CLOSED-FORM ERROR PROBABILITY FOR 𝑀 -ARY CHIRP MODULATION INFREQUENCY-SELECTIVE AND -NONSELECTIVE FADING CHANNELS

Muhammad Ajmal Khan, Raveendra K. Rao, Xianbin Wang

Department of Electrical and Computer EngineeringWestern University, London, Ontario, Canada, N6A 5B9.

ABSTRACT

A class of chirp signals for 𝑀 -ary data transmission, referredto as 𝑀 -ary chirp modulation (MCM), is considered. Theperformance analysis of this MCM over frequency-selectiveand -nonselective fading channels is carried out and closed-form error probability results that are easy for numerical com-putation are derived. The parameters that affect error per-formance are identified and illustrated. These analytical re-sults are then compared using extensive Monte Carlo simula-tions. Specifically, the results are presented over Rayleigh andNakagami-𝑚 flat fading and over Nakagami-𝑚 frequency-selective channels. It is shown that analytical results comparewell with simulations.

Index Terms— Chirp signals, Nakagami-𝑚 fading, closed-form BER, Rayleigh fading, multipath fading.

1. INTRODUCTION

Chirp modulation is used for data transmission as it possessesinherent advantages in terms of anti-eavesdropping, anti-inter-ference and low-Doppler sensitivity properties. Chirp mod-ulation does not necessarily employ coding and produces atransmitted signal bandwidth much greater than the bandwidthof the information signal being sent. Furthermore, spreadspectrum type of chirp modulation, by using wider bandwidththan the minimum required, can realize higher processing gai-ns. In recent years, chirp modulation has been used in IEEE802.15.4a standard [1].

Berni and Gregg [2] compared the binary chirp modula-tion with PSK and FSK over AWGN analytically. Tsai andChang [3] analyzed the chirp modulation performance overRayleigh and Rician fading channels using computer sim-ulations. Kocian and Dahlhaus [4] analyzed the combina-tion of chirp modulation and code-division multiple access(CDMA) over both frequency-nonselective and -selective fad-ing channels using Monte Carlo simulations with zero cross-correlation assumption between two chirp signals which ledthe results to lower bound of the performance. In a similarfashion, Liu [5] proposed to employ chirp modulation withzero cross-correlation in ultra-wideband (UWB) scheme. El-Khamy et. al. [6] evaluated chirp modulation performances

but presented closed-form results over AWGN channel only.Gupta et. al. [7] analyzed chirp modulation signalling in afrequency-hopped CDMA using computer simulations. Duttaet. al. [8] proposed chirp modulation along with FSK andPSK schemes to obtain interference robust radio communica-tion in the presence of partial band interference over AWGNchannel. Alsharef [9] evaluated the performance of chirp mod-ulation and continuous phase chirp modulation for both co-herent and non-coherent cases over AWGN channel and pre-sented the closed-form results. Several other studies [10, 11]have focused on chirp modulation. However, in the literatureclosed-form analytical bit error rate (BER) results for MCMare not available particularly over frequency-selective and-nonselective fading channels. The closed-form BER perfor-mance of chirp modulation over these fading channels is im-portant particularly in the system design and analysis.

This paper presents a rigorous derivation of BER perfor-mance of chirp modulation for coherent detection over fadingchannels. To obtain the precise BER results, we include thecross-correlation between chirp signals and also consider ex-ponentially decaying power delay profile (PDF) in frequency-selective channel. Numerical results are obtained throughMonte Carlo simulations and compared with closed-form re-sults in order to confirm the accuracy and exactness of ob-tained analytical results.

The paper is organized as follows. The next section ex-plains the system model. Sections 3, 4 and 5 present theBER derivations over Rayleigh, Nakagami-𝑚 and frequency-selective Nakagami-𝑚 fading channels, respectively. Resultsare presented in Section 6 and conclusions in Section 7.

