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On the Fourier transform of bi-valued function with modulated transitions : frequency synthesis
N. Yahyabey, S. Hassani
To cite this version:
N. Yahyabey, S. Hassani. On the Fourier transform of bi-valued function with modulated transi- tions : frequency synthesis. Journal de Physique III, EDP Sciences, 1994, 4 (10), pp.2047-2055.
�10.1051/jp3:1994256�. �jpa-00249241�
/ Phy~<. III Fiance 4 (1994) 2047-2055 OCTOBER 1994, PAGE 2047
Classification Phj'.wc..I Ah-in act-v
02.30 02.90 06.30
On the Fourier transform of hi-valued function with modulated transitions : frequency synthesis
N. Yahyabey (') and S. Hassani (2)
('1 Univer~itd Fran~oi~ Rabelais de Tour~, U. F-R- Science & Techniques, Antenne Univer~itaire de Blois, Place Jean Jaurb~, 41000 Bloi~, France
(2) Centre de ddveloppement des techniques nucldaires, 2 Bd Frantz Fanon, Alger, Algeria
(Re<en.efi 15.Iwie 1993, iei.rued 15 Malt-h 1994, a«eptefi 1.Tell>' 1994)
Rksumk. Nous consid6ron~ une mdthode originate de gdndration d'harmonique~ aux frdquences fractionnaires obtenue par la modulation des transitions d'une fonction carrde. Le spectre produit
montre une ~tructure po,sddant des bifurcation~ dontle nombre est ddpendant du nombre de figure~
de la frdquence de l'onde modulante. Plus la partie fractionnaire de la frdquence de l'onde modulante est tongue, plus le nombre de figures de la frdquence de l'harrnonique produite croit dans la mdme proportion. Les rdsultats montrent que cette propridtd peut dtre mise en
wuvre pour la synthkse de frdquence.
Abstract. we con~ider a novel mode of fractional frequency harmonic generation obtained by
the ~inu;oidal modulation of the transition in~tants of a ~quare wave reference ~ignal. The
produced ~pectrum exhibits a bifurcation structure created by the number of figures contained in the frequency of the tran~ition modulating wave. The longer the fractional part of the frequency of the modulating wave I;, the more figures the i,alue of the frequency of the produced harmonic ha,.
Re,ult~ ,how that thi; property can be carried out in frequency synthesi~ operation.
1. Introduction.
In a previously published work [I( a method'ba~ed on the concept of the overall period wa~
derived for the analy~is of a non Fourier-transformable function in the u~ual sense [?, 3(. In this context, we propose the spectral analy~is theory of a signal represented by a bi-valued
function who~e tran;ition~ are modulated by a sinusoidal waveform function. On the other hand, it is customary to use the harmonic approach [4] when simultaneous generation of small number of harmonics i~ required. These approache~ u~e a~ building blocks a ~et of multipliers,
divider~ and mixers. The encountered problems are spurious output~ with an a~wciated
degradation in pha~e noise perforniance~. An other design approach of the component frequency generator block based on the spectral propertie~ derived from the above analy~is is
proposed.
2. Theory.
Let us define the following square waveform function
a (t, ~, co as follows,
« (t, ~, CO)
=
( jolt T~(~)) 2. Hit Tm+1/~(~)i + o(t Tm +1(~))i (ii
The transitions T~,(~), T~,~ j~~(~), T~,
~
(~) are given by, T,(~
=
,<T A sin (4 7r ~x) (2)
x is an integer or half-integer. A and T are constants.
The signal switches between two levels, the time of the x-th transition is varied from that of the reference square waveform by the delay A sin (4 7r ~x), ~ is a parameter. t ~ [0, co [, the co
symbol in the argument of the function refers to the upper limit of the sum. This form of
m
assures, when the sum is truncated to N, to have no tail (I.e., m (t
~ NT, F, N )
=
0). Other forms are however possible. (t) is the heaviside function. The constant A is chosen to be at most TM in order to avoid overlapping of adjacent T, for finite ~. We note that if the frequency
2 F
= I/p where p is an integer, m (t, ~, co ) becomes a periodic function repeating itself every p switches. The frequency 2 F is then of the form,
2 ~
=
f # ; f, q
are mutual prime numbers (3)
When ~ sati~fies the above equation, the period of the signal m (t, F, co is fT, where f is the
above defined integer. This periodicity is however lost when F becomes irrational,
m (t, F, co is hence a non Fourier-transformable function. Our using of (3) is due to the fact that every irrational number is in fact a rational number in any finite precision computation.
