Performance of Differential Frequency Hopping Systems in a Fading Channel with Partial-Band Noise

Jamming

ZHU Yichao, GAN Liangcai, LIN Jing, XIONG Junqiao School of Electronic Information

Wuhan University Wuhan, China

webboyy2004@yahoo.com.cn

Abstract—In this paper, an error probability analysis is performed for the differential frequency hopping (DFH) communication system in a frequency-nonselective, slowly fading channel with partial-band noise jamming. Each transmitted hop is assumed to fade independently according to a Rician process. The partial-band noise jamming is modeled as a Gaussian process. Thermal noise is neglected in the analysis. In the DFH receiver, the noncoherent square-law energy detector is combined with the Viterbi decoder to determine a soft-decision estimate of the transmitted data. The DFH bit error probability is obtained via Union-Chernoff bounds. Results show that the inherent dependence of adjacent transmitted frequencies provides good partial-band jamming rejection capability for DFH systems.

Keywords- differential frequency hopping, DFH, partial-band, Rician fading

I. INTRODUCTION A differential frequency hopping (DFH) waveform was

originally proposed by Herrick and Lee for operation in High Frequency (HF) bands [1] [2]. For DFH waveforms, the frequency of the transmitted hop depends on both the current data symbol and the previous transmitted hop. That is, given a data symbol nX and frequency of the previous hop 1nf − , the frequency of the current hop is determined as ( )1,n n nf G f X−= where the functionG can be viewed as a directed graph which has nodes corresponding to frequencies and vertices labeled with input data [3-6]. An example showing the directed graph for a frequency set of 8 and one bit per data symbol is shown in Fig. 1. Because the series of transmitted tones form a trellis, the receiver is able to make soft decisions on the data symbol estimates. Viterbi or maximum a posteriori (MAP) decoding techniques can be used. The bit error rate (BER) upperbound of Viterbi decoded DFH in an additive white Gaussian noise (AWGN) channel is given in [3] [5].

Partial-band noise jamming is a severe detriment to military communications [7]. If DFH is applied in military environments, it is necessary to investigate its anti-jamming capabilities. This paper presents an error probability analysis of the DFH system with partial-band noise jamming in a frequency nonselective, slowly Rician fading channel.

II. SYSTEM MODEL AND DESCRIPTION The DFH system block diagram is shown in Fig. 2. Bit-

symbol converter unit combinesb input data bits to form a data symbol. The value of b is the number of bits encoded within a frequency hop. Function G unit uses both the current b-bit symbol and the previously transmitted tone to determine the current frequency selection. The current frequency selection is then sent to the tone generator, which transmits the selected tone. The transmitted DFH tone fades in the channel and is jammed by partial-band noise. At the receiver, the energy in the received signal at each available frequency is noncoherent square-law detected. The Viterbi decoder uses these energy values to make soft-decision estimates of the transmitted data symbols. The symbols are then split to data bits, which are the system output.

We assume that each transmitted tone fades independently; that is, we assume the smallest spacing between frequency slots is larger than the coherence bandwidth of the channel [8]. We also model the channel as a frequency-nonselective, slowly fading Rician process; thus we assume the signal bandwidth is much smaller than the coherence bandwidth of the channel and the hop duration is much smaller than the coherence time of the channel [8]. As a result, after passing through the channel, the amplitude of the transmitted hop is a Rician random variable and it can be considered as the sum of two components: a nonfaded (direct) component and a Rayleigh-faded (diffuse) component.

The partial-band noise jamming is modeled as additive Gaussian noise. We assume the available orthogonal frequency set is { }1 2, ,..., Mf f fF , where M is the number of total available frequencies; thus the full frequency hopping band ssW consists ofM subbands with equal bandwidth. We assume the jammer spreads its total noise power J evenly over q ( q M≤ ) subbands so that a fraction q Mρ = of the frequency slots contain jamming power. Consequently, ρ represents the probability that jamming is present on a certain frequency slot, and the probability that jamming is not present on a certain frequency slot is 1 ρ− . Let J ssN J W be the single-sided jamming power spectral density over the entire DFH

This paper is supported by National Natural Science Foundation of China (No.60372056)

1-4244-0517-3/06/$20.00 ©2006 IEEE 1

bandwidth, then JN ρ is the single-sided jamming power spectral density of partial-band noise jamming when it is present. We assume the background thermal noise power spectral density is small compared with JN , so the effect of the background thermal noise is neglected in the analysis.

