3477:
2529:
203:
3460:
systems, a delay is incurred because incoming samples need to fill the filter window before the filter can be applied to the signal. While no amount of delay can make a theoretical
Gaussian filter causal (because the Gaussian function is non-zero everywhere), the Gaussian function converges to zero so rapidly that a causal approximation can achieve any required tolerance with a modest delay, even to the accuracy of
2305:
2622:
and would theoretically require an infinite window length. However, since it decays rapidly, it is often reasonable to truncate the filter window and implement the filter directly for narrow windows, in effect by using a simple rectangular window function. In other cases, the truncation may introduce
2505:
yields a magnitude of -2.986 dB, which represents an error of only ~0.8% from the desired -3.010 dB. This error will decrease as the number of orders increases. In addition, the error at higher frequencies will be more pronounced for all
Gaussian filters, bug will also decrease as the order of the
3451:
of that element's neighborhood. The focal element receives the heaviest weight (having the highest
Gaussian value), and neighboring elements receive smaller weights as their distance to the focal element increases. In Image processing, each element in the matrix represents a pixel attribute such as
2514:
Although
Gaussian filters exhibit desirable group delay, as described in the opening description, the steepness of the cutoff attenuation may be less than desired. To work around this, tables have been developed and published that preserve the desirable Gaussian group delay response and the lower
3459:
The
Gaussian filter is non-causal, which means the filter window is symmetric about the origin in the time domain. This makes the Gaussian filter physically unrealizable. This is usually of no consequence for applications where the filter bandwidth is much larger than the signal. In real-time
2019:
1510:
2056:
2455:
1827:
2688:(a rectangular pulse). For example, four iterations of a moving average yield a cubic B-spline as a filter window, which approximates the Gaussian quite well. A moving average is quite cheap to compute, so levels can be cascaded quite easily.
1704:
985:
565:
1879:
3299:
1385:
2836:
2754:
661:
349:
3047:
2931:
1141:
1251:
849:
437:
1708:
The ability of the filter to simulate a true
Gaussian function depends on how many terms are taken from the series. The number of terms taken beyond 0 establishes the order N of the filter.
3056: = 2 the constant before the standard deviation in the frequency domain in the last equation equals approximately 1.1774, which is half the Full Width at Half Maximum (FWHM) (see
2300:{\displaystyle F_{3}((j\omega )^{2})={\frac {1}{1.33333a^{3}(j\omega )^{6}+2a^{2}(j\omega ^{4})+2a(j\omega )^{2}+1}}={\frac {1}{-1.33333a^{3}\omega ^{6}+2a^{2}\omega ^{4}-2a\omega ^{2}+1}}}
2338:
1592:
1552:
1063:
2620:
1713:
3217:
3147:
2333:, and building the polynomials using only the left half plane poles yields the transfer function for a third order Gaussian filter with the required -3.010 dB cutoff attenuation..
1346:
2023:
Since only half the poles are located in the left half plane, selecting only those poles to build the transfer function also serves to square root the equation, as is seen above.
3609:
3345:
3427:
3374:
798:
3398:
2866:
760:
2966:
However, it is more common to define the cut-off frequency as the half power point: where the filter response is reduced to 0.5 (−3 dB) in the power spectrum, or 1/
2503:
1378:
1167:
1872:
1599:
2049:
1849:
704:
189:
2961:
733:
246:
to a step function input while minimizing the rise and fall time. This behavior is closely connected to the fact that the
Gaussian filter has the minimum possible
3167:
3096:
2327:
1271:
1187:
1083:
684:
460:
2657:
of the
Gaussian function yields a Gaussian function, the signal (preferably after being divided into overlapping windowed blocks) can be transformed with a
250:. A Gaussian filter will have the best combination of suppression of high frequencies while also minimizing spatial spread, being the critical point of the
864:
2638:. The filter function is said to be the kernel of an integral transform. The Gaussian kernel is continuous. Most commonly, the discrete equivalent is the
2646:
which has superior characteristics for some purposes. Unlike the sampled
Gaussian kernel, the discrete Gaussian kernel is the solution to the discrete
472:
2014:{\displaystyle F_{N}(j\omega )={\sqrt {\frac {1}{\sum _{k=0}^{\mathbb {N} }{\frac {(2a)^{k}(j\omega )^{2k}}{k!}}}}}{\bigg |}_{\text{left half plane}}}
3646:
3225:
855:
where the standard deviations are expressed in their physical units, e.g. in the case of time and frequency in seconds and hertz, respectively.
182:
1505:{\displaystyle F(\omega )=\epsilon ^{-a\omega ^{2}}{\sqrt {(\epsilon ^{-a\omega ^{2}})^{2}}}={\sqrt {\frac {1}{\epsilon ^{2a\omega ^{2}}}}}}
3600:
3671:
2773:
2701:
3447:
is precomputed from the formula and convolved with two-dimensional data. Each element in the resultant matrix new value is set to a
175:
355:
576:
287:
2987:
2874:
1088:
2963:
is the sample rate. The response value of the
Gaussian filter at this cut-off frequency equals exp(−0.5) ≈ 0.607.
1192:
806:
800:
it can be shown that the product of the standard deviation and the standard deviation in the frequency domain is given by
1253:, or 0.34657359 for an approximate -3.010 dB cutoff attenuation. If an attenuation of other than -3.010 dB is desired,
363:
3790:
3834:
3806:
3741:
3725:
2568:
2763:
is the total number of samples. The standard deviation of a filter can be interpreted as a measure of its size. The
2661:, multiplied with a Gaussian function and transformed back. This is the standard procedure of applying an arbitrary
2546:
2450:{\displaystyle F_{3}(j\omega )={\frac {1}{0.2355931(j\omega )^{3}+1.0078328(j\omega )^{2}+1.6458471(j\omega )+1}}}
3432:
When applied in two dimensions, this formula produces a Gaussian surface that has a maximum at the origin, whose
3075:
1822:{\displaystyle F_{N}(\omega )={\sqrt {\frac {1}{\sum _{k=0}^{\mathbb {N} }{\frac {(2a)^{k}\omega ^{2k}}{k!}}}}}}
3869:
2550:
1557:
1517:
1028:
3429:
of 2.42. It remains to be seen where the advantage is over using a gaussian rather than a poor approximation.
2584:
3172:
3101:
2665:
filter, with the only difference being that the Fourier transform of the filter window is explicitly known.
1276:
251:
100:
3635:
2515:
and mid frequencies, but switches to a higher steepness Chebyshev attenuation at the higher frequencies.
