ELECTRONICS AND ELECTRICAL ENGINEERING
ISSN 1392 – 1215
2011. No. 1(107)
ELEKTRONIKA IR ELEKTROTECHNIKA
ELECTRONICS
T 170
ELEKTRONIKA
Robust Fingerprint Enhancement by Directional Filtering in Fourier
Domain
B. M. Popovic
Academy of Criminalistic and Police Studies,
Cara Dušana 196, 11080 Belgrade, Serbia, phone: +381 11 3161444, e-mail: [email protected]
M. V. Bandjur, A. M. Raicevic
The Faculty of Technical Sciences, University of Priština,
Kneza Miloša 7, 38220 K. Mitrovica, Serbia, phone: +381 28 425320, e-mails: [email protected],
[email protected]
area of fingerprint image, ridges and valleys have welldefined frequency and orientation, majority of proposed
enhancement techniques utilize this contextual information.
Several filters were proposed in spatial [4, 5] and frequency
[6–9] domain.
In this paper, a robust enhancement algorithm in
Fourier domain is presented. It is based first on amplifying
the dominant frequencies by a factor of raised power
spectrum (noise reduction) and then by appropriate filtering
with Log-Gabor filter (directional enhancement). We
extended our research [10] with fingerprint images taken by
ink to images obtained by inkless scanners, and method
proves to be effective for both cases.
Introduction
Reliable
and
accurate
automatic
personal
authentication introduces biometric technology as a
necessary component of any ID management system, both
in government and civil sector. Amongst available
biometrics, fingerprints are one of the most researched and
used method of authentication [1, 2]. There were different
realizations of automatic fingerprint identification systems
(AFIS) in the last 50 years. We can either digitalize
fingerprint image taken by ink or use inkless scanners to
provide input image for AFIS. A number of operations are
then applied in order to extract features (mostly minutiae)
later used in matching process, so it is obvious that the
AFIS accuracy relies heavily on the reliability of those
extracted features. As Pankanti et al. [3] have shown
detected spurious minutia can be more harmful to reliable
AFIS performance than missed genuine one, so it is
imperative to minimize the number of spurious minutia to
maximal extent.
In order to utilize AFIS in law enforcement agencies
first step was (is) to digitalize an archive of fingerprints
obtained by ink method. Quality of those fingerprint images
greatly depends on equipment and technique that is used,
but there are also a number of factors (postnatal marks,
occupational marks, creases, etc.) that contribute to large
number of spurious minutiae detected. To minimize that
number, enhancement is usually performed before feature
extraction. When dealing with a particular fingerprint
image, i.e. latent, we can apply any of many digital
techniques and filters, even combining and adjusting them
for that particular case. But in case of making an database
of fingerprint templates for AFIS from an archive of cards
of fingerprints taken by ink, the process of automatic
minutia extraction is rarely intervened by human expert, so
an enhancement method need to be robust and applicable to
a wide range of input images. Due to the fact that in local
Directional filtering in Fourier domain
When dealing with filtering in Fourier domain we
have two different cases. First is when the filter is designed
in spatial domain and filtering is performed in Fourier
domain (instead of spatial convolution methods). Another
approach is to design filter and perform filtering both in
Fourier domain which was the case in our research. Filter
designed in frequency domain is described with separable
radial (Hr) and angular (Ha) components and is tuned
specifically to the distribution of orientation and
frequencies in the local region of the fingerprint image.
Radial component depends only on local ridge spacing
(r=1/f) and the angular depends only on local ridge
orientation (), so overall filter function is described as
H r ,    H r r   H a   .
(1)
Positional dependence of filters requires defining and
applying appropriate filter for each pixel which is
computationally very expensive. Instead, a finite number of
predefined filters are usually applied (regarding to finite
number of discrete orientations, and fixed or average
frequencies).
37
Band-pass Butterworth filter as radial component, and
raised cosine filter as angular component proposed by
Sherlock et al. [6], and adopted in [7, 8] are found to give
very good results. Gabor based filtering [5], as the most
popular method in fingerprint enhancement, can also be
performed in frequency domain, but has some limitations
(maximum bandwidth of a Gabor filter is limited to
approximately one octave). To overcome that limitation and
promote filtering in frequency domain, the Log-Gabor filter
is proposed [9].
Log-Gabor functions have Gaussian transfer functions
when viewed on the logarithmic frequency scale. 2D LogGabor filter is constructed in the frequency domain, with
radial and angular component given by (2), where (r,)
represents the polar coordinates, f0 is the center frequency
of the filter, 0 is the orientation angle of the filter, σr
determines the scale bandwidth and σ determines the
angular bandwidth
2



