3644:
2274:
3636:
3652:
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20:
3668:
2266:
3940:
3177:
Therefore, when two periodic signals have the same frequency, they are always in phase, or always out of phase. Physically, this situation commonly occurs, for many reasons. For example, the two signals may be a periodic soundwave recorded by two microphones at separate locations. Or, conversely,
3713:
If the two frequencies were exactly the same, their phase relationship would not change and both would appear to be stationary on the oscilloscope display. Since the two frequencies are not exactly the same, the reference appears to be stationary and the test signal moves. By measuring the rate of
3717:
Vertical lines have been drawn through the points where each sine signal passes through zero. The bottom of the figure shows bars whose width represents the phase difference between the signals. In this case the phase difference is increasing, indicating that the test signal is lower in frequency
3266:
For sinusoidal signals (and a few other waveforms, like square or symmetric triangular), a phase shift of 180° is equivalent to a phase shift of 0° with negation of the amplitude. When two signals with these waveforms, same period, and opposite phases are added together, the sum
3757:
3698:
is a comparison of the phase of two waveforms, usually of the same nominal frequency. In time and frequency, the purpose of a phase comparison is generally to determine the frequency offset (difference between signal cycles) with respect to a reference.
3623:
2217:
2480:
The phase difference is particularly important when two signals are added together by a physical process, such as two periodic sound waves emitted by two sources and recorded together by a microphone. This is usually the case in
2476:
In the clock analogy, each signal is represented by a hand (or pointer) of the same clock, both turning at constant but possibly different speeds. The phase difference is then the angle between the two hands, measured clockwise.
1513:
1701:
1635:
2104:
Since phases are angles, any whole full turns should usually be ignored when performing arithmetic operations on them. That is, the sum and difference of two phases (in degrees) should be computed by the formulas
4423:
relative to some standard (strictly speaking, phase is always relative). To the extent that this standard is accepted by all parties, one can speak of an absolute phase in a particular field of application.
2355:
3512:
1114:
23:
Plot of one cycle of a sinusoidal function. The phase for each argument value, relative to the start of the cycle, is shown at the bottom, in degrees from 0° to 360° and in radians from 0 to 2π.
3762:
1018:
3074:
504:
2817:
4260:
3537:
2108:
3663:
moving from top to bottom. Right: the same wave after a central section underwent a phase shift, for example, by passing through a glass of different thickness than the other parts.
1987:
1843:
4383:
4045:
2843:
3683:
of same long-held note on the flute come into dominance at different points in the phase cycle. The phase difference between the different harmonics can be observed on a
4072:
2691:
2643:
2571:
1552:
1224:
869:
536:
332:
288:
4141:
3935:{\displaystyle {\begin{aligned}x(t)&=A\cos(2\pi ft+\varphi )\\y(t)&=A\sin(2\pi ft+\varphi )=A\cos \left(2\pi ft+\varphi -{\tfrac {\pi }{2}}\right)\end{aligned}}}
3258:
30° west of that point, then the phase difference between the two signals will be 30° (assuming that, in each signal, each period starts when the shadow is shortest).
3148:
2903:
743:
619:
3128:
1058:
723:
4283:
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4000:
3752:
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2255:
2054:
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1390:
1367:
1344:
1278:
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419:
212:
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1395:
173:
3440:
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1642:
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3532:
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3414:
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3354:
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3172:
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2534:
2510:
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2007:
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1903:
1883:
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933:
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815:
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392:
372:
352:
259:
232:
140:
112:
88:
64:
1577:
4092:
3980:
3960:
4666:
3178:
they may be periodic soundwaves created by two separate speakers from the same electrical signal, and recorded by a single microphone. They may be a
3293:
is either identically zero, or is a sinusoidal signal with the same period and phase, whose amplitude is the difference of the original amplitudes.
4581:
2965:. In the clock analogy, this situation corresponds to the two hands turning at the same speed, so that the angle between them is constant.
3706:. The oscilloscope will display two sine signals, as shown in the graphic to the right. In the adjacent image, the top sine signal is the
1517:
The phase expressed in degrees (from 0° to 360°, or from −180° to +180°) is defined the same way, except with "360°" in place of "2π".
938:
3015:
3185:
A well-known example of phase difference is the length of shadows seen at different points of Earth. To a first approximation, if
3182:
signal that reaches the receiving antenna in a straight line, and a copy of it that was reflected off a large building nearby.
2287:
538:
depends on the arbitrary choice of the start of each period, and on the interval of angles that each period is to be mapped to.
3465:
4537:
2512:
when the phase difference is zero, the two signals will have the same sign and will be reinforcing each other. One says that
3296:
The phase shift of the co-sine function relative to the sine function is +90°. It follows that, for two sinusoidal signals
1935:
1063:
4660:
4444:
2658:
2278:
4492:, the quality of a wave to display a well defined phase relationship in different regions of its domain of definition
451:
2602:
2513:
2768:
4232:
4692:
2009:
is a "canonical" function of a phase angle in 0 to 2π, that describes just one cycle of that waveform; and
2269:
Illustration of phase shift. The horizontal axis represents an angle (phase) that is increasing with time.
1795:
4504:, a phase change that happens when a wave is reflected off of a boundary from fast medium to slow medium
3676:
4649:
1143:
is an arbitrary "origin" value of the argument, that one considers to be the beginning of a cycle.
