2
Decoing with CSI but without JSI
. . . 294.3.3
Decoidng with JSI but without CSI
. . . 314.3.4
Design Criteria for Constructing Good Space-Time Codes
32 4.4Design Criteria for Constructing Good Space-Time Codes with
FSK Modulation
. . . 354.5
Simulation Result
. . . 375
Conclusion
46
A
Derivation of the ML Decoding of STC/WFHSS System
without JSI
48
B
Derivation of the ML Decoding of STC/WFHSS System with
JSI but without CSI
50
C
Derivation of the Design Criteria of STC/WFHSS System
57
D
Derivation of the ML Decoding of STC/OFHSS System
with-out JSI
60
E
Derivation of the ML Decoding of STC/OFHSS System with
JSI but without CSI
63
F
Derivation of the Design Criteria of STC/OFHSS System
66
List of Figures
2.1 FH/MFSK system model. . . 4
2.2 Power spectral density of broadband noise jammer. . . 5
2.3 Power spectral density of partial-band noise jammer. . . 7
2.4 Performance of FH/MFSK system in partial-band noise jamming environment. 7 2.5 Power spectral density of multitone noise jammer. . . 8
2.6 Band multitone noise jammer and independent multitone noise jammer strate-gies. . . 9
2.7 Performance of FH/MFSK in several different jamming environment. . . 10
3.1 STC system model. . . 11
3.2 The encoder of STTC system for two transmitter antennas. . . 13
4.1 The proposed STC/FHSS system. . . 17
4.2 The STC/WFHSS system model. . . 18
4.3 The STC/OFHSS system model. . . 20
4.4 Performance plots of STC/WFHSS with CSI and JSI available for Eb/N0 = 5dB. . . 39
4.5 Performance plots of STC/WFHSS with CSI and JSI available for Eb/N0 = 5dB. . . 40
4.6 Performance plots of STC/WFHSS with JSI available but without CSI avail-able for Eb/N0 = 10dB. . . 40
4.7 Performance plots of STC/WFHSS with JSI available but without CSI avail-able for Eb/N0 = 10dB. . . 41
4.8 Performance plots of STC/WFHSS with CSI available but without JSI avail-able for Eb/N0 = 15dB. . . 41
4.9 Performance plots of STC/WFHSS with CSI available but without JSI avail-able for Eb/N0 = 15dB. . . 42
4.10 Performance of STC/WFHSS with CSI and JSI available and STC/WFHSS with CSI available but without JSI available for Eb/N0 = 12dB and µ = 1. . 42
4.11 Performance of STC/WFHSS with CSI and JSI available and STC/WFHSS with CSI available but without JSI available for Eb/N0 = 12dB and µ = 0.3. 43
4.12 Performance of STC/WFHSS, original STC, and CC/Alamouti systems for Eb/N0 = 15dB and µ = 0.1. . . 43
4.13 Performance of STC/WFHSS, original STC, and CC/Alamouti systems for Eb/N0 = 15dB and µ = 0.4. . . 44
4.14 Performance of STC/WFHSS, original STC, and CC/Alamouti systems for Eb/N0 = 15dB and µ = 0.7. . . 44
4.15 Performance of STC/WFHSS, original STC, and CC/Alamouti systems for Eb/N0 = 15dB and µ = 1. . . 45
List of Tables
4.1 Optimal Space-time codes of the STC/WFHSS system with 4FSK and 2 transmitter antennas for wireless jamming channels. . . 28 4.2 Optimal Space-time codes of the STC/OFHSS system with 4FSK and 2
trans-mitter antennas for wireless jamming channels. . . 35 4.3 Optimal Space-time codes of the STC/FSK system with 4FSK and 2
Chapter 1
Introduction
Wireless communication systems have been used for a long time and undergone a notable development. Wireless communication technology is moving towards higher mobility and higher data rates. A communication system emploied in wireless channel consists of three main components: transmitter, receiver, and channel. In general, the signals are transmitted through a wireless channel by using electromagnetic wave forms from the transmitter to the receiver. The singals arrived the receiver from different directions with different delays, that causes the variations in the amplitude and the phase of the composite received signals. That phenomenon is called multipath fading. The fading channel might bring significant degradation in the performance of a communication system. In addition, the received signals are also distorted by channel impairments and the intentional or unintentional interference signals, such as, thermal noise and the signals transmitted from other users. We can regard pratial band noise jammer as the unintentional interference, and regard multitone noise jammer as intentional interference. The thermal noise is caused by the random motion of the electrons in conductors at the receiver. These factors make the transmitted signals distort seriously.
Frequency-hopping spread spectrum (FHSS) systems are typically used to against the jammers in wireless channel environments [1]. The M -ary frequency-shift-keying modulation is usually utilized with the FHSS system. The M FSK signals are hopped with a pseudo-random sequence, and the pseudo-pseudo-random sequence is used to select a set of the carrier frequency. Therefore, the signals are pseudo-randomly hopped over the total bandwidth, and the jammer can not generate the same pseudo-random numbers and frequency hopping bands which are emploied by the FHSS system. That reduces the effect of the jammers. FHSS systems usually combine with ordinary singal-input and singal-output channel codes [2]-[8]. The performance analyses of combining the Reed-Solomon(RS) code scheme and the fast frequecy hopping spread spectrum system are shown in [2]. In [7][8], convolutional
codes (CC) are combined with the noise-normalized method in FHSS systems to improve the system performance. However, the overall performance is not as satisfactory as the performance with the fading effect considered.
The design of channel codes for providing high date rate and high quality of communi-cations over fading channels using multiple transmitter antennas have been investigated in recent years. Tarohk, Seshadri, and Calderbank et al [9][10], first proposed the space-time coding (STC) scheme, which is an effective way to to make the system data rate closer to the capacity of multiple-input and multiple-output wireless channels. The STC scheme introduces a temproal and spatial correlation into the transmitted signals by using multiple antennas and has been shown to provide excellent performance against multipath fading. It can achieve the transmit diversity as well as a coding gain without sacrificing the bandwidth. Generally, the FHSS system is the most effective anti-jamming communication techniques, and the STC scheme can minimize the effects of multipath fading. Therefore, we combine the FHSS system with the STC scheme to construct a power transmission scheme which can mitgate the effect of the multipath fading and the jamming interferences.
An overview of the FHSS system and the jamming environments are given in Chapter 2. The STC schemes are introduced in Chapter 3, and the design criteria for STC system with M FSK modulation over fading channel is also shown in Chapter 3. The STC/FHSS systems which combine STC scheme with FHSS systems are proposed in Chapter 4. In Chapter 4, we focuse on two kinds of FHSS systems. One is the worst case frequency hopping spread spectrum system which hops the signals from all transmitter antennas into the same M -ary band, and the other is the optimal case frequency hopping spread spectrum system which hops the signals from any transmitter antennas into different M -ary band. The ML decoding schemes with respect to both two STC/FHSS systems are also presented. The design criteria for constructing good space-time codes and the simulation results are alse given in this chapter. The conclusions for this thesis are in Chapter 5.
Chapter 2
Overview of Frequency-Hopping Spread Spectrum Systems and
Jamming Environments
In wireless channels, sometimes the signals we transmitted are interfered by the jammers [1][11]. FHSS system is one of the most effective anti-jamming communication techniques. In this chapter, we will discribe FHSS system and the jamming environment.
2.1
FHSS System
Spread spectrum techinques are usually used for anti-jamming [11][12]. For spread spectrum systems, the bit singal-to-jammer noise ratio is define as
Eb
NJ
= WsS RbJ
(2.1) where Ws is the total spread spectrum signal bandwidth, S is the singal power, Rb is the
data rate for bit per second, Eb = S/Rb is the energy per bit, J is the jamming power, and
NJ = J/Ws is the signal-sided jammer noise power spectral density. We can also define the
processing gain (PG)
P G = Ws Rb
. (2.2)
The bit signal-to-jammer noise ratio represented in decibels (dB) is Eb NJ (dB) = (P G)(dB)− J S(dB) (2.3) where JS is the jammer-to-signal power ratio, and we can find that when PG is increasing, the value of Eb/NJ also increases. Figure 2.1 shows the block diagram of the uncoded FH
system with MFSK modulation. The binary data are fed into the MFSK modulator, then the modulated signal is hopped pseudo-randomly over the total system bandwidth Wsunder
Figure 2.1: FH/MFSK system model.
periodically, such that the jammers do not konw where to jam. FHSS systems are classified into slow frequency hopping (SFH) and fast frequency hopping (FFH) [12]. Hop rate Rh
of the FFH system is an multiple of the MFSK symbol rates Rs, and the SFH hops serval
symbols each time. Each symbol of FFH system is hopped into serval chips, and each chip is transmitted in distinct M -ary band. Symbol can be demodulated after all the chips of this symbol is being collected and dehopped. Every symbol of SFH system is hopped into only one chip, and each chip is also transmitted in distinct M -ary band. The complexity of receiver of the FFH system is much higher than the receiver of the SFH system.