 H  r   exp( log  r f 0   ),
 r
2 r2



  0 2 ).
 H a    exp(
2 2

utilized Gradient–based approach for orientation
estimation [11]. Orientation image is seldom computed at
full resolution. Instead, one (dominant) orientation is
assigned for each non-overlapping block of size B×B of
the image in following way: for each pixel (i,j) of the
block, horizontal and vertical component of the gradient,
Gx and Gy respectively, are calculated. We can use any
gradient operator (mostly simple Sobel operator). Then,
dominant local orientation O(i,j) for the block is given by
Oi, j  
  2GxG y  
1
B

tan 1 
  Gx2  G y2  2 .
2

 B
(3)
Additional smoothing (low pass filtering) must be
applied on the orientation values in order to eliminate
inconsistencies in distorted and noisy regions containing
creases and scars. It is done by converting orientation
image into a continuous vector field and by performing
vector averaging with some smoothening kernel [8].
The ridge frequency as a slowly varying property can
be computed only once for each non-overlapping block of
the image. We adopted method presented in [5], based on
the projection sum taken along a line oriented orthogonal
to the ridges which forms a sinusoidal signal, and the
distance between any two peaks provides the inter-ridge
distance K. The frequency f is computed as f= 1/K. The
local frequency and orientation estimated above
correspond to the center frequency f0 and the orientation
angle 0 of the Log-Gabor filter.
(2)
Comparing with Gabor filters, Log-Gabor filters can
be constructed with arbitrary bandwidth which can be
optimized to have minimal spatial extent and are allowed to
reduce the over-representation of low frequencies, which is
the reason why we choose them for our enhancement
method.
Local filtering
Firstly we transformed the original fingerprint image
to frequency domain. To reduce the computational cost,
the windowed Fourier transform (2D WFT) is applied to
each non-overlapping block of size B×B in order to extract
corresponding frequency spectrum. To eliminate the
effects result from the block dividing of the image, the
window size W should be larger than the block size B. This
means that the neighboring windows will overlay each
other with OV number of pixels. In order to meet the
requirement for recovering the image from frequency
domain, a raised cosine window is employed and defined
as in [7]:
Proposed algorithm
In order to perform fingerprint enhancement in
frequency domain proposed algorithm consists of
following steps: local normalization, local orientation and
frequency estimation and local filtering which are briefly
described. Previously, input fingerprint image is divided
into a number of B×B (16×16 in our experiment) square
non-overlapping blocks since each of mentioned steps
regards to a block.
Local normalization