2. COMMUNICATION SYSTEM MODEL

The block diagram of 𝑀 -ary chirp communication system isshown in Fig. 1. The input data is assumed to be a sequence ofbinary digits from an equally likely and statistically indepen-dent data source. This data is then passed through the chirpmodulator that maps each 𝑘-bit chunk to one of the 𝑀 = 2𝑘

waveforms, i.e. 𝑠𝑖(𝑡), where 𝑖 = 1, 2, 3, ...,𝑀 . Mathemati-cally, 𝑀 -ary chirp modulated signal is given by

2013 26th IEEE Canadian Conference Of Electrical And Computer Engineering (CCECE)

978-1-4799-0033-6/13/$31.00 ©2013 IEEE

Input

Data

Chirp

Modulator

s1*(t)

Fading

Channel

+

AWGN

+

+

( )0

T

dt∫

( )0

T

dt∫

sM*(t)

Choose

Largest

t=T

t=T

Output

DataDecision

Fig. 1: 𝑀 -ary Chirp Communication System.

𝑠𝑖(𝑡) =

√2𝐸𝑠

𝑇cos [2𝜋𝑓𝑐𝑡+ 𝑑𝑖𝑔(𝑡) + 𝜙0] , 0 ≤ 𝑡 ≤ 𝑇 (1)

where 𝐸𝑠 is the energy of 𝑠(𝑡) in the symbol duration 𝑇 , 𝑓𝑐is the carrier frequency, 𝜙0 is the starting phase of the carriersignal at 𝑡 = 0, and 𝑑𝑖 is the information data taking one ofthe values ±1, ±3, ±5, ...,±(𝑀 − 1) and 𝑀 = 2, 4, 8, ....The phase function 𝑔(𝑡) is given by

𝑔(𝑡) =

⎧⎨⎩0 , 𝑡 ≤ 0, 𝑡 > 𝑇

𝜋{ℎ(

𝑡𝑇

)− 𝑤(

𝑡𝑇

)2}, 0 ≤ 𝑡 ≤ 𝑇

𝜋(ℎ− 𝑤) = 𝜋𝑞 , 𝑡 = 𝑇

(2)

where the parameters ℎ and 𝑤 represent the peak-to-peak fre-quency deviation divided by the symbol rate, and the fre-quency sweep width divided by the symbol rate, respectively.Since 𝑞 = ℎ − 𝑤, the pair 𝑞 and 𝑤, (𝑞, 𝑤), represents the setof independent modulation parameters.

The signal 𝑠(𝑡) is then transmitted over a fading channelℎ(𝑡), hence received signal can be written as:

𝑟(𝑡) = ℎ(𝑡) ∗ 𝑠(𝑡) + 𝑛(𝑡) (3)

where 𝑛(𝑡) is additive white Gaussian noise (AWGN) withzero-mean and power spectral density (PSD) 𝑁0/2. Coherentreception is assumed and the received signal is passed throughthe correlators to decide which data bit was sent based on theamplitude output of each correlator.

3. BER IN RAYLEIGH FLAT FADING

The average BER (𝑃𝑏) of binary chirp modulation over a fad-ing channel can be calculated by averaging the instantaneouserror rate, which is the error rate of binary chirp over anAWGN channel 𝑃𝑏(𝛾) with SNR 𝛾, over the distribution func-tion 𝑝𝛾(𝛾) of 𝛾. Mathematically,

𝑃𝑏 =

∫ ∞

0

𝑃𝑏(𝛾)𝑝𝛾(𝛾)𝑑𝛾, (4)

where 𝑃𝑏(𝛾) is given by [9]

𝑃𝑏(𝛾) = 𝑄

(√𝐸𝑏

𝑁0(1− 𝜌𝑖𝑗)

)= 𝑄 (

√𝛾) (5)

where 𝜌𝑖𝑗 is the normalized cross-correlation coefficient be-tween the chirp signals and is given by [9].

