PERFORMANCE ANALYSIS OF MRC-CHIRP SYSTEM OVER INDEPENDENTAND CORRELATED FADING CHANNELS

Muhammad Ajmal Khan, Raveendra K. Rao, Xianbin Wang

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

ABSTRACT

In this paper, chirp modulation is proposed to employ in Max-imal Ratio Combining (MRC) diversity, referred to as MRC-Chirp system. Moment generating function (MGF) approachis used to derive easy-to-compute expressions for average biterror probability (ABEP) for two fading situations. Firstly,independent fadings with Rayleigh and Nakagami-𝑚 statis-tics are considered. Next, an exponentially correlated fadingenvironment with Nakagami-𝑚 statistics is considered. TheABEP performance of the proposed system is illustrated usinganalytical expressions and using extensive Monte Carlo simu-lations. Numerical results show close agreement of analyticalwork with those of simulations. A discussion of numerical re-sults on the performance of MRC-Chirp system as a functionof diversity order 𝐿, chirp modulation parameters, and fadingparameters is presented.

Index Terms— MRC diversity, chirp modulation, Naka-gami-𝑚 fading, independent fading, correlated fading.

1. INTRODUCTION

Chirp signals, also known as linear frequency-modulated sig-nals, have been widely used in radar and sonar applications.These signals are robust to multipath interference and can re-alize higher processing gains and have been used in wirelesscommunications to improve the performance due to its anti-jamming characteristic [1]. Chirp signals have also been usedin various other applications such as combating multipath in-terference [2], spread spectrum techniques [3], multiple-accessschemes [4], equalization [5], and channel estimation [6].

Although the error performance of linear modulation sch-emes such as PSK, QAM, etc. has been extensively analyzedin independent [7] and correlated fading channels with MRCdiversity reception [8], there is hardly any work in the lit-erature devoted to the chirp modulation particularly in cor-related fading channels. In [9], authors mainly focused onchannel estimation using chirp signals and briefly discussedorthogonal chirp signals in multipath diversity using com-puter simulations only. In [10], authors suggested two RAKEreceiver architectures for chirp spread spectrum system and

evaluated performance using computer simulations. It is im-portant to understand the bit error probability performance ofchirp modulation in diversity reception as well as in correlatedfading channels. In this paper, we propose and analyze a wire-less communication system using chirp modulation scheme inconjunction with Maximal Ratio Combining (MRC) diversity.The main contribution of the paper are: (i) derivation of errorprobability performance of MRC-Chirp system using MGFapproach under independent Rayleigh and Nakagami-𝑚 sta-tistical channels, (ii) analysis of MRC-Chirp system in ex-ponentially correlated Nakagami-𝑚 fading channel, and (iii)validation of the ABEP through Monte Carlo simulations.

The paper is organized as follows: Section 2 describes theproposed MRC-Chirp system, followed by theoretical perfor-mance analysis of the system over fading channels in Section3. Sections 4 and 5 present performance analysis in indepen-dent and correlated fading channels, respectively. Numericalresults are presented in Section 6 and the paper is concludedin Section 7.

2. MRC-CHIRP SYSTEM MODEL

Input

Data

Chirp

Modulator

+

s(t)

+

h1(t)

+

h2(t)

+

hL(t)

+

+

n1(t)

n2(t)

nL(t)

r1(t)

r2(t)

rL(t)

MRC-Chirp

Receiver

Output

DataDecision

Fig. 1: MRC-Chirp Communication System.

The proposed MRC-chirp communication system with 𝐿fading paths for coherent detection is shown in Fig. 1. The in-put data is assumed to be a sequence of binary digits from anequally likely and statistically independent data source. Thisdata is then passed through the chirp modulator that mapseach 𝑘-bit chunk to one of the 𝑀 = 2𝑘 waveforms, i.e. 𝑠𝑖(𝑡),where 𝑖 = 1, 2, 3, ...,𝑀 . Mathematically, chirp modulated

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

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

signal is represented by

𝑠𝑖(𝑡) =

√2𝐸𝑠

𝑇cos [2𝜋𝑓𝑐𝑡+ 𝑑𝑖𝑔(𝑡) + 𝜓0] , 0 ≤ 𝑡 ≤ 𝑇 (1)

where 𝐸𝑠 is the energy of symbol over the duration of 𝑇 , 𝑓𝑐is the carrier frequency, 𝜓0 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 set ofindependent modulation parameters. The optimized values of𝑞 and 𝑤 (𝑞 = 0.28, 𝑤 = 1.85) [11] are used for binary chirpmodulation throughout this paper for illustration of numericalresults.