1,
if  x , y   B 2,

W  x, y    1 
   x  B 2  
otherwise,
  ,
  1  cos 

OV
 2 


First step is to normalize fingerprint image in each
block separately to a constant mean and variance. The
main purpose of normalization is to have input images with
similar characteristics, to remove the effects of sensor
noise and also to reduce the variation in gray-level values
along the ridges and valleys (without changing the ridge
and valley structures). If normalization is performed on the
entire image as suggested in [5], then it cannot compensate
for the intensity variations in different parts of the image
due to the elastic nature of the finger. Separate
normalization of each individual block alleviates this
problem. Local normalization of input fingerprint image is
done as proposed in [8].
(4)
 W W
where  x, y     ,  .
 2 2
To reach a compromise between performance and
complexity, the window size W×W was set to 32×32. By
moving the location of the window, the frequency
spectrum corresponding to each block of the image can be
obtained. Then the power spectrum is estimated, raised to a
power α (0.5 was used) and multiplied by the spectrum
elements (F1=F•|F|α). This has the effect of amplifying the
dominant frequencies in the block, which, presumably, are
those corresponding to the ridges, thereby increasing the
ratio of ridge information to non-ridge noise and adapting
Local orientation and frequency estimation
Reliable ridge orientation estimation is very
important when directional filtering is employed. We
38
to variations in ridge frequency from one block to the next.
Then, each spectrum is filtered by a Log-Gabor filter tuned
according to the orientation and frequency of the
corresponding block (E=F1•Hr•Ha). Apparently, the filter
is constructed with the same size of the spectrum. Finally,
the inverse Fourier transform of the filtered spectrums is
computed (Benh=IFT(E)), and its real part becomes the
enhanced block.
larger fingerprint database (including public) in order to
estimate more relevant statistical parameters for
performance. Nevertheless, we can say that proposed
method proves to be an adequate enhancement solution for
AFIS since it provides significant reduction of extracted
spurious minutiae.
Experimental results
Our goal was to see if proposed enhancement
algorithm is suitable to be implemented for various ranges
of fingerprint images in process of creating AFIS template
both for images obtained by ink and inkless scanners.
Experiment was conducted on our own fingerprint
database containing 100 fingerprint images (10 cards)
taken by ink and then digitized with optical scanner using
spatial resolution of 500 dpi and amplitude resolution of 8
bit per pixel. Second database [12] contains 168 fingerprint
images obtained by optical scanner with same resolution (8
instances of 21 finger). Testing software was implemented
using MATLAB® development environment (The
Mathworks Inc., USA).
Result of proposed image enhancement method for
test image from our database is shown in Fig. 1(a). We
choose image containing scars (one quite wide) and
smudgy regions in order to emphasize the enhancement
results. For comparison the figure also shows two other
images, enhanced with STFT method [7] (matlab code
available from http://www.cubs.buffalo.edu/code.shtml)
and with Gabor filter enhancement method [5] (Peter
Kovesi’s
implementation
code
available
at
http://www.csse.uwa.edu.au/~pk/Research/MatlabFns/).
Example demonstrates that proposed algorithm has
preferable performance. Similar results were obtained for
our entire database set with obviously enhanced ridge
structure. In cases of images of relatively good quality all
tested algorithms provide similar (comparable) results, as
shown in Fig. 1(b) with sample fingerprint image taken
from [12].
In order to quantify effect of enhancement, rather
than to rely on subjective opinion (visual examination), we
compared number of automatically extracted minutiae
from original image and from enhanced one. Fingerprint
Expert was previously asked to extract true minutiae from
original fingerprint. We used algoritam presented in [13].
Results for image presented in Fig. 1 are compared in
Table 1. E represents number of minutiae determined by
the expert; A is the number of automatically extracted
minutiae; P is the number of paired minutiae; M is number
of missing minutiae and F is the number of false minutiae.
As result of proposed enhancement process there is
significant decrease of number of automatically extracted
minutiae and false minutiae (it goes up to 67% for some
images from the database). Although directional filtering
can reconnect ridges, some broad creases remains,
resulting in false minutiae extraction. Rests of false
minutiae are due to boundary effect and bad segmentation
(especially for smudgy regions) and that will be the focus
of our future work. Testing shall also be performed on
(a) Original image from our database
(b) Original image taken from [12]
Fig. 1. Examples of enhanced images. From left to right: original
image, image enhanced with STFT method, image enhanced with
Gabor filter and proposed enhanced image