III. UNION-CHERNOFF BOUNDS When the DFH waveform is Viterbi decoded, the symbol

error probability is dominated by the minimum free distance of its decoding trellis [5] [8]. The minimum free distance is defined as the minimum path length of all the incorrect paths that diverge from the correct path at a node and merge with the correct path at another node. The symbol error probability can be upperbounded by Union-Chernoff bounds as [7]

1

12

freel

ds i

i

P D N=

≤ ∑ (1)

where freed is the minimum free distance of the decoding trellis, l is the number of incorrect paths having the minimum free distance, iN is the number of symbol errors of the i-th incorrect path andD is the Chernoff parameter given by [7]

( )0

minD Dλ

λ≥

= (2)

and

( ) ( ) ( )( )( ){ } ˆˆ ˆexp , ; , ;

f fD E m r f z m r f z fλ λ

≠= − . (3)

Here f is a transmitted frequency in the correct path, f is the corresponding frequency in an incorrect path, ˆ,f f ∈F and

ˆf f≠ . ( ), ;m r f z is the decoding metric corresponding to f , given that { }1 2, ,..., Mr r r r is the noncoherent energy detector output vector and the jammer state variable is z where 1z = indicates that f is jammed while 0z = indicates that f is not jammed. The expectation E is over r , z and z . In this paper, we use the weighted energy detector output as the decoding metric, i.e.

( ) ( ), ;m r f z c z r= (4)

where r is the component of r corresponding to f and ( )c z is the weighting function.

IV. BIT ERROR PROBABILITY ANALYSIS Consider a frequency 1f in the correct path and the

corresponding frequency 2f in one of the incorrect paths having the minimum free distance. Assume 1z and 2z are the jammer state variable corresponding to 1f and 2f . Let H denote the event

1 1z = and H denote the event 1 0z = . Let S denote the event

2 1z = and S denote the event 2 0z = . The joint probability of H and S can be computed as

{ } { } { } ( ) ( ) ( )Pr , Pr Pr 1 1H S H S H q M q M= ⋅ = ⋅ − − . (5.1)

Similarly, the other three joint probabilities can be derived as { } ( ) ( )Pr , 1H S q M q M M= − − (5.2)

{ } ( ) ( )Pr , 1H S q M q M M= − − (5.3)

{ } ( )( ) ( )Pr , 1 1H S M q M q M M= − − − − . (5.4)

Averaging ( )D λ in (3) over all the possible combinations of 1z and 2z , and making use of the assumption that all the available frequencies are orthogonal to each other, we have

( ) { } ( )( ){ } ( )( ){ }2 1Pr , exp 1 exp 1D H S E c r S E c r Hλ λ λ= −

{ } ( )( ){ } ( )( ){ }2 1Pr , exp 0 exp 1H S E c r S E c r Hλ λ+ −

{ } ( )( ){ } ( )( ){ }2 1Pr , exp 1 exp 0H S E c r S E c r Hλ λ+ −

{ } ( )( ){ } ( )( ){ }2 1Pr , exp 0 exp 0H S E c r S E c r Hλ λ+ − (6) where 1r and 2r are the components of r corresponding to

1f and 2f , respectively. The expectation E here is over 1r and 2r .

Let 2s be the transmitted signal energy. According to the channel model, s is a Rician random variable. Its probability density function (PDF) is [8]

( )2 2

02 2 2exp2S

s s sp s Iα ασ σ σ

+ = −

(7)

where 2α is the average energy of the direct component and 22σ is average energy of the diffuse component. 2 22α σ+ is the

average received symbol energy and is assumed to remain constant from hop to hop. Given s andH , the conditional PDF of 1r is [8]

( )1

211

1 02 2 2

1, exp2 2R

J J J

s rr sp r s H Iσ σ σ

+= − (8)

where 2 2J JNσ ρ= is the partial-band noise jamming variance, ( )0I • is the zeroth order modified Bessel function of the first

kind. The conditional PDF of 1r given H can be found by integrating (8) respect to s

( ) ( ) ( )1 11 1

0

,R R Sp r H p r s H p s ds∞

= ∫ . (9)

Substituting (7) and (8) into (9), we obtain

( ) ( ) ( )1

211

1 0 2 22 2 2 2

1 exp2 2R

JJ J

rrp r H Iαα

σ σσ σ σ σ

+ = − ++ + . (10)