𝜌𝑖𝑗 =cos(Ω)√2𝜉𝑤

[C(𝑥+)−C(𝑥−)]+sin(Ω)√2𝜉𝑤

[S(𝑥+)− S(𝑥−)]

where 𝑥+ =√

𝜉2𝑤−𝑞√

𝑤, 𝑥− = −

√𝜉2𝑤+𝑞√

𝑤, Ω = 𝜋

4 𝜉(𝑞+𝑤)2

𝑤 ,

𝜉 = ∣𝑑𝑗 −𝑑𝑖∣, 𝑑𝑗 and 𝑑𝑖 are data symbols ±1,±3, ...,±(𝑀 −1), and C(⋅) and S(⋅) are the standard Fresnel cosine and sineintegrals respectively and are given by

C(𝑥) =

∫ 𝑥

0

cos

(𝜋𝑡2

2

)𝑑𝑡, S(𝑥) =

∫ 𝑥

0

sin

(𝜋𝑡2

2

)𝑑𝑡

Gaussian Q-function is defined by 𝑄(𝑥) = 1√2𝜋

∫∞𝑥

𝑒−𝑡2/2𝑑𝑡

and its alternate form is given by [12]

𝑄(𝑥) =1

𝜋

∫ 𝜋/2

0

exp

(− 𝑥2

2 sin2 𝜙

)𝑑𝜙 (6)

For Rayleigh fading, the PDF of the received SNR 𝛾 is givenby

𝑝𝛾(𝛾) =1

𝛾exp

(−𝛾

𝛾

), 𝛾 ≥ 0 (7)

where 𝛾 = 𝛼2𝐸𝑏(1 − 𝜌𝑖𝑗)/𝑁0 is the instantaneous SNR perbit, 𝛾 = 𝛼2𝐸𝑏(1− 𝜌𝑖𝑗)/𝑁0 is the average SNR per bit, and 𝛼

is the fading amplitude with mean square value of 𝛼2. Substi-tuting 𝑄(

√𝛾) according to (6) and 𝑝𝛾(𝛾) from (7) in (4), and

rearranging variables, we get

𝑃𝑏 =1

𝜋𝛾

∫ 𝜋/2

0

[∫ ∞

0

exp

(− 𝛾

2 sin2 𝜙− 𝛾

𝛾

)𝑑𝛾

]𝑑𝜙 (8)

Integrating (8) by using [13] and rearranging, we get

𝑃𝑏 =1

𝜋

∫ 𝜋/2

0

1

1 + 𝛾/2sin2 𝜙

𝑑𝜙 (9)

Integrating (9) by using [13], we get

𝑃𝑏 =1

2

(1−

√𝛾/2

1 + 𝛾/2

)(10)

Hence, the average BER of the binary chirp modulation overRayleigh flat fading channel can be written in closed-form as

𝑃𝑏 =1

2

⎛⎝1−

√√√⎷ 𝛼2 𝐸𝑏

2𝑁0(1− 𝜌𝑖𝑗)

1 + 𝛼2 𝐸𝑏

2𝑁0(1− 𝜌𝑖𝑗)

⎞⎠ (11)

Using union bound, averaging over all equally likely trans-mitted signals, the SER of MCM can be expressed as

𝑃𝑠 =1

𝑀

𝑀∑𝑗=1

𝑀∑𝑖=1𝑖∕=𝑗

𝑃𝑏 (12)

4. BER IN NAKAGAMI-𝑚 FLAT FADING

Now, we derive BER in Nakagami-𝑚 fading, where the PDFof the received SNR 𝛾 is given by

𝑝𝛾(𝛾) =𝑚𝑚𝛾𝑚−1

𝛾𝑚Γ(𝑚)exp

(−𝑚𝛾

𝛾

)(13)

Substituting 𝑄(√𝛾) according to (6) and 𝑝𝛾(𝛾) from (13) in

(4) and re-arranging variables we obtain

𝑃𝑏 =𝑚𝑚

𝜋𝛾𝑚Γ(𝑚)