2.2
Jamming Environments
There are a lot of jamming waveforms that could distort the transmitted signal. A class of jamming waveforms are selected to illustrate in this section, such as broadband noise jammer, partial-band noise jammer, and multitone noise jammer[1][11].
2.2.1
Broaband Noise Jammer
A broadband noise jammer spreads it’s total power J over the frequency range of the system bandwidth Ws. The broadband noise jammer can be regard as the additive white
Gaussian nosie (AWGN) channel with zero mean shows in Figure 2.2, but the one-sided noise power spectral density (PSD) is
NJ =
J Ws
. (2.4)
A slow frequency hopping with noncoherent MFSK modulation system is used in AWGN channel without any jammers, and the bit error probability is
Ps = 1 Mexp − Es 2N0 M X q=2 M q (−1)qexp Es(2 − q) 2N0q (2.5)
Figure 2.2: Power spectral density of broadband noise jammer.
where N0 is the one-sided PSD of AWGN and Es is the energy per symbol. When a symbol
error occurs, the error probability can be regarded as the probability of choosing any other M − 1 orthogonal symbols. Then the number of bit errors corresponding to a symbol error is 1 M − 1 1 X i=1 M q i = l2 l−1 M − 1 = M 2(M − 1)l (2.6) where l is the number of bits per symbol. By (2.5) and (2.6), the bit error probability is
Pb Eb N0 = M 2(M − 1) = M 2(M − 1)exp − lEb 2N0 M X q=2 M q (−1)qexp Eb(2 − q) 2N0q . (2.7) In a AWGM channel with power J broadband jammer, the one-sided PSD is replaced by N0+ NJ.Then the bit error probability could be written
Pb = M 2(M − 1)exp − lEb 2(N0 + NJ) M X q=2 M q (−1)qexp Eb(2 − q) 2(N0+ NJ)q (2.8)
and it is defined as Pb(N0E+Nb J). For a special case l = 1, equation (2.8) becomes
Pb = 1 2exp − Eb 2(N0+ NJ) (2.9) when NJ decreases, the performance could be better.
2.2.2
Partial-Band Noise Jammer
The partial-band noise jammers can be regarded as the signals which are transmitted by other users and occupy a fraction of the frequency bandwidth. Power of partial-band
noise jammer is restricted over the frequency range of bandwidth WJ = ρWs, which is a
fraction ρ (0 ≤ ρ ≤ 1) of the total system bandwidth Ws. The power spectral density of
the partial-band nosie jammer is
NJ0 = J WJ
= J ρWS
. (2.10) Assume that the partial-band noise jammer can be regarded as AWGN then the average probability is Pb = (1 − ρ)Pb Eb N0 + ρPb Eb N0+ NJ (2.11) In general, we assume the power of the partial-band noise jammer is much larger than the power of thermal noise, such that NJ is much larger than N0. The bit error probability is
Pb = ρPb( ρEb NJ ) = ρ 2(M − 1) M X q=2 M q (−1)qexp lρEb(1 − q) N0q (2.12)
The worst case partial-band noise jammer chooses ρ to maximize Pb with a given M and Eb
NJ, and the average performance can be expressed as
(Pb)max = max0<ρ≤1 " ρ 2(M − 1) M X q=2 M q (−1)qexp lρEb(1 − q) N0q # (2.13)
Let ρ0 denote the worst case partial-band noise jammer [1][13] and maximize Pb
ρ0 = ( 2 Eb/NJ, for Eb NJ > 2 1, for Eb NJ ≥ 2 (2.14)
From (2.14), the miximum Pb is
Pb max = 0.3679 Eb/NJ, for Eb NJ > 2 1 2exp − Eb 2NJ , for Eb NJ ≥ 2 (2.15)
Figure 2.4 shows the performance curves of an FH/BFSK system in partial-band noise jammer environment with different factors ρ. When Eb/NJ is small, partial-band jammer
with value of ρ = 1 which can be view as broadband noise jammer has the best efficency to interfere with the signal. However, when Eb/NJ exceeds a threshold level, the partial-band
Figure 2.3: Power spectral density of partial-band noise jammer.
Figure 2.5: Power spectral density of multitone noise jammer.
2.2.3
Multitone Noise Jammer
We can consider multitone noise jammer as the signals transmitted from other users, and the frequencies of the carriers are in the range of the system bandwidth. Multitone noise jammer divides its total power into Q tone jammers with equal power and random continuous wave. The waveform of multitone noise jammer is
J (t) = Q X l=1 s 2J Q cos [ω0t + φl] (2.16) where φl is a random varible in (0, 2π] for ∀l. Figure 2. illustrates the PSD of the multitone
noise jammer.
We assume that there is at most one multitone jammer per frequency slot. In general, there are two kinds of multitone jammers. One is band multitone jammer which places n jamming tones in each jammed M -ary band. The fraction of the jammed FH slots is defined as
ρ = Q
M N (2.17) where N is the number of frequency band. The probability of n jamming tones in each jammed M -ary band is
µ = Q/n
N . (2.18) The other one is called independent multitone noise jammer which places Q equal power jamming tones into N M FH frequency slots pseudo-randomly. The jamming noise could be independently hopped over the entire spread-spectrum bandwidth.
Assume the signal power is S, and the fraction of signal power to the power of each jamming tone is
α = S
Figure 2.6: Band multitone noise jammer and independent multitone noise jammer strate-gies.
When the data symbol is not jammed and any of the other slots in this M -ary band is hit by a jamming tone, an error will occur if α < 1. In contrast, no error will occur if α > 1. Therefore, choosing Q appropariately could determine the worst case of α and seriously degrade the performance of FH/MFSK systems [1][14]. For slow frequency hopping, the bandwidth of a M -ary band is
Wb = M Rs =
M Rb
log2M = M Rb
k (2.20) where Rb is the bit rate and Rs = Rb/ log2 is the symbol rate. Then the probability of a
M -ary band being jamming is
µ = Q W/Wb
. (2.21)
By (2.4), (2.19), (2.21), and Eb = S/Rb, we can reweite µ as following form
µ = αM nkEb/NJ
. (2.22)
When the data symbol is not hit and any other frequency slots in this M -ary band are jammed, the symbol error probability for α < 1 is
Ps = µ
M − 1
Figure 2.7: Performance of FH/MFSK in several different jamming environment. The relation between Ps and Pb is
Ps=
M
2(M − 1)Ps, (2.24) so the bit error probability is
Pb = M 2(M − 1)µ M − 1 M = αM 2nkEb/NJ . (2.25)
We can make the system achieve the worst case performance by adjusting α, and we restrict the number of jamming tones to be smaller than the number of M -ary bands. The worst case band multitone jammer sets αwc to be
αwc= ( kE b M NJ, for Eb NJ < M k 1, for Eb NJ ≤ M k (2.26) Figure 2.7 shows that the partial-band and multitone jammers are both significantly more effective than the broadband noise jammer to against the FH/MFSK system. And the n = 1 band multitone is the most effective to against the FH/MFSK system.
Chapter 3
Review of Space-Time Coding
The multiple transmitter antennas system can be used to increase the transmitted data rate and against the multipath fading [15]. Tarohk, Seshadri, and Calderbank et al proposed the space-time coding scheme in 1998. Space-time coding scheme is an effective way to make the system data rate closer to the capacity of multiple-input and multiple-output wireless channels [9]. Temporal and spatial correlation are introduced into transmitted signals to achieve transmit diversity and coding gain without sacrificing system bandwith. This chapter introduces the encoding scheme, the decoding scheme, and the design criteria over fading channels of space-time coding system.