2822:
4353:
4017:
4687:
2667:
2619:
2547:
2486:
1528:
1200:
824:
512:
308:
264:
4050:
3618:{\displaystyle C={\sqrt {A^{2}+B^{2}}}\quad \quad {\text{ and }}\quad \quad \sin(\varphi )=B/C.}
2212:{\displaystyle 360\,\left\!\!\right]\quad \quad {\text{ and }}\quad \quad 360\,\left\!\!\right]}
1865:
is a function of an angle, defined only for a single full turn, that describes the variation of
4527:
4501:
4105:
3727:
3133:
2888:
728:
604:
4655:
3101:
1025:
696:
176:
4663:— Discusses the time-domain sources of phase shift in simple linear time-invariant circuits.
4484:
2971:
2237:
2032:
1708:
1372:
1349:
1326:
1256:
1173:
1119:
424:
397:
194:
4268:
4179:
3985:
3737:
3188:
2576:
149:
8:
4489:
4449:
4391:
3419:
3270:
4645:
4459:
3517:
3445:
3399:
3379:
3359:
3339:
3319:
3299:
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3217:
3157:
3081:
2991:
2948:
2928:
2908:
2868:
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2748:
2728:
2696:
2519:
2495:
2448:
2428:
2408:
2380:
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2079:
2059:
2012:
1992:
1911:
1888:
1868:
1848:
1775:
1755:
1735:
1557:
1303:
1283:
1229:
1226:
is then the angle from the 12:00 position to the current position of the hand, at time
1153:
918:
896:
874:
800:
780:
752:
676:
656:
636:
584:
564:
544:
377:
357:
337:
244:
217:
125:
97:
73:
49:
4317:
4288:
4203:
4150:
4637:
4533:
4495:
302:
may be used instead of sine, depending on where one considers each period to start.)
44:
2713:. The periodic changes from reinforcement and opposition cause a phenomenon called
1060:
denotes the fractional part of a real number, discarding its integer part; that is,
4553:
4077:
3965:
3945:
3643:
3009:
2714:
2273:
143:
4573:
2029:
is a scaling factor for the amplitude. (This claim assumes that the starting time
4454:
4386:
119:
4474:
4469:
4434:
4416:
4410:
2482:
184:
40:
2725:
The phase difference is especially important when comparing a periodic signal
4681:
4464:
3635:
2661:
of a composite signal or even different signals (e.g., voltage and current).
2536:
when the phases are different, the value of the sum depends on the waveform.
1508:{\displaystyle \varphi (t)=2\pi \left(\left\!\!\right]-{\frac {1}{2}}\right)}
115:
3714:
motion of the test signal the offset between frequencies can be determined.
2227:, minus one full turn), and subtracting 50° from 30° gives a phase of 340° (
1696:{\displaystyle \varphi (t_{0}+kT)=0\quad \quad {\text{ for any integer }}k.}
3721:
3703:
3651:
180:
3684:
1630:{\displaystyle \varphi (t+T)=\varphi (t)\quad \quad {\text{ for all }}t.}
818:
67:
32:
3627:
19:
4532:(3 ed.). Focal Press, Gulf Professional Publishing. p. 1499.
3660:
1323:
The formula above gives the phase as an angle in radians between 0 and
601:
is expressed as a fraction of the period, and then scaled to an angle
4671:
3710:, and the bottom sine signal represents a signal from the reference.
3707:
3680:
3656:
3255:
1247:
295:
305:
Usually, whole turns are ignored when expressing the phase; so that
94:-like quantity representing the fraction of the cycle covered up to
4609:
4439:
4420:
1929:
1150:
with a hand that turns at constant speed, making a full turn every
238:
4350:
In the context of communication waveforms, the time-variant angle
3667:
541:
The term "phase" is also used when comparing a periodic function
188:
28:
4479:
3675:
A real-world example of a sonic phase difference occurs in the
3442:
is a sinusoidal signal with the same frequency, with amplitude
506:) if the difference between them is a whole number of periods.
299:
4594:
This content has been copied and pasted from an NIST web page
4571:
4554:"Federal Standard 1037C: Glossary of Telecommunications Terms"
3702:
A phase comparison can be made by connecting two signals to a
2265:
3179:
3098:
is a "canonical" representative for a class of signals, like
1147:
91:
2465:
when the difference is zero, the two signals are said to be
3731:
2968:
In this case, the phase shift is simply the argument shift
2865:. Suppose also that the origin for computing the phase of
2350:{\displaystyle \varphi (t)=\varphi _{G}(t)-\varphi _{F}(t)}
1554:
of a periodic signal is periodic too, with the same period
1300:. For example, for a sinusoid, a convenient choice is any
291:
2885:
has been shifted too. In that case, the phase difference
1320:
where the function's value changes from zero to positive.
4014:
of the sinusoid. These signals are periodic with period
2219:
respectively. Thus, for example, the sum of phase angles
394:
is said to be "at the same phase" at two argument values
4656:
Phase angle, phase difference, time delay, and frequency
3722:
Formula for phase of an oscillation or a periodic signal
3507:{\displaystyle -90^{\circ }<\varphi <+90^{\circ }}
4607:
2664:
If the frequencies are different, the phase difference
1639:
The phase is zero at the start of each period; that is
693:
is a "canonical" function for a class of signals, like
4356:
4320:
4291:
4271:
4235:
4206:
4182:
4153:
4108:
4080:
4053:
4047:, and they are identical except for a displacement of
4020:
3988:
3968:
3948:
3912:
3740:
1109:{\displaystyle \!]=x-\left\lfloor x\right\rfloor \!\,}
16:
The elapsed fraction of a cycle of a periodic function
3760:
3540:
3520:
3468:
3448:
3422:
3402:
3382:
3362:
3342:
3322:
3302:
3273:
3240:
3220:
3191:
3160:
3136:
3104:
3084:
3018:
3012:) of the two signals and then scaled to a full turn:
2994:
2974:
2951:
2931:
2911:
2891:
2871:
2851:
2825:
2771:
2751:
2731:
2699:
2670:
2622:
2579:
2550:
2522:
2498:
2451:
2431:
2411:
2383:
2363:
2290:
2240:
2111:
2082:
2062:
2035:
2015:
1995:
1938:
1914:
1891:
1871:
1851:
1798:
1778:
1758:
1738:
1711:
1645:
1580:
1560:
1531:
1398:
1375:
1352:
1329:
1306:
1286:
1259:
1232:
1203:
1176:
1156:
1122:
1066:
1028:
941:
921:
899:
877:
827:
803:
783:
755:
731:
699:
679:
659:
639:
607:
587:
567:
547:
515:
454:
427:
400:
380:
360:
340:
334:
is also a periodic function, with the same period as
311:
267:
247:
220:
197:
152:
128:
100:
76:
52:
3261:
3130:
is for all sinusoidal signals, then the phase shift
354:, that repeatedly scans the same range of angles as
4377:
4347:but are relative to their own specific references.