∫ 𝜋/2

0

∫ ∞

0𝛾𝑚−1 exp

{−𝛾

(1

2 sin2 𝜙+

𝑚

𝛾

)}𝑑𝛾𝑑𝜙

Integrating by using the method of substitution and replacingvariables [13], the average BER of binary chirp modulationover Nakagami-𝑚 flat fading in closed-form is obtained as

𝑃𝑏 =1

2

[1− 𝜇

𝑚−1∑𝑘=0

(2𝑘

𝑘

)(1− 𝜇2

4

)𝑘]

(14)

This equation is valid for integer values of fading figure ‘𝑚’and 𝜇 is given by

𝜇 =

√√√⎷ 𝛼2 𝐸𝑏

2𝑁0(1− 𝜌𝑖𝑗)

𝑚+ 𝛼2 𝐸𝑏

2𝑁0(1− 𝜌𝑖𝑗)

(15)

For 𝑚 = 1, (14) becomes (11), which is evident that BERin Nakagami-𝑚 at 𝑚 = 1 equals to BER in Rayleigh fading.The average SER can be obtained follwoing (12).

5. BER IN FREQUENCY-SELECTIVE FADING

Radio signals while propagating through frequency-selectivefading channel undergo time dispersion and severe distortiondue to inter-symbol interference (ISI). The frequency-selectivechannel is characterized by the following impulse response

ℎ(𝑡) =

𝐿𝑝∑𝑙=1

𝛼𝑙𝑒−𝑗𝜃𝑙𝛿(𝑡− 𝜏𝑙) (16)

where 𝑙 is the channel index, 𝐿𝑝 is the number of resolvablepaths, 𝛿(.) is the Dirac delta function, and 𝛼, 𝜃 and 𝜏 arethe random channel amplitude, phase and delay, respectively.Hence, the received signal becomes

𝑟(𝑡) =

𝐿𝑝∑𝑙=1

𝛼𝑙𝑒−𝑗𝜃𝑙𝑠𝑙(𝑡− 𝜏𝑙) + {𝑛(𝑡) + 𝑖𝑠(𝑡)} (17)

where 𝑖𝑠(𝑡) is ISI component. In this paper, slow fading isassumed and 𝛼, 𝜃 and 𝜏 are all constant over a symbol pe-riod. The fading amplitude 𝛼 is a random variable with mean-square value 𝛼2 (also denoted by Ω). The instantaneous SNRper bit of the 𝑙th channel (𝛾𝑙 = 𝛼2𝐸𝑏(1 − 𝜌𝑖𝑗)/𝑁0) is con-sidered as Nakagami-𝑚 distribution (given in (13)) and theaverage SNR per bit of the 𝑙th channel is given by 𝛾𝑙 =Ω𝑙𝐸𝑏(1 − 𝜌𝑖𝑗)/𝑁0. Ω𝑙 accounts for the decay of channelstrength as a function of channel delay (or power decay factor𝛿) and its shape is referred to as power delay profile (PDF),(also multipath intensity profile (MIP)). In this work, Ω𝑙 isconsidered as exponentially decaying PDP, as validated by theexperimental measurements for indoor office buildings andcongested urban areas [14].

Ω𝑙 = Ω1𝑒−(𝑙−1)𝛿 , 𝛿 ≥ 0, 𝑙 = 1, 2, 3, ..., 𝐿𝑝 (18)

where Ω1 is the average fading power of the first path. Thevariance of ISI component is 𝜎2

𝑆 = (Ω𝑇 − 1)Ω1𝐸𝑏/2𝑃𝑔 [15],

where 𝑃𝑔 is the processing gain and Ω𝑇 =∑𝐿𝑝

𝑙=1 𝑒−(𝑙−1)𝛿

is the normalized total average fading power. Therefore, theequivalent interference plus noise power spectral density canbe written as

𝑁𝑒

2=

Ω𝑇 − 1

2𝑃𝑔Ω1𝐸𝑏 +

𝑁0

2(19)