3.1
STC System Model
A space-time coding system with n transmitter antennas and m receiver antennas is shown in Figure 3.1. First, the information bits fed into the space-time encoder. After encoding, the encoded data is divided into n codeword symbols, and the symbols are passed into the modulator and transmitted by n transmitter antennas. The signal are degraded by multipath fading at the each m receiver antenna. The received signal is a superposition of the signals from n transmitter antennas with noise. Assume the wireless channels are a quasic-static flat fading and memoryless channels. Let Si
t with energy Es be the symbol
which is transmitted by the ith antenna at time t. The received signal rtq of the qth receiver antenna at time t fot all 0 ≤ q ≤ m and 0 ≥ t ≤ L is given by
rqt = n X i=1 αi,qSti+ η q t (3.1)
where αi,q is the fading gain of the multipath from the ith transmitter antenna to qth
receiver antenna and ηtq is the thermal noise of the qth receiver antenna at time t. Assume αi,q is a constant during a frame L of information sequences and vary from one frame to
another. Assume ηqt are independent Gaussian distribution with zero mean and one-sided power spectral density N0 for ∀q and ∀t.
STC systems differ with respect to distinct coding schemes, such as space-time block coding [16][17], space- time trellis coding [18][19], unitary space-time modulation [20][21], time turbo trellis coding [22], differential time coding [23][24], layered space-time coding [25][26], and space-space-time frequency coding [27][28], etc. The following section focuses on space-time trellis coding scheme (STTC).
3.2
Encoder Structure and Maximum Likelihood Decoding for
STTC
Space-time trellis codes are provided by Tarohk, Seshadri, and Calderbank et al. STTC scheme combines the modulation and the trellis coding scheme to transmit data over multiple antennas. The generator sequences of the system are shown in Figure 3.2
(xt1, xt2) = bt−1(1, 1) ⊕4at−1(2, 2) ⊕4bt(2, 1) ⊕4 at(3, 2) (3.2)
where (xt
1, xt2) stand for 2 coded QPSK symbols transmitted through the first antenna and
the second antenna. at and bt represent a pair of input data bits at time t,and ⊕4 is an
operation to take added module 4. For example, assume (at, bt) = (1, 1) and (at−1, bt−1) =
(0, 1) then the output sequence generated by (3.2) at time t is (xt1, xt2) = (2, 0).
Let the received signals r = (rtq ∀q, t), the fading gain α = (αi,q ∀q, t), and the
estimated symbols ˆS = ( ˆSi
t) ∀i, t. Assume α is available at the receiver, and then the ML
decoding is given by f r|α, ˆS = L Y t=1 m Y q=1 f ηqt = rqt − n X i=1
αi,qSti|αi,q, Sti ∀i, q, t
! = L Y t=1 m Y q=1 " 1 √ πN0 exp −|r q t − Pn i=1αi,qS i t| 2 N0 !# . (3.3)
Figure 3.2: The encoder of STTC system for two transmitter antennas. Drop the factors of √1
πN0 and
1
N0 in (3.4), and apply the log-domain metric:
minSˆ L X t=1 m X q=1 rtq− n X i=1 αi,qSti 2 . (3.4)
Use the Viterbi algorithm to select the minimum path metric as the decoding sequence when this ML decoding is used.
3.3
Design Criteria for Constructing Good Space-Time Codes
Conisder the coded communication system with ML decoding shown in (3.5) [29]. A block of transmitted symbols is denoted by
S = Sti| ∀ i, 1 ≤ t ≤ L
(3.5) and an erroneous sequence selected by the decoder is denoted by
ˆ S = ˆSi
t| ∀ i, 1 ≤ t ≤ L
. (3.6)
Assume αi,q is available at the receiver for ∀i, q, and then the pairwise error probability is
PrS → ˆS|αi,q, ∀i, q = Pr L X t=1 m X q=1 rtj− n X i=1 αi,q p EsSti 2 ≥ L X t=1 m X q=1 rtj− n X i=1 αi,q p EsSˆti 2 = L X t=1 m X q=1 2Re ( ηtj n X i=1 αi,q p Es Sti− ˆSi t ) ≥ L X t=1 m X q=1 n X i=1 αi,q p Es sit− ˆSi t 2 = Q r d2S, ˆS Es 2N0 ! (3.7) where d2S, ˆS= L X t=1 m X q=1 n X i=1 αi,q p Es Sti− ˆSi t 2 (3.8) and Q(x) is the complementary error function defined by
Q(x) = √1 2π
Z ∞
x
exp −x2/2 dx. (3.9) Use the Chernoff Bound inequality
Q(x) ≤ 1 2e
−x2/2
(3.10) and then the conditional pairwise error probability can be upper bounded by
PrS → ˆS|αi,q ∀i, q ≤ 1 2exp −d2S, ˆS Es 4N0 (3.11) Assume the fading coeficients αi,q are independent Gaussian random variables with zero
mean and varance 1/2. Let “∗” denote the operator of taking complex conjugate, and H denotes the operator of taking Hermirian, and Ωj = (α1,q, α2,q, . . . , αn,q). Then we can
rewrite equation (3.8) as d2S, ˆS= m X q=1 n X i=1 n X l=1 αi,qα∗i,q L X t=1 sit− ˆsit slt− ˆslt∗ = m X q=1 ΩqB S, ˆS BH S, ˆS ΩHq = m X q=1 ΩqA S, ˆS ΩHq (3.12)
where B S, ˆS = S1 1 − ˆS11 S21− ˆS21 . . . SL1 − ˆSL1 S1 1 − ˆS12 S22− ˆS22 . . . SL2 − ˆSL2 .. . .... .. ... S11− ˆS1n S2n− ˆS2n . . . SLn− ˆSLn
and AS, ˆS = BS, ˆSBHS, ˆS. AS, ˆS is nonnegative definite and Hermitian, and the eigenvalues of A
S, ˆS
are real numbers. Then we have
AS, ˆS= V DVH (3.13) where V = (v1, v2, . . . , vn) is a unitary matrix and D is a diagonal matrix, where vi’s are
the eigenvectors of AS, ˆS. Let λi be the diagonal elements of D, where 1 ≤ i ≤ n, and
ΩqVH = (β1,q, . . . , βn,q) . (3.14)
From (3.13) and (3.14), we can rewrite the equation (3.8) as following d2S, ˆS= m X q=1 n X i=1 λi|βi,q| 2 . (3.15)
Use equation (3.15) to replace d2S, ˆSin (3.11), then we have
PrS → ˆS|αi,q ∀i, q ≤ 1 2exp − Es 4N0 m X q=1 n X i=1 λi|βi,q|2 ! (3.16)
Obviously, all of βi,q are independent complex Gaussian random variables with mean µi,q
and variance 1/2 per dimension. The µi,q is given by
µi,q= E [Ωqvi]
= [α1,q, α2,q, . . . , αn,q]vi (3.17)
where E[ ] denotes the expectation. |βi,q| is a Rician distribution demonstrated by following
probability density function
where I0 represents the zero-order modified Bessel function of the first kind. The pairwise
error probability is derived by averaging |βi,q|, then the pairwise error probability is
PrS → ˆS= Z ∞ 0 · · · Z ∞ 0 PrS → ˆS|αi,q ∀i, q p(α1,1)p(α1,2) . . . p(αn,m)dα1,1 dα1,2. . . dαn,m = Z ∞ 0 · · · Z ∞ 0 PrS → ˆS| |βi,q| ∀i, q p (|β1,1|) p (|β1,2|) . . . p (|βn,m|) d |β1,1| d |β1,2| . . . d |βn,m| ≤ Z ∞ 0 · · · Z ∞ 0 1 2exp − Es 4N0 m X q=1 n X i=1 λi|βi,q|2 ! p (|β1,1|) p (|β1,2|) . . . p (|βn,m|) d |β1,1| d |β1,2| . . . d |βn,m| ≤ 1 2 n Y i=1 1 1 + Es 4N0λi exp −|βi,q| 2 Es 4N0λi 1 + Es 4N0λ !! . (3.19) Assume µi,q = 0, then βi,q become a Rayleigh distribution random variable and the
proba-bility density function is
PrS → ˆS≤ 1 2 n Y i=1 1 1 + Es 4N0λi !m . (3.20)
When SNR is a big number, (3.20) can be expressed as
Pr S → ˆS ≤ 1 2 r Y i=1 λi !−m Es 4N0 −rm . (3.21)
where r is the rank of AS, ˆS. The exponent of SNR term, rm, is called the diversity gain, and the product of eigenvalues is called the coding gain. In order to minimize the error probability, to make the diversity gain and the coding gain as large as posible is necessary. These are the two criteria which are called rank cirteria and determinant criteria.