4335:
4306:
4277:
4254:
4221:
4188:
4168:
4135:
4086:
4066:
4039:
3994:
3974:
3954:
3934:
3746:
3617:
3526:
3506:
3454:
3434:
3408:
3388:
3368:
3348:
3328:
3308:
3285:
3246:
3226:
3206:
3166:
3142:
3122:
3090:
3068:
3000:
2980:
2957:
2937:
2917:
2897:
2877:
2857:
2837:
2811:
2757:
2737:
2705:
2685:
2653:), sinusoidal signals are sometimes said to be in
2637:
2585:
2565:
2544:For sinusoidal signals, when the phase difference
2528:
2504:
2457:
2437:
2417:
2389:
2369:
2349:
2249:
2211:
2088:
2068:
2048:
2021:
2001:
1981:
1920:
1897:
1877:
1857:
1837:
1784:
1764:
1744:
1724:
1695:
1629:
1566:
1546:
1507:
1384:
1361:
1338:
1312:
1292:
1272:
1238:
1218:
1189:
1162:
1135:
1108:
1052:
1012:
927:
905:
883:
863:
809:
789:
761:
737:
717:
685:
665:
645:
613:
593:
573:
553:
530:
498:
440:
413:
386:
366:
346:
326:
282:
253:
226:
206:
167:
134:
106:
82:
58:
3057:
3056:
3037:
3036:
2925:), called the 'phase shift' or 'phase offset' of
2203:
2202:
2175:
2174:
2150:
2149:
2122:
2121:
1481:
1480:
1431:
1430:
1253:The phase concept is most useful when the origin
1104:
1080:
1070:
1048:
1044:
1032:
1013:{\displaystyle \varphi (t)=2\pi \left\!\!\right]}
1004:
1003:
969:
968:
797:be a periodic function of one real variable, and
4679:
237:This convention is especially appropriate for a
4102:It can refer to a specified reference, such as
3069:{\displaystyle \varphi =2\pi \left\!\!\right].}
2988:, expressed as a fraction of the common period
499:{\displaystyle \varphi (t_{1})=\varphi (t_{2})}
4582:National Institute of Standards and Technology
2099:
1146:This concept can be visualized by imagining a
817:be its period (that is, the smallest positive
175:goes through each complete cycle). It may be
4572:Time and Frequency from A to Z (2010-05-12).
2601:. Then the signals have opposite signs, and
1705:Moreover, for any given choice of the origin
1525:With any of the above definitions, the phase
1170:seconds, and is pointing straight up at time
294:of the phase, multiplied by some factor (the
4521:
4519:
2745:with a shifted and possibly scaled version
2357:between the phases of two periodic signals
772:
241:function, since its value at any argument
4516:
3254:is the length seen at the same time at a
2790:
2168:
2115:
1957:
1105:
1049:
1040:
1036:
4567:
4565:
4563:
3666:
3650:
3642:
3634:
3626:
2812:{\displaystyle G(t)=\alpha \,F(t+\tau )}
2272:
2264:
2234:Similar formulas hold for radians, with
1346:. To get the phase as an angle between
18:
4255:{\textstyle \varphi -{\frac {\pi }{2}}}
4098:can refer to several different things:
2593:radians), one says that the phases are
4680:
4525:
2720:
1982:{\displaystyle F(t)=A\,w(\varphi (t))}
4560:
2693:increases linearly with the argument
2645:is a quarter of turn (a right angle,
4667:Open Source Physics JavaScript HTML5
725:is for all sinusoidal signals, then
621:spanning a whole turn, one gets the
4498:, a method of changing phase by 90°
4002:are constant parameters called the
3690:
3647:Representation of phase comparison.
3336:with same frequency and amplitudes
1838:{\displaystyle F(t)=f(\varphi (t))}
13:
4608:Clint Goss; Barry Higgins (2013).
4445:In-phase and quadrature components
3687:of the sound of a warbling flute.
2659:in-phase and quadrature components
2613:implies a 180-degree phase shift.
14:
4704:
4630:
4403:
4143:, in which case we would say the
3677:warble of a Native American flute
3396:has phase shift +90° relative to
3262:For sinusoids with same frequency
374:goes through each period. Then,
4409:This section is an excerpt from
2539:
4661:ECE 209: Sources of Phase Shift
3582:
3581:
3575:
3574:
2164:
2163:
2157:
2156:
2056:chosen to compute the phase of
1908:In fact, every periodic signal
1905:ranges over a single period.