Then ( )( ){ }1exp 1E c r Hλ− is obtained by integrating

( )( )1exp 1c rλ− respect to 1r

( )( ){ } ( )( ) ( )11 1 1 1

0

exp 1 exp 1 RE c r H c r p r H drλ λ∞

− = −∫ . (11)

Substituting (10) into (11), we obtain

( )( ){ } ( )( )( ) ( )( ){ }

12 21

2 2 2

exp 1 1 2 1

exp 1 1 2 1

J

J

E c r H c

c c

λ λ σ σ

λ α λ σ σ

− − = + + ⋅

− + +

. (12)

The conditional PDF of 2r given S is [8]

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( )2

22 2 2

1 exp2 2R

J J

rp r Sσ σ

= −

. (13)

Then ( )( ){ }2exp 1E c r Sλ is obtained by integrating ( )( )2exp 1c rλ respect to 2r

( )( ){ } ( )( ) ( )22 2 2 2

0

exp 1 exp 1 RE c r S c r p r S drλ λ∞

= ∫ . (14)

Substituting (13) into (14), we obtain

( )( ){ } ( )( )22exp 1 1 1 2 1 JE c r S cλ λ σ= − . (15)

Since H and S indicate that jamming is not present on frequency 1f and 2f , let 2 0Jσ = in (12) and (15), we obtain

( )( ){ } ( )( ) ( ) ( )( ){ }12 2 21exp 0 1 2 0 exp 0 1 2 0E c r H c c cλ λ σ λ α λ σ

−− = + − + (16)

( )( ){ }2exp 0 1E c r Sλ = . (17) Combine (5.1)-(5.4), (12), (15), (16) and (17) into (6).

Let ( )2 22bE bα σ+ be the average received bit energy, where b is the number of bits per symbol. Change q into Mρ , substitute λ in place of ( )22 1Jcσ λ and let ( ) ( )0 1c c c= , we obtain

( ) ( ) ( )( )( )

( )

( )( )( ) ( )

11 exp1 1 1 1 1 1 1

1 11 exp , 0 11 1 1 1 1

MM ADA M M A

MM c Ac A M M c A

ρρ ρ λγρλλ ρ λ λ ρ

ρρ ρ λ γρ λλ ρ λ λ ρ

−− = + − + + − − − + + − − −+ + − ≤ < + − − − +

(18)

where ( ) ( )1 b JA b E Nγ= + and 2 22γ α σ is the direct-to-diffuse ratio of a Rician variable.

When jammer state information is unavailable at the receiver, ( ) ( )0 1c c= and 1c = . Then

( ) ( ) ( )( )( )

( )

( )( )( )

11 exp1 1 1 1 1 1 1

1 11 exp1 1 1 1 1

MM ADA M M A

MM AA M M A

ρρ ρ λγρλλ ρ λ λ ρ

ρρ ρ λγρλρ λ λρ

−− = + − + + − − − + + − − −+ + − + − − − +

. (19)

When jammer state information is available at the receiver, we should choose c to minimize ( )D λ in (18). Note that only the second term of ( )D λ includes c . It is easily seen that the second term of ( )D λ is a decreasing function of c , so we may choose c large enough to make the second term of ( )D λ negligible. Then

( ) ( ) ( )( )( )

( )11

exp1 1 1 1 1 1 1

MM AD

A M M Aρρ ρ λγρλ

λ ρ λ λ ρ −− = + − + + − − − + +

. (20)

Substituting (19) and (20) into (2), we derive the expressions of D with and without the jammer sate information. The parameter λ minimizing D can be found through numerical search. Then from (1), we obtain the symbol error probability bound. The bit error probability error is

( )12 2 1b bb sP P−= − . (21)

V. NUMERICAL RESULTS AND DISCUSSION In this section, the numerical results are presented

according to the analysis in Section IV. We consider a function G (DFH encoder) described in [5] [9]. The schematic diagram of the encoder is illustrated in Fig. 3. As shown in the diagram, the encoder consists of a K-stage shift register all of whose stages are connected to a tone selector. In every symbol interval sT , all the b-bit data symbols in the register are shifted one stage to the right, the content of the rightmost stage is discarded and a new symbol is shifted into the leftmost stage. Immediately after each shift, the tone selector produces a frequency if of duration sT seconds, where i is the integer between 0 and 2 1bK − whose 2b -ary expansion corresponds to the contents of the shift register, read from left to right, at the given time. From the structure of the encoder, we set the parameters as 2b = , 3K = and 2 64bKM = = ; thus 3freed = ，

3l = and 1iN = ( )1 i l≤ ≤ .