The Fourier coefficients of n (t, ~, co ), calculated independently from the nature of ~ in the
appendix, are given by,
C(b)=~ ~f
J,,( ~"~b) .~(b+2fi~). (1-Exp17r(b+2nF)) (4)
17rb,,~_~ T
J,,(.< is the Bessel function of the first kind of integral order fi and argument ~. The coefficients C (b) are different from zero for frequencies b satisfying the following fundamental relation,
b
= 2 ~p + p' p is an integer, p' in an odd integer (5) Relation (5) shows that the produced spectrum exhibits a particular structure. The produced frequencies are not given by the classical rule stating that harmonic frequencies are integral multiples of the fundamental frequency J, 3] and used in the process of harmonic generation
from a sinusoidal waveform reference function [4]. For q/f
=
0.9 and q/f
=
0.93 (e.g.) only
harmonics of frequencies containing respectively one figure and two figures are created with harmonics of integral frequencies (see the Appendix). With 2 ~ an integer, only integral frequencies are available but when 2
~ possesses one figure (0.9), the empty segment between every successive harmonic frequencies of the former amplitude spectrum will be occupied by
new harmonics possessing one figure according to relation (5). This bifurcation scheme will repeat itself for the two figures (0.93) and for every new figure that 2 ~ may possess (see Fig. I). This spreading is tenfold because we are using a decimal base. For another base
number B, it is B-fold. Figure 2 displays the spectra for 2 ~ =
0.9 and 2 ~
=
0.91 the lofiger
N° lo ON THE FOURIER TRANSFORM OF BI-VALUED FUNCTION 2049
em E
~ ~~ I
em 0,ab
~o~a~ci /~ ~l ~
'~
~~
em 0.abode...xy...
Fig. I. The ~tructure of the produced ~pectrum for different figures (a, b, c, ...) of the modulation
frequency 2 F. when 2 ~ posse~se, one figure (2 F
=. a ), the empty ~egment between every ~uccessive integer harmonic frequencie, of the former spectrum with 2 F E (E is an integral value), will be
occupied by new harmonic~ pmse~sing one figure according to the relation h 2 pF + p p is an integer, p' is an odd integer. we have a bifurcation proces~ dependent on the number of figure, that 2 F may posse~,.
the fi.actioiia/ part of 2 F is, the mfire figm.es the i>a/ue cij'the pioduc.ed haimoni<. b can hai'e.
The effect of the figures of the frequency 2 F independently form the nominal value of
2 ~ is shown in figure I. Due to the nominal value of 2 F, ~ome harmonics having the required
number of figures as 2~ may not appear; this happens when the last figure of
2 F is 5 or even. The mis~ing peaks in figure ~ are due to their small magnitude.
3. The harmonic approach in frequency synthesis.
In frequency synthesis operation, the output frequency f~, is intended to be a rational multiple
of a single standard frequency f~ so that f,,/f,
=
N/M where N and M are integers. When the
zemo.9
o
b
j z 3
a)
ze=o.91 o
1
16
z~t
i z 3
b)
Fig. 2.-Typical calculated spectra in,a logarithmic scale for different modulation frequencies 2 F with their different figures (Fig. ?a and Fig 2b). Frequency i~ in arbitrary units.
simultaneous generation of'~mall number of frequencies is required, it is customary to use the brute-force or the harmonic approach [4]. These approaches use as bui [din g blocks a reference
frequency source with a set of multipliers, dividers and mixer~. The synthesizer depicted in
figure 3 generates each output frequency as a programmed rational multiple of a reference
frequency f,. We emphasize its component frequency generator block. The latter con~ists of
one divider and 4 multipliers producing the following set R of the 4frequencies,
R [45.5984, 44.2368, 40.9600, 39.3216 for an input frequency j',
=
8 (frequencies are expres~ed in arbitrary units). The basic problems of the component frequency generator block
are the spurious outputs generated in frequency multiplication and division processes with an
important degradation in terms of phase noise performances. The phase noise is amplified by the multiplication factor of each frequency multiplier that feeds the
« Mix & Divide » stages of the synthesizer. The global performances of the instrument are then reduced. We propose in the sequel a component frequency generator block based on the developed theory in section 2.
N° lo ON THE FOURIER TRANSFORM OF Bl-VALUED FUNCTION 2051
blot
x Cl
Mix
~ C2 &
A
B D V DE
FREQUENCY C3
S TAG E 5
x C4
bin)
OUTPUT FREQUENCY
COMPONENT FREQUENCY GENERATOR BLOCK
Fig. 3.-Block diagram of the component frequency generator block of a synthesizer using the
harrnonic approach. h(0) 2°, b(n)
=
2".
