Performance Analysis of a JTIDS/Link-16-type Waveform ...

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second layer of transmission security, each 32-chip CCSK sequence is scrambled with a 32-chip pseudo-noise sequence. The resulting 32-chip sequence is modulated for transmission with MSK to generate analog pulses. Each pulse is then upconverted to one of the 51 possible carrier frequencies, which contributes a third layer of transmission security. Normally, the starting point of the pulse train is pseudo-randomly jittered, which provides a fourth layer of transmission security since it can make it difficult for a jammer to decide when to turn on the jamming signal. After up-conversion, the signal is amplified, filtered, and transmitted over the channel [6]. B. JTIDS-type Receiver At the receiver (Figure 2), the receiving process is the reverse of the transmission process. After frequency dehopping, MSK chip demodulation, and de-scrambling by the 32-chip PN sequence, each 5-bit channel symbol is recovered by a CCSK symbol demodulator. The determination of which 5-bit channel symbol was received is done by computing the cross-correlation between the de-scrambled 32-chip sequence and all possible 32 sequences, and the decision is made by choosing the 5-bit channel symbol corresponding to the branch with the largest cross-correlation value. After symbol de-interleaving, the channel data symbols are decoded by a (31,15) RS decoder. If the decoding is successful, the data symbols are converted into a bit stream which are sent to the upper layer. Link-16 message (31,15) RS Decoder Symbol De-interleaver CCSK 32-ary Symbol Demodulator Fig. 2. Model of a JTIDS-type receiver. 32-Chip PN Sequence De-scrambling MSK Chip Demodulator Frequency De-hopping III. PERFORMANCE ANALYSIS Given the assumptions that frequency de-hopping is perfectly synchronized with the frequency-hopped waveform and that the signal-to-noise ratio (SNR) is large, the MSK chip demodulator recovers the original scrambled 32-chip sequence. Given that de-scrambling is perfectly synchronized, the CCSK symbol demodulator detects the original 5-bit coded symbol. As seen in Figure 2, to evaluate the probability of symbol error of a JTIDS/Link-16-type waveform, the probability of channel chip error at the output of the MSK chip demodulator and the probability of channel symbol error at the output of the CCSK symbol demodulator are both required. A. Probability of Channel Chip Error MSK can be considered as a special case of offset quadrature phase-shift keying (OQPSK) with sinusoidal pulse shaping [7]. When a coherent matched filter or correlator is used to recover the data chips, MSK has the same performance as binary PSK and OQPSK; that is, ⎛ Pc= Q ⎜ ⎝ 2E ⎞ c ⎟, N ⎟ 0 ⎠ (1) where E c is the average energy per chip. Since each 5-bit symbol is converted into 32 chips, E = 5E = 32E ; where s b c E S is the average energy per symbol and E b is the average energy per bit. Therefore, (1) can be expressed as ⎛ 10E ⎞ b Pc= Q⎜ ⎟. (2) ⎜ 32N ⎟ ⎝ 0 ⎠ The JTIDS waveform is currently received noncoherently at the chip level, but the performance with coherent detection is evaluated in order to determine performance if coherent chip demodulation were practical. The analysis presented in this paper can easily be modified to evaluate noncoherent chip demodulation. From (2), when forward error correction (FEC) coding is used, the probability of channel chip error for a JTIDS/Link-16-type waveform in AWGN is given by ⎛ 10E ⎞ b ⎛ 10rE ⎞ c b pc= Q⎜ ⎟= Q⎜ ⎟, (3) ⎜ 32N ⎟ ⎜ 0 32N ⎟ ⎝ ⎠ ⎝ 0 ⎠ where Eb = rE c b is the average energy per coded bit, and r is the code rate. The Link-16 message data can be sent with either a singlepulse structure or a double-pulse structure [8]. The doublepulse structure increases the anti-jam capability of the link since it provides a diversity of L = 2 . Note that (3) is only valid for the single-pulse structure. To recover the data sent with the double-pulse structure, soft decision (SD) combining on a chip-by-chip basis is assumed in the coherent MSK chip demodulator as shown in Figure 3. rt () πt cos cos 2π ft c 2T c c ( 2k 1) Tc ( 2k 1) T dt + − ∫ ( 2k+ 2) Tc ∫ dt 2kTc πt sin sin 2π fct 2T t = ( 2k+ 2) Tc c t = ( 2k+ 1) Tc 1 st pulse even chips 2 nd pulse even chips 1 st pulse odd chips 2 nd pulse odd chips SD combining + Σ + + Σ + SD combining Fig. 3. MSK coherent chip demodulator with double-pulse/SD combining. When the double-pulse structure is used, JTIDS is a hybrid DS/FFH spread spectrum system with sequential diversity L = 2 since each symbol is transmitted twice on two different carrier frequencies. Thus, the average energy per symbol is Es = LEp (4) where L = 2 , and E p is the average energy per pulse. Since Es = 5Eb and E p = 5Eb', from (4) we get Eb = LEb' (5) where E b ' is the average energy per bit per pulse. Using (5) in (3), we obtain an expression for the probability of channel chip error of a JTIDS/Link-16-type waveform in AWGN as + - + - S/P ˆS