2051:= 1 requires the use of terms k=0 to k=3 in the Taylor series to produce the squared Gaussian function.
3891:
3579:
2628:
227:
3488:
2974:
239:
219:
117:
70:
65:
33:
3314:
2642:
that is produced by sampling points from the continuous Gaussian. An alternate method is to use the
3720:. New York, Chichester, Brisbane, Toronto, Singapore: John Wiley & Sons, Inc. pp. 70, 71.
3554:
3403:
3350:
2643:
765:
3674:," IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 39, pp 723–727, March 1991.
255:
3886:
3379:
2844:
2662:
2639:
2539:
2330:
1699:{\displaystyle \epsilon ^{2a\omega ^{2}}=\sum _{k=0}^{\infty }{\frac {(2a)^{k}\omega ^{2k}}{k!}}}
1022:
738:
111:
3057:
2462:
1354:
2658:
1146:
123:
83:
3715:
1854:
3376:
of 3, it needs a kernel of length 17. A running mean filter of 5 points will have a sigma of
2669:
2034:
1834:
689:
270:
3850:
Lindeberg, T., "Scale-space for discrete signals," PAMI(12), No. 3, March 1990, pp. 234–254.
3444:
2939:
709:
8:
3849:
3525:
3461:
2868:
is measured in samples, the cut-off frequency (in physical units) can be calculated with
2767:
of a Gaussian filter might be defined by the standard deviation in the frequency domain:
2676:), the Gaussian can be approximated by several runs of a very simple filter such as the
1554:
may be approximated with a Taylor Series expansion about 0. The full Taylor series for
3824:
3800:
3780:
3735:
3539:
3437:
3152:
3081:
2692:
2647:
2312:
1256:
1172:
1068:
1003:
669:
463:
445:
243:
128:
40:
3865:
3830:
3786:
3721:
2764:
2654:
1015:
980:{\displaystyle g(x,y)={\frac {1}{2\pi \sigma ^{2}}}e^{-(x^{2}+y^{2})/(2\sigma ^{2})}}
266:
235:
215:
90:
22:
3766:
3754:
3569:
3549:
3448:
231:
95:
45:
3559:
2624:
560:{\displaystyle g(x)={\frac {1}{{\sqrt {2\pi }}\sigma }}e^{-x^{2}/(2\sigma ^{2})}}
50:
2623:
significant errors. Better results can be achieved by instead using a different
2677:
858:
In two dimensions, it is the product of two such Gaussians, one per direction:
3071:
this constant equals approximately 0.8326. These values are quite close to 1.
3880:
3683:
3564:
3453:
105:
55:
3686:& Stockman, G. C: "Computer Vision", page 137, 150. Prentence Hall, 2001
3294:{\displaystyle \sigma ={\sqrt {\frac {n_{1}^{2}+\cdots +n_{m}^{2}-m}{12}}}}
3433:
3672:
A Class of Fast Gaussian Binomial Filters for Speech and Image Processing
3574:
3544:
3441:
2681:
2635:
1018:
262:
247:
211:
161:
3476:
2759:
where the standard deviations are expressed in a number of samples and
2673:
2553: in this section. Unsourced material may be challenged and removed.
462:
the ordinary frequency. These equations can also be expressed with the
156:
151:
146:
238:(or an approximation to it, since a true Gaussian response would have
281:
The one-dimensional Gaussian filter has an impulse response given by
2528:
3862:
Noise and Signal Interference in Optical Fiber Transmission Systems
3305:
2685:
2031:
A 3rd order Gaussian filter with a -3.010 dB cutoff attenuation at
3829:(3rd ed.). US: McGraw-Hill, Inc. pp. 2.56, 2.65, 11.62.
2981:
for the response of the filter, the cut-off frequency is given by
202:
3785:. Beaverton, Oegon, US: Matrix Publishers, Inc. pp. 45–73.
3452:
brightness or color intensity, and the overall effect is called
2841:
where all quantities are expressed in their physical units. If
2831:{\displaystyle f_{c}=\sigma _{f}={\frac {1}{2\pi \sigma _{t}}}}
2749:{\displaystyle \sigma _{t}\cdot \sigma _{f}={\frac {N}{2\pi }}}
261:
Mathematically, a Gaussian filter modifies the input signal by
3608:(1 ed.). Beaverton, Oregon: Tektronix Circuit Concepts.
1380:
must be of the form obtained below, with no stop band zeros,
3762:
3758:
3519:
3512:
1009:
656:{\displaystyle {\hat {g}}(f)=e^{-f^{2}/(2\sigma _{f}^{2})}}
998:
is the distance from the origin in the vertical axis, and
206:
Shape of the impulse response of a typical Gaussian filter
3508:
1025:
expansion of the square of Gaussian function of the form
344:{\displaystyle g(x)={\sqrt {\frac {a}{\pi }}}e^{-ax^{2}}}
3636:"Low-Pass Risetime Filters for Time Domain Applications"
3042:{\displaystyle f_{c}={\sqrt {2\ln(c)}}\cdot \sigma _{f}}
994:
is the distance from the origin in the horizontal axis,
2926:{\displaystyle f_{c}={\frac {F_{s}}{2\pi \sigma _{t}}}}
1136:{\displaystyle \epsilon ^{-a\omega ^{2}}={\sqrt {1/2}}}
3406:
3382:
3353:
3317:
3228:
3175:
3155:
3104:
3084:
2990:
2942:
2877:
2847:
2776:
2704:
2587:
2465:
2341:
2315:
2059:
2037:
1882:
1857:
1837:
1716:
1602:
1560:
1520:
1388:
1357:
1279:
1259:
1246:{\displaystyle a=-log{\bigg (}{\sqrt {1/2}}{\bigg )}}
1195:
1175:
1149:
1091:
1071:
1031:
867:
844:{\displaystyle \sigma \sigma _{f}={\frac {1}{2\pi }}}
809:
768:
741:
712:
692:
672:
579:
475:
448:
366:
290:
242:). Gaussian filters have the properties of having no
2695:(in the time and frequency domains) are related by
1273:may be recalculated using a different attenuation,
254:. These properties are important in areas such as
3823:Williams, Arthur Bernard; Taylor, Fred J. (1995).
3421:
3392:
3368:
3339:
3304:(Note that standard deviations do not sum up, but
3293:
3211:
3161:
3141:
3090:
3041:
2973: ≈ 0.707 in the amplitude spectrum (see e.g.