The average signal to interference plus noise ratio (SINR) is

𝛾𝑒 =

(1

𝛾1/2+

Ω𝑇 − 1

2𝑃𝑔

)−1

(20)

Following the same procedure as in sections 3 and 4, the aver-age BER of binary chirp modulation over frequency-selectiveNakagami-𝑚 fading in closed-form is obtained as

𝑃𝑏 =1

2

[1− 𝜇𝑒

𝑚−1∑𝑘=0

(2𝑘

𝑘

)(1− 𝜇2

𝑒

4

)𝑘]

(21)

where 𝜇𝑒 =√

𝛾𝑒

𝑚+𝛾𝑒.

6. NUMERICAL RESULTS

The closed-form BER derived in this paper is mainly a func-tion of SNR and set of modulation parameters (𝑞, 𝑤). TheSER performance of MCM over Rayleigh flat fading is il-lustrated in Fig. 2, which shows that SER increases as ‘𝑀 ’increases for a fixed SNR. Fig. 3 depicts SER for 4-chirpmodulation over Nakagami-𝑚 flat fading channel, and clearlyconfirms the improvement in SER as ‘𝑚’ increases.

0 10 20 30 40 5010

−6

10−5

10−4

10−3

10−2

10−1

100

101

Es/N

0 [dB]

Sym

bo

l Err

or

Pro

bab

ility

Binary Chirp4−Chirp8−Chirp16−Chirp

Fig. 2: Performance of 𝑀 -ary Chirp Modulation overRayleigh fading channel.

0 10 20 30 40 5010

−6

10−5

10−4

10−3

10−2

10−1

100

101

Es/N

0 [dB]

Sym

bo

l Err

or

Pro

bab

ility

m=1m=2m=4

Fig. 3: Performance of4-Chirp Modulation overNakagami-𝑚 fading channel.

Figs. 4 and 5 illustrate the effects of modulation parame-ters (𝑞, 𝑤) on BER. These results are obtained for binary chirpmodulation over Rayleigh fading for various sets of (𝑞, 𝑤). Itis evident from these figures that the optimized values of 𝑞

and 𝑤 (𝑞 = 0.28,𝑤 = 1.85) [9] allows an SNR gain of upto 5dB for any BER. Thus, the optimized values have been usedin all the work.

0 10 20 30 40 5010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/N

0 [dB]

Bit

Err

or

Pro

bab

ility

q=0.28, w=1.85q=0.28, w=1q=0.28, w=3q=0.28, w=4q=0.28, w=5

Fig. 4: Effect of ‘𝑤’ on BER.

0 10 20 30 40 5010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/N

0 [dB]

Bit

Err

or

Pro

bab

ility

w=1.85, q=0.28w=1.85, q=0.1w=1.85, q=0.4w=1.85, q=0.6w=1.85, q=0.8

Fig. 5: Effect of ‘𝑞’ on BER.

The performance of binary chirp over 3-path frequency-selective Nakagami-𝑚 (𝑚 = 1) fading channel is shown inFig. 6, which explains that BER increases as decay factor 𝛿increases.

Numerical results are also obtained by Monte Carlo simu-lations to compare with analytical results. Fig. 7 depicts bothanalytical and simulation results for binary chirp modulationover Nakagami-𝑚 (𝑚 = 1) flat fading which confirms thevalidity and accuracy of our closed-form expressions.

0 10 20 30 40 5010

−6

10−5

10−4

10−3

10−2

10−1

100

SINR [dB]

Bit

Err

or

Pro

bab

ility

δ=0.1δ=0.5δ=0.9δ=2δ=4δ=8

Fig. 6: Performance ofBinary Chirp Modulationover Frequency-SelectiveNakagami-𝑚 at different 𝛿.

0 10 20 30 40 5010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/N

0 [dB]

Ave

rag

e B

it E

rro

r P

rob

abili

ty

SimulationAnalytical

m=1

m=2

m=4

Fig. 7: Comparison of An-alytical and Simulation Re-sults (Binary Chirp Modula-tion over Nakagami-𝑚).

