Interference Avoidance in Spectrally Encoded Multiple …cooper/SPOT/Reference/SPE/Interference... · Interference Avoidance in Spectrally Encoded Multiple Access Communications Using

Interference Avoidance in Spectrally Encoded Multiple AccessCommunications Using MPSK Modulation

A.S. Nunez, M.A. Temple, R.F. Mills and R.A. RainesDepartment of Electrical and Computer Engineering

Air Force Institute of TechnologyWright-Patterson AFB, OH 45433

Abstract— Spectral encoding is employed to provide interfer-ence avoidance and multiple access capability using M-ary phaseshift keyed (MPSK) data modulation. Communication symbolsare formed using composite phase modulation (independent dataand coded multiple access) on selected spectral components andthen inverse Fourier transforming to obtain time domain wave-forms. This technique enables multiple access and adaptive chan-nel interference suppression. One inherent advantage is ana-lytic tractability of phase modulation components across domainswhich enables robust theoretical performance prediction withvariation in multiple access phase value assignment. Detectionand estimation is accomplished using conventional correlation re-ceiver techniques and error performance is shown to be consistentwith conventional MPSK signaling. Analytic and simulated re-sults are provided for multiple access bit error performance andinterference suppression demonstrated using randomly assigned,uniformly distributed multiple access phase values.

I. INTRODUCTION

Spectrally encoded communication techniques have beenshown capable of providing considerable protection from var-ious types of interference while providing effective multipleaccess capability [1, 2]. Such techniques have generally em-ployed M-ary cyclic shift keying, a form of orthogonal sig-naling, as the primary data modulation. Expanding uponprevious work, a form of spectrally encoded M-ary PhaseShift Keying (MPSK) is introduced whereby phase modula-tion (independent data and coded multiple access) is appliedin the frequency domain, and demodulation is performed inthe time domain [3]. In conjunction with providing effec-tive multiple access communications with interference avoid-ance mechanisms, the proposed technique is attractive becausethe phase modulation factors are analytically tractable betweenfrequency and time domains. This tractability permits detaileddevelopment, analysis and characterization of communicationperformance. Analytic development of frequency and timedomain representations of spectrally encoded MPSK signalsis provided, and multiple access communication performance(bit error rate) validated via simulation. Spectrally encodedMPSK performance is shown to be consistent with conven-tional MPSK signaling while providing interference avoidance(suppression) commensurate with that of previous orthogonaltransform domain techniques.

II. SPECTRALLY ENCODED MPSK SYMBOL

DEVELOPMENT

Spectrally encoded MPSK signaling begins in the frequencydomain where composite phase modulation, comprised of in-dependent data and coded multiple access components, are ap-plied to sinusoidal components. Although this spectral modu-lation process is conceptually similar to orthogonal frequencydivision multiplexing (OFDM) [4], the modulation processhere differs in that the data phase modulation is constant acrossall spectral components. The mth spectrally encoded MPSKcommunication symbol for user v is generated from

S(v)m (f) =

K∑k=1

A(v)k

[∆−

k + ∆+k

](1)

∆−k = δ(f − kfsb)e

+j[φ

(v)k +θ(v)

m

](2)

∆+k = δ(f + kfsb)e

−j[φ

(v)k +θ(v)

m

](3)

where superscript v in (1) denotes a specific user,v ∈ [1, 2, ..., Nu], subscript m denotes a specific communica-tion symbol, m ∈ [1, 2, ...,M ], K is the number of sinusoidalcomponents, fsb is sinusoidal component spacing (symbol fre-quency), Tsb = 1/fsb is symbol duration, Ak is the ampli-tude weight of the kth component, Ak ≥ 0, θ

(v)m is the Data

Phase Modulation for the mth symbol of user v, and φ(v)k is

the Coded Multiple Access (MA) Phase Modulation for user v.The time domain representation of spectrally encoded

MPSK symbols is obtained by taking the inverse Fourier trans-form of (1), yielding symbols of the form [3]

s(v)m (t) = 2

K∑k=1

A(v)k cos

[2πfsbkt + φ

(v)k + θ(v)

m

](4)

for 0 ≤ t ≤ Tsb and having total symbol energy over intervalTsb equaling

Es(v)m

= 2Tsb

K∑k=1

[A

(v)k

]2

∀ m ∈ [1, 2, ...,M ] (5)