In the proposed MRC-Chirp system, the signal 𝑠(𝑡) istransmitted over 𝐿 independent channels, where each channelhas an impulse response of ℎ𝑙(𝑡) = 𝛼𝑙𝑒

−𝑗𝜙𝑙𝛿(𝑡 − 𝜏𝑙). Eachreplica of the signal is perturbed by the additive white Gaus-sian noise (AWGN) 𝑛(𝑡) with zero-mean and power spectraldensity (PSD) 𝑁0/2. We assume AWGN to be statisticallyindependent from channel to channel and independent of thefading amplitude. The received signal at 𝑙th branch can bewritten as

𝑟𝑙(𝑡) = 𝛼𝑙𝑒−𝑗𝜙𝑙𝑠(𝑡− 𝜏𝑙) + 𝑛𝑙(𝑡), 𝑙 = 1, 2, ..., 𝐿 (3)

where 𝑙 is the branch index, and 𝛼, 𝜙 and 𝜏 are the randomchannel amplitude, phase and delay, respectively. In this pa-per, slow fading is assumed, so 𝛼, 𝜙 and 𝜏 are all constantover a symbol period. The perfect knowledge of the chan-nel state information (CSI) is also assumed to be available atthe receiver but transmitter has no information of the channel.At the receiver, each received replica of the signal is passedthrough the correlators and combined at the MRC and thendecision is made based on the amplitude output of each com-biner branch.

3. ANALYSIS OF MRC-CHIRP SYSTEM

In this paper, we consider binary case (𝑀 = 2) for our pro-posed MRC-Chirp system and derive the average bit errorprobability (ABEP) 𝑃𝑏(𝑒) over a generalized fading channelin this section. The instantaneous signal-to-noise ratio (SNR)per bit per path for binary chirp modulation is defined by [11]

𝛾𝑙 =𝛼2𝑙𝐸𝑏

𝑁𝑙(1− 𝜌𝑖𝑗) (4)

where 𝜌𝑖𝑗 is the normalized cross-correlation coefficient be-tween the chirp signals and is given by [11].

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

Now, the total conditional SNR per bit 𝛾𝑡 at the output ofthe MRC combiner for equally likely transmitted data is ex-pressed as

𝛾𝑡 =

𝐿∑𝑙=1

𝛾𝑙 (5)

The conditional BEP 𝑃𝑏(𝑒∣{𝛾𝑙}𝐿𝑙=1) (i-e. BEP over AWGNchannel) of binary chirp modulation is given by [11]

𝑃𝑏(𝑒∣{𝛾𝑙}𝐿𝑙=1) = 𝑄 (√𝛾𝑡) (6)

where 𝑄(⋅) is the Gaussian Q-function defined by

𝑄(𝑥) =1√2𝜋

∫ ∞

𝑥

𝑒−𝑡2/2𝑑𝑡 (7)

For mathematical analysis, the alternate form of 𝑄(⋅) is moreconvenient and is given by [12]

𝑄(𝑥) =1

𝜋

∫ 𝜋/2

0

exp

(− 𝑥2

2 sin2 𝜃

)𝑑𝜃 (8)

Using (8) in (6), the conditional BEP (6) can be written as

𝑃𝑏(𝑒∣{𝛾𝑙}𝐿𝑙=1) =1

𝜋

∫ 𝜋/2

0

exp

(− 𝛾𝑡

2 sin2 𝜃

)𝑑𝜃 (9)

Using (5), the conditional BEP (9) can be expressed as

𝑃𝑏(𝑒∣{𝛾𝑙}𝐿𝑙=1) =1

𝜋

∫ 𝜋/2

0

𝐿∏𝑙=1

exp

(− 𝛾𝑙

2 sin2 𝜃

)𝑑𝜃 (10)

The ABEP 𝑃𝑏(𝑒) of the proposed MRC-Chirp system is ob-tained by averaging conditional BEP 𝑃𝑏(𝑒∣{𝛾𝑙}𝐿𝑙=1) over thejoint PDF of the instantaneous SNR sequence 𝑝𝛾1,𝛾2,...,𝛾𝐿

(alsoequal to

∏𝐿𝑙=1 𝑝𝛾𝑙

(𝛾𝑙) for independent identically distributedpaths), hence

𝑃𝑏(𝑒) =

∫ ∞

0

∫ ∞

0

...

∫ ∞

0

𝑃𝑏({𝛾𝑙}𝐿𝑙=1)𝐿∏

𝑙=1

𝑝𝛾𝑙(𝛾𝑙)𝑑𝛾1𝑑𝛾2...𝑑𝛾𝐿

𝑃𝑏(𝑒) =

∫ ∞

0

∫ ∞

0

...