7. CONCLUSIONS

In this paper, we developed closed-form BER expressions forchirp modulation over fading channels - Rayleigh, Nakagami-𝑚 and frequency-selective Nakagami-𝑚, one of our majorcontribution. Intensive mathematical and analytical tools wereused to derive the closed-form results. The cross-correlation

between chirp signals was considered in the analysis and theeffect of modulation parameter (𝑞, 𝑤) was also presented. TheBER performance was obtained for various 𝑀 over Rayleighfading and various Nakagami-𝑚 parameter (𝑚). Moreover,ISI was considered in frequency-selective analysis and re-sults showed the irreducible error, characterized by the decayfactor and ISI. One possible extension of this work is to de-sign an equalizer for chirp modulation to remove ISI. Finally,Monte Carlo simulation was performed and compared withthe closed-form results obtained which clearly validated ouranalytical work in the paper.

8. REFERENCES

[1] [Online] http://www.ieee802.org/15/pub/TG4a.html.

[2] A. Berni and W. Gregg, “On the utility of chirp modulation fordigital signaling,” IEEE Trans. on Comm., vol. 21, no. 6, pp.748–751, June 1973.

[3] Y. Tsai and J. Chang, “The feasibility of combating multipathinterference by chirp spread spectrum techniques over rayleighand rician fading channels,” in IEEE Third ISSSTA, vol.1, pp.282–286, July 1994.

[4] A. Kocian and D. Dahlhaus, “Downlink performance analysisof a CDMA mobile radio system with chirp modulation,” inIEEE VTC, vol. 1, pp. 238–242, July 1999.

[5] H. Liu, “Multicode ultra-wideband scheme using chirp wave-forms,” IEEE J. on Selected Areas in Comm., vol. 24, no. 4,pp. 885–891, 2006.

[6] S. E. El-Khamy, S. E. Shaaban, and E. A. Tabet, “Efficientmultiple-access communications using multi-user chirp mod-ulation signals,” in IEEE 4th ISSSTA, vol. 3, pp. 1209–1213.Sep. 1996.

[7] C. Gupta, T. Mumtaz, M. Zaman, and A. Papandreou-Suppappola, “Wideband chirp modulation for FH-CDMAwireless systems: coherent and non-coherent receiver struc-tures,” in IEEE ICC, vol. 4, pp. 2455–2459, May 2003.

[8] R. Dutta, A. B. J. Kokkeler, R. A. R. van der Zee, and M. J.Bentum, “Performance of chirped-FSK and chirped-PSK inthe presence of partial-band interference,” in 18th IEEE SCVT,pp. 1–6, 2011.

[9] M. A. Alsharef, “M-ary chirp modulation for data transmis-sion,” M.E.Sc. thesis, The Univ. of Western Ontario, 2011.

[10] X. Wang, M. Fei, and X. Li, “Performance of chirp spreadspectrum in wireless communication systems,” in 11th IEEEICCS, pp. 466–469, Nov. 2008.

[11] Y. Lee, T. Yoon, S. Yoo, S. Y. Kim, and S. Yoon, “PerformanceAnalysis of a CSS System with M-ary PSK in the Presence ofJamming Signals,” in IEEE VTC, pp. 1–5, 2009.

[12] J. W. Craig, “A new, simple and exact result for calculating theprobability of error for two-dimensional signal constellations,”in IEEE MILCOM, vol. 2, pp. 571–575, Nov. 1991.

[13] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series,and Products, vol. 6, California, Academic Press, 2000.

[14] COST 207. Management Committee, COST 207: Digital LandMobile Radio Communications: Final Report, 1989.

[15] T. Eng and L. B. Milstein, “Coherent DS-CDMA performancein Nakagami multipath fading,” IEEE Trans. on Comm., vol.43, no. 2-4, pp. 1134–1143, 1995.