4. Possible realisation of the method.
4.I COMPONENT FREQUENCY GENERATOR BLOCK. m(t, F, cc) given by equation ill
appears as a bi-valued function with transitions T,(F) modulated by a sinusoidal waveform function (Eq. (2)). Figure 4 shows the block diagram of the proposed component frequency generator block. The square wave signal represented by the function m (t, p, co where p i~ an
integer may be obtained from sinusoidal reference frequency source by means of a limiter and
an amplifier. The sinusoidal modulation signal who~e frequency has the appropriate chosen
figures may be derived from the sinusoidal reference source by means of a divider. The modulation process as defined by equations II and (2) may be obtained by various means.
4.I.I Som.c.e and proc.e~<.<iii~q c.h~iin noise ejfb<.ts. Let us assume the transitions T, of the bi-valued function a (t, ~, co) when the noise i~ pre~ent to be given by,
T,(F) =,<T A sin (4 7rF< + A,) (6a)
or
T,(~
= ,iT A ~in (4 7r ~,i + T, (6b)
depending on the type of noise we are considering. ~ ~tands for an integer or half integer. The random variables T, and measure re~pectively the total periodicity fluctuations and the total
phase fluctuation~ introduced by the conjribution of the source and the processing chain : limiter, amplifier, divider and the modulator. We call the periodicity noi~e the T-tj'pe noise and the phase noise the A-tj,pe noise. T, are assumed uniformly distributed in the interval
[- AT/2, AT/2 around nominal times i-T of frequency switch. A, are al~o assumed uniformly
distributed in the interval [-A~b/2, A~b/2] around nominal time~ of the frequency
~ ~ switch. In these conditions the periodicity of the function m (t, F, co is lost. Its spectral analysis follow~ the same treatment as for the free noise signal (see the Appendix). The Fourier
LIMITER
F IL T
&
AMPLIFIER
INPUT FILTER2
FREQUENCY
MODUL ArOR
Fl TE R3
FREQUENCY
DIVIDER FILT
Fig. 4. The proposed design approach of the block diagram of the component frequency generator block. The limiter and the amplifier generates the ~quare wave ;ignal a it, p, co where p i~ an integer.
The modulation signal of frequency 2 F is available at the output of the frequency divider.
coefficients are modified as follows,
E(c~(bj)
=
=~ jj J,,(~"~b.fl~(A~b)) .~(b+2n~).(1-Exp17r(b+211~)j (7a)
E(C~(b))
= fl~(AT, b).C16) (7b)
where,
sin (A~b/2) HA (~lb ) (8a)
~ ~~b/2
sin (b7r AT/T)
(8b)
"T (~T, b
"
bgr &T/T
C(b) is given by equation (4). E is the expectation operator. fl~(AT, bl is the Fourier transform of the distribution function of the random variables T,. The T-t~>pe noise induce a reduction of spectral lines of the free-noise square waveform function
m it, ~, co directly by the amount fl~ (AT, b), contrary to the A-type noise. This allows to distinguish between both
N° lo ON THE FOURIER TRANSFORM OF BI-VALUED FUNCTION 2053
types of noise just by considering the envelope function of the amplitude spectrum. The distinction is achieved by defining the following ratio-function,
Q(b
=
~~~'~~~~
i or T (9)
C (b)
If Q(b) matches the form (8a), the T-tvpe noise is dominant. C(b) i~ given then by the
theoretical model of m(t, ~, co). The factor Q(b) when measured at the output of the
modulator, expresses the quality factor of the proces~ing chain limiter, amplifier, divider and the modulator.
4.1.2 Fieqiiefic.y gefieiatifig sow.c'e. Section 2 shows that a large range of harmonics with the desired frequency figures are made available by an adequate choice of the modulation
frequency 2 F. The amplitude spectrum of m (t, F, co will contain the spectral line~ whose
frequencie~ are in the following set S clo~e to the ~et R of ~ection 3, S
=
[44.2426, 4~.5978, 40.9094, 39.3 l16] when we choose 2 F
= 0.4??1 with an input frequency wurce F~ =
=
f/8 (frequencies are expressed in arbitrary units). F~ is much lower than j'~ present at the input of the component frequency generator block of section 3 (Fig. 3). The~e line~ can be
extracted by various filtering techniques. The proposed block diagram combines a small
number of electronic functions and does not include any frequency multiplication proces~.