Chapter 4
Design of Space-Time Coding with FHSS Technique in Wireless
Channels
The transmitted signals are commonly distorted by some intentional or unintentional jamming noise in wireless channels. As disscussed in Chapter 2 and Chapter 3, we know that spread spectrum systems are the most effective anti-jamming communication techniques, and the space-time coding schemes effectively minimize the effects of multipath fading. So, we propose the design schemes combin with space- time coding scheme and the spread spectrum system. Two kinds of FHSS systems are discussed in this chapter, one is the worst case frequency hopping spread spectrum (WFHSS) system which hops the symbols from all transmitter antennas into the same M -ary band. Another is the optimum case frequency hopping spread spectrum (OFHSS) system which hops the symbols from any transmitter antennas into different M -ary band. The two system are called STC/WFHSS and STC/OFHSS systems.
The detailed description of the STC/FHSS system model and the ML decoding are given in this chapter. The criteria for constructing good space-time codes are also proposed. Some simulation results are also presented in the last section.
Figure 4.2: The STC/WFHSS system model.
4.1
STC/FHSS System Model
The STC/FHSS system model is shown in Figure 4.1. There are n transmitter antennas and m receiver antennas. Interleaver is inserted to break bust channel errors to guarantee memoryless channels, and the M FSK modulation is utilized with the FHSS system. The slow frequency hopping with one hop per symbol is assumed for simplicity, and the hopping patterns generated from the transmitter are available to the receiver.
The STC/WFHSS system is shown in Figure 4.2. Let the signal of the qth receiver antenna be rq(t) = n X i=1 Ai,q(t) si(t) + Bq(t) nJ(t) + n (t) (4.1)
where Ai,q(t) is the fading gain of the multipath from the ith transmitter antenna to qth
receiver antenna, Bq(t) is the fading gain of the multipath from the jamming transmitter
to the qth receiver antenna, nJ(t) is the jammer , and n (t) is statistically independent low
pass white Gaussian noise process with one-side spectral density N0. For the slow fading
channel, assume the fading coefficients are the same during a frame L and vary from one to frame another. Due to the system has no perfect synchronization, the received signal which is dehopped and demodulated is composed of cos part and sin part. The cos part of the recived signal of the qth receiver antenna in the kth frequency slot at time t rkR,q,t is
rR,q,tk = Z t+Ts t " M X k0=1 n X i=1 Ai,q r 2 Ts ski,t00cos ((ωb+ ωk0) t0+ θ) + xkt0Bq s 2J Q cos ((ωb+ ωk0) t 0 + φq,t) ! + n (t0) # r 2 Ts cos ((ωb+ ωk0) t0) dt0
=
n
X
i=1
(Ai,qcos (θ)) ski,t+ x k tBqcos (φq,t) s 2J Q + n k R,q,t = n X i=1 αR,i,qski,t + x k tBqcos (φq,t) s 2J Q + n k R,q,t = n X i=1 αR,i,qski,t + n k J,R,q,t+ n k R,q,t (4.2)
where ski,t is the symbol transmitted by the ith antenna in kth slot at time t, θ is the phase error of signal, φ is the phase error of the jammer, Ts is the bit interval, Q is the number
of tone jammer, and J is the total jamming power, ωk is the particular carrier frequency
for modulation, ωb is the particular carrier frequency for hopping, and xkt is the jamming
state information (JSI) of the multitone noise jammer (MTNJ) taking value from 1 and 0 with probability M Q/Nt and 1 − M Q/Nt. xkt = 1 means the kth slot of the band which the
signal is transmitted in is jammed at time t. The sin part of the recived signal of the qth receiver antenna in the kth frequency slot at time t rk
I,q,t is rkI,q,t = Z t+Ts t " M X k0=1 n X i=1 Ai,q r 2 Ts ski,t00cos ((ωb + ωk0) t0+ θ) + xkt0Bq s 2J Q cos ((ωb + ωk0) t 0 + φq,t) ! + n (t0) # r 2 Ts sin ((ωb+ ωk0) t0) dt0 = n X i=1
(Ai,qsin (θ)) ski,t+ x k tBqsin (φq,t) s 2J Q + n k I,q,t = n X i=1 αI,i,qski,t+ x k tBqsin (φq,t) s 2J Q + n k I,q,t = n X i=1 αI,i,qski,t+ n k J,I,q,t + n k I,q,t. (4.3)
Therefore the received signal can be expressed as following rkq,t = rR,q,tk + jrI,q,tk = n X i=1 αi,qski,t + x k tn k J,q,t+ n k q,t = n X i=1 αi,qski,t + η k q,t (4.4) The noise ηk
Figure 4.3: The STC/OFHSS system model. (4.4), the MTNJ nkJ,q,t can be written as
nkJ,q,t = nkJ,R,q,t+ jnkJ,I,q,t
= xkt (Bqcos (φq,t) + jBqsin (φq,t))
s 2J
Q. (4.5) In Rayleigh fading channel, Bq is a Rayleigh random variable. Then nkJ,q,t is a complex
Gaussian random variable with zero mean and variance J Ts
Q σ 2
J,q. Assume AWGN nkq,t and
MTNJ nk
J,q,t are independent for ∀ q, t, k. The probability of ηq,tk conditioned on xkt is
f ηq,tk |xkt = r 1 πN0+ xkt2JQTsσJ,q2 exp − ηk q,t N0+ xkt2 J Ts Q σ 2 J,q . (4.6)
The equation (4.6) can be used to derive the likelihood function of the decoding scheme with respect to the STC/WFHSS system.
The STC/OFHSS is shown in Figure 4.3. The received signal rk
i,q,t of the qth receiver
antenna and from the ith transmitter antenna in kth slot at time t is rki,q,t = αi,qski,t+ η
k
i,q,t (4.7)
where
ηki,q,t= nki,q,t+ xki,tnkJ,i,q,t (4.8) where xki,t is the jamming state information (JSI) of the MJNJ taking value from 1 and 0 with probability M Q/Nt and 1 − M Q/Nt, but xki,t and xki,t0 are not independent for t 6= t0.
The probability density function of ηk
i,q,t conditioned on xki,t is
f ηi,q,tk |xki,t = r 1 πN0+ xki,t2 J Ts Q σ 2 J,q exp − ηi,q,tk N0+ xki,t2 J Ts Q σ 2 J,q . (4.9)
The likelihood function of the decoding scheme for the STC/OFHSS system can be derived by the equation (4.9).
Space-time codes achieve the transmit diversity as well as a coding gain. In addition, the signal transmitted by the frequency hopping system avoid the multitone jammers effec-tively. Therefore, the STC/FHSS combines with temporal, frequency, and spatial domain to against the multipath fadding and the multitone jamming interferences. With respect to the two types of STC/FHSS systems, the performance variation is observed for comparison.
4.2
STC Combined with the Worst Case Frequency Hopping
For STC/WFHSS system which is shown in Figure 4.2, the encoded codewords form all transmitter antennas are hopped into the same M -ary band at time t. The received symbols form any receiver antennas are dehopped with the same hopping pattern. Assume the symbols are transmited in the slow fading channel, and the fading coefficients αi,q are
complexe Gaussian random variable with zero mean and variance σ2 i,q.
4.2.1
Decoing with CSI and JSI Available
The ML decoding scheme will be derived in this section. The derived result is shown here for discussion and compairson with respect to the proposed system. Let the received signals r = rkq,t| ∀q, k, 1 ≤ t ≤ L, the jamming state infromation x = xkt| ∀k, 1 ≤ t ≤ L, the fading coefficients α = (αi,q| ∀i, q), and the estimated symbols ˆs = ski,t| ∀i, 1 ≤ t ≤ L.