1681:
1680:
1615:
1614:
1520:
1280:is chosen based on features of
509:The numeric value of the phase
4601:
4546:
4330:
4324:
4301:
4295:
4216:
4210:
4163:
4157:
4130:
4115:
3870:
3849:
3830:
3824:
3814:
3793:
3774:
3768:
3595:
3589:
3201:
3195:
3117:
3111:
2905:is a constant (independent of
2806:
2794:
2781:
2775:
2680:
2674:
2632:
2626:
2560:
2554:
2344:
2338:
2322:
2316:
2300:
2294:
2260:
1976:
1973:
1967:
1961:
1948:
1942:
1832:
1829:
1823:
1817:
1808:
1802:
1671:
1649:
1611:
1605:
1596:
1584:
1541:
1535:
1408:
1402:
1213:
1207:
1081:
1077:
1071:
1067:
1045:
1041:
1033:
1029:
951:
945:
858:
852:
843:
831:
712:
706:
525:
519:
493:
480:
471:
458:
321:
315:
277:
271:
162:
156:
1:
4509:
4378:{\textstyle 2\pi ft+\varphi }
4285:, in which case we would say
4040:{\textstyle T={\frac {1}{f}}}
3679:. The amplitude of different
2838:{\displaystyle \alpha ,\tau }
2765:of it. That is, suppose that
2076:corresponds to argument 0 of
1772:depends only on its phase at
191:, thus increasing by 360° or
114:. It is expressed in such a
4529:Handbook for sound engineers
3754:in the following functions:
2516:is occurring. At arguments
7:
4596:and is in the public domain
4427:
4067:{\textstyle {\frac {T}{4}}}
3728:simple harmonic oscillation
3214:is the length seen at time
2686:{\displaystyle \varphi (t)}
2638:{\displaystyle \varphi (t)}
2597:, and that the signals are
2566:{\displaystyle \varphi (t)}
2100:Adding and comparing phases
1684: for any integer
1547:{\displaystyle \varphi (t)}
1219:{\displaystyle \varphi (t)}
864:{\displaystyle F(t+T)=F(t)}
531:{\displaystyle \varphi (t)}
327:{\displaystyle \varphi (t)}
283:{\displaystyle \varphi (t)}
118:that it varies by one full
10:
4709:
4408:
4136:{\textstyle \cos(2\pi ft)}
2616:When the phase difference
1732:, the value of the signal
2651:−90° = 270° = −π/2 = 3π/2
2514:constructive interference
261:then can be expressed as
234:completes a full period.
4640:". Prof. Jeffrey Hass. "
3704:two-channel oscilloscope
3143:{\displaystyle \varphi }
2898:{\displaystyle \varphi }
2605:occurs. Conversely, a
2603:destructive interference
1792:. Namely, one can write
738:{\displaystyle \varphi }
614:{\displaystyle \varphi }
581:of it. If the shift in
3123:{\displaystyle \sin(t)}
2487:superposition principle
2231:, plus one full turn).
1053:{\displaystyle \!]\!\,}
773:Mathematical definition
718:{\displaystyle \sin(t)}
561:with a shifted version
298:of the sinusoid). (The
4502:Reflection phase shift
4379:
4337:
4308:
4279:
4256:
4223:
4190:
4170:
4137:
4088:
4068:
4041:
3996:
3976:
3956:
3936:
3748:
3672:
3664:
3648:
3640:
3632:
3619:
3528:
3508:
3456:
3436:
3410:
3390:
3370:
3350:
3330:
3310:
3287:
3248:
3228:
3208:
3168:
3144:
3124:
3092:
3070:
3002:
2982:
2959:
2939:
2919:
2899:
2879:
2859:
2839:
2813:
2759:
2739:
2707:
2687:
2639:
2587:
2567:
2530:
2506:
2459:
2439:
2419:
2391:
2371:
2351:
2281:
2270:
2251:
2213:
2090:
2070:
2050:
2023:
2003:
1983:
1922:
1899:
1879:
1859:
1839:
1786:
1766:
1746:
1726:
1697:
1631:
1568:
1548:
1509:
1386:
1363:
1340:
1314:
1294:
1274:
1240:
1220:
1191:
1164:
1137:
1110:
1054:
1014:
929:
907:
885:
865:
811:
791:
763:
739:
719:
687:
667:
647:
615:
595:
575:
555:
532:
500:
442:
415:
388:
368:
348:
328:
284:
255:
228:
208:
169:
136:
108:
84:
60:
24:
4526:Ballou, Glen (2005).
4380:
4338:
4309:
4280:
4278:{\textstyle \varphi }
4257:
4224:
4191:
4189:{\textstyle \varphi }
4171:
4138:
4089:
4069:
4042:
3997:
3995:{\textstyle \varphi }
3977:
3957:
3937:
3749:
3747:{\textstyle \varphi }
3670:
3654:
3646:
3638:
3630:
3620:
3529:
3509:
3457:
3437:
3411:
3391:
3371:
3351:
3331:
3311:
3288:
3249:
3229:
3209:
3169:
3145:
3125:
3093:
3071:
3003:
2983:
2981:{\displaystyle \tau }
2960:
2940:
2920:
2900:
2880:
2860:
2840:
2814:
2760:
2740:
2708:
2688:
2640:
2588:
2568:
2531:
2507:
2460:
2440:
2420:
2392:
2372:
2352:
2276:
2268:
2252:
2250:{\displaystyle 2\pi }
2214:
2091:
2071:
2051:
2049:{\displaystyle t_{0}}
2024:
2004:
1984:
1932:can be expressed as
1923:
1900:
1880:
1860:
1840:
1787:
1767:
1747:
1727:
1725:{\displaystyle t_{0}}
1698:
1632:
1569:
1549:
1510:
1387:
1385:{\displaystyle +\pi }
1364:
1362:{\displaystyle -\pi }
1341:
1339:{\displaystyle 2\pi }
1315:
1295:
1275:
1273:{\displaystyle t_{0}}
1241:
1221:
1192:
1190:{\displaystyle t_{0}}
1165:
1138:
1136:{\displaystyle t_{0}}
1111:
1055:
1015:
930:
908:
886:
866:
812:
792:
764:
740:
720:
688:
668:
648:
616:
596:
576:
556:
533:
501:
443:
441:{\displaystyle t_{2}}
416:
414:{\displaystyle t_{1}}
389:
369:
349:
329:
285:
256:
229:
209:
207:{\displaystyle 2\pi }
170:
137:
109:
90:(such as time) is an
85:
61:
39:(symbol φ or ϕ) of a
22:
4485:Polarization (waves)
4389:, is referred to as
4354:
4318:
4289:
4269:
4233:
4204:
4180:
4151:
4106:
4078:
4051:
4018:
3986:
3966:
3946:
3758:
3738:
3718:than the reference.