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It was noted earlier that sinusoidal amplitudes are assignedsuch that Ak ≥ 0. The inclusion of zero amplitude valuesis specifically permitted such that 1) it is possible to avoidtransmitting energy in selected spectral regions such that in-terference with other systems may be intentionally avoided, or2) the transmitted communication signals of interest “avoid”crowded spectral regions and interference effects are mini-mized during receiver detection and estimation, i.e., the re-ceiver avoids extracting energy (noise and other interfering sig-nals) from spectral regions where no signal energy was inten-tionally placed; this is the fundamental spectral notching andinterference avoidance mechanism demonstrated previously intransform domain communication systems [1, 2]. In the in-terference avoidance case where symbols are specifically de-signed to avoid spectrally crowded regions, symbol energy isgenerally scaled following spectral notching to ensure all com-munication symbols are transmitted with equal energy (assum-ing of course that equal energy signaling is the goal).

Communication performance using the spectrally encodedwaveform of (4) is characterized by considering the cross cor-relation between two communication symbols for user v. Con-sidering the mth and nth communication symbols of (4), andthe symbol energy expression of (5), integration over symbolinterval Tsb yields symbol cross correlation R

(v)mn given by

R(v)mn = E(v)

s cos[θ(v)

m − θ(v)n

](6)

assuming equal energy signaling such thatEs(v) = E

s(v)m

= Es(v)n

for all m and n. Cross correla-tion results of (6) clearly indicate how symbol phase valueassignment dictates communication performance.

A. Communication Performance

Using data phase modulation of θm = (2πm/M) forM = 2l communication symbols, produces R

(v)mn characteris-

tics similar to conventional MPSK signaling such that (6) dic-tates correlation receiver performance. Thus, the spectrally en-coded MPSK technique achieves theoretic bit error probability(Pb) equivalent to conventional MPSK. An analytic Pb expres-sion for binary phase shift keying, using optimum coherent de-tection over an AWGN channel, is given by (7) where Eb isaverage energy per bit and No is noise power spectral density.For M > 2 and large energy to noise ratios, Pb is well approx-imated by (8) assuming Gray code bit-to-symbol mapping [5].

Pb = Q

[√2(

Eb

No

) ](7)

Pb ≈ 2Q

l

[√2l

(Eb

No

)sin

( π

M

)](8)

Communication results from Monte Carlo simulation ofM = 2, 4, 8 and 16 spectrally encoded MPSK signaling areshown in Fig. 1 for K = 255 sinusoidal components. Note that

0 5 10 15 2010

−5

10−4

10−3

10−2

10−1

100

Eb/N

0 (dB)

Prob

abili

ty o

f B

it E

rror

(P b)

Analytic BPSKAnalytic QPSKAnalytic 8PSKAnalytic 16PSKTD−BPSKTD−QPSKTD−8PSKTD−16PSK

Fig. 1. Communication Performance: Pb versus Eb/No for Spectrally En-coded MPSK Using K = 255 Sinusoidal Components [3]

the M = 2 and M = 4 data is virtually coincident. Less than2.7% variation between simulated and analytic results is ex-hibited for all modulations at Eb/No > 0 dB and Pb > 10−5.

B. Multiple Access Performance

Introducing a second network user, the cross correlationover one symbol period between the mth symbol of user vgiven by (4) and the nth symbol of user w given by (9) can beexpressed as given by (10) and (11) [3].

s(w)n (t) = 2

K∑k=1

A(w)k cos

[2πfsbkt + φ

(w)k + θ(w)

n

](9)

R(vw)mn = 2Tsb

K∑k=1

A(v)k A

(w)k cos

[Φ(vw)

mn (k)]

(10)

Φ(vw)mn (k) = φ

(v)k − φ

(w)k + θ(v)

m − θ(w)n − 2πfsbkτ (11)

Delay parameter τ in (11) is the relative received time differ-ence between communication symbol boundaries of user v andw. Under synchronized network conditions, (τ = 0) and usersymbol boundaries are perfectly aligned during receiver corre-lation and symbol estimation. For the synchronized MPSKsystem, assuming 1) data and multiple access phase modu-lations in (11) are mutually independent ∀ m, n, v, and w,and 2) data phase modulations (θ) are discrete, uniformly dis-tributed random variables over [0, 2π], it can be shown that

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E{

R(vw)mn

}= 0 for all M and the variance of (10) can be

written as shown in (12) where Fφ(j, k) equals (13) and (14)for the binary (M = 2) and M-ary (M > 2) cases, respectively.