Enhancement
method
Without
enhancement
Gabor-based
STFT
Our proposed
enhancement
Fig.
A
P
M
F
E
1(a)
222
53
22
169
75
1(b)
102
32
1
70
33
1(a)
151
61
14
90
75
1(b)
82
31
2
51
33
1(a)
138
63
12
75
75
1(b)
60
33
0
27
33
1(a)
105
62
13
43
75
1(b)
50
30
3
20
33
Table 1. Comparison of number of minutiae for various
enhancement approaches
39
4. Gorman L. O’ and Nickerson J. V. An approach to
fingerprint filter design // Pattern Recognition, 1989. – Vol.
22(1). – P. 29–38.
5. Hong L., Wan Y., Jain A. K. Fingerprint image
enhancement: Algorithm and performance evaluation // IEEE
Tran. Pattern Ana. Machine Intel, 1998. – Vol. 20, No. 8. – P.
777–789.
6. Sherlock B. G., Monro D. M., and Millard K. Fingerprint
enhancement by directional Fourier filtering // Visual Image
Signal Process, 1994. – Vol. 141. – P. 87–94.
7. Chikkerur S., Cartwright A. N., and Govindaraju V.
Fingerprint enhancement using STFT analysis // Pattern
Recognition, 2007. – Vol. 40. – No. 1. – P. 198–211.
8. Raicevic A., Popovic B. An Effective and Robust Fingerprint
Enhancement by Adaptive Filtering in Frequency Domain //
Facta Universitatis Series: Electronics and Energetics. – Niš,
2009. – Vol. 22. – No. 1. – P. 91–104.
9. Wang W., Li J. W., Huang F. F., Feng H. L. Design and
implementation of Log–Gabor filter in fingerprint image
enhancement // Pattern Recognition Letters, 2008. – Vol. 29.
– P. 301–308.
10. Popovic B., Bandjur M., Raicevic A., Robust enhancement
of fingerprint images obtained by ink method // Electronic
Letters, 2010. – Vol. 46(20). – P. 1379–1380.
11. Kass M., Witkin A. Analyzing oriented patterns // Comput.
Vision Graph. Image Processing, 1987. – Vol. 37. – P. 362–
385.
12. Biometric System Lab., University of Bologna, Cesena
Italy. Online: www.csr.unibo.it/research/biolab/.
13. Popovic B., Maskovic Lj, Bandjur M., Spurious
Fingerprint Minutiae Detection Based on Multiscale
Directional Information // Electronics and Electrical
Engineering. – Kaunas: Technologija. – 2007. – No. 7(79). –
P. 23–28.
Received 2010 07 05
Conclusions
Fingerprint enhancement is a common and critical
step in AFIS, and directional filtering technique, a general
image-processing operation, is widely used for that
purpose. Due to the characteristics of fingerprint image in
frequency domain it seems natural to use directional band
pass filter to enhance fingerprint image.
In this paper, method based first on amplifying the
dominant frequencies by a factor of raised power spectrum
(noise reduction) and then by appropriate filtering with
Log-Gabor filter (directional enhancement), was found to
produce preferable results for the fingerprints considered in
this study. It is shown that filtering techniques work well in
a broad range of cases, and is suitable for the quantity of
fingerprints to be enhanced. As a result of enhancement
process more reliable feature extraction is obtained, less
spurious minutiae are extracted, improving the overall
AFIS accuracy.
References
1. Maltoni D., Maio D., Jain A. K., Prabhakar S. Handbook
of Fingerprint Recognition. – USA: Springer, 2003. –348 p.
2. Ivanovas E., Navakauskas D. Development of Biometric
Systems for Person Recognition: Biometric Feature Systems,
Traits and Acquisition // Electronics and Electrical
Engineering. – Kaunas: Technologija. – 2010. – No. 5(101). –
P. 87–90.
3. Pankanti S., Prabhakar S., Jain A. On the individuality of
fingerprints // IEEE Trans Pattern Anal. Machine Intell, 2002.
– Vol. 24. – No. 8. – P. 1010–1025.
B. M. Popovic, M. V. Bandjur, A. M. Raicevic. Robust Fingerprint Enhancement by Directional Filtering in Fourier Domain //
Electronics and Electrical Engineering. – Kaunas: Technologija, 2011. – No. 1(107). – P. 37–40.
Reliable extraction of true minutiae in fingerprint image is critical to the performance of an automated fingerprint identification
system (AFIS). In order to utilize AFIS in law enforcement agencies first step is to digitalize an archive of fingerprints obtained by ink
method. For improving the quality of automatically extracted minutiae (both in number and type) enhancement is previously performed,
but, nevertheless, quite a number of spurious minutiae (especially in blurry or regions containing scars and creases) are extracted. It is
crucial for AFIS performance that number of extracted (and consequently saved in fingerprint template) spurious minutia is minimized
to maximal extent. In order to do so, we propose directional Log-Gabor filtering in frequency domain. Results proved to be preferable
for a wide range of input digitized fingerprint images. Ill. 1, bibl. 13, tabl. 1 (in English; abstracts in English and Lithuanian).
B. M. Popovic. M. V. Bandjur, A. M. Raicevic. Tiesiogino filtravimo Furje transformacijose taikymas pirštų antspaudų
atpažinimui // Elektronika ir elektrotechnika. – Kaunas: Technologija, 2011. – Nr. 1(107). – P. 37–40.
Patikimas individualių požymių išskyrimas yra labai svarbus automatizuotose pirštų antspaudų sistemose. Norint panaudoti tokią
sistemą, reikia surinkti antspaudus skaitmeniniu pavidalu taikant „rašalo“ metodą. Raukšlėtose ar randuotose vietose bandoma