∫ ∞

0

1

𝜋

∫ 𝜋/2

0

𝐿∏𝑙=1

exp(− 𝛾𝑙2 sin2 𝜃

)

⋅ 𝑝𝛾𝑙(𝛾𝑙)𝑑𝛾1𝑑𝛾2...𝑑𝛾𝐿

(11)

The moment generating function (MGF) ℳ𝛾 of SNR per bitfor 𝑙th path is defined by [7]

ℳ𝛾𝑙≜∫ ∞

0

𝑝𝛾𝑙(𝛾𝑙)𝑒

𝑠𝛾𝑙𝑑𝛾𝑙 (12)

Therefore, ABEP of binary chirp modulation with MRC di-versity of 𝐿-paths over a generalized fading channel can bewritten as

𝑃𝑏(𝑒) =1

𝜋

∫ 𝜋/2

0

𝐿∏𝑙=1

ℳ𝛾𝑙

(− 1

2 sin2 𝜃

)𝑑𝜃 (13)

The MGF of Rayleigh fading channel is given by [7]

ℳ𝛾𝑙(𝑠) = (1− 𝑠𝛾𝑙)

−1 (14)

The MGF of Nakagami-𝑚 fading channel is given by [7]

ℳ𝛾𝑙(𝑠) =

(1− 𝑠𝛾𝑙

𝑚

)−𝑚

(15)

4. ANALYSIS IN INDEPENDENT FADING

In this section, we consider the case of independent identi-cally distributed fading with the same average SNR per bit 𝛾for all 𝐿 channels, hence (13) can be written as

𝑃𝑏(𝑒) =1

𝜋

∫ 𝜋/2

0

(ℳ𝛾

(− 1

2 sin2 𝜃

))𝐿

𝑑𝜃 (16)

Using (14), ABEP (16) over Rayleigh fading channel is givenby

𝑃𝑏(𝑒) =1

𝜋

∫ 𝜋/2

0

(sin2 𝜃

sin2 𝜃 + 𝛾/2

)𝐿

𝑑𝜃 (17)

Integrating using [13], ABEP of binary chirp modulation inMRC diversity of 𝐿-paths over Rayleigh fading channel isexpressed as

𝑃𝑏(𝑒) = (𝒜𝛾)𝐿

𝐿−1∑𝑙=0

(𝐿− 1 + 𝑙

𝑙

)[1−𝒜𝛾 ]

𝑙 (18)

where

𝒜𝛾 =1

2

(1−

√𝛾

2 + 𝛾

)(19)

Similarly, using (15) and integrating using [13], we can haveABEP of binary chirp modulation in MRC diversity of 𝐿-paths over Nakagami-𝑚 fading channel as

𝑃𝑏(𝑒) = (ℬ𝛾)𝑚𝐿

𝑚𝐿−1∑𝑙=0

(𝑚𝐿− 1 + 𝑙

𝑙

)[1− ℬ𝛾 ]

𝑙 (20)

where

ℬ𝛾 =1

2

(1−

√𝛾

2𝑚+ 𝛾

)(21)

When 𝑚 = 1, (20) becomes (18), it is evident that ABEPunder Nakagami-𝑚 is identical to under Rayleigh fading.

5. ANALYSIS IN CORRELATED FADING

Generally, diversity branches are assumed to be independentof one another, but it is not the case in real-world situationsdue to antenna spacing particularly [14]. Thus, correlationalways exists among diversity branches and various channelcorrelation models have been proposed in the literature. Oneof the commonly used model is Exponential correlation model,which is proposed by Aalo [14] for identically distributedNakagami-𝑚 channels and is represented by the followingMGF [7]

ℳ(𝑠) =

(1− 𝑠 𝑟𝜌 𝛾

𝑚𝐿

)−𝑚𝐿2/𝑟𝜌

(22)

where [14]

𝑟𝜌 = 𝐿+2√𝜌

1−√𝜌

(𝐿− 1− 𝜌𝐿/2

1−√𝜌

)(23)

𝜌𝑖𝑗 =cov

(𝑟2𝑖 , 𝑟

2𝑗

)√

var(𝑟2𝑖 )var(𝑟2𝑗 ), 𝑖, 𝑗 = 1, 2, ..., 𝐿 (24)

Using (22) in (13), we obtain ABEP of binary chirp modula-tion in MRC diversity over exponentially correlated Nakagami-𝑚 fading channel as

𝑃𝑏(𝑒) =1

𝜋

∫ 𝜋/2

0

𝐿∏𝑙=1

(sin2 𝜃

sin2 𝜃 +𝑟𝜌𝛾2𝑚𝐿

)𝑚𝐿2/𝑟𝜌

𝑑𝜃 (25)

6. NUMERICAL RESULTS

Numerical results of average bit error probability (ABEP) wereobtained using simulations and using analytical expressionsderived in Sections 4 and 5 as a function of average SNR perbit. In both cases, binary chirp modulation in conjunctionwith MRC diversity were considered over Rayleigh/Nakagami-𝑚 fading channels for different diversity order 𝐿 and channelcorrelation parameter 𝜌. Numerical results were obtained byMonte Carlo simulation using over 107 samples for informa-tion bits and the generation of the fading envelopes at eachSNR. Coherent detection is assumed in all results as well aschannel state information (CSI) was assumed to be availableat the receiver.