Var{

R(vw)mn

}= 4T 2

sb

K∑j=1

K∑k=1

A(v)j A

(w)j A

(v)k A

(w)k Fφ(j, k)

(12)

FM=2φ (j, k) = cos

[φ

(v)j − φ

(w)j

]cos

[φ

(v)k − φ

(w)k

](13)

FM>2φ (j, k) = cos

[φ

(v)j − φ

(w)j − φ

(v)k + φ

(w)k

](14)

Multiple access interference (MAI) effects are analyzedby introducing an interfering term (NI ) into bit error analy-sis as an additional noise contribution, e.g., if the MAI canbe approximately characterized as relatively wide band thenEb/(No + NI) can replace Eb/No in non-network Pb expres-sions to obtain reasonable approximations. For the spectrallyencoded MPSK system under consideration, multiple accessinterference can be effectively reduced by minimizing (12).

Assuming the multiple access phase modulations (φ) in(13) and (14) are uniformly distributed random variables over[0, 2π], and equal power is received from all users, it can beshown that the variance expression of (12) identically reducesto the expression in (15) which is valid for all M [3].

Var{

R(vw)mn

}= 2T 2

sb

K∑k=1

A4k (15)

In this case, the amount of MAI power (NI/2) impacting de-tection and estimation is proportional to the variance of R

(vw)mn

(cross correlation power) [6] and may be expressed as

NI

2≈ 1

ES

NU−2∑v=1,v �=w

Var{

R(vw)mn

}(16)

NI

2≈ 2T 2

sb

ES(NU − 2)

K∑k=1

A4k (17)

For M = 2l symbols having Es = lEb average energy, theNI multiple access interference of (17) can be substituted intoEb/(No + NI) as shown in (18) assuming the multiple accessinterference effects are accurately incorporated as wide bandinterference. The result of (18) is then substituted into Pb ex-pressions of (7) and (8) as shown in (19) and (20) to obtainanalytic MPSK Pb estimates for multiple access networks us-ing M = 2 and M > 2 communication symbols, respectively.

Eb

No + NI=

[No

Eb+

4T 2sb

lEb2 (NU − 2)

K∑k=1

A4k

]−1

(18)

2 7 12 17 22 27 32

10−5

10−4

10−3

10−2

10−1

Number of Users (NU

)

Pro

babi

lity

of B

it E

rror

(P b)

Analytic BPSKAnalytic QPSKAnalytic 8PSKAnalytic 16PSKTD−BPSKTD−QPSKTD−8PSKTD−16PSK

Fig. 2. Multiple Access Communication Performance using Uniformly Dis-tributed Phase Modulation with fixed Eb/No = 10.0 dB (all modulations)and K = 255 Sinusoidal Components [3]

Pb = Q

[√2(

Eb

No + NI

) ](19)

Pb ≈ 2Q

l

[√2l

(Eb

No + NI

)sin

( π

M

)](20)

To verify the theoretical multiple access Pb results presentedin (18) through (20), Monte Carlo simulations were conductedusing M = 2, 4, 8, and 16 spectrally encoded MPSK net-works containing up to NU = 32 users. For all cases con-sidered, simulated results agreed very favorably with analyticresults of (19) and (20) when incorporating multiple access in-terference according to (18). Representative results of Pb ver-sus NU for Eb/No = 10.0 dB are shown in Fig. 2. Analysisof data presented in Fig. 2 revealed that simulated and analyticresults are nearly identical; approximately 2.5% average error(averaged across NU ) between simulated and analytic resultsfor all modulations and Pb > 10−5.

III. ADAPTIVE INTERFERENCE AVOIDANCE

Adaptive interference avoidance first involves estimatingspectral locations where interference is occurring. In Fourier-based spectral estimation, the Discrete Fourier Transform(DFT) can be used as a computational tool to effectively de-scribe the spectral content of a time-sampled electromagneticenvironment. Using a given number of equally spaced DFTcomponents, each frequency component is described by fre-quency, amplitude, and phase parameters [7]. Frequency and

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amplitude information can be extracted from the DFT and usedto determine which spectral region(s) are clear of interference.The frequency spectrum can be adaptively estimated using anyavailable technique, e.g., periodogram, autoregressive linearpredictive filtering, etc, as demonstrated with other interfer-ence avoiding transform domain processing techniques [1,2,8].