paryškinti ir išskirti individualius požymius. Maksimalų automatizuotos pirštų anspaudų sistemos našumą siekiama padidinti mažinant
individualių požymių kiekį. Pasiūlytas Logo ir Gaboro dažnių filtravimo metodas. Gauti rezultatai yra geresni lyginant su didelio
skaičiaus pirštų antspaudų duomenų baze. Il. 1, bibl. 13, lent. 1 (anglų kalba; santraukos anglų ir lietuvių k.).
40
ODREĐIVANJE USLOVA ZA POJAVU HISTEREZISA U PLL PETLJI
Anđelija M. Raičević1, Brankica M.Popović 2
1
Fakultet Tehničkih Nauka, Univerzitet u Prištini, Kosovska Mitrovica, Srbija
2
Policijska Akademija, Beograd, Srbija
DETERMINING THE CONDITIONS FOR OCCURRENCE OF JUMP
PHENOMENON IN PHASE LOCKED LOOPS
Anđelija M. Raičević1, Brankica M.Popović2
1 The Faculty of Technical Sciences, University of Priština, Kosovska. Mitrovica, Serbia
2 Academy of Criminalistic and Police Studies, Beograd, Srbija
Abstract: Nonlinear response of PLL (Phase Locked Loops) occurs as a consequence of sinusoidal characteristics of
phase detector, whereas each nonlinearity causes a distortion in response. This mode is not desirable and in it, in one part
of frequency characteristics a large discontinuity occurs, known as a jump phenomenon. The area of the jump
phenomenon and critical values of PLL parameter can be determined by analyzing frequency characteristics.
Key words: PLL (Phase Locked Loops), jump phenomenon, nonlinear distortion, frequency characteristics.
Uvod
Kolo koje čini da jedan sistem po frekvenciji i fazi prati drugi je PLL petlja (Phase Locked
Loops). Preciznije, to je kolo koje sinhronizuje izlazni signal oscilatora sa referentnim ili ulaznim
signalom. Prve fazno zatvorene petlje je primenio Bellesize još 1932. godine i ovaj francuski inženjer
se smatra pronalazačem koherentne komunikacije. Zahvaljujući širokj primeni u detekciji FM signala
u satelitskoj televiziji, u TV i radio prijemnicima, u sintezi frekvencija, u digitalnim komunikacijama
PLL petlji je posvećen veliki broj radova i knjiga. Njena široka primena uticala je i na razvoj
tehnologije u ovoj oblasti, pa danas na tržištu postoji širok spektar integrisanih kola PLL petlje za
različite namene [1].
Ulazni signal petlje je referentni, dok petlja ima dva izlaza. Jedan izlaz je napon greške kojim se
upravlja naponsko kontrolisani oscilator, dok drugi je visokofrekventni signal naponsko kontrolisanog
oscilatora. Na osnovu ovoga moguća je i različita primena petlje. Prva na kojoj je i zasnovan koncept
PLL-a [2], jeste primena za detekciju frekfentno odnosno fazno modulisanih signala. Zahvaljujući
činjenici da se signal koji generiše naponsko kontrolisani oscilator dobija sa istom frekvencijom kao i
referentni signal i tačno definisanim
***Rad je delimično realizovan iz projekta Ministarstva nauke Srbije i to iz oblasti Tehnološkog
razvoja- TR35026***
41
faznim stavom u odnosu na fazni stav referentnog signala proistekla je i druga, ne manje važna
primena PLL petlje u sintezi frekvencija.
Detekcija fazno i frekventno modulisanih signala moguća je na više načina [3] a detekcija
pomoću PLL-a je nezamenljiva kada je odnos signal-šem mali tj. predstavlja jedino rešenje za slabe
ulazne signale kao što su satelitski signali. Zavisno od parametara petlje, i parametara ulaznog signala,
petlja može imati manju ili veću faznu grešku tj može u svom radu biti linearna ili nelinearna.
Osnovne karakteristike koje opisuju rad PLL petlje kao FM demodulatora su frekvencijske
karakteristike koje predstavljaju zavisnost rezultujuće fazne greške (detektovanog napona) od
modulacione frekvencije FM signala. Naravno, nelinearni režim rada PLL petlje nije poželjan, jer se
pri takvom režimu rada dobijaju velika izobličenja. Kao granica primenljivosti PLL kao demodulator
u linearnom režimu rada je pojava histerezisa u frekvencijskim karakteristikama.
Harmonijska izobličenja PLL FM demodulatora
PLL petlja je sistem u kome se učestanost naponsko kontrolisanog oscilatora (VCO) menja pod
dejstvom kontrolnog napona faznog detektora sve dotle dok se ne izjednači sa učestanošću ulaznnog
signala. Posmatraćemo sinhronizovanu petlju, a faza ulaznog signala se menja pod dejstvom
frekvencijske modulacije. U opštem slučaju, karakteristika faznog detektora je nelinearna (najčešće
sinusna), što analizu čini komplikovanijom, obzirom da se sistem tada opisuje nelinearnom
diferencijalnom jednačinom.
Input
signal
Phase
output1
LFfilter
det ector
Voltage
controlled
oscillator
output 2
Slika 1. Blok šema PLL petlje
Prema blok dijagramu PLL petlje na slici 1., ulazni signal i signal na izlazu naponsko kontrolisanog
oscilatora su:
y i  A sint  i (t )
(1)
y 0  B cos t   0 (t )
Fazni detektor ima sinusnu karakteristiku i napon na njegovom izlazu je:
u1  k1 sini (t )   0 (t )
(2)
Ako je F(j) prenosna funkcija filtra u sistemu a f(t) impulsni odziv i čine Fourier-ov transformacioni
par pa je izlazni napon filtra:
u 2 (t )  u1 (t ) * f (t )
(3)
gde znak * konvolucioni proizvod. Naponsko kontrolisani oscilator je definisan kao integrator:
d 0
 k 3 u 2 (t )
(4)
dt
Na osnovu gornjih jednačina dobija se opšta jednačina u vremenskom domenu PLL petlje sa sinusnom
karakteristikom faznog detektora:
d 0
 k sin i (t )   0 (t )* f (t ) 
(5)
dt