2955:
2925:
2860:
2830:
2748:
2614:
2497:
2449:
2321:
2299:
2043:
2013:
1866:
1843:
1821:
1698:
1586:
1546:
1504:
1372:
1340:
1265:
1245:
1181:
1161:
1135:
1077:
1057:
979:
843:
792:
754:
727:
698:
678:
655:
559:
454:
432:{\displaystyle {\hat {g}}(f)=e^{-\pi ^{2}f^{2}/a}}
431:
343:
2000:
1238:
1216:
3878:
2509:
3822:
3677:
3440:with the origin as center. A two-dimensional
1189:may be calculated with this constraint to be
3778:
183:
3782:Filter Theory and Design: Active and Passive
3779:Sedra, Adel S.; Brackett, Peter O. (1978).
3664:
2680:. The simple moving average corresponds to
2026:
354:and the frequency response is given by the
269:; this transformation is also known as the
16:Filter in electronics and signal processing
3689:
190:
176:
3074:A simple moving average corresponds to a
2569:Learn how and when to remove this message
2518:
2329:into the coefficients, factoring using a
1932:
1763:
1587:{\displaystyle \epsilon ^{2a\omega ^{2}}}
1547:{\displaystyle \epsilon ^{2a\omega ^{2}}}
1058:{\displaystyle \epsilon ^{-a\omega ^{2}}}
1010:Synthesizing Gaussian filter polynomials
201:
3697:Feature Extraction and Image Processing
3633:
3598:
2615:{\displaystyle x\in (-\infty ,\infty )}
570:and the frequency response is given by
258:and digital telecommunication systems.
3879:
3864:, p. 242, John Wiley & Sons, 2008
3713:
3212:{\displaystyle {n}_{1},\dots ,{n}_{m}}
3142:{\displaystyle {\sqrt {(n^{2}-1)/12}}}
3843:
3695:Mark S. Nixon and Alberto S. Aguado.
3400:. Running it three times will give a
3149:. Thus the application of successive
1341:{\displaystyle a=log(10^{(|dB|/20)})}
3818:
3816:
3709:
3707:
3705:
3615:from the original on 14 October 2011
3518:the Gaussian filter is also used in
3471:
2977:). For an arbitrary cut-off value 1/
2551:adding citations to reliable sources
2522:
3757:filter design lecture notes, 1985,
2691:In the discrete case, the filter's
2459:A quick sanity check of evaluating
1514:To complete the transfer function,
13:
3078:and thus its filter width of size
2606:
2600:
1645:
14:
3903:
3826:Electronic Filter Design Handbook
3813:
3772:
3748:
3702:
3652:from the original on 21 July 2016
3602:Oscilloscope Vertical Amplifiers
3475:
3076:uniform probability distribution
2527:
3854:
3467:
2538:needs additional citations for
3699:. Academic Press, 2008, p. 88.
3670:R.A. Haddad and A.N. Akansu, "
3627:
3592:
3340:{\displaystyle 6\sigma _{t}-1}
3219:yield a standard deviation of
3126:
3107:
3021:
3015:
2609:
2594:
2491:
2487:
2481:
2467:
2435:
2426:
2411:
2401:
2386:
2376:
2361:
2352:
2190:
2180:
2168:
2152:
2127:
2117:
2092:
2083:
2073:
2070:
1970:
1960:
1951:
1941:
1902:
1893:
1782:
1772:
1733:
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1663:
1653:
1456:
1429:
1398:
1392:
1367:
1361:
1335:
1330:
1318:
1307:
1303:
1295:
1006:of the Gaussian distribution.
972:
956:
948:
922:
883:
871:
787:
781:
775:
722:
716:
648:
627:
598:
592:
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552:
536:
485:
479:
385:
379:
373:
300:
294:
1:
3805:: CS1 maint: date and year (
3740:: CS1 maint: date and year (
3586:
3462:floating point representation
3422:{\displaystyle {\sigma _{t}}}
3369:{\displaystyle {\sigma _{t}}}
2581:The Gaussian function is for
2510:Gaussian Transitional Filters
793:{\displaystyle {\hat {g}}(f)}
276:
66:Optimum "L" (Legendre) filter
3717:Handbook of Filter Synthesis
1351:To meet all above criteria,
7:
3532:
3393:{\displaystyle {\sqrt {2}}}
3311:A gaussian kernel requires
3169:moving averages with sizes
2861:{\displaystyle \sigma _{t}}
1143:(equivalent of -3.01dB) at
1021:may be synthesized using a
762:with the two equations for
755:{\displaystyle \sigma _{f}}
706:with the two equations for
10:
3908:
3714:Zverev, Anatol I. (1967).
3580:Scale space implementation
2629:scale space implementation
2498:{\displaystyle |F_{3}(j)|}
1373:{\displaystyle F(\omega )}
3645:. Picosecond Pulse Labs.
3634:Andrews, James R (1999).
3528:used in image processing.
1162:{\displaystyle \omega =1}
240:infinite impulse response
220:digital signal processing
118:Bridged T delay equaliser
34:Network synthesis filters
3759:Montana State University
3555:Discrete Gaussian kernel
2644:discrete Gaussian kernel
2027:Simple 3rd order example
1867:{\displaystyle j\omega }
1831:For the frequency axis,
3098:has standard deviation
2663:finite impulse response
2640:sampled Gaussian kernel
2044:{\displaystyle \omega }
1844:{\displaystyle \omega }
699:{\displaystyle \sigma }
84:Image impedance filters
51:Elliptic (Cauer) filter
3423:
3394:
3370:
3341:
3295:
3213:
3163:
3143:
3092:
3043:
2957:
2927:
2862:
2832:
2750:
2659:fast Fourier transform
2616:
2519:Digital implementation
2499:
2451:
2331:root finding algorithm
2323:
2301:
2045:
2015:
1937:
1868:
1845:
1823:
1768:
1700:
1649:
1588:
1548:
1506:
1374:
1342:
1267:
1247:
1183:
1163:
1137:
1079:
1059:
981:
845:
794:
756:
729:
700:
680:
657:
561:
456:
433:
345:
207:
124:Composite image filter
3599:Orwiler, Bob (1969).