Better noise performances can therefore be expected.
5. Conclusion.
The developed theory of a novel mode of fractionnal frequency harmonic generation i~
presented. Results show that the produced spectrum possesses a bifurcation ~tructure created by the number of figures of the transition modulating wave frequency. The longer the fractional part of the frequency of the modulating wave is, the more figures the value of the produced
harmonic frequency has. We show that this property can open the way to a new generation of
frequency synthe~izers. It can provide significant advantages in terms of noise performances of component frequency generator blocks of frequency synthesizers.
Appendix.
In this appendix, we show the ~teps leading to relation (3). Let us consider an overall period
NT where N is an integer. The function m(t, ~, N) in the restricted interval (0, NT] is
rewritten as follows, a(t, ~, N )
=
,-i
= jj [H(t-mT-Asin47r~m)-2H(t- (m+1/2)T-Asin27r~(2m+1))+
,>,=o
+H(t-(m+I)T-Asin27r~(2m+2))] (A.I)
a (t, ~, N ) = m (t, ~, N ) + a~(t, ~, N + a,(t, ~, N ) (A.2)
m, (t, ~, N ) are the three ~ums in (A. ). The Fourier coefficients of m (t, ~, N are given by,
NT
R~ = exp(- 2 wkt/NT) x
NT
~
x
jj~ ( ~~ ~
Al' sinP (4
gr ~m ~ 'l(t mT) dt (A.3 )
n> oi, =o P.
In the above expres~ion, m (t, ~, N is Taylor series expanded around t mT. ~"~(t) is the n-th derivative of the Dirac distribution function. The Fourier coefficients become,
R '
~
' f (- ' Y'
~/, I 2 grk
j/,- ' ~
~ 4igrk NT pl NT
""
11 -1
x ~ sin" (4 7r~m ). Exp (- I 2 7rkm/N ). (A 4
,,, -n
Let us compute the coefficients R~ for the orders p = 0 and p = I, thiq gives,
N ~
integer
~ N 2 F ~
integer
~<u. i1
~
N
~
N
4 7rk 2 7rk
,
otherwise 2 iNT G (- 2 ~
,
otherwise
~
)~~ )
~
~
j~~/j~)~~~~~~j
(A.5
where,
~~ ~ ~,~v
(A.6)
~ ~, ~ EXP~i ~ )~j)~) /(,i ~~N
~~
and the superscripts 0 and refers to the considered orders being summed.
The result in (A.5) is obtained for a(t, F,N) defined in the restricted interval
[0, NT]. The dummy index I. has to change when N varies in order to keep the ratio
(IN finite [I]. We assume that the mean spectrum of is given by (A.5) when N extends to
infinity. This infinity limit means the more figures 2 F has, the larger N must be. For 2F ii./.atifiiia/, N i~ iiififiite. Let b= (IN, we perform this limit on the coefficient R(°. '', the latter is relabelled Rl~'. '(b). We get,
The ii-th order (p
= ii is given by, R~"'(b)
=
~'~ "~' ~' jj (-
)~ ~ (b + 2 (n 21 ) ~ (A.8
T'217rfi!
~~,,
Whe~e,
,~ RI
(A.9)
~j L!(n L)!
Relation (A.7) shows the spectral lines created by the succes~ive orders. We note that the different even order~ contribute to the creation of harmonics of integral frequency. Every higher order contribute to harmonics created by lower orders and create new ones with their
appropriate figures. R(b) is obtained by summing all orders,
R (b)
=
( J,,(2 7rAb/T). ~ (b + 2 nF) (A.10)
where the series representation of the Bessel function is used. The same calculations are
performed for m~(t, F, N and m i(t, F, N ). The collection of result~ leads to the coefficient C16) given by (3).
N° lo ON THE FOURIER TRANSFORM OF BI-VALUED FUNCTION 2055
References
II Ha+ani S. and Yahyabey N., Globally periodic signal analy,is. IEEE Tiaii.I. hi.<tiuJ1i Mea-v 39 (1990) 574-577 The last line of equation (C.3b) ~hould read (ii iiT + AT/? )/AT instead of
(ii iiT + AT/2 )/AT.
[2] Shwartz L., Mdthodes mathdmat<que~ pour les ~ciences physiques (Hermann & Cie, Pari~, France, 1983).
[3] Papouli~ A., The Fourier integral and it~ application~ (McGraw-Hill, New York, 1962).
[4] Manassevitch v., Frequency synthesis theory and de,ign (John Wiley & son~ Inc., 1976).
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