Assume the fading coefficients αi,q and the jamming state information xkt are available at
the receiver, the likelihood of r given ˆs, x, and α can be express as
f {r|ˆs, x, α} = L Y t=1 M Y k=1 m Y q=1 f ( ηq,tk = rkq,t− n X i=1
αi,qˆski,t|ˆski,t, αi,q, xkt
) = L Y t=1 M Y k=1 m Y q=1 1 r πN0+ xkt2J TQsσ 2 J,q exp − rk q,t− Pn i=1αi,qˆs k i,t 2 N0 + xkt2J TQsσ 2 J,q . (4.10)
We can decode the codeword in ML decoding sense by minimizing the following metric
L X t=1 M X k=1 m X q=1 rk q,t− Pn i=1αi,qsˆ k i,t N0+ xkt2J TQsσ 2 J,q . (4.11)
4.2.2
Decoing with CSI but without JSI
Assume the fading coefficients αi,q are available at the receiver, but the jamming state
information xk
t are not available at the receiver. The likelihood function of r given ˆs and α
can be obtained by averaging (4.10) with respect to x. f (r|ˆs, α) = Ex[f (r|ˆs, x, α)] = Ex L Y t=1 M Y k=1 m Y q=1 1 r π N0+ xkt2J TQsσ 2 J,q exp − rk q,t− Pn i=1αi,qˆs k i,t 2 N0+ xkt2J TQsσ 2 J,q (4.12) where xk
t takes value from 1 and 0 with probability M Q/Nt and 1 − M Q/Nt. xkt are
independent for different t , but are not independent for different k. Assume there are n = 1 band multitone jammers in the channel, the probability density function of xt =
xkt|1 ≤ k ≤ M is Pr x1t = 1, x2t = 0, · · · , xMt = 0 = Q Nt Pr x1t = 0, x2t = 1, · · · , xMt = 0 = Q Nt .. . Pr x1t = 0, x2t = 0, · · · , xMt = 1 = Q Nt Pr x1t = 0, x2t = 0, · · · , xMt = 0 = 1 − M Q Nt . (4.13) After averaging (4.10) with respect to x, the likelihood function of r given ˆs and α is derived in Appendix A and can be express as
f (r|ˆs, α) = L Y t=1 m Y q=1 ( exp " − M X i=1 rkq,t−Pn
i=1αi,qsˆki,t
2 N0 # ( M X i=1 Q Nt 1 √ πa 1 √ πN0 M −1 · exp " (a − N0) rkq,t−Pn i=1αi,qˆs k i,t 2 N0a # + 1 − M Q Nt 1 √ πN0 2)) (4.14)
where a = N0+ 2J TQsσ2J,q. By taking logarithm on that likelihood function, codewords can
be decoded in the ML decoding sense by maximizing the following metric
L X t=1 m X q=1 ( − M X k=1 rk q,t− Pn i=1αi,qˆs k i,t 2 N0 + ln ( M X k=1 Q Nt 1 √ πa 1 √ πN0 M −1 · exp " (N0− a) rq,tk −Pn
i=1αi,qsˆki,t
2 N0a # + 1 −M Q Nt 1 √ πN0 2)) (4.15)
4.2.3
Decoing with JSI but without CSI
Suppose the fading coefficients αi,q are not available at the receiver, and the fading
coefficients are modeled as independent complex Gaussian random variables with zero mean and variance σi,q2 per dimension with respect to Rayleigh fading channels. In order to simplify mathematics, we assume σ2i,q= 1/2 and σJ,q2 = 1/2 for ∀i, q in this section.
Let akt = N0+ xktJ TQs, then the likelihood function f (r|ˆs, α, x) can be rewritten as
f (r|ˆs, α, x) = L Y t=1 M Y k=1 m Y q=1 1 p πak t exp − 1 ak t rq,tk − n X i=1 αi,qsˆki,t 2 = L Y t=1 M Y k=1 m Y q=1 1 p πak t exp ( − L X t=1 M X k=1 m X q=1 1 ak t " rq,tk 2 − 2Re rkq,t n X i=1 αi,qsˆki,t ! + n X i=1 αi,qsˆki,t n X l=1 α∗l,qsˆk∗l,t #) (4.16) and the fading coefficients αi,q can be presented as
αi,q = αR,i,q+ jαI,i,q (4.17)
where αR,i,q and αI,i,q are statistically independent Gaussian random variables with zero
mean and variance σi,q = 1/2. Rewrite Re rkq,t
Pn
i=1αi,qˆski,t and P n
i=1αi,qsˆki,t
Pn l=1α ∗ l,qsˆk∗l,t in (4.16) as Re rq,tk n X i=1 αi,qˆski,t ! = Re rkq,t n X i=1
(αR,i,q+ jαI,i,q) ˆski,t
! = Re rkq,t n X i=1 αR,i,qˆski,t ! + Re rkq,t n X i=1 jαI,i,qˆski,t ! = Re rkq,t n X i=1 αR,i,qˆski,t ! + Im rq,tk n X i=1 αI,i,qˆski,t ! (4.18)
and n X i=1 αi,qsˆki,t n X l=1 αl,q∗ sˆk∗l,t = n X i=1 n X l=1
(αR,i,q+ jαI,i,q) ˆski,t(αR,l,q + jαI,l,q) ˆskl,t
= n X i=1 n X l=1
αR,i,qα∗R,l,q+ αI,i,qαI,l,q ˆski,tsˆ k l,t = n X i=1 α2R,i,q ˆski,t 2 + n X i=1 α2I,i,q ˆski,t 2 + n X i=1 i6=l n X l=1 l6=i αR,i,qαR,l,qˆski,tsˆ k l,t+ n X i=1 i6=l n X l=1 l6=i
αI,i,qαI,l,qsˆki,tsˆ k
l,t. (4.19)
Then (4.15) can be expressed as f (r|ˆs, α, x) = L Y t=1 M Y k=1 m Y q=1 1 p πak t exp ( − L X t=1 M X k=1 m X q=1 1 ak t " rkq,t 2 − Re rkq,t n X i=1 αR,i,qˆski,t ! − Im rq,tk n X i=1 αI,i,qsˆki,t ! + n X i=1 α2R,i,q ˆski,t 2 + n X i=1 α2I,i,q ˆski,t 2 + n X i=1 i6=l n X l=1 l6=i αR,i,qαR,l,qsˆki,tsˆkl,t+ n X i=1 i6=l n X l=1 l6=i
αI,i,qαI,l,qsˆki,tˆskl,t
. (4.20)
We can get the likelihood function of r given ˆs and x by averaging (4.20) with respect to the probability density function of αR,i,q and αI,i,q. f (r|ˆs, x) can be written as
f (r|ˆs, x) = " n Y i=1 m Y q=1 M Y k=1 akt− 1 2 exp − rk q,t 2 ak t !# " m Y q=1 n Y i=1 L X t=1 M X k=1 1 ak t λki,t+ 1 !#−1 · exp m X q=1 n X i=1 PL t=1 PM k=1 1 ak t zk i,q,t 2 +PL t=1 PM k=1 1 ak t wk i,q,t 2 4 PM k=1 Pn i=1 1 ak tλ k i,t+ 1 (4.21) where
zi,q,t=2Re rq,tk sˆk1,t , 2Re rq,tk sˆk2,t , . . . , 2Re rkq,tˆskn,t vki,t
wi,q,t =2Im rkq,tˆs k 1,t , 2Im r k q,tsˆ k 2,t , . . . , 2Im r k q,tsˆ k n,t v k i,t
vk
i,t and λki,t are the eigenvectors and the eigenvalues of the following matrix, respectively:
ˆsk1,t 2 ˆ sk1,tˆsk2,t . . . ˆsk1,tˆskn,t ˆ sk2,tsˆk1,t ˆsk2,t 2 . . . ˆsk2,tˆskn,t .. . ... . .. ... ˆ sk n,tsˆk1,t sˆkn,tsˆk2,t . . . ˆsk n,t 2 . (4.22)
The derivation of (4.21) is in Appendix B. The ML decoding makes the decision by maxi-mizing (4.21). Take logarithm on this likelihood function, then the codewords can also be decoded by maximizing the following metric
m X q=1 n X i=1 PL t=1 PM k=1 1 ak t zi,q,tk 2 +PL t=1 PM k=1 1 ak t wki,q,t 2 4PM k=1 Pn i=1 1 ak t λk i,t+ 1 − m X q=1 n X i=1 L X t=1 M X k=1 1 ak t λki,t + 1 ! (4.23)
4.2.4
Decoing without JSI and CSI
Suppose the fading coefficients αi,q and the jamming state information xkt are not available