3424:
3395:
3371:
3342:
3296:
3214:
3164:
3144:
3093:
3044:
2958:
2956:{\displaystyle F_{s}}
2928:
2863:
2833:
2751:
2670:central limit theorem
2617:
2500:
2452:
2324:
2302:
2046:
2016:
1915:
1869:
1846:
1824:
1746:
1701:
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1589:
1549:
1507:
1375:
1343:
1268:
1248:
1184:
1164:
1138:
1080:
1060:
982:
846:
795:
757:
735:and as a function of
730:
701:
681:
658:
562:
457:
434:
346:
271:Weierstrass transform
252:uncertainty principle
205:
101:General image filters
71:Linkwitz–Riley filter
3404:
3380:
3351:
3315:
3226:
3173:
3153:
3102:
3082:
2988:
2940:
2875:
2845:
2774:
2702:
2585:
2547:improve this article
2463:
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2313:
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2035:
1880:
1855:
1835:
1714:
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1518:
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1029:
865:
807:
766:
739:
728:{\displaystyle g(x)}
710:
690:
670:
577:
473:
446:
364:
288:
3860:Stefano Bottacchi,
3755:Dr. Byron Bennett's
3526:Canny Edge Detector
3347:values, e.g. for a
3277:
3253:
2693:standard deviations
2634:Filtering involves
647:
108:(constant R) filter
3540:Butterworth filter
3487:. You can help by
3438:concentric circles
3419:
3390:
3366:
3337:
3291:
3263:
3239:
3209:
3159:
3139:
3088:
3039:
2975:Butterworth filter
2953:
2923:
2858:
2828:
2746:
2684:with the constant
2648:diffusion equation
2612:
2506:filter increases.
2495:
2447:
2319:
2297:
2041:
2011:
1864:
1841:
1819:
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1584:
1544:
1502:
1370:
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1263:
1243:
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1133:
1075:
1055:
1004:standard deviation
977:
841:
790:
752:
725:
696:
676:
653:
633:
557:
464:standard deviation
452:
429:
341:
208:
41:Butterworth filter
25:electronic filters
3892:Gaussian function
3511:since it applies
3505:
3504:
3388:
3289:
3288:
3162:{\displaystyle m}
3137:
3091:{\displaystyle n}
3058:Gaussian function
3024:
2921:
2826:
2765:cut-off frequency
2744:
2655:Fourier transform
2579:
2578:
2571:
2445:
2322:{\displaystyle a}
2295:
2206:
2008:
1995:
1994:
1991:
1817:
1816:
1813:
1694:
1500:
1499:
1465:
1266:{\displaystyle a}
1234:
1182:{\displaystyle a}
1131:
1085:is set such that
1078:{\displaystyle a}
1016:transfer function
912:
839:
778:
686:as a function of
679:{\displaystyle a}
589:
511:
505:
455:{\displaystyle f}
376:
356:Fourier transform
316:
315:
267:Gaussian function
236:Gaussian function
216:signal processing
200:
199:
91:Constant k filter
3899:
3872:
3858:
3852:
3847:
3841:
3840:
3820:
3811:
3810:
3804:
3796:
3776:
3770:
3752:
3746:
3745:
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3731:
3711:
3700:
3693:
3687:
3681:
3675:
3668:
3662:
3661:
3659:
3657:
3651:
3640:
3631:
3625:
3624:
3622:
3620:
3614:
3607:
3596:
3570:Gaussian pyramid
3550:Chebyshev filter
3500:
3497:
3479:
3472:
3449:weighted average
3428:
3426:
3425:
3420:
3418:
3417:
3416:
3399:
3397:
3396:
3391:
3389:
3384:
3375:
3373:
3372:
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3365:
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3363:
3346:
3344:
3343:
3338:
3330:
3329:
3300:
3298:
3297:
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3290:
3284:
3276:
3271:
3252:
3247:
3237:
3236:
3218:
3216:
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3210:
3208:
3207:
3202:
3187:
3186:
3181:
3168:
3166:
3165:
3160:
3148:
3146:
3145:
3140:
3138:
3133:
3119:
3118:
3106:
3097:
3095:
3094:
3089:
3070:
3069:
3048:
3046:
3045:
3040:
3038:
3037:
3025:
3005:
3000:
2999:
2972:
2971:
2962:
2960:
2959:
2954:
2952:
2951:
2932:
2930:
2929:
2924:
2922:
2920:
2919:
2918:
2902:
2901:
2892:
2887:
2886:
2867:
2865:
2864:
2859:
2857:
2856:
2837:
2835:
2834:
2829:
2827:
2825:
2824:
2823:
2804:
2799:
2798:
2786:
2785:
2755:
2753:
2752:
2747:
2745:
2743:
2732:
2727:
2726:
2714:
2713:
2621:
2619:
2618:
2613:
2574:
2567:
2563:
2560:
2554:
2531:
2523:
2504:
2502:
2501:
2496:
2494:
2480:
2479:
2470:
2456:
2454:
2453:
2448:
2446:
2444:
2419:
2418:
2394:
2393:
2368:
2351:
2350:
2328:
2326:
2325:
2320:
2306:
2304:
2303:
2298:
2296:
2294:
2287:
2286:
2268:
2267:
2258:
2257:
2242:
2241:
2232:
2231:
2212:
2207:
2205:
2198:
2197:
2167:
2166:
2151:
2150:
2135:
2134:
2116:
2115:
2099:
2091:
2090:
2069:
2068:
2050:
2048:
2047:
2042:
2020:
2018:
2017:
2012:
2010:
2009:
2006:
2004:
2003:
1996:
1993:
1992:
1990:
1982:
1981:
1980:
1959:
1958:
1939:
1936:
1935:
1929:
1910:
1909:
1892:
1891:
1873:
1871:
1870:
1865:
1851:is replace with
1850:
1848:
1847:
1842:
1828:
1826:
1825:
1820:
1818:
1815:
1814:
1812:
1804:
1803:
1802:
1790:
1789:
1770:
1767:
1766:
1760:
1741:
1740:
1726:
1725:
1705:
1703:
1702:
1697:
1695:
1693:
1685:
1684:
1683:
1671:
1670:
1651:
1648:
1643:
1625:
1624:
1623:
1622:
1594:is shown below.