⎛ 10rLE ⎞ b' pc= Q⎜ ⎟, (6) ⎜ 32N ⎟ ⎝ 0 ⎠ where L = 1 for the single-pulse structure, and L = 2 for the double-pulse structure. Note that if p c is expressed in terms of b E instead of LE b' , then p c is the same for both the singleand the double-pulse structure in AWGN. B. Probability of Channel Symbol Error The probability of symbol error for the 32-ary CCSK sequence chosen for JTIDS is given by [9] 32 Ps= ∑ P{ symbol error N = j} P{ N = j} (7) j = 0 where N is a binomial random variable which represents the total number of chip errors occur at the output of the MSK chip demodulator. Since the demodulation of CCSK is independent of the FEC coding, the probability of channel symbol error can be obtained from (7) as 32 ⎛32⎞ j 32− j ps = ∑ ζ j ⎜ ⎟ pc ( 1− pc) (8) j= 0 ⎝ j ⎠ where ζ j are the conditional probabilities of channel symbol error given that N = j chip errors occur at the output of the MSK chip demodulator, and p c is the probability of channel chip error. In [9], the values of ζ j were obtained both analytically and by Monte Carlo simulation. Since the analytic result ζ UB yields a tight upper bound, it is used to evaluate j JTIDS’s performance in this paper. Therefore, from (8), 32 ⎛32⎞ j 32− j ps < ∑ ζ UB p ( 1 ) . j ⎜ ⎟ c − pc (9) j = 0 ⎝ j ⎠ C. Performance Analysis in AWGN As mentioned earlier, JTIDS uses a RS code for FEC coding. A RS code is a linear, nonbinary block code. For a tsymbol error correcting, nonbinary block code, the probability of decoder, or block, error is upper bounded by [10] n ⎛n⎞ i n−i PE ≤ ∑ ⎜ ⎟ ps( 1− ps) (10) i=+ t 1⎝i⎠ where the equality holds for either a perfect code or a bounded distance decoder, and p s is the probability of channel symbol error. Since each block consists of n coded symbols, the probability of symbol error for a linear, nonbinary block code is obtained by approximating the probability of information symbol error given i channel symbol errors per block by in [11]. Substituting this into (10), we get n 1 ⎛n⎞ i n−i PS ≈ ∑ i⎜ ⎟ ps( 1 − ps) . (11) n i=+ t 1 ⎝i⎠ Now, using (6) with r = 15 31 and either L = 1 or 2 in (9) along with ζ UB from [9], we obtain j s p . Next, using p s in (11), we obtain the probability of symbol error of a JTIDS/Link-16-type waveform for either the single- or the double-pulse structure over AWGN. D. Performance Analysis in both AWGN and PNI When a JTIDS/Link-16-type waveform is subjected to both AWGN and PNI, (11) can still be used to evaluate the probability of symbol error since (11) is independent of the types of noise and/or fading channels. Since the probability of channel symbol error is determined at the symbol level instead of the chip level and since the PNI is assumed, the probability of channel symbol error shown in (8) must be modified as L ⎛L⎞ L− ps = ∑⎜ ⎟ ρ1 ( 1−ρ1) ps (12) = 0 ⎝⎠ where L = 1 for a single-pulse, L = 2 for a double-pulse, and 0< ρ1 ≤ 1 represents the fraction of time the PNI is turned on. Note that ρ 1 = 1 represents barrage noise interference (BNI). The probability of channel symbol error given that pulses are jammed s p is upper-bounded by 32 ⎛32⎞ 32− j j ps < ∑ζ UB p ( 1 ) , j ⎜ ⎟ c − pc (13) j= 0 ⎝ j ⎠ where = 0,..., L . The probability of channel chip error given that pulses are jammed c p is given by ⎛ 0.3125rLE ⎞ b ' pc= Q⎜ ⎟. (14) ⎜ N0 + ( NILρ1) ⎟ ⎝ ⎠ Now, using (12) through (14) in (11) with r = 15 31, and either L = 1 or 2, we obtain the probability of symbol error of a JTIDS/Link-16-type waveform for either the single- or the double-pulse structure in both AWGN and PNI. E. Performance Analysis over Nakagami Fading Channels The process of evaluating the performance of a JTIDS/Link-16-type waveform for the single-pulse structure in both AWGN and PNI over Nakagami fading channels is similar to that for channels with no fading. The only difference is that (14) with L = 1 is now a conditional probability of channel chip error since the received signal amplitude a c fluctuates and is modeled as a Nakagami random variable with a probability density function (pdf) m 2 2 ⎛ m ⎞ ⎛ 2m−1 −ma ⎞ c ACc ⎜ ⎟ ( ) 2 c ⎜ ⎟ 2 c Γ m ⎜a ⎟ ⎜ c a ⎟ c f ( a ) = a exp , a ≥0 (15) ⎝ ⎠ ⎝ ⎠ where Γ() i is the Gamma function, and m is the fading figure. When m < 1 , the fading is more severe than Rayleigh fading; m = 1 is Rayleigh fading; when m > 1 , there is a lineof-sight (LOS) component to the received signal, and when 2 m →∞, there is no fading. Replacing E b ' with aT c b' in (14) with L = 1 and = 0 , we get ⎛ 2 0.3125racT ⎞ b' pc ( a ) , 0 c = Q⎜ ⎟ (16) ⎜ N ⎟ ⎝ 0 ⎠ where T b ' is the bit duration per pulse. Similarly, with L = 1 and = 1 , we get
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Performance Analysis of a JTIDS/Link-16-type Waveform ...

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