3639:Out-of-phase signals
3538:
3518:
3466:
3446:
3420:
3400:
3380:
3360:
3340:
3320:
3300:
3271:
3238:
3218:
3207:{\displaystyle F(t)}
3189:
3158:
3134:
3102:
3082:
3016:
2992:
2972:
2949:
2929:
2909:
2889:
2869:
2849:
2823:
2769:
2749:
2729:
2697:
2668:
2620:
2586:{\displaystyle \pi }
2577:
2548:
2520:
2496:
2469:otherwise, they are
2449:
2429:
2409:
2381:
2361:
2288:
2277:Phase shifter using
2238:
2109:
2080:
2060:
2033:
2013:
1993:
1936:
1912:
1889:
1869:
1849:
1796:
1776:
1756:
1736:
1709:
1643:
1578:
1558:
1529:
1396:
1373:
1350:
1327:
1304:
1284:
1257:
1230:
1201:
1174:
1154:
1120:
1064:
1026:
939:
919:
897:
875:
825:
801:
781:
753:
729:
697:
677:
657:
637:
605:
585:
565:
545:
513:
452:
425:
398:
378:
358:
338:
309:
265:
245:
218:
195:
168:{\displaystyle F(t)}
150:
126:
98:
74:
50:
4693:Physical quantities
4648:, 2003. See also: (
4642:An Acoustics Primer
4490:Coherence (physics)
4450:Instantaneous phase
4392:instantaneous phase
3681:harmonic components
3435:{\displaystyle F+G}
3286:{\displaystyle F+G}
2819:for some constants
2721:For shifted signals
1618: for all
1392:, one uses instead
4646:Indiana University
4460:Phase cancellation
4419:is the phase of a
4375:
4333:
4304:
4275:
4252:
4219:
4186:
4166:
4133:
4084:
4064:
4037:
3992:
3972:
3952:
3932:
3930:
3921:
3744:
3673:
3665:
3649:
3641:
3633:
3615:
3524:
3504:
3452:
3432:
3406:
3386:
3366:
3346:
3326:
3306:
3283:
3244:
3224:
3204:
3164:
3150:called simply the
3140:
3120:
3088:
3066:
2998:
2978:
2955:
2935:
2915:
2895:
2875:
2855:
2835:
2809:
2755:
2735:
2703:
2683:
2635:
2583:
2563:
2526:
2502:
2485:systems, when the
2455:
2435:
2415:
2387:
2367:
2347:
2282:
2271:
2247:
2209:
2086:
2066:
2046:
2019:
1999:
1979:
1918:
1895:
1875:
1855:
1835:
1782:
1762:
1742:
1722:
1693:
1627:
1564:
1544:
1505:
1382:
1359:
1336:
1310:
1290:
1270:
1236:
1216:
1187:
1160:
1133:
1106:
1050:
1010:
925:
903:
881:
861:
807:
787:
759:
735:
715:
683:
663:
643:
611:
591:
571:
551:
528:
496:
438:
411:
384:
364:
344:
324:
280:
251:
224:
204:
165:
142:goes through each
132:
104:
80:
56:
25:
4539:978-0-240-80758-4
4496:Hilbert transform
4336:{\textstyle y(t)}
4307:{\textstyle x(t)}
4250:
4222:{\textstyle y(t)}
4169:{\textstyle x(t)}
4062:
4035:
3920:
3732:sinusoidal signal
3579:
3572:
3527:{\displaystyle F}
3455:{\displaystyle C}
3409:{\displaystyle F}
3389:{\displaystyle G}
3369:{\displaystyle B}
3349:{\displaystyle A}
3329:{\displaystyle G}
3309:{\displaystyle F}
3247:{\displaystyle G}
3234:at one spot, and
3227:{\displaystyle t}
3167:{\displaystyle G}
3091:{\displaystyle F}
3050:
3008:(in terms of the
3001:{\displaystyle T}
2958:{\displaystyle F}
2938:{\displaystyle G}
2918:{\displaystyle t}
2878:{\displaystyle G}
2858:{\displaystyle t}
2758:{\displaystyle G}
2738:{\displaystyle F}
2706:{\displaystyle t}
2529:{\displaystyle t}
2505:{\displaystyle t}
2473:with each other.