at the receiver. f (r|ˆs) = n Y i=1 m Y q=1 M X k0=1 Q Nt L Y t=1 M Y k=1 k6=k0 1 √ N0 exp − rkq,t 2 N0 ! +√1 aexp − rkq,t 2 a ! exp 1 4 L X t M X k=1 k6=k0 1 N0 uki,q,t+ 1 au k0 i,q,t 2 − L X t=1 M X k=1 k6=k0 1 N0 uki,q,t+1 au k0 i,q,t 2 · L X t=1 M X k=1 k6=k0 1 N0 λki,t+ 1 aλ k0 i,t 1 − L X t=1 M X k=1 k6=k0 1 N0 λki,t+1 aλ k0 i,t 1 − M Q Nt " L Y t=1 M Y k=1 1 √ N0 exp − rk q,t 2 N0 !# exp 1 4 L X t=1 M X k=1 1 N0 uki,q,t !2 −1 4 L X t=1 M X k=1 1 N0 uki,q,t !2 L X t=1 M X k=1 1 N0 λk i,t ! 1 − L X t=1 M X k=1 1 N0 λk i,t ! (4.23) where uk
4.2.5
Design Criteria for Constructing Good Space-Time Codes
We propose a design criteria for constructing good space-time codes of the STC/WFHSS system with respect to wireless multitone jamming channels. Let s = sk
i,t|1 ≤ i ≤ n, 1 ≤ t ≤
L, 1 ≤ k ≤ M ) be the codeword sequence transmitted from transmitter, and ˜s = ˜sk
i,t|1 ≤ i ≤
n, 1 ≤ t ≤ L, 1 ≤ k ≤ M ) be the error codeword sequence decided at the receiver. Assume the perfect estimation of αi,q and xkt are available for ∀ i, q, t, k at the receiver. The
con-ditional pairwise error probability that the decoder decides in favor of ˜s than s is given by Pr (s → ˜s|α, x) = Pr L Y t=1 M Y k=1 m Y q=1 1 r πN0+ xktJ TQs exp − rk q,t− Pn i=1αi,qsˆ k i,t 2 N0+ xktJ TQs ≤ L Y t=1 M Y k=1 m Y q=1 1 r πN0+ xktJ TQs exp − rq,tk −Pn
i=1αi,qs˜ki,t
2 N0+ xktJ TQs = Q v u u u t PL t=1 Pm q=1 PM k=1 Pn
i=1αi,q ski,t − ˜ski,t
2 2N0+ xktJ TQs (4.24) where Q(a) is the complementary error function defined by
Q(a) = √1 2π
Z ∞
a
e−x2/2dx. (4.25) Accroding to the inequality Q(a) ≤ 12exp (−a2/2) ∀a ≥ 0, (4.24) can be upper bounded by
P r (s → ˜s|α, x) ≤ 1 2exp − L X t=1 m X q=1 M X k=1 1 bk t n X i=1 αi,q ski,t− ˜s k i,t 2 (4.26) where bkt = 4N0+ xktJ TQs
. By averaging (4.26) with respect to α, the conditional pairwise error probability given x is
P r (s → ˜s|x) ≤ 1 2 n Y i=1 m Y q=1 1 + L X t=1 M X k=1 λk i,t bk t !−1 . (4.27)
where λk
i,t ∀i are the eigenvalues of the matrix Akt, and matrix Akt is expressed as
sk 1,t − ˜sk1,t 2 sk 1,t− ˜sk1,t sk 2,t− ˜sk2,t . . . sk 1,t− ˜sk1,t sk n,t− ˜skn,t sk 2,t− ˜sk2,t sk 1,t− ˜sk1,t sk 2,t− ˜sk2,t 2 . . . sk 2,t− ˜sk2,t sk n,t− ˜skn,t .. . ... . .. ... sk n,t− ˜skn,t sk 1,t− ˜sk1,t sk n,t− ˜skn,t sk 2,t− ˜sk2,t . . . sk n,t− ˜skn,t 2 . (4.28) Then averaging (4.27) with respect to x, the pairwise error probability is approxmated as
P r (s → ˜s) ≤ 1 2 n Y i=1 n Y q=1 1 − L X t−1 1 − Q Nt M X k=1 1 4N0 λki,t+ Q Nt M X k=1 1 4N0+ J TQs λ k i,t . (4.29) From (4.29), we konw that we would construct different good space-time codes with different SNR and SJR. In order to simplify (4.29), assume there are only two transmitter antennas, the pairwise error probability is
P r (s → ˜s) ≤ 1 2 n Y q=1 1 + 1 − Q Nt 1 4N0 + Q Nt 1 4N0+J TQs 2 2 Y i=1 L X t=1 M X k=1 λki,t ! − 1 − Q Nt 1 4N0 + Q Nt 1 4N0+J TQs L X t=1 2 X i=1 M X k=1 λki,t (4.30) Let v1 = 1 − Q Nt 1 4N0 + Q Nt 1 4N0+ J TQs 2 2 Y i=1 L X t=1 M X k=1 λki,t ! v2 = 1 − 1 − Q Nt 1 4N0 + Q Nt 1 4N0+ J TQs L X t=1 2 X i=1 M X k=1 λki,t. (4.31) Assume the multitone jamming power is much larger than the thermal noise power, then N0+ J TQS N0 and 1
N0+J TSQ
1
N0 and we also have following inequality functions
2 Y i=1 L X t=1 M X k=1 λki,t ! ≤ (LM Es)2 L X t=1 2 X i=1 M X k=1 λki,t ≤ 2LM Es. (4.32)
Base on the above inequality functions, v1 and v2 can be upper bounded by v1 ≤ 1 − Q Nt LM 4 SNR 2 v2 ≤ 1 − 1 − Q Nt LM 2 SNR. (4.33) After doing the simulation, we konw that |v1| |v2|. Then good codes could be constructed by maximizing m (s, ˜s) for all possiable s and ˜s, and m (s, ˜s) can be expressed as
m (s, ˜s) = 2 Y i=1 L X t=1 M X k=1 λki,t ! . (4.34) According to the design criteria, good space-time codes are searched by the computer are given in table (4.1)
Table 4.1: Optimal Space-time codes of the STC/WFHSS system with 4FSK and 2 trans-mitter antennas for wireless jamming channels.
Memory Generator Sequences
2 (xt1, xt2) = bt−1(1, 0) ⊕4 at−1(3, 0) ⊕4bt(1, 3) ⊕4at(2, 2) 3 (xt 1, xt2) = at−2(1, 2) ⊕4bt−1(1, 1) ⊕4at−1(1, 0) ⊕4bt(2, 1) ⊕4at(3, 2) 4 (xt 1, xt2) = bt−2(1, 1) ⊕4 at−2(2, 2) ⊕4bt−1(0, 2) ⊕4 at−1(3, 0) ⊕4bt(2, 0) ⊕4at(3, 0)
4.3
STC Combined with the Optimum Case Frequency Hopping
The other system is STC/OFHSS system is shown in Figure4.3. The encoded codewords from any transmitter antennas are hopped into distinct M -ary bands. The ML decoding schemes with respect to STC/OFHSS system are derived in this section, and the criteria of constructing good space-time codes for OFHSS system is also proposed.
4.3.1
Decoing with CSI and JSI Available
The ML decoding of STC/OFHSS system is derived as follow, and the system are assumed to transmit the signal in slow fading channel with n=1 band multitone jam-mers. Let the received signals r = ri,q,tk | ∀q, k, 1 ≤ t ≤ L, the jamming state infromation x = xki,t| ∀i, k, 1 ≤ t ≤ L, the fading coefficients α = (αi,q| ∀i, q), and the estimated
sym-bols ˆs = sk
information xk
t are available at the receiver, the likelihood of r given ˆs, x, and α can be
express as f {r|ˆs, x, α} = L Y t=1 M Y k=1 m Y q=1 n Y i=1
fηi,q,tk = rki,q,t− αi,qsˆki,t|ˆs k i,t, αi,q, xkt = L Y t=1 M Y k=1 m Y q=1 n Y i=1 1 r π N0+ xki,t2J TQsσ 2 J,q exp − rk
i,q,t− αi,qsˆki,t
2 N0+ xki,t2J TQsσ 2 J,q . (4.36)
The codeword can also be decoded in ML decoding sense by minimizing the following metric
L X t=1 M X k=1 m X q=1 n X i=1 rk
i,q,t− αi,qsˆki,t
N0+ xki,t2J TQsσ 2 J,q . (4.37)
4.3.2
Decoing with CSI but without JSI
Assume the fading coefficients αi,q are available at the receiver, but the jamming state
information xki,t are not available at the receiver. The likelihood function of r given ˆs and α can be obtained by averaging (4.36) with respect to x.