1593:
1591:
1590:
1585:
1583:
1582:
1581:
1580:
1553:
1551:
1550:
1545:
1543:
1542:
1541:
1540:
1511:
1509:
1508:
1503:
1501:
1498:
1497:
1496:
1495:
1472:
1471:
1466:
1464:
1463:
1454:
1453:
1452:
1451:
1428:
1426:
1425:
1424:
1423:
1379:
1377:
1376:
1371:
1347:
1345:
1344:
1339:
1334:
1333:
1326:
1321:
1310:
1272:
1270:
1269:
1264:
1252:
1250:
1249:
1244:
1242:
1241:
1235:
1230:
1222:
1220:
1219:
1188:
1186:
1185:
1180:
1169:. The value of
1168:
1166:
1165:
1160:
1142:
1140:
1139:
1134:
1132:
1127:
1119:
1114:
1113:
1112:
1111:
1084:
1082:
1081:
1076:
1064:
1062:
1061:
1056:
1054:
1053:
1052:
1051:
986:
984:
983:
978:
976:
975:
971:
970:
955:
947:
946:
934:
933:
913:
911:
910:
909:
890:
850:
848:
847:
842:
840:
838:
827:
822:
821:
799:
797:
796:
791:
780:
779:
771:
761:
759:
758:
753:
751:
750:
734:
732:
731:
726:
705:
703:
702:
697:
685:
683:
682:
677:
662:
660:
659:
654:
652:
651:
646:
641:
626:
621:
620:
591:
590:
582:
566:
564:
563:
558:
556:
555:
551:
550:
535:
530:
529:
512:
510:
506:
498:
492:
461:
459:
458:
453:
438:
436:
435:
430:
428:
427:
423:
418:
417:
408:
407:
378:
377:
369:
350:
348:
347:
342:
340:
339:
338:
337:
317:
308:
307:
232:impulse response
192:
185:
178:
96:m-derived filter
46:Chebyshev filter
19:
18:
3907:
3906:
3902:
3901:
3900:
3898:
3897:
3896:
3877:
3876:
3875:
3859:
3855:
3848:
3844:
3837:
3821:
3814:
3798:
3797:
3793:
3777:
3773:
3753:
3749:
3733:
3732:
3728:
3712:
3703:
3694:
3690:
3682:
3678:
3669:
3665:
3655:
3653:
3649:
3638:
3632:
3628:
3618:
3616:
3612:
3605:
3597:
3593:
3589:
3584:
3560:Elliptic filter
3535:
3501:
3495:
3492:
3485:needs expansion
3470:
3412:
3408:
3407:
3405:
3402:
3401:
3383:
3381:
3378:
3377:
3359:
3355:
3354:
3352:
3349:
3348:
3325:
3321:
3316:
3313:
3312:
3272:
3267:
3248:
3243:
3238:
3235:
3227:
3224:
3223:
3203:
3198:
3197:
3182:
3177:
3176:
3174:
3171:
3170:
3154:
3151:
3150:
3129:
3114:
3110:
3105:
3103:
3100:
3099:
3083:
3080:
3079:
3067:
3065:
3033:
3029:
3004:
2995:
2991:
2989:
2986:
2985:
2969:
2967:
2947:
2943:
2941:
2938:
2937:
2914:
2910:
2903:
2897:
2893:
2891:
2882:
2878:
2876:
2873:
2872:
2852:
2848:
2846:
2843:
2842:
2819:
2815:
2808:
2803:
2794:
2790:
2781:
2777:
2775:
2772:
2771:
2736:
2731:
2722:
2718:
2709:
2705:
2703:
2700:
2699:
2625:window function
2586:
2583:
2582:
2575:
2564:
2558:
2555:
2544:
2532:
2521:
2512:
2490:
2475:
2471:
2466:
2464:
2461:
2460:
2414:
2410:
2389:
2385:
2372:
2367:
2346:
2342:
2340:
2337:
2336:
2314:
2311:
2310:
2282:
2278:
2263:
2259:
2253:
2249:
2237:
2233:
2227:
2223:
2216:
2211:
2193:
2189:
2162:
2158:
2146:
2142:
2130:
2126:
2111:
2107:
2103:
2098:
2086:
2082:
2064:
2060:
2058:
2055:
2054:
2036:
2033:
2032:
2029:
2007:left half plane
2005:
1999:
1998:
1997:
1983:
1973:
1969:
1954:
1950:
1940:
1938:
1931:
1930:
1919:
1914:
1908:
1887:
1883:
1881:
1878:
1877:
1856:
1853:
1852:
1836:
1833:
1832:
1805:
1795:
1791:
1785:
1781:
1771:
1769:
1762:
1761:
1750:
1745:
1739:
1721:
1717:
1715:
1712:
1711:
1686:
1676:
1672:
1666:
1662:
1652:
1650:
1644:
1633:
1618:
1614:
1607:
1603:
1601:
1598:
1597:
1576:
1572:
1565:
1561:
1559:
1556:
1555:
1536:
1532:
1525:
1521:
1519:
1516:
1515:
1491:
1487:
1480:
1476:
1470:
1459:
1455:
1447:
1443:
1436:
1432:
1427:
1419:
1415:
1408:
1404:
1387:
1384:
1383:
1356:
1353:
1352:
1322:
1317:
1306:
1302:
1298:
1278:
1275:
1274:
1258:
1255:
1254:
1237:
1236:
1226:
1221:
1215:
1214:
1194:
1191:
1190:
1174:
1171:
1170:
1148:
1145:
1144:
1123:
1118:
1107:
1103:
1096:
1092:
1090:
1087:
1086:
1070:
1067:
1066:
1047:
1043:
1036:
1032:
1030:
1027:
1026:
1012:
966:
962:
951:
942:
938:
929:
925:
918:
914:
905:
901:
894:
889:
866:
863:
862:
831:
826:
817:
813:
808:
805:
804:
770:
769:
767:
764:
763:
746:
742:
740:
737:
736:
711:
708:
707:
691:
688:
687:
671:
668:
667:
642:
637:
622:
616:
612:
608:
604:
581:
580:
578:
575:
574:
546:
542:
531:
525:
521:
517:
513:
497:
496:
491:
474:
471:
470:
447:
444:
443:
419:
413:
409:
403:
399:
395:
391:
368:
367:
365:
362:
361:
333:
329:
322:
318:
306:
289:
286:
285:
279:
224:Gaussian filter
196:
167:
166:
142:
134:
133:
129:mm'-type filter
86:
76:
75:
61:Gaussian filter
36:
24:
17:
12:
11:
5:
3905:
3895:
3894:
3889:
3887:Linear filters
3874:
3873:
3853:
3842:
3835:
3812:
3792:978-0916460143
3791:
3771:
3747:
3726:
3701:
3688:
3684:Shapiro, L. G.