2458:{\displaystyle t}
2438:{\displaystyle F}
2418:{\displaystyle G}
2390:{\displaystyle G}
2370:{\displaystyle F}
2196:
2161:
2143:
2089:{\displaystyle w}
2069:{\displaystyle F}
2022:{\displaystyle A}
2002:{\displaystyle w}
1921:{\displaystyle F}
1898:{\displaystyle t}
1878:{\displaystyle F}
1858:{\displaystyle f}
1785:{\displaystyle t}
1765:{\displaystyle t}
1752:for any argument
1745:{\displaystyle F}
1685:
1619:
1567:{\displaystyle T}
1498:
1473:
1460:
1313:{\displaystyle t}
1293:{\displaystyle F}
1239:{\displaystyle t}
1163:{\displaystyle T}
997:
928:{\displaystyle t}
906:{\displaystyle F}
884:{\displaystyle t}
810:{\displaystyle T}
790:{\displaystyle F}
762:{\displaystyle G}
686:{\displaystyle F}
666:{\displaystyle F}
646:{\displaystyle G}
594:{\displaystyle t}
574:{\displaystyle G}
554:{\displaystyle F}
387:{\displaystyle F}
367:{\displaystyle t}
347:{\displaystyle F}
254:{\displaystyle t}
227:{\displaystyle t}
135:{\displaystyle t}
107:{\displaystyle t}
83:{\displaystyle t}
59:{\displaystyle F}
45:periodic function
4700:
4672:Phase Difference
4638:What is a phase?
4624:
4623:
4621:
4620:
4605:
4599:
4593:
4591:
4589:
4569:
4558:
4557:
4550:
4544:
4543:
4523:
4384:
4382:
4381:
4376:
4342:
4340:
4339:
4334:
4313:
4311:
4310:
4305:
4284:
4282:
4281:
4276:
4265:It can refer to
4261:
4259:
4258:
4253:
4251:
4243:
4228:
4226:
4225:
4220:
4195:
4193:
4192:
4187:
4175:
4173:
4172:
4167:
4142:
4140:
4139:
4134:
4094:axis. The term
4093:
4091:
4090:
4085:
4073:
4071:
4070:
4065:
4063:
4055:
4046:
4044:
4043:
4038:
4036:
4028:
4001:
3999:
3998:
3993:
3981:
3979:
3978:
3973:
3961:
3959:
3958:
3953:
3941:
3939:
3938:
3933:
3931:
3927:
3923:
3922:
3913:
3753:
3751:
3750:
3745:
3734:is the value of
3696:Phase comparison
3691:Phase comparison
3631:In-phase signals
3624:
3622:
3621:
3616:
3608:
3580:
3577:
3573:
3571:
3570:
3558:
3557:
3548:
3533:
3531:
3530:
3525:
3513:
3511:
3510:
3505:
3503:
3502:
3481:
3480:
3462:and phase shift
3461:
3459:
3458:
3453:
3441:
3439:
3438:
3433:
3415:
3413:
3412:
3407:
3395:
3393:
3392:
3387:
3375:
3373:
3372:
3367:
3355:
3353:
3352:
3347:
3335:
3333:
3332:
3327:
3315:
3313:
3312:
3307:
3292:
3290:
3289:
3284:
3253:
3251:
3250:
3245:
3233:
3231:
3230:
3225:
3213:
3211:
3210:
3205:
3173:
3171:
3170:
3165:
3149:
3147:
3146:
3141:
3129:
3127:
3126:
3121:
3097:
3095:
3094:
3089:
3075:
3073:
3072:
3067:
3062:
3058:
3055:
3051:
3043:
3010:modulo operation
3007:
3005:
3004:
2999:
2987:
2985:
2984:
2979:
2964:
2962:
2961:
2956:
2944:
2942:
2941:
2936:
2924:
2922:
2921:
2916:
2904:
2902:
2901:
2896:
2884:
2882:
2881:
2876:
2864:
2862:
2861:
2856:
2844:
2842:
2841:
2836:
2818:
2816:
2815:
2810:
2764:
2762:
2761:
2756:
2744:
2742:
2741:
2736:
2712:
2710:
2709:
2704:
2692:
2690:
2689:
2684:
2652:
2648:
2644:
2642:
2641:
2636:
2592:
2590:
2589:
2584:
2572:
2570:
2569:
2564:
2535:
2533:
2532:
2527:
2511:
2509:
2508:
2503:
2464:
2462:
2461:
2456:
2445:. At values of
2444:
2442:
2441:
2436:
2424:
2422:
2421:
2416:
2399:phase difference
2396:
2394:
2393:
2388:
2376:
2374:
2373:
2368:
2356:
2354:
2353:
2348:
2337:
2336:
2315:
2314:
2257:instead of 360.