After normalizing the estimated spectral response, a thresh-old is established such that all frequency components exceed-ing the threshold (those containing interference) are assigneda value of zero (0). Frequency components whose magnitudesfall below the threshold are deemed interference free and as-signed a value of one (1) [8]; this process yields a vector rep-resenting an ideal rectangular spectrum which has been selec-tively “notched.” Assuming the transmitter and receiver ob-serve identical electromagnetic environments, the spectral es-timation and thresholding process yields identical “notched”magnitude vectors. An achievable condition assuming all in-terfering sources are sufficiently removed from the geographi-cal region containing the transmitter and receiver of interest.

Using the “notched” spectrum vector, phase values are as-signed to those components which are available for waveformgeneration, i.e., those with a magnitude of one. This phasecoding represents the form of spectral encoding employed hereand provides the degree of freedom required to enable effectiveinterference avoidance and multiple access capability.

After spectral estimation, thresholding, and spectral encod-ing, the remaining frequency components are scaled to ensurethat equal energy symbols are transmitted, independent of thenumber of frequency components being used. For example, ifonly one-third of the available frequency components remainafter thresholding, the amount of required energy contained ineach remaining component is tripled relative to the case whenno interference is present.

IV. SPECTRAL RESPONSE AND NOTCHING OF NARROW

BAND INTERFERENCE (NBI)

For demonstrating interference suppression capability, nar-row band interference (NBI) is introduced using P = 31 sinu-soids. Figure 3 shows the spectral response of the NBI con-sidered and the resultant spectral notching mask (dashed line)obtained for a threshold value of 1.0. Equations (21) through(23) analytically describe the NBI where amplitude weightsAp are shown in Fig. 3 and phase values φp were randomlyassigned for each iteration of the simulation.

I(f) =P∑

p=1

A(v)p

[∆−

p + ∆+p

](21)

∆−p = δ(f − pfsb)e+j[φp] (22)

∆+p = δ(f + pfsb)e−j[φp] (23)

For the NBI in Fig. 3, 8 of 31 (≈ 26%) sinusoidal com-ponents were notched out using the spectral notching mask

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

3

3.5

4

Frequency index (k)

Ap

Jamming SpectrumNotching Mask

Fig. 3. Spectral Response of Narrow Band Interferer (P = 31 Sinusoids) andSpectral Notching Mask (Dashed Line) for Threshold of 1.0

shown. The amplitudes of remaining frequency compo-nents were raised by approximately 16% (relative to the non-interferece case) for this 26% NBI case to maintain equal en-ergy conditions. Figure 4 shows NBI effects on MPSK perfor-mance for NU = 3 and an interference-to-signal power ratio(I/S) of 3.14 dB. Comparison of this data with Fig. 1 clearlyshows how the addition of MAI (one additional transmitter)and NBI have severely degraded communication performance.

−5 0 5 10 1510

−4

10−3

10−2

10−1

100

Eb/N

o (dB)

Prob

abili

ty o

f B

it E

rror

(P B

)

Analytic BPSKAnalytic QPSKAnalytic 8PSKAnalytic 16PSKSimulated BPSKSimulated QPSKSimulated PSKSimulated 16PSK

Fig. 4. Narrow Band Interference and NU = 3 Users Present With NoSpectral Notching: PB vs Eb/N0 for TD-MPSK Signaling with I/S = 3.14dB and P = 31 Sinusoids [3]

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By way of illustrating the interference avoidance capabil-ity of the proposed technique, representative results for QPSKmodulation is provided in Fig. 5 for the same I/S of 3.14 dBas used for Fig. 4 data. The significance of each line pre-sented in Fig. 5 is as follows: (1) The unfilled diamond datapoints represent baseline performance with no multiple ac-cess (MAI) or narrow band interference (NBI) present and nospectral components notched, (2) The filled diamond data il-lustrates performance degradation (relative to the unfilled dia-mond data) due solely to MAI resulting from one equal poweradditional user with no NBI present or notching used, (3) Thedashed line illustrates performance degradation (relative to thefilled diamond data) due to introducing the spectral notch (notethat the jammer is NOT present in this case and the degradationbetween the dashed line and filled diamond data is due solelyto corruption of communication symbol correlation propertiesbetween the two network users even though equal energy sig-naling is maintained for both); this dashed line represents thebest case performance that can be expected when NBI is intro-duced and notching applied, (4) The filled circle data pointsrepresent the case with both MAI and NBI present and nonotching applied, and (5) The unfilled circle data representsthe case with both MAI and NBI present and spectral notchingapplied (interference avoidance). The fact that the suppresseddata doesn’t completely improve to the dashed line is dueto NBI energy which remains in spectral components/regionswhich are outside the notched location(s), i.e., residual NBI en-ergy in sinusoidal components which falls below the thresholdand remains after spectral notching. There is approximately7.5 dB of processing gain (defined here as the reduction in re-quired Eb/No to achieve a fixed Pb) achieved at Pb = 10−3

when interference avoidance is applied. Similar suppressionperformance was achieved for all MPSK modulations consid-ered in Fig. 4.

V. CONCLUSION

Analytic development and simulated multiple access perfor-mance results are provided for an interference avoiding, spec-trally encoded M-Ary Phase Shift Keyed (MPSK) technique.As developed, the composite phase modulation, consisting ofindependent data and coded multiple access components, en-ables effective multiple access capability while spectral esti-mation and thresholding permits adaptive interference avoid-ance. In conjunction with providing effective multiple ac-cess communications and interference avoidance, the proposedtechnique is attractive because the phase modulation factorsare analytically tractable between frequency and time domains.This permits reliable theoretical performance characterizationfor variation in multiple access phase modulation assignment.Analytic and simulated results are provided using randomlyassigned, uniformly distributed multiple access phase values.Both multiple access communication and narrow band inter-ference avoidance (suppression) are demonstrated. Analyticand simulated Pb versus Eb/No performance of a non-network

−4 −2 0 2 4 6 8 10 12 14 1610

−4

10−3

10−2

10−1

100

Eb/N

o (dB)

Prob

abili

ty o

f B

it E

rror

(P B

)

No MAI, No NBI, No NotchingMAI Present, No NBI, No NotchingMAI Present, NBI Present, No NotchingMAI Present, No NBI, NotchingMAI Present, NBI Present, Notching

Fig. 5. Illustration of Narrow Band Interference (NBI) Avoidance for QPSKModulation Using P = 31 Sinusoids and I/S = 3.14 dB [3]

(single channel) spectrally encoded MPSK system is shownconsistent with conventional MPSK signaling. For a multi-ple access network, approximately 2.5% average error betweensimulated and analytic results is demonstrated in a multiple ac-cess environment containing up to 32 users. Up to 7.5 dB ofprocessing gain is demonstrated when interference suppressionmechanisms are employed.

REFERENCES

[1] R. W. Klein, M. A. Temple, R. A. Raines, and R. L. Claypoole, “Inter-ference avoidance communications using wavelet domain transformationtechniques,” IEE Electronics Letters, vol. 37, pp. 987–989, January 2001.

[2] P. J. Swackhammer, M. A. Temple, and R. A. Raines, “Performance sim-ulation of a transform domain communication system for multiple accessapplications,” Military Communications Conference Proceedings, MIL-COM 1999, vol. 2, pp. 1055–1059, October 1999.

[3] A. S. Nunez, “Interference suppression in multiple access communica-tions using m-ary phase shift keying generated via spectral encoding,”Master’s thesis, Air Force Institute of Technology, 2004.

[4] R. Van Nee and R. Prasad, OFDM for Wireless Multimedia Communica-tions. Boston, Massachusetts: Artech House, 2000.

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

[6] T. Rappaport, Wireless Communications. Upper Saddle River, New Jer-sey: Prentice Hall PTR, 1996.

[7] J. Proakis and D. Manolakis, Digital Signal Processing, Principles, Algo-rithms, and Applications. Saddle River, New Jersey: Prentice-Hall, 1996.

[8] M. Roberts, “Initial acquisition performance of a transform domain com-munication system: Modeling and simulation results,” MILCOM 2000Proceedings. 21st Century Military Communications. Architectures andTechnologies for Information Superiority, vol. 2, pp. 1119–1123, 2000.

“The views expressed in this article are those of the author(s)and do not reflect official policy of the United States Air Force,Department of Defense or the U.S. Government.”

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