3676:
3663:
3626:
3590:
3588:
3585:
3583:
3582:
3577:
3572:
3567:
3562:
3557:
3552:
3547:
3542:
3536:
3534:
3531:
3530:
3529:
3523:
3516:
3503:
3502:
3482:
3480:
3469:
3466:
3415:
3411:
3387:
3362:
3358:
3336:
3333:
3328:
3324:
3320:
3302:
3301:
3287:
3283:
3280:
3275:
3270:
3266:
3262:
3259:
3256:
3251:
3246:
3242:
3234:
3231:
3206:
3201:
3196:
3193:
3190:
3185:
3180:
3158:
3136:
3132:
3128:
3125:
3122:
3117:
3113:
3109:
3087:
3050:
3049:
3036:
3032:
3028:
3023:
3020:
3017:
3014:
3011:
3008:
3003:
2998:
2994:
2950:
2946:
2934:
2933:
2917:
2913:
2909:
2906:
2900:
2896:
2890:
2885:
2881:
2855:
2851:
2839:
2838:
2822:
2818:
2814:
2811:
2807:
2802:
2797:
2793:
2789:
2784:
2780:
2757:
2756:
2742:
2739:
2735:
2730:
2725:
2721:
2717:
2712:
2708:
2678:moving average
2611:
2608:
2605:
2602:
2599:
2596:
2593:
2590:
2577:
2576:
2559:September 2013
2535:
2533:
2526:
2520:
2517:
2511:
2508:
2493:
2489:
2486:
2483:
2478:
2474:
2469:
2443:
2440:
2437:
2434:
2431:
2428:
2425:
2422:
2417:
2413:
2409:
2406:
2403:
2400:
2397:
2392:
2388:
2384:
2381:
2378:
2375:
2371:
2366:
2363:
2360:
2357:
2354:
2349:
2345:
2318:
2293:
2290:
2285:
2281:
2277:
2274:
2271:
2266:
2262:
2256:
2252:
2248:
2245:
2240:
2236:
2230:
2226:
2222:
2219:
2215:
2210:
2204:
2201:
2196:
2192:
2188:
2185:
2182:
2179:
2176:
2173:
2170:
2165:
2161:
2157:
2154:
2149:
2145:
2141:
2138:
2133:
2129:
2125:
2122:
2119:
2114:
2110:
2106:
2102:
2097:
2094:
2089:
2085:
2081:
2078:
2075:
2072:
2067:
2063:
2040:
2028:
2025:
2002:
1989:
1986:
1979:
1976:
1972:
1968:
1965:
1962:
1957:
1953:
1949:
1946:
1943:
1934:
1928:
1925:
1922:
1918:
1913:
1907:
1904:
1901:
1898:
1895:
1890:
1886:
1863:
1860:
1840:
1811:
1808:
1801:
1798:
1794:
1788:
1784:
1780:
1777:
1774:
1765:
1759:
1756:
1753:
1749:
1744:
1738:
1735:
1732:
1729:
1724:
1720:
1692:
1689:
1682:
1679:
1675:
1669:
1665:
1661:
1658:
1655:
1647:
1642:
1639:
1636:
1632:
1628:
1621:
1617:
1613:
1610:
1606:
1579:
1575:
1571:
1568:
1564:
1539:
1535:
1531:
1528:
1524:
1494:
1490:
1486:
1483:
1479:
1475:
1469:
1462:
1458:
1450:
1446:
1442:
1439:
1435:
1431:
1422:
1418:
1414:
1411:
1407:
1403:
1400:
1397:
1394:
1391:
1369:
1366:
1363:
1360:
1337:
1332:
1329:
1325:
1320:
1316:
1313:
1309:
1305:
1301:
1297:
1294:
1291:
1288:
1285:
1282:
1262:
1240:
1233:
1229:
1225:
1218:
1213:
1210:
1207:
1204:
1201:
1198:
1178:
1158:
1155:
1152:
1130:
1126:
1122:
1117:
1110:
1106:
1102:
1099:
1095:
1074:
1050:
1046:
1042:
1039:
1035:
1011:
1008:
988:
987:
974:
969:
965:
961:
958:
954:
950:
945:
941:
937:
932:
928:
924:
921:
917:
908:
904:
900:
897:
893:
888:
885:
882:
879:
876:
873:
870:
853:
852:
837:
834:
830:
825:
820:
816:
812:
789:
786:
783:
777:
774:
749:
745:
724:
721:
718:
715:
695:
675:
664:
663:
650:
645:
640:
636:
632:
629:
625:
619:
615:
611:
607:
603:
600:
597:
594:
588:
585:
568:
567:
554:
549:
545:
541:
538:
534:
528:
524:
520:
516:
509:
504:
501:
495:
490:
487:
484:
481:
478:
451:
440:
439:
426:
422:
416:
412:
406:
402:
398:
394:
390:
387:
384:
381:
375:
372:
352:
351:
336:
332:
328:
325:
321:
314:
311:
305:
302:
299:
296:
293:
278:
275:
198:
197:
195:
194:
187:
180:
172:
169:
168:
165:
164:
159:
154:
149:
143:
141:Simple filters
140:
139:
136:
135:
132:
131:
126:
121:
115:
112:Lattice filter
109:
103:
98:
93:
87:
82:
81:
78:
77:
74:
73:
68:
63:
58:
53:
48:
43:
37:
32:
31:
28:
27:
15:
9:
6:
4:
3:
2:
3904:
3893:
3890:
3888:
3885:
3884:
3882:
3871:
3867:
3863:
3857:
3851:
3846:
3838:
3836:0-07-070441-4
3832:
3828:
3827:
3819:
3817:
3808:
3802:
3794:
3788:
3784:
3783:
3775:
3769:, Montana, US
3768:
3764:
3763:EE Department
3760:
3756:
3751:
3743:
3737:
3729:
3727:0 471 98680 1
3723:
3719:
3718:
3710:
3708:
3706:
3698:
3692:
3685:
3680:
3673:
3667:
3648:
3644:
3637:
3630:
3611:
3604:
3603:
3595:
3591:
3581:
3578:
3576:
3573:
3571:
3568:
3566:
3565:Gaussian blur
3563:
3561:
3558:
3556:
3553:
3551:
3548:
3546:
3543:
3541:
3538:
3537:
3527:
3524:
3521:
3517:
3514:
3510:
3507:
3506:
3499:
3490:
3486:
3483:This section
3481:
3478:
3474:
3473:
3465:
3463:
3457:
3455:
3454:Gaussian blur
3450:
3446:
3443:
3439:
3435:
3430:
3413:
3409:
3385:
3360:
3356:
3334:
3331:
3326:
3322:
3318:
3309:
3307:
3285:
3281:
3278:
3273:
3268:
3264:
3260:
3257:
3254:
3249:
3244:
3240:
3232:
3229:
3222:
3221:
3220:
3204:
3199:
3194:
3191:
3188:
3183:
3178:
3156:
3134:
3130:
3123:
3120:
3115:
3111:
3085:
3077:
3072:
3064: =
3063:
3059:
3055:
3034:
3030:
3026:
3018:
3012:
3009:
3006:
3001:
2996:
2992:
2984:
2983:
2982:
2980:
2976:
2964:
2948:
2944:
2915:
2911:
2907:
2904:
2898:
2894:
2888:
2883:
2879:
2871:
2870:
2869:
2853:
2849:
2820:
2816:
2812:
2809:
2805:
2800:
2795:
2791:
2787:
2782:
2778:
2770:
2769:
2768:
2766:
2762:
2740:
2737:
2733:
2728:
2723:
2719:
2715:
2710:
2706:
2698:
2697:
2696:
2694:
2689:
2687:
2683:
2679:
2675:
2671:
2666:
2664:
2660:
2656:
2651:
2649:
2645:
2641:
2637:
2632:
2631:for details.