2256:
2254:
2253:
2248:
2230:
2226:
2222:
2218:
2216:
2215:
2210:
2208:
2204:
2201:
2197:
2192:
2181:
2162:
2159:
2155:
2151:
2148:
2144:
2139:
2128:
2095:
2093:
2092:
2087:
2075:
2073:
2072:
2067:
2055:
2053:
2052:
2047:
2045:
2044:
2028:
2026:
2025:
2020:
2008:
2006:
2005:
2000:
1988:
1986:
1985:
1980:
1928:with a specific
1927:
1925:
1924:
1919:
1904:
1902:
1901:
1896:
1884:
1882:
1881:
1876:
1864:
1862:
1861:
1856:
1844:
1842:
1841:
1836:
1791:
1789:
1788:
1783:
1771:
1769:
1768:
1763:
1751:
1749:
1748:
1743:
1731:
1729:
1728:
1723:
1721:
1720:
1702:
1700:
1699:
1694:
1686:
1683:
1661:
1660:
1636:
1634:
1633:
1628:
1620:
1617:
1573:
1571:
1570:
1565:
1553:
1551:
1550:
1545:
1514:
1512:
1511:
1506:
1504:
1500:
1499:
1491:
1486:
1482:
1479:
1475:
1474:
1466:
1461:
1456:
1455:
1454:
1438:
1391:
1389:
1388:
1383:
1368:
1366:
1365:
1360:
1345:
1343:
1342:
1337:
1319:
1317:
1316:
1311:
1299:
1297:
1296:
1291:
1279:
1277:
1276:
1271:
1269:
1268:
1245:
1243:
1242:
1237:
1225:
1223:
1222:
1217:
1196:
1194:
1193:
1188:
1186:
1185:
1169:
1167:
1166:
1161:
1142:
1140:
1139:
1134:
1132:
1131:
1115:
1113:
1112:
1107:
1103:
1059:
1057:
1056:
1051:
1019:
1017:
1016:
1011:
1009:
1005:
1002:
998:
993:
992:
991:
975:
934:
932:
931:
926:
912:
910:
909:
904:
890:
888:
887:
882:
870:
868:
867:
862:
816:
814:
813:
808:
796:
794:
793:
788:
768:
766:
765:
760:
744:
742:
741:
736:
724:
722:
721:
716:
692:
690:
689:
684:
672:
670:
669:
664:
652:
650:
649:
644:
631:phase difference
620:
618:
617:
612:
600:
598:
597:
592:
580:
578:
577:
572:
560:
558:
557:
552:
537:
535:
534:
529:
505:
503:
502:
497:
492:
491:
470:
469:
447:
445:
444:
439:
437:
436:
420:
418:
417:
412:
410:
409:
393:
391:
390:
385:
373:
371:
370:
365:
353:
351:
350:
345:
333:
331:
330:
325:
289:
287:
286:
281:
260:
258:
257:
252:
233:
231:
230:
225:
214:as the variable
213:
211:
210:
205:
174:
172:
171:
166:
141:
139:
138:
133:
122:as the variable
113:
111:
110:
105:
89:
87:
86:
81:
65:
63:
62:
57:
4708:
4707:
4703:
4702:
4701:
4699:
4698:
4697:
4678:
4677:
4633:
4628:
4627:
4618:
4616:
4606:
4602:
4587:
4585:
4570:
4561:
4552:
4551:
4547:
4540:
4524:
4517:
4512:
4507:
4455:Lissajous curve
4430:
4425:
4424:
4414:
4406:
4387:principal value
4355:
4352:
4351:
4319:
4316:
4315:
4290:
4287:
4286:
4270:
4267:
4266:
4242:
4234:
4231:
4230:
4205:
4202:
4201:
4181:
4178:
4177:
4152:
4149:
4148:
4107:
4104:
4103:
4079:
4076:
4075:
4054:
4052:
4049:
4048:
4027:
4019:
4016:
4015:
3987:
3984:
3983:
3967:
3964:
3963:
3947:
3944:
3943:
3929:
3928:
3911:
3889:
3885:
3833:
3818:
3817:
3777:
3761:
3759:
3756:
3755:
3739:
3736:
3735:
3726:The phase of a
3724:
3693:
3671:Out of phase AE
3604:
3578: and
3576:
3566:
3562:
3553:
3549:
3547:
3539:
3536:
3535:
3519:
3516:
3515:
3498:
3494:
3476:
3472:
3467:
3464:
3463:
3447:
3444:
3443:
3421:
3418:
3417:
3401:
3398:
3397:
3381:
3378:
3377:
3361:
3358:
3357:
3341:
3338:
3337:
3321:
3318:
3317:
3301:
3298:
3297:
3272:
3269:
3268:
3264:
3239:
3236:
3235:
3219:
3216:
3215:
3190:
3187:
3186:
3159:
3156:
3155:
3135:
3132:
3131:
3103:
3100:
3099:
3083:
3080:
3079:
3042:
3038:
3035:
3031:
3017:
3014:
3013:
2993:
2990:
2989:
2973:
2970:
2969:
2950:
2947:
2946:
2930:
2927:
2926:
2910:
2907:
2906:
2890:
2887:
2886:
2870:
2867:
2866:
2850:
2847:
2846:
2824:
2821:
2820:
2770:
2767:
2766:
2750:
2747:
2746:
2730:
2727:
2726:
2723:
2698:
2695:
2694:
2669:
2666:
2665:
2650:
2646:
2621:
2618:
2617:
2611:phase inversion
2578:
2575:
2574:
2549:
2546:
2545:
2542:
2521:
2518:
2517:
2497:
2494:
2493:
2450:
2447:
2446:
2430:
2427:
2426:
2410:
2407:
2406:
2382:
2379:
2378:
2362:
2359:
2358:
2332:
2328:
2310:
2306:
2289:
2286:
2285:
2284:The difference
2263:
2239:
2236:
2235:
2228:
2225:190 + 200 = 390
2224:
2220:
2182:
2180:
2176:
2173:
2169:
2160: and
2158:
2129:
2127:
2123:
2120:
2116:
2110:
2107:
2106:
2102:
2081:
2078:
2077:
2061:
2058:
2057:
2040:
2036:
2034:
2031:
2030:
2014:
2011:
2010:
1994:
1991:
1990:
1937:
1934:
1933:
1913:
1910:
1909:
1890:
1887:
1886:
1870:
1867:
1866:
1850:
1847:
1846:
1797:
1794:
1793:
1777:
1774:
1773:
1757:
1754:
1753:
1737:
1734:
1733:
1716:
1712:
1710:
1707:
1706:
1682:
1656:
1652:
1644:
1641:
1640:
1616:
1579:
1576:
1575:
1559:
1556:
1555:
1530:
1527:
1526:
1523:
1490:
1465:
1450:
1446:
1439:
1437:
1436:
1432:
1429:
1425:
1424:
1420:
1397:
1394:
1393:
1374:
1371:
1370:
1351:
1348:
1347:
1328:
1325:
1324:
1305:
1302:
1301:
1285:
1282:
1281:
1264:
1260:
1258:
1255:
1254:
1231:
1228:
1227:
1202:
1199:
1198:
1181:
1177:
1175:
1172:
1171:
1155:
1152:
1151:
1127:
1123:
1121:
1118:
1117:
1093:
1065:
1062:
1061:
1027:
1024:
1023:
987:
983:
976:
974:
970:
967:
963:
940:
937:
936:
920:
917:
916:
898:
895:
894:
876:
873:
872:
826:
823:
822:
802:
799:
798:
782:
779:
778:
777:Let the signal
775:
754:
751:
750:
730:
727:
726:
698:
695:
694:
678:
675:
674:
658:
655:
654:
638:
635:
634:
606:
603:
602:
586:
583:
582:
566:
563:
562:
546:
543:
542:
514:
511:
510:
487:
483:
465:
461:
453:
450:
449:
432:
428:
426:
423:
422:
405:
401:
399:
396:
395:
379:
376:
375:
359:
356:
355:
339:
336:
335:
310:
307:
306:
266:
263:
262:
246:
243:
242:
219:
216:
215:
196:
193:
192:
151:
148:
147:
127:
124:
123:
99:
96:
95:
75:
72:
71:
51:
48:
47:
17:
12:
11:
5:
4706:
4696:
4695:
4690:
4688:Wave mechanics
4676:
4675:
4669:
4664:
4658:
4653:
4650:pages 1 thru 3
4644:", Section 8.