f (r|ˆs, α) = Ex[f (r|ˆs, x, α)] = Ex L Y t=1 M Y k=1 m Y q=1 n Y i=1 1 r πN0+ xki,t2J TQsσ 2 J,q exp − rk q,t− αi,qˆski,t 2 N0+ xki,t2J TQsσ 2 J,q . (4.38)
The probability of x with respect to two transmitter antennas is P r (x1,t = (1, 0, . . . , 0), x2,t = (1, 0, . . . , 0)) = φ1 M2 P r (x1,t = (1, 0, . . . , 0), x2,t = (0, 1, . . . , 0)) = φ1 M2 .. . P r (x1,t = (0, 0, . . . , 1), x2,t = (0, 0, . . . , 1)) = φ1 M2 P r (x1,t = (0, 0, . . . , 0), x2,t = (1, 0, . . . , 0)) = φ2 M P r (x1,t = (0, 0, . . . , 0), x2,t = (0, 1, . . . , 0)) = φ2 M
.. . P r (x1,t = (0, 0, . . . , 0), x2,t = (0, 0, . . . , 1)) = φ2 M P r (x1,t = (1, 0, . . . , 0), x2,t = (0, 0, . . . , 0)) = φ3 M P r (x1,t = (0, 1, . . . , 0), x2,t = (0, 0, . . . , 0)) = φ3 M .. . P r (x1,t = (0, 0, . . . , 1), x2,t = (0, 0, . . . , 0)) = φ3 M P r (x1,t = (0, 0, . . . , 0), x2,t = (0, 0, . . . , 0)) = φ4 (4.39)
where xi,t = x1i,t, x2i,t, . . . , xMi,t, φ1 = NtQ/MNtQ−1/M −1, φ2 = NtQ/M
1 − NQ−1 t/M −1 , φ3 = 1 −NQ t/M ·N Q t/M −1, and φ4 = 1 − NQ t/M 1 − N Q t/M −1
. Therefore, a close-form expression of f (r|ˆs, α) with respect to two transmitter antennas is derived in Appendix D
f (r|ˆs, α) = L Y t=1 m Y q=1 ( exp " − n X i=1 M X k=1
rki,q,t− αi,qˆski,t
2 N0 # ( φ1 M2 1 √ πN0 2M −2 1 √ πa 2 · M X k=1 M X k0=1 exp a − N0 aN0 rk1,q,t− α1,qsˆk1,t 2 + r2,q,tk − α2,qsˆk2,t 2 + φ2 M · 1 √ πN0 2M −1 1 √ πa M X k=1 exp a − N0 aN0 rk1,q,t− α1,qsˆk1,t 2 + φ3 M · 1 √ πN0 2M −1 1 √ πa M X k=1 exp a − N0 aN0 r2,q,tk − α2,qsˆk2,t 2 + φ4 1 √ πN0 2M)) (4.40) where a = N0+ 2J TQsσ2J,q. By taking logarithm on that likelihood function, codewords can
be decoded in the ML decoding sense by maximizing the following metric :
L X t=1 m X q=1 ( − n X i=1 M X k=1 rk
i,q,t− αi,qsˆki,t
2 N0 + ln ( φ1 M2 1 √ πN0 2M −2 1 √ πa 2 · M X k=1 M X k0=1 exp a − N0 aN0 rk1,q,t− α1,qsˆk1,t 2 + r2,q,tk − α2,qsˆk2,t 2 + φ2 M · 1 √ πN0 2M −1 1 √ πa M X k=1 exp a − N0 aN0 rk1,q,t− α1,qsˆk1,t 2 + φ3 M
· 1 √ πN0 2M −1 1 √ πa M X k=1 exp a − N0 aN0 r2,q,tk − α2,qsˆk2,t 2 + φ4 · 1 √ πN0 2M)) (4.41)
4.3.3
Decoidng with JSI but without CSI
Suppose the fading coefficients αi,q’s are not available at the receiver, and the fading
coefficients are modeled as independent complex Gaussian random variables with zero mean and variance σ2
i,qper dimension with respect to Rayleigh fading channels. In order to simplify
mathematics, we assume σ2i,q= 1/2 and σJ,q2 = 1/2 for ∀i, q in this section. Let aki,t = N0+ xki,tJ TQs, then (4.36) can be rewritten as
f {r|ˆs, x, α} = L Y t=1 M Y k=1 m Y q=1 n Y i=1 1 q πak i,t exp ( − L X t=1 M X k=1 m X q=1 n X i=1 1 ak i,t h rki,q,t 2 −2Re rk
i,q,tαi,qsˆki,t +
αi,qˆski,t 2io (4.42) and the fading gain αi,q can be presented as
αi,q = αR,i,q+ jαI,i,q
where αR,i,q and αI,i,q are statistically independent Gaussian random variables with zero
mean and variance σi,q= 1/2. Rewrite Re ri,q,tk αi,qsˆki,t and
αi,qˆski,t 2 in (4.42) as Re ri,q,tk αi,qsˆki,t = Re r
k
i,q,t(αR,i,q+ jαI,i,q) ˆski,t
= Re ri,q,tk αR,i,qsˆki,t + Re r
k
i,q,tjαI,i,qsˆki,t
= Re ri,q,tk αR,i,qsˆki,t + Im rki,q,tαI,i,qsˆki,t
(4.43) and αi,qsˆki,t 2
= (αR,i,q+ jαI,i,q) (αR,i,q− jαI,i,q)
ˆski,t
2
= α2R,i,q+ αI,i,q2 ˆski,t
2
The likelihood of r given ˆs and x can be derived by averaging α, and can be expressed as = f {r|ˆs, x} = Z ∞ −∞ f {r|ˆs, x, α} dα = L Y t=1 M Y k=1 m Y q=1 n Y i=1 1 q πak i,t exp − 1 rk i,q,t 2 ! ( m Y q=1 n Y i=1 L X t=1 M X k=1 ˆsk i,t 2 ak i,t + 1 !)−1 exp m X q=1 n X i=1 PL t=1 PM k=1 2 ak i,t Re rki,q,tˆski,t 2 + PL t=1 PM k=1 2 ak i,t Im ri,q,tk sˆki,t 2 4 PL t=1 PM k=1 |ˆsk i,t| 2 ak i,t + 1 . (4.45)
The derivation of (4.45) is in Appendix E. The codewords can also be decoded by minimizing the followind metric:
m X q=1 n X i=1 PL t=1 PM k=1 2 aki,tRe r k i,q,tsˆki,t 2 + PL t=1 PM k=1 2 aki,tIm r k i,q,tsˆki,t 2 4 PL t=1 PM k=1 |ˆsk i,t| 2 ak i,t + 1 − m X q=1 n X i=1 ln ( L X t=1 M X k=1 ˆski,t 2 ak i,t + 1 !) . (4.46)
4.3.4
Design Criteria for Constructing Good Space-Time Codes
We proposed a design criteria for contructing good space-time codes of the STC/OFHSS system with respect to the wireless channels. To evaluate the performance of the ML decoding, two transmitted sequences s = sk
i,t∀i, k, 1 ≤ t ≤ L and ˆs = ˆski,t∀i, k, 1 ≤ t ≤ L.