42
Zbog težnje rešavanja u frekventnom domenu i korišćenjem osobine da dve veličine vezane
konvolucionim produktom, predstavljaju običan proizvod njihovih Fourier-ovih transformacija i ako
na ulazu deluje frekventno modulisani signal, onda se PLL petlja koja se doristi kao FM demodulator
opisuje diferencijalnom jednačinom:
d
d
 KF ( s ) sin    
(6)
dt
dt
gde je:    sin  m t , β- indeks modulacije, m- frekvencija modulisanog signala,
  c  0 razlika noseće frekvencije  c i frekvencije  0 naponsko kontrolisanog oscilatora (VCO)
u režimu slobodnog oscilovanja, K- slabljenje ili pojačanje filtra, F(s)– prenosna karakteristika filtra.
Jednačina (6) spada u grupu nelinearnih jednačina, i njeno rešenje je numeričko, odnosno
aproksimativno, a greška koja iz toga sledi zavisi od usvojene aproksimacije sinusne karakteristike
faznog detektora. Kako je  periodična kada je  prostoperiodična funkcija , onda je rešenje
jednačine (6) oblika:
N
     ri sin(i n   i )
(7)
i 1
gde je  - jednosmerna komponenta a ri i  i amplituda i faza i-tog harmonika odziva.
Određivanje graničnih uslova za pojavu histerezisa
Nelinearni odziv PLL petlje nastaje kao posledica nelinearne karakteristike faznog detektora,
što se manifestuje pojavom histerezisa u odzivu PLL petlje. Obzirom da je radni režim praćen ovim
fenomenom veoma nepovoljan, određuju se parametri kritičnog režima rada petlje. Matematički to
znači da izbegavamo histerezis ako za prvi harmonik r1 imamo jedno realno a da su druga dva rešenja
konjugovano kompleksna. Kada za r1 postoji trostruko realno rešenje, onda je to granični slučaj za
pojavu histerezisa i tada r1(ωn) ima jednu vertikalnu tangentu. Rešavanjem jednačine (7), ali
zanemarivanjem drugog i trećeg harmonika [1] dobija se jednačina:


2
2
(r1 ) (a3  a 3 x n2 )  (r1 ) 2 (a 3 n  a1 a 3  a1 a 3 x 2 n ) 


2
2
2
4
2
2 2
2
4
2
2
2 2
r1 a1  2a1 n  (1  x ) n    n  a1 x     n   2 (1  x) 2  n
2
2
2

3
2
2

(8)
gde su a1 i a3 koficijenti polinoma kojim se aproksimira sinusna funkcija faznog detektora. Rešenja za
osnovni-prvi harmonik, koji i predstavlhja korisni signal, za različite vrednosti ulaznog signala i
parametara
petlje
data
su
tabelarno.
Uočljivo
je
da
u
Tabeli
1
( a1  0.9862, a3  0,10665, x  0.03,   1,   0.2 ) u opsegu frekvencije od 0.59-0.7, imamo tri realna
rešenja tj pojavu histerezisa, a van ovog opsega, jedno realno i dva konjugovano-kopleksna, dok u
Tabeli 2 ( a1  0.9862, a3  0,10665, x  0.05,   0.5,   0.2 ) u celom opsegu imamo po jedno realno
i dva konjugovano-komleksna rešenja.
Tabela 1
Tabela 2
realni koreni
konjugovano
realni koreni konjugovano
n
n
kopleksni
kopleksni
0.3
0.10
5.1±j0.8
0.3
0.22
6.6±j1.38
0.5
0.13
6.7±j0.8
0.5
0.03
6.6±j1.42
0.59
0.59
0.08
5.6±j1.46
0.38; 5.46; 5.71 ; ------0.6
------0.6
0.09
5.57±j1.45
0.40; 5.20; 5.82;
0.62
0.62
0.11
5.33±j1.44
0.52; 4.53; 5.92; -----0.65
------0.65
0.15
4.94±j1.40
0.79; 3.57; 5.88;
0.68
0.68
0.21
4.54±j1.34
1.71; 1.99; 5.78; -----0.7
5.70
1.6±j1.0
0.7
0.27
4.24±j1.28
0.8
5.20
0.4±j2.6
0.8
1.43
1.43±j0.88
1.0
4.08
0.7±j3.4
1.0
2.29
1.25±j2.71
43
Da bi uopšte postojala vertikalna tangenta treba da je:
1
 dr1 2 


(9)
 d 2   0
 n 
Diferenciranjem jednačine (8) i primenom uslova (9) dobija se jednačina :
4
2
Ar 1  Br1  C  0
(10)


A  3(a3  a 3 x n2 ); B  4(a1 a3  a1 a3 x 2 n2 )


2
2
2
4
2
2 2 n
2
C  a1  2a1 n  (1  x) n    n  a1 x
2
2
2
2
2
2
Bikvadratna jednačina (10) ima samo jedno realno rešenje ako je njena diskriminanta jednaka nuli tj.:
B 2  4 AC  0
2
2
2
2
2
 2
3a3 x 2
a1 a3 x 2
6a1 a3 x 2 3a1 a3 x 4  4
6
2 2
2 2
2
(1  x) n  3a3 (1  x)  4(
 a3 )  3a3 x 

 n 
2
2
4
4 

2
2
 2 2
6a1 a3 x 2
 2
 a1 a3 x 2
2
2
2

3
a

6
a
a
8
a
a



 a3  n  a1 a3  0
 3
1 3
1 3
2
2


 

Rešenje jednačine po ωn daje vrednost frekvencije signala u normalizovanom obliku, ωnc, pri kojoj
karakteristika r1 ima vertikalnu asimtotu.
Zaključak
U radu su razmatrane frekventne karakteristike nelinearne PLL petlje , koja se koristi za FM
detekciju. Nelinearni rad PLL petlje se javlja pri velikim faznim greškama kada sinusnu funkciju
faznog detektora aproksimiramo polinomom trećeg reda (Taylorovim redom ili koeficijente
određujemo iz uslova minimalne greške) i tada u odzivu imamo više harmonike. Nelinearni režim
rada nije poželjan, jer se pri takvom režimu rada dobijaju velika izobličenja. Na osnovu Tabele 2
vidimo da se histerzis može izbeći biranjem parametara ulaznog signala i same petlje. Nelinearna
izobličenja su najveća u oblasti histerezisa, i u praktičnoj primeni PLL petlje tu oblast treba
izbegavati.
Literatura
1. V.Krupa, J.Štursa, V.Čizek, Direct Digital Frequency Synthesizer PLL Systems, IEE2001, pp.799801.
2. Best R. Phase-Locked-loops.Desing and Aplications, New York,1993.
3. Krstić D.Radiokomunikaciona elektronika i sistemi, Elektronski fakultet,Niš,2004.
4. A.Raičević, B.Prica, One Solution For Diferential Equation For Non/linear Mode PLL loop,
MIT2009,pp.342-346
5. S.Gupta, Phase Locked-Loops, Proccedings of the IEEE,vol 63. pp.291-306,2005.
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