Fig. 2 depicts the error performance of binary chirp mod-ulation in conjunction with MRC diversity for different or-ders 𝐿 operating over independent and identically distributed(i.i.d.) Rayleigh fading channel. The performance over AWGNchannel is also shown for comparison. From this figure, it isnoted that ABEP decreases with the increase in diversity or-der 𝐿.

For example, a fixed ABEP of 𝑃𝑏 = 10−5 can be achievedat SNR=45 dB with 𝐿 = 1, or SNR=25.4 dB with 𝐿 = 2, orSNR=19.6 dB with 𝐿 = 3 or SNR=14.6 dB with 𝐿 = 4. Thus

0 5 10 15 20 25 30 35 40 4510

−6

10−5

10−4

10−3

10−2

10−1

100

Average SNR per bit [dB]

Ave

rag

e B

it E

rro

r P

rob

abili

ty

SimulationAnalytical

L=1

L=2

L=3L=4

AWGN

Fig. 2: Binary MRC-Chirpover Rayleigh Fading Chan-nel.

−10 −5 0 5 10 15 20 25 3010

−6

10−5

10−4

10−3

10−2

10−1

100

Average SNR per Bit [dB]

Ave

rag

e B

it E

rro

r P

rob

abili

ty

SimulationAnalytical

L=1

L=2

L=3

L=4

AWGN

Fig. 3: Binary MRC-Chirpover Nakagami-𝑚 FadingChannel (𝑚 = 2).

comparing to single antenna system, MRC-Chirp system al-lows an SNR gain of more than 19.6 dB, 25.4 dB and 30.4dB using two, three and four receive antennas, respectively.Also, it is observed from the figure that both simulation andanalytical results are in close agreement.

Error performance over Nakagami-𝑚 channel with fadingparameter 𝑚 = 2 for different 𝐿 is illustrated in Fig. 3. Itis apparent from this figure that diversity order 𝐿 providesimprovement in ABEP and the figure also verifies the derivedanalytical results.

The effects of channel correlation in MRC-Chirp systemare shown in Figs. 4 and 5, where ABEP of binary chirpmodulation with diversity orders, 𝐿 = 3 and 𝐿 = 5, overan exponential correlation profile across Nakagami-𝑚 fadingchannel (𝑚 = 1) is represented. From Fig. 5, for a fixedABEP of 𝑃𝑏 = 10−5, an SNR loss of 3.6 dB with correla-tion 𝜌 = 0.5 is noticed as compared to uncorrelated fading.Both figures depict that channel correlation degrades the errorperformance of MRC-Chirp system.

−10 −5 0 5 10 15 20 25 3010

−6

10−5

10−4

10−3

10−2

10−1

100

Average SNR per Bit [dB]

Ave

rag

e B

it E

rro

r P

rob

abili

ty

Uncorrelatedρ = 0.2ρ = 0.5ρ = 0.9

Fig. 4: Binary MRC-Chirpwith 𝐿 = 3 over CorrelatedNakagami-𝑚 Fading Chan-nel (𝑚 = 1).

−10 −5 0 5 10 15 20 25 3010

−6

10−5

10−4

10−3

10−2

10−1

100

Average SNR per Bit [dB]

Ave

rag

e B

it E

rro

r P

rob

abili

ty

Uncorrelatedρ = 0.2ρ = 0.5ρ = 0.9

Fig. 5: Binary MRC-Chirpwith 𝐿 = 5 over CorrelatedNakagami-𝑚 Fading Chan-nel (𝑚 = 1).

7. CONCLUSIONS

This paper investigated the error performance of MRC-Chirpsystem in independent and correlated fading channels. Theanalysis included both Rayleigh and Nakagami-𝑚 fading chan-nels. MGF based approach was employed and error probabil-ity expressions were derived, one of our major contribution

in this paper. Moreover, an SNR gain of more than 30.4 dBwas noticed with the aid of diversity order 𝐿 = 4 as com-pared to without diversity. Furthermore, exponential channelcorrelation model was employed to examine the error perfor-mance of the MRC-Chirp system in correlated fading and anSNR loss of 3.6 dB with correlation 𝜌 = 0.5 was observedfor binary chirp modulation with 𝐿 = 5. It was found that thechannel correlation degrades the performance of MRC-Chirpsystem. Finally, Monte Carlo simulation was carried out andthe simulation results were compared with the obtained ana-lytical results. It is observed that both analytical and simula-tion results compare very well thereby validating the former.This work can be further extended for 𝑀 -ary chirp modula-tion and the challenge may be in deriving accurate symbolerror probability, one of our future goal.

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