Assume perfect estimation of αi,q and xki,t are available at the receiver. The conditional
pairwise error probability that the decoder decides in favor of ˆs than s is given by Pr {s → ˆs|x, α} = Pr L X t=1 L X k=1 m X q=1 n X i=1 ln 1 r πN0+ xki,t J Ts Q exp −
rki,q,t− αi,qsˆki,t
2 N0+ xki,t J Ts Q ≤ L X t=1 L X k=1 m X q=1 n X i=1 ln 1 r πN0+ xki,t J Ts Q exp −
rki,q,t− αi,qsˆki,t
2 N0+ xki,t J Ts Q
= Q v u u u t PL t=1 PM k=1 Pm q=1 Pn i=1
αi,q ski,t− ˆski,t
2 2N0 + xki,t J Ts Q . (4.47) Refer to the inequality Q(a) ≤ 12exp (−a2/2) ∀a ≥ 0, the conditional pairwise error
proba-bility (4.47) can be upper bounded by
Pr {s → ˆs|x, α} ≤ 1 2exp − PL t=1 PM k=1 Pm q=1 Pn i=1
αi,q ski,t− ˆski,t
2 4N0+ xki,t J Ts Q . (4.48) By averaging (4.48) with respect to α, the conditional pairwise error probability given x is approximated as Pr {s → ˆs|x} ≤ 1 2 n Y i=1 m Y q=1 1 + L X t=1 M X k=1
aki,t ski,t− ˆski,t
2 !−1 (4.49) where ak i,t = N0+ xki,t J Ts Q −1
. Assume there are only two transmitter antennas, then the conditional pairwise error probability can be approximated as
Pr {s → ˆs|x} ≤ 1 2 n Y i=1 m Y q=1 1 + L X t=1 M X k=1
aki,t ski,t− ˆski,t
2 !−1 = 1 2 m Y q=1 n Y i=1 ∞ X j=0 − L X t=1 M X k=1
aki,t ski,t− ˆski,t
2 !j ∼ = 1 2 m Y q=1 n Y i=1 " 1 − L X t=1 M X k=1
aki,t ski,t − ˆski,t
2
#
(4.50) The probability density function of x for two transmitter antennas is shown in (4.39), then the pairwise error probability is derived by averaging (4.50) with respect to x. The pairwise error probability for two transmitter antennas can be written as
Pr {s → ˆs} ≤ 1 2 m Y q=1 ( 1 − L X t=1 M X k=1 1 − Q Nt 1 N0 sk1,t− ˆsk1,t 2 + Q Nt 1 a sk1,t− ˆsk1,t 2 − L X t=1 M X k=1 1 − Q Nt 1 N0 sk2,t− ˆsk2,t 2 + Q Nt 1 a sk2,t − ˆsk2,t 2 +τ L X t=1 M X k=1 L X t0=1 M X k0=1 sk1,t− ˆsk1,t 2 s k0 2,t0 − ˆsk 0 2,t0 2) (4.51)
where τ = Q Nt− M · Q Nt 1 a 2 + 1 − Q Nt− M Q Nt 1 aN0 + Q − 1 Nt− M Q Nt (M − 1) + Q Nt− M 1 −M Q Nt 1 aN0 + 1 − Q − 1 Nt− M Q Nt (M − 1) + 1 − Q Nt− M 1 −M Q Nt 1 N0 2 . Let W1 = 1 − L X t=1 M X k=1 1 − Q Nt 1 N0 sk1,t− ˆsk1,t 2 + Q Nt 1 a sk1,t− ˆsk1,t 2 − L X t=1 M X k=1 1 − Q Nt 1 N0 sk2,t− ˆsk2,t 2 + Q Nt 1 a sk2,t− ˆsk2,t 2 W2 = τ L X t=1 M X k=1 L X t0=1 M X k0=1 sk1,t − ˆsk1,t 2 s k0 2,t0 − ˆsk 0 2,t0 2 . (4.52) Assume the power of the multitone jammer is much larger than the power of the thermal noise, then a N0 and N10 1a. Base on the above inequations we have
τ ∼= 1 − Q − 1 Nt− M Q Nt (M − 1) + 1 − Q Nt− M 1 −M Q Nt 1 N0 2 (5.53) and L X t=1 M X k=1 1 − Q Nt 1 N0 sk1,t− ˆsk1,t 2 + Q Nt 1 a sk1,t − ˆsk1,t 2 ∼ = L X t=1 M X k=1 1 − Q Nt 1 N0 sk1,t− ˆsk1,t 2 . (5.54) Cause sk i,t− ˆski,t 2
∈ (0, Es), then W1 and W2 can be bounded as
W1 ≤ 1 − Q Nt 1 N0 LM Es = 1 − Q Nt LM (SN R) W2 ≤ 1 − Q − 1 Nt− M Q Nt (M − 1) + 1 − Q Nt− M 1 −M Q Nt 1 N0 2 2LM Es2 = 1 − Q − 1 Nt− M Q Nt (M − 1) + 1 − Q Nt− M 1 − M Q Nt 2LM (SN R)2 (4.55)
After doing the simulation, we konw that |w1| |w2|. Then good codes could be con-structed by maximizing m (s, ˜s) for all possiable s and ˜s, and m (s, ˜s) can be expressed as m (s, ˜s) = L X t=1 M X k=1 L X t0=1 M X k0=1 sk1,t− ˆsk1,t 2 s k0 2,t0 − ˆsk 0 2,t0 2 . (4.56) According to the design criteria, good space-time codes are searched by the computer, and are given in following table.
Table 4.2: Optimal Space-time codes of the STC/OFHSS system with 4FSK and 2 trans-mitter antennas for wireless jamming channels.
Memory Generator Sequences 2 (xt
1, xt2) = bt−1(1, 0) ⊕4at−1(1, 0) ⊕4 bt(0, 1) ⊕4at(0, 2)
3 (xt1, xt2) = at−2(1, 3) ⊕4bt−1(1, 1) ⊕4 at−1(1, 0) ⊕4bt(2, 1)
⊕4at(0, 1)
4.4
Design Criteria for Constructing Good Space-Time Codes
with FSK Modulation
Consider a coded communication system with M FSK modulation and ML decoding. A block of transmitted symbols is denoted by
s = ski,t|∀i, k, 1 ≤ t ≤ L
(4.57) and an erroneous sequence selected by the decoder is
ˆ
s = ˆski,t|∀i, k, 1 ≤ t ≤ L . (4.58) We know that the likelihood function can be expressed as
f (r|α, s) = L Y t=1 m Y q=1 f ηtq = rtq− n X i=1
αi,qski,t|αi,q, ski,t ∀i, q, t
! = L Y t=1 m Y q=1 m Y q=1 " 1 √ πN0 exp − rtq−Pn
i=1αi,qski,t
2 N0 !# . (4.59)
Assume the fading coefficients are available at the receiver, then the pairwise error proba-bility is Pr (s → ˆs|αi,q, ∀i, q) = Pr L X t=1 m X q=1 M X k=1 rtj− n X i=1 αi,q p Esski,t 2 ≥ L X t=1 m X q=1 M X k=1 rtj− n X i=1 αi,q p Esˆski,t 2 ≤ 1 2exp −d2(s, ˆs) Es 4N0 (4.60) where d2(s, ˆs) = L X t=1 m X q=1 M X k=1 n X i=1 αi,q p Es ski,t− ˆski,t 2 . (4.61)
Assume the fading coeficients αi,q are independent Gaussian random variables with zero
mean and varance 1/2. Let “∗” denote the operator of taking complex conjugate, and H denotes the operator of taking Hermirian, and Ωj = (α1,q, α2,q, . . . , αn,q). Then we can
rewrite equation (4.61) as d2(s, ˆs) = m X q=1 n X i=1 n X l=1 αi,qα∗i,q L X t=1 M X k=1 ski,t− ˆski,t skl,t− ˆskl,t∗ = m X q=1 ΩqB (s, ˆs) BH(s, ˆs) ΩHq = m X q=1 ΩqA (s, ˆs) ΩHq (4.62) where B (s, ˆs) = s11,1− ˆs11,1 . . . sM1,1− ˆS1,1M . . . s11,L− ˆs11,L . . . sM1,L− ˆsM1,L s1 2,1− ˆs12,1 . . . sM2,1− ˆS2,1M . . . s12,L− ˆs12,L . . . sM2,L− ˆsM2,L .. . ... ... ... ... ... ... s1 n,1− ˆs1n,1 . . . sMn,1− ˆSn,1M . . . s1n,L− ˆs1n,L . . . sMn,L− ˆsMn,L = s1,1− ˆs1,1 s1,2− ˆs1,2 . . . s1,L− ˆs1,L s2,1− ˆs2,1 s2,2− ˆs2,2 . . . s2,L− ˆs2,L .. . ... . .. ... sn,1− ˆsn,1 sn,2− ˆsn,2 . . . sn,L− ˆsn,L
where sn,1 = s1i,t, s2i,t, · · · , sMi,t, ˆsn,1= ˆs1i,t, ˆs2i,t, · · · , ˆsMi,t , and A (s, ˆs) = B (s, ˆs) BH(s, ˆs).
A (s, ˆs) is nonnegative definite and Hermitian, and the eigenvalues of A (s, ˆs) are real num-bers. Then we have
A (s, ˆs) = V DVH (4.63) where V = (v1, v2, . . . , vn) is a unitary matrix and D is a diagonal matrix, where vi’s are
the eigenvectors of A (s, ˆs). Let λi be the diagonal elements of D, where 1 ≤ i ≤ n, and
ΩqVH = (β1,q, . . . , βn,q) . (4.64)
From (4.63) and (4.64), we can rewrite the equation (4.61) as following d2(s, ˆs) = m X q=1 n X i=1 λi|βi,q|2. (4.65)
Use equation (4.65) to replace d2(s, ˆs) in (4.60), then we have Pr (s → ˆs|αi,q ∀i, q) ≤ 1 2exp − Es 4N0 m X q=1 n X i=1 λi|βi,q| 2 ! (4.66)
By Using the same derivation in (3.19), we have the pairwise error probability
Pr (s → ˆs) ≤ 1 2 n Y i=1 1 1 + Es 4N0λi !m . (4.67)
When SNR is a big number, (4.67) can be expressed as
Pr (s → ˆs) ≤ 1 2 r Y i=1 λi !−m Es 4N0 −rm . (4.68)
where r is the rank of A (s, ˆs). In order to minimize the error probability, to make rm and the the product of eigenvalues as large as posible is necessary. A good space time code with memory 2 is searched by the computer with these cirteria, and the generator sequence is Table 4.3: Optimal Space-time codes of the STC/FSK system with 4FSK and 2 transmitter antennas for wireless jamming channels.
Memory Generator Sequences