2630:
2626:
2603:
2597:
2591:
2588:
2573:
2570:
2562:
2552:
2548:
2542:
2541:
2536:This section
2534:
2530:
2525:
2524:
2516:
2507:
2484:
2476:
2472:
2457:
2441:
2438:
2432:
2429:
2423:
2420:
2415:
2407:
2404:
2398:
2395:
2390:
2382:
2379:
2373:
2369:
2364:
2358:
2355:
2347:
2343:
2334:
2332:
2316:
2307:
2291:
2288:
2283:
2279:
2275:
2272:
2269:
2264:
2260:
2254:
2250:
2246:
2243:
2238:
2234:
2228:
2224:
2220:
2217:
2213:
2208:
2202:
2199:
2194:
2186:
2183:
2177:
2174:
2171:
2163:
2159:
2155:
2147:
2143:
2139:
2136:
2131:
2123:
2120:
2112:
2108:
2104:
2100:
2095:
2087:
2079:
2076:
2065:
2061:
2052:
2038:
2024:
2021:
1987:
1984:
1977:
1974:
1966:
1963:
1955:
1947:
1944:
1926:
1923:
1920:
1916:
1911:
1905:
1899:
1896:
1888:
1884:
1875:
1861:
1858:
1838:
1829:
1809:
1806:
1799:
1796:
1792:
1786:
1778:
1775:
1757:
1754:
1751:
1747:
1742:
1736:
1730:
1722:
1718:
1709:
1706:
1690:
1687:
1680:
1677:
1673:
1667:
1659:
1656:
1640:
1637:
1634:
1630:
1626:
1619:
1615:
1611:
1608:
1604:
1595:
1577:
1573:
1569:
1566:
1562:
1537:
1533:
1529:
1526:
1522:
1512:
1492:
1488:
1484:
1481:
1477:
1473:
1467:
1460:
1448:
1444:
1440:
1437:
1433:
1420:
1416:
1412:
1409:
1405:
1401:
1395:
1389:
1381:
1364:
1358:
1349:
1327:
1323:
1314:
1311:
1299:
1292:
1289:
1286:
1283:
1280:
1260:
1231:
1227:
1223:
1211:
1208:
1205:
1202:
1199:
1196:
1176:
1156:
1153:
1150:
1128:
1124:
1120:
1115:
1108:
1104:
1100:
1097:
1093:
1072:
1048:
1044:
1040:
1037:
1033:
1024:
1023:Taylor series
1020:
1017:
1014:The Gaussian
1007:
1005:
1001:
997:
993:
967:
963:
959:
952:
943:
939:
935:
930:
926:
919:
915:
906:
902:
898:
895:
891:
886:
880:
877:
874:
868:
861:
860:
859:
856:
835:
832:
828:
823:
818:
814:
810:
803:
802:
801:
784:
772:
747:
743:
719:
713:
693:
673:
643:
638:
634:
630:
623:
617:
613:
609:
605:
601:
595:
583:
573:
572:
571:
547:
543:
539:
532:
526:
522:
518:
514:
507:
502:
499:
493:
488:
482:
476:
469:
468:
467:
466:as parameter
465:
449:
424:
420:
414:
410:
404:
400:
396:
392:
388:
382:
370:
360:
359:
358:
357:
334:
330:
326:
323:
319:
312:
309:
303:
297:
291:
284:
283:
282:
274:
272:
268:
264:
259:
257:
256:oscilloscopes
253:
249:
245:
241:
237:
233:
229:
225:
221:
217:
213:
204:
193:
188:
186:
181:
179:
174:
173:
171:
170:
163:
160:
158:
155:
153:
150:
148:
145:
144:
138:
137:
130:
127:
125:
122:
119:
116:
113:
110:
107:
106:Zobel network
104:
102:
99:
97:
94:
92:
89:
88:
85:
80:
79:
72:
69:
67:
64:
62:
59:
57:
56:Bessel filter
54:
52:
49:
47:
44:
42:
39:
38:
35:
30:
29:
26:
23:Linear analog
21:
20:
3861:
3856:
3845:
3825:
3781:
3774:
3750:
3716:
3696:
3691:
3679:
3666:
3654:. Retrieved
3642:
3629:
3617:. Retrieved
3601:
3594:
3493:
3489:adding to it
3484:
3468:Applications
3458:
3431:
3310:
3303:
3073:
3061:
3053:
3051:
2978:
2965:
2935:
2840:
2760:
2758:
2690:
2667:
2652:
2633:
2580:
2565:
2556:
2545:Please help
2540:verification
2537:
2513:
2458:
2335:
2308:
2053:
2030:
2022:
1876:
1830:
1710:
1707:
1596:
1513:
1382:
1350:
1013:
999:
995:
991:
989:
857:
854:
665:
569:
441:
353:
280:
260:
223:
218:, mainly in
209:
60:
3656:17 November
3619:17 November
3575:Scale space
3545:Comb filter
3442:convolution
2682:convolution
2668:Due to the
2636:convolution
1019:polynomials
666:By writing
263:convolution
248:group delay
212:electronics
3881:Categories
3870:047051681X
3643:kh6htv.com
3587:References
3515:modulation
2674:statistics
2653:Since the
2309:Absorbing
277:Definition
162:RLC filter
120:(all-pass)
114:(all-pass)
3801:cite book
3736:cite book
3410:σ
3357:σ
3332:−
3323:σ
3306:variances
3279:−
3258:⋯
3230:σ
3192:…
3121:−
3031:σ
3027:⋅
3013:
2912:σ
2908:π
2850:σ
2817:σ
2813:π
2792:σ
2741:π
2720:σ
2716:⋅
2707:σ
2607:∞
2601:∞
2598:−
2592:∈
2433:ω
2424:1.6458471
2408:ω
2399:1.0078328
2383:ω
2374:0.2355931
2359:ω
2280:ω
2270:−
2261:ω
2235:ω
2218:−
2187:ω
2160:ω
2124:ω
2080:ω
2039:ω
1967:ω
1917:∑
1900:ω
1862:ω
1839:ω
1793:ω
1748:∑
1731:ω
1674:ω
1646:∞
1631:∑
1616:ω
1605:ϵ
1574:ω
1563:ϵ
1534:ω
1523:ϵ
1489:ω
1478:ϵ
1445:ω
1438:−
1434:ϵ
1417:ω
1410:−
1406:ϵ
1396:ω
1365:ω
1203:−
1151:ω
1105:ω
1098:−
1094:ϵ
1045:ω
1038:−
1034:ϵ
964:σ
920:−
903:σ
899:π
836:π
815:σ
811:σ
776:^
744:σ
694:σ
635:σ
610:−
587:^
544:σ
519:−
508:σ
503:π
401:π
397:−
374:^
324:−
313:π
244:overshoot
157:LC filter
152:RL filter
147:RC filter
3647:Archived
3610:Archived
3533:See also
3496:May 2012
3434:contours
2686:B-spline
3767:Bozeman
3066:√
3060:). For
2968:√
2221:1.33333
2105:1.33333
1002:is the
265:with a
3868:
3833:
3789:
3724:
3445:matrix
2936:where
2672:(from
2627:; see
1065:where
990:where
230:whose
228:filter
3650:(PDF)
3639:(PDF)
3613:(PDF)
3606:(PDF)
3308:do.)