4632:
4631:External links
4629:
4626:
4625:
4600:
4559:
4545:
4538:
4514:
4513:
4511:
4508:
4506:
4505:
4499:
4493:
4487:
4482:
4477:
4475:Phase velocity
4472:
4470:Phase spectrum
4467:
4462:
4457:
4452:
4447:
4442:
4437:
4435:Absolute phase
4431:
4429:
4426:
4417:Absolute phase
4415:
4411:Absolute phase
4407:
4405:
4404:Absolute phase
4402:
4401:
4400:
4374:
4371:
4368:
4365:
4362:
4359:
4348:
4343:have the same
4332:
4329:
4326:
4323:
4303:
4300:
4297:
4294:
4274:
4263:
4249:
4246:
4241:
4238:
4218:
4215:
4212:
4209:
4185:
4165:
4162:
4159:
4156:
4132:
4129:
4126:
4123:
4120:
4117:
4114:
4111:
4087:{\textstyle t}
4083:
4061:
4058:
4034:
4031:
4026:
4023:
3991:
3975:{\textstyle f}
3971:
3955:{\textstyle A}
3951:
3926:
3919:
3916:
3910:
3907:
3904:
3901:
3898:
3895:
3892:
3888:
3884:
3881:
3878:
3875:
3872:
3869:
3866:
3863:
3860:
3857:
3854:
3851:
3848:
3845:
3842:
3839:
3836:
3834:
3832:
3829:
3826:
3823:
3820:
3819:
3816:
3813:
3810:
3807:
3804:
3801:
3798:
3795:
3792:
3789:
3786:
3783:
3780:
3778:
3776:
3773:
3770:
3767:
3764:
3763:
3743:
3723:
3720:
3708:test frequency
3692:
3689:
3614:
3611:
3607:
3603:
3600:
3597:
3594:
3591:
3588:
3585:
3569:
3565:
3561:
3556:
3552:
3546:
3543:
3523:
3501:
3497:
3493:
3490:
3487:
3484:
3479:
3475:
3471:
3451:
3431:
3428:
3425:
3405:
3385:
3365:
3345:
3325:
3305:
3282:
3279:
3276:
3263:
3260:
3243:
3223:
3203:
3200:
3197:
3194:
3163:
3139:
3119:
3116:
3113:
3110:
3107:
3087:
3065:
3061:
3054:
3049:
3046:
3041:
3034:
3030:
3027:
3024:
3021:
2997:
2977:
2954:
2934:
2914:
2894:
2874:
2854:
2834:
2831:
2828:
2808:
2805:
2802:
2799:
2796:
2793:
2789:
2786:
2783:
2780:
2777:
2774:
2754:
2734:
2722:
2719:
2702:
2682:
2679:
2676:
2673:
2634:
2631:
2628:
2625:
2607:phase reversal
2582:
2562:
2559:
2556:
2553:
2541:
2538:
2525:
2501:
2492:For arguments
2454:
2434:
2414:
2397:is called the
2386:
2366:
2346:
2343:
2340:
2335:
2331:
2327:
2324:
2321:
2318:
2313:
2309:
2305:
2302:
2299:
2296:
2293:
2262:
2259:
2246:
2243:
2207:
2200:
2195:
2191:
2188:
2185:
2179:
2172:
2167:
2154:
2147:
2142:
2138:
2135:
2132:
2126:
2119:
2114:
2101:
2098:
2085:
2065:
2043:
2039:
2018:
1998:
1978:
1975:
1972:
1969:
1966:
1963:
1960:
1956:
1953:
1950:
1947:
1944:
1941:
1917:
1894:
1874:
1854:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1781:
1761:
1741:
1719:
1715:
1692:
1689:
1679:
1676:
1673:
1670:
1667:
1664:
1659:
1655:
1651:
1648:
1626:
1623:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1563:
1543:
1540:
1537:
1534:
1522:
1519:
1503:
1497:
1494:
1489:
1485:
1478:
1472:
1469:
1464:
1459:
1453:
1449:
1445:
1442:
1435:
1428:
1423:
1419:
1416:
1413:
1410:
1407:
1404:
1401:
1381:
1378:
1358:
1355:
1335:
1332:
1309:
1289:
1267:
1263:
1235:
1215:
1212:
1209:
1206:
1184:
1180:
1159:
1130:
1126:
1102:
1099:
1096:
1092:
1089:
1086:
1083:
1079:
1076:
1073:
1069:
1047:
1043:
1039:
1035:
1031:
1008:
1001:
