New Sign Hadamard-γ-Haar-χ Transform inTernary Communication System

Bogdan J. Falkowski and Shixing YanSchool of Electrical and Electronic Engineering, Nanyang Technological University

Block S1, 50 Nanyang Avenue, Singapore 639798

Abstract— The application of a novel sign Hadamard-γ-Haar-χ transform in designing a digital communication system isconsidered. The nonlinear transform converts binary/ternaryvectors into digital spectral domain and is invertible. With itsunique and isomorphic properties, the sign Hadamard-γ-Haar-χ transform is suitable for security coding in a communicationsystem. Ternary amplitude frequency shift keying is proposedfor the implementation of the system. Power spectral densityof the resultant signalling is formulated and analyzed. Thenon-coherent receiver is designed and proposed as a ternarycommunication system. New transform is extremely effective interms of computational costs when compared with known signWalsh transform in application inside ternary communicationsystem.

I. INTRODUCTION

In many applications of computer engineering and science,where logic functions need to be analyzed or synthesized,it is useful to transform such functions to the correspondingspectral domain that provides various new insights into solvingsome important problems. The most popular ternary transformis Reed-Muller transform over GF(3) [1]. It operates on ternarylogic functions and provides 3n ternary spectra for a giventernary function. Another family of invertible nonlinear trans-forms, which uniquely map ternary logic functions into ternarytransform space, are sign transforms. The first transform underthe name of ‘sign transform’ was based on Walsh functions[1], [2] and is known as sign Walsh transform [1].

The rationalized Hadamard-Haar transform was proposedin [3]. In [4], an application based on rationalized Hadamard-Haar transform was developed for multirate filter bank struc-tures. In this paper, a new transform called sign Hadamard-γ-Haar-χ transform that is related to sign Hadamard-Haar trans-form introduced in [5] is applied in ternary communicationsystem. The basic definitions for this transform have beengiven here. In the new recursive sign transform, γ and χ showthe number of butterflies in Hadamard and Haar parts of signHadamard-γ-Haar-χ transform. Besides applications in ternarylogic design as shown for sign Hadamard-Haar transform in[5], a new sign Hadamard-γ-Haar-χ transform can be usedwhen there is a need for a unique coding of ternary vectorsinto the spectral domain of the same dimensions. In this articlewe are showing one such application in the form of ternarycommunication system and compare computational advantagesof our new transform over known sign Walsh transform thatis just one case of our new transform for γ = n − 1 and

χ = 1. Another area in which our new transform could beused is development of cryptographic functions that need tobe immune to input transformations.

The structure of this article is as follows. For comparisonpurpose, Section II covers basic definitions of Sign Walshas well as sign Hadamard-γ-Haar-χ transforms. Section IIIshows computational advantages of our new transform oversign Walsh transform in ternary communication system whileSection IV concludes the paper.

II. DEFINITIONS OF SIGN WALSH AND SIGN

HADAMARD-γ-HAAR-χ TRANSFORMS

The following symbols are used: Let −→x n={xn, xn−1, . . . ,xp, . . . , x2, x1}, −→α n = {αn, αn−1, . . . , αp, . . . , α2, α1} and−→ω n={ωn, ωn−1, . . . , ωp, . . . , ω2, ω1} be n-tuples over GF(2).The symbol xp stands for a data variable, αp representsa sign Walsh transform variable, and ωp a sign Hadamard-γ-Haar-χ transform variable, p is an integer and 1≤ p≤n,γ and χ are integers where 0 ≤ γ < n, 0 < χ ≤ n andγ+χ = n. Let

−→F = [F0, F1, . . . , Fj , . . . , FN−2, FN−1] be

a ternary vector. For example, it can be the S-coded truthvector [1] of f : (0, 1)n → (−1, 0, 1) where the value ofFj (1≤ j < N) is given by F (−→x n) when

∑np=1 xp2p−1=j.

Let−→WF = [w0, w1, . . . , wj , . . . , wN−2, wN−1] and

−−→HHF =

[hh0, hh1, . . . , hhj , . . . , hhN−2, hhN−1] be the vector corre-sponding to sign Walsh spectrum of

−→F and sign Hadamard-

γ-Haar-χ spectrum of−→F , accordingly. The value of wj (0 ≤

j < N) is given by−→WF (α) when

∑np=1 αp2p−1 = j. The

value of hhj (0 ≤ j < N) is given by−−→HHF (ω) when∑n

p=1 ωp2p−1 = j. Let−→O i represent the vector of i zeros,

1 ≤ i ≤ n. Let the symbol ⊕c represent cyclic addition,the symbol ⊕d represent dyadic addition, and the symbol ∧represent bit-by-bit logical AND.

When the above operations are applied to two vectors−→A l

and−→B k, 1 ≤ l < k, l and k are two different integer numbers,

they result in the vector−→C k of the length k. Only l elements

of−→B k and all elements of

−→A l are manipulated on, remaining

(k−1) elements of the resulting vector−→C k are not affected by

the applied operation and are simply the same as the elementsof the vector

−→B k between positions k and l + 1.

Definition 1: An invertible sign Hadamard-γ-Haar-χ trans-form hh and its inverse transform hh−1 are the map-

1230-7803-9197-7/05/$20.00 © 2005 IEEE.

pings hh : {+1, 0,−1}N → {+1, 0,−1}Nhh and hh−1 :

{+1, 0,−1}Nhh→ {+1, 0,−1}N , where N= 2n. In the above

equations, {+1, 0,−1}Nhh represents a set with the elements

from {+1, 0,−1}N permuted by the mapping hh of all theelements of the set {+1, 0,−1}N . The cardinality of theoriginal data set {+1, 0,−1}N and its transformed spectrumis equal to 3N .

Definition 2: An invertible forward sign Walsh transform wis [1]:

w (−→α n) = sign1∑

xn=0

[sign1∑

xn−1=0

[sign1∑

xn−2=0

[. . .

sign1∑

x1=0

[f (−→x n) (−1)

n∑p=1

αpxp

] . . .]]] (1)

The inverse sign Walsh transform is [1]:

f (−→x n) = sign1∑

α1=0

[sign1∑

α2=0

[sign1∑

α3=0

[. . .

sign1∑

αn=0

[w (−→α n) (−1)

n∑p=1

αpxp

] . . .]]] (2)

In (1) and (2), 1 ≤ p ≤ n.

Definition 3: An invertible forward sign Hadamard-γ-Haar-χ transform hh is defined as:

hh(−→On ⊕d ω1 ⊕d ωχ+12χ ⊕d ωχ+22χ+1 ⊕d . . . ⊕d

ωn−12n−2 ⊕d ωn2n−1) = sign1∑

xn=0

[(−1)xnωn [sign

1∑xn−1=0

[(−1)xn−1ωn−1 [. . . sign1∑

xχ+2=0

[(−1)xχ+2ωχ+2 [

sign1∑

xχ+1=0

[(−1)xχ+1ωχ+1 [sign1∑

xχ=0

[sign1∑

xχ−1=0

[. . . sign

1∑x2=0

[sign1∑

x1=0

[(−1)xχω1f(−→x n)]] . . .]]]]]] . . .]]]] (3)

and

hh(−→On ⊕d

−→ω i ⊕d 2i ⊕d ωχ+12χ ⊕d ωχ+22χ+1 ⊕d

. . . ⊕d ωn−12n−2 ⊕d ωn2n−1)= sign1∑

xn=0

[(−1)xnωn [

sign1∑

xn−1=0

[(−1)xn−1ωn−1 [. . .sign1∑

xχ+1=0

[(−1)xχ+1ωχ+1[

sign1∑

xχ−i=0

[sign1∑

xχ−i−1=0

[. . . sign1∑

x2=0

[sign1∑

x1=0

[(−1)xχ−1

f{[(−→Oχ ⊕d−→ω i) ⊕c (χ−i)] ⊕d

−→x χ−i}]] . . .]]]] . . .]]]] (4)

where 1 ≤ i < χ.The inverse sign Hadamard-γ-Haar-χ transform is:

f(−→x n)=sign

{(−1)x1 [sign

1∑ωχ+1=0

(−1)xχ+1ωχ+1 [sign

1∑ωχ+2=0

(−1)xχ+2ωχ+2 [. . . sign1∑

ωn−1=0

(−1)xn−1ωn−1 [sign

1∑ωn=0

(−1)xnωn [hh{[(−→O 1 ∧ −→x χ) ⊕c 1] ⊕d 2χ−1 ⊕d

ωχ+12χ ⊕d ωχ+22χ+1 ⊕d . . . ⊕d ωn−12n−2 ⊕d

ωn2n−1}]] . . .]]] + sign

{(−1)x2 [sign

1∑ωχ+1=0

(−1)xχ+1ωχ+1 [. . . sign1∑

ωn=0

(−1)xnωn [hh{[(−→O 2 ∧ −→x χ)

⊕c2] ⊕d 2χ−2 ⊕d ωχ+12χ ⊕d ωχ+22χ+1 ⊕d . . . ⊕d

ωn−12n−2 ⊕d ωn2n−1}] . . .]] + . . . + sign

{

(−1)xχ−1 [sign1∑

ωχ+1=0

(−1)xχ+1ωχ+1 [. . . sign1∑

ωn=0

(−1)xnωn [hh{[(−→Oχ−1 ∧ −→x χ) ⊕c (χ − 1)] ⊕d 2 ⊕d

ωχ+12χ ⊕d ωχ+22χ+1 ⊕d . . . ⊕d ωn−12n−2 ⊕d

ωn2n−1}] + sign

{(−1)xχ [sign

1∑ωχ+1=0

(−1)xχ+1ωχ+1 [

. . . sign1∑

ωn=0

(−1)xnωn [hh(−→Oχ ⊕d ω1 ⊕d ωχ+12χ

⊕dωχ+22χ+1 ⊕d . . . ⊕d ωn−12n−2 ⊕d

ωn2n−1)] . . .]]}

. . .]]}

. . .

}}. (5)

In (1) - (5), sign z =

−1 if z < 0,

0 if z = 0,

1 if z > 0.

III. APPLICATION OF SIGN HADAMARD-γ-HAAR-χTRANSFORM IN TERNARY COMMUNICATION SYSTEM

Similarly to known sign Walsh and sign Hadamard-Haartransforms, sign Hadamard-γ-Haar-χ transform exhibits non-linear properties. Though non-linear, these transforms areunique and invertible. With intrinsic coding property, thesetransforms reveal possible application in secured communi-cation systems [6]. In this Section, the application of signHadamard-γ-Haar-χ as the sequence for ternary communica-tion system will be considered. For comparison the well knownsign Walsh transform will also be used in the same application.In such a system, the incoming binary/ternary data is first

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SerialParallel

Converter

ParallelSerial

Converter

signHadamard- .-Haar- -qtransform

BandpassFilter

centers at acf

V

VCO

q-control

N N

)(tsBinary/Ternary

Data

γχ

Fig. 1. Block diagram of TAFSK transmitter.

encoded by performing sign Hadamard-γ-Haar-χ transform onit. The digital modulation technique responsible for carryinginformation in sign Hadamard-γ-Haar-χ spectra is TernaryAmplitude Frequency Shift Keying (TAFSK) [6], [7]. In thissignalling, a ternary +1 is transmitted by a Radio Frequency(RF) pulse of carrier cos ω1t, a ternary −1 is transmitted byan RF pulse of carrier cos ω2t, and a 0 corresponds to no RFpulse. The technique combines Binary Amplitude Shift Keyingand Binary Frequency Shift Keying for the ternary case. ThePower Spectral Density (PSD) of the resultant signalling isgiven by

S(ω) =12

[A1(ω + ω1) + A1(ω − ω1)+ A1(ω + ω2)

+ A1(ω − ω2)] (6)

where A1(ω)= 29T0sinc2

(ωT02π

)[1+ π

T0

∞∑m=−∞

δ(ω− 2πm

T0

)].

Proof Let Sign Hadamard-γ-Haar-χ transform of bi-nary/ternary data streams be represented by

A(t) =∞∑

k=−∞akp(t − kT0) =A1(t) + A2(t)

where p(t) represents a full rectangular pulse whichrepeats every T0 seconds, and it is assumed that ak isequally likely to be +1, 0 or −1, i.e., P (ak = 1) =P (ak = −1) = P (ak = 0) = 1

3 . Furthermore A1(t) =∞∑

k=−∞a〈1〉

kp(t − kT0) and A2(t) =

∞∑k=−∞

a〈−1〉k

p(t − kT0)

with P(a〈1〉

k= 1

)= P

(a〈−1〉

k= −1

)= 1

3 andP

(a〈1〉

k= 0

)= P

(a〈−1〉

k= 0

)= 2

3 . The PSD of ON-

OFF signalling [7] is A0 (ω) = |P (ω)|2T0

[ ∞∑m=−∞

Rme−jmωT0

]

where Rm is the coefficient of the time-autocorrelationfunction of the signalling and j =

√−1. Thereforethe PSD of A1(t) is A1(ω) = A0(ω), with

R0 = limNT →∞

1NT

NT∑k=1

(a〈1〉k )2 = 1

3 and Rm =

limNT →∞

1NT

NT∑k=1

a〈1〉k a

〈1〉k+m = 1

9 , if m �= 0.

BandpassFilter

centers at acf

BandpassFilter

centers at a2f

BandpassFilter

centers at a1f

noisy

)(ts

EnvelopeDetector

EnvelopeDetector

NegativeThreshold

Device

PositiveThreshold

Device

0Tt =

0Tt =

∑ParallelSerial

ConverterN

Inverse signHadamard--Haar- -qtransform

q-control

NSerial

ParallelConverterrecovered

message

χγ

Fig. 2. Block diagram of TAFSK receiver.

Since P (ω) = T0sinc(ωT02π ) and using

∞∑m=−∞

e−jmωT0 =

2πT0

∞∑m=−∞

δ(ω − 2πmT0

), then

A1(ω) =29T0sinc2(

ωT0

2π)

[1 +

π

T0

∞∑m=−∞

δ(ω − 2πm

T0)

].

Since (a〈1〉k )2 = (a〈−1〉

k )2 and a〈1〉k a

〈1〉k+m = a

〈−1〉k a

〈−1〉k+m,

therefore A2(ω) = A1(ω). Using the frequency shiftingproperty and since s(t) = A1(t) cos ω1t + A2(t) cos ω2t, theproof of (6) is complete.

For ω2 > ω1, if ω2 − ω1 = 2ω0 then the transmissionbandwidth of TAFSK signalling is 4f0 (where f0 = 1/T0 isthe clock frequency).

A new recursive transform can also be developed basedon sign Hadamard-γ-Haar-χ transform. As an example, signHadamard-γ-Haar-χ transform can be applied twice onto aternary truth column vector

−→F such that a new transform

space is developed. The overall transform is named as signHadamard-γ-Haar-χ-2 transform. In general, there are together3N different sign Hadamard-γ-Haar-χ transform spaces, de-noted as sign Hadamard-γ-Haar-χ-q transforms with 1 ≤ q ≤3N , where when q = 1, the transform yields the original signHadamard-γ-Haar-χ transform. For comparison purpose, signWalsh-q transform can be defined similarly.

Fig. 1 shows the block diagram of a TASFK transmitter.The continuous streams of binary/ternary data are convertedto parallel words of length N by means of a serial-parallelconverter. Sign Hadamard-γ-Haar-χ-q transform is applied toeach word before converting back to the format of serial data.The output signal V of the parallel-serial converter controlsthe output frequency of the voltage-controlled oscillator, andboth outputs are fed together into the mixer. The output ofthe mixer is TAFSK signalling. The output of the oscillator ismathematically given by

V CO = V0 cos [2π (fc + V fm) t] (7)

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TABLE I

COMPARISON OF COMPUTATIONAL COSTS IN TERNARY COMMUNICATION

SYSTEM

q n

sign Hadamard-γ-Haar-χ-qtransform

sign Walsh-q transform

[(γ + 2) 2n+1 − 2γ+2] × q qn2n+1

31 12 (γ=0) 122 36 (γ=0) 48 (γ=1) 483 84 (γ=0) 120 (γ=1) 144

54 300 (γ=0) 560 (γ=2) 6405 920 (γ=1) 1440 (γ=3) 16006 1880 (γ=1) 3520 (γ=4) 3840

87 11776 (γ=4) 14336 (γ=6) 143368 16256 (γ=2) 27648 (γ=5) 327689 40704 (γ=3) 69632 (γ=7) 73728

1010 142080 (γ=5) 179200 (γ=7) 20480011 163680 (γ=2) 325120 (γ=6) 45056012 245680 (γ=1) 409280 (γ=3) 983040

1213 1178880 (γ=4) 1374720 (γ=5) 255590414 2750976 (γ=5) 3532800 (γ=7) 550502415 3145536 (γ=2) 6288384 (γ=6) 11796480

where V ∈ {−1, 0, 1} and V0 is an arbitrary amplitude. Iffm = f0, then the resultant transmission bandwidth will be4f0, and fc + f0 = f2, fc − f0 = f1.

Fig. 2 shows a block diagram of a TAFSK receiver. Theincoming noisy RF signal is bandpass filtered centered atfrequency fc. The bandpass filters centered at f2 and f1 arematched to the two RF pulses corresponding to ternary logicof −1 and +1, accordingly. The outputs of the two matchedfilters are detected by two envelope detectors. The envelopedetector is sampled at t = T0 to make the ternary decision of−1 or 0 and 1 or 0 by negative and positive threshold devices,respectively. The output of summer is ternary, which is fed toa serial-parallel converter, an inverse sign Hadamard-γ-Haar-χ-q transform block and a parallel-serial converter to extractthe original message.

The proposed non-coherent system is the simplest imple-mentation of a ternary communication system. Other possibil-ities include the complicated M-ARY communication systems[6]. The addition of a sign Hadamard-γ-Haar-χ transformprovides security in the digital communication system. Thelevel of security is easily adjustable by controlling γ, χ andq, which corresponds to sign Hadamard-γ-Haar-χ-q transformapplied q times. If γ, χ and q are varied for each wordtransformed in a manner transparent to a friendly receiver,the level of security in the communication system will befurther enhanced. It is obvious that sign Hadamard-γ-Haar-χ transform provides better security properties by its designand is suitable for cryptographic systems.

The application of sign Hadamard-γ-Haar-χ transform ina ternary communication system has been considered. Suchapplication may be used by other quantized transforms aswell. When the well known sign Walsh transform is appliedto the ternary communication system, sign Hadamard-γ-Haar-χ-q transform will be replaced by sign Walsh-q transform inFig. 1 and Fig. 2. Table I shows some results about compu-

tational costs of two systems where sign Hadamard-γ-Haar-χ-q transform and sign Walsh-q transform are implementedrespectively. The computational costs which mean the numberof additions and subtractions required for the generation offorward and inverse transforms represent the computationalcomplexity of the calculations for the transforms. From Table1, it is obvious that the system based on sign Hadamard-γ-Haar-χ-q transform is much more efficient than the one basedon sign Walsh-q transform due to much smaller number ofcomputations especially for higher q and n.

IV. CONCLUSION

A non-linear transform called sign Hadamard-γ-Haar-χtransform has been applied in ternary communication system.It has been shown that the transform exhibits a non-linearproperty. Essentially, it transforms ternary data into ternaryspectrum. Though non-linear, the transform is unique, andhence invertible. Recursive definition of the transform hasbeen given. When a fast flow diagram is directly implementedin software, there is no need to keep the original data, andeach consecutive butterfly requires a geometrically smallernumber of operations on the transformed data. Each time,2n binary/ternary data need to be stored in the memory.The sign Hadamard-γ-Haar-χ transform block may also beimplemented as a parallel dedicated processor.

Sign Hadamard-γ-Haar-χ transform is just another repre-sentative of a family of quantized transforms based on theusage of sign function as the quantizer and as shown in thisarticle it has advantage in terms of computational costs overwell known sign Walsh transform in the ternary communica-tion system. In addition, sign Walsh transform as well as signHadamard-Haar transform belong to our novel sign Hadamard-γ-Haar-χ transform as they are just one possible case of thenew transform. By introducing such general ternary quantizedtransform we can also improve spectral representation ofternary logic functions by finding the best γ and χ for a givenfunction.

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