442:with
234:is a
226:is a
3866:ISBN
3831:ISBN
3807:link
3787:ISBN
3742:link
3722:ISBN
3658:2022
3621:2022
3520:GFSK
3513:GMSK
3436:are
3052:For
222:, a
214:and
3765:,
3509:GSM
3491:.
2549:by
210:In
3883::
3815:^
3803:}}
3799:{{
3761:,
3738:}}
3734:{{
3704:^
3641:.
3464:.
3456:.
3286:12
3135:12
3010:ln
2650:.
1874:.
1348:.
1328:20
1300:10
273:.
3839:.
3809:)
3795:.
3744:)
3730:.
3660:.
3623:.
3522:.
3498:)
3494:(
3414:t
3386:2
3361:t
3335:1
3327:t
3319:6
3282:m
3274:2
3269:m
3265:n
3261:+
3255:+
3250:2
3245:1
3241:n
3233:=
3205:m
3200:n
3195:,
3189:,
3184:1
3179:n
3157:m
3131:/
3127:)
3124:1
3116:2
3112:n
3108:(
3086:n
3068:2
3062:c
3054:c
3035:f
3022:)
3019:c
3016:(
3007:2
3002:=
2997:c
2993:f
2979:c
2970:2
2949:s
2945:F
2916:t
2905:2
2899:s
2895:F
2889:=
2884:c
2880:f
2854:t
2821:t
2810:2
2806:1
2801:=
2796:f
2788:=
2783:c
2779:f
2761:N
2738:2
2734:N
2729:=
2724:f
2711:t
2610:)
2604:,
2595:(
2589:x
2572:)
2566:(
2561:)
2557:(
2543:.
2492:|
2488:)
2485:j
2482:(
2477:3
2473:F
2468:|
2442:1
2439:+
2436:)
2430:j
2427:(
2421:+
2416:2
2412:)
2405:j
2402:(
2396:+
2391:3
2387:)
2380:j
2377:(
2370:1
2365:=
2362:)
2356:j
2353:(
2348:3
2344:F
2317:a
2292:1
2289:+
2284:2
2276:a
2273:2
2265:4
2255:2
2251:a
2247:2
2244:+
2239:6
2229:3
2225:a
2214:1
2209:=
2203:1
2200:+
2195:2
2191:)
2184:j
2181:(
2178:a
2175:2
2172:+
2169:)
2164:4
2156:j
2153:(
2148:2
2144:a
2140:2
2137:+
2132:6
2128:)
2121:j
2118:(
2113:3
2109:a
2101:1
2096:=
2093:)
2088:2
2084:)
2077:j
2074:(
2071:(
2066:3
2062:F
2001:|
1988:!
1985:k
1978:k
1975:2
1971:)
1964:j
1961:(
1956:k
1952:)
1948:a
1945:2
1942:(
1933:N
1927:0
1924:=
1921:k
1912:1
1906:=
1903:)
1897:j
1894:(
1889:N
1885:F
1859:j
1810:!
1807:k
1800:k
1797:2
1787:k
1783:)
1779:a
1776:2
1773:(
1764:N
1758:0
1755:=
1752:k
1743:1
1737:=
1734:)
1728:(
1723:N
1719:F
1691:!
1688:k
1681:k
1678:2
1668:k
1664:)
1660:a
1657:2
1654:(
1641:0
1638:=
1635:k
1627:=
1620:2
1612:a
1609:2
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1538:2
1530:a
1527:2
1493:2
1485:a
1482:2
1474:1
1468:=
1461:2
1457:)
1449:2
1441:a
1430:(
1421:2
1413:a
1402:=
1399:)
1393:(
1390:F
1368:)
1362:(
1359:F
1336:)
1331:)
1324:/
1319:|
1315:B
1312:d
1308:|
1304:(
1296:(
1293:g
1290:o
1287:l
1284:=
1281:a
1261:a
1239:)
1232:2
1228:/
1224:1
1217:(
1212:g
1209:o
1206:l
1200:=
1197:a
1177:a
1157:1
1154:=
1129:2
1125:/
1121:1
1116:=
1109:2
1101:a
1073:a
1049:2
1041:a
1000:σ
996:y
992:x
973:)
968:2
960:2
957:(
953:/
949:)
944:2
940:y
936:+
931:2
927:x
923:(
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907:2
896:2
892:1
887:=
884:)
881:y
878:,
875:x
872:(
869:g
851:,
833:2
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785:f
782:(
773:g
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720:x
717:(
714:g
674:a
649:)
644:2
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628:(
624:/
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602:=
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593:(
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537:(
533:/
527:2
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500:2
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489:=
486:)
483:x
480:(
477:g
450:f
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421:/
415:2
411:f
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393:e
389:=
386:)
383:f
380:(
371:g
335:2
331:x
327:a
320:e
310:a
304:=
301:)
298:x
295:(
292:g
191:e
184:t
177:v
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