Fig. 4 and Fig. 5 show the performance of DFH system in partial-band noise jamming for various jamming ratios with and without jammer state information. The Rician fading direct-to-diffuse ratio γ is set to 10. As can be seen in Fig. 4, the bit error probability upperbound increases as the jamming ratio ρ decreases, e.g., for the case of 1ρ = (full band case), DFH requires approximately 6dB average signal-to-noise ratio (SNR) to achieve a bit error probability upperbound of

310− ,while for 1 64ρ = , DFH requires 14dB more average SNR to achieve the same bit error probability upperbound. It means that the DFH receiver using soft decision energy metric is vulnerable to narrow-band jamming if jammer state information is unavailable. But when jammer state information is available, as seen in Fig. 5, when the average SNR is less than 6dB, the largest bit error probability bound is given with 1ρ = , which means the jammer is forced to jam the full band to do the most harm to DFH system and excellent narrow-band jamming rejection capability is achieved. That is, jammer sate information is essential for DFH receiver using soft decision metric to combat narrow-band jamming.

VI. CONCLUSION In this paper, we analyzed the performance of DFH system

in partial-band noise jamming. The inherent dependence of adjacent transmitted frequencies makes the system possible to be decoded with trellis decoder, which significantly improves the system performance and provides the system with good narrow band jamming rejection capability with the aid of jammer state information. The DFH system is suitable in military communication environment.

REFERENCES [1] D. L. Herrick, P. K. Lee, “CHESS a New Reliable High Speed HF

Radio,” MILCOM’96, Vol.3, pp. 684-690, 1996. [2] D. L. Herrick, P. K. Lee and L. L. Ledlow, Jr, “Correlated Frequency

Hopping-An Improved Approach to HF Spread Spectrum,” Tactical communication conference, pp. 319-324, 1996.

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[3] D. G. Mills, G. S. Edelson and D. E. Egnor, “A Multiple Access Differential Frequency Hopping System,” MILCOM 2003, Vol.2, pp.1184-1189, 2003.

[4] D. G. Mills, D. E. Egnor and G. S. Edelson, “A Performance Comparioson of Differential Frequency Hopping and Fast Frequency Hopping,” MILCOM 2004, Vol.1, pp.445-450, 2004.

[5] D. G. Mills and G. S. Edelson, “CHESS Study Final Report,” report for DARPA and ARFL, 2001.

[6] D. G. Mills, C. S. Myers, G. S. Edelson and D. L. Herrick, “Soft-decision Trellis-coded Differential Frequency-hopped Spread Spectrum,” United States Patent, No. US 6,954,482 B2, 2005.

[7] M. K. Simon, J. K. Omura, R. A. Scholtz and B. K. Levitt, Spread Spectrum Communications Handbook, McGraw-Hill, New York, 2002

[8] J. G. Proakis, Digital Communications, 4th ed, McGraw-Hill, New York, 2001

[9] PAN Wu, ZHOU Shi-dong and YAO Yan, “Performance analysis of differential frequency hopping communication system,” Chinese Journal of Electronics, Vol.27, No.11A, pp.102-104, 1999.

Figure 1. Example of a directed graph for Function G

Figure 2. DFH system block diagram

if( )0,1,..., 2 1bKi = −

Figure 3. DFH encoder schematic diagram

0 2 4 6 8 10 12 14 16 18 2010

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

10γ =

(1)

(2)

(3)(4)

(5)

(1) 1 64ρ =(2) 4 64ρ =(3) 16 64ρ =(4) 32 64ρ =(5) 64 64ρ =

bP, U

PPER

BOU

ND

S

b JE N , dB

Figure 4. Performance of DFH for various partial-band noise jamming ratio without jammer state information

0 2 4 6 8 10 12 14 1610

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

bP, U

PPER

BOU

ND

S

10γ =

1 64ρ =

4 64ρ =

16 64ρ =

32 64ρ =64 64ρ =

b JE N , dB

Figure 5. Performance of DFH for various partial-band noise jamming ratio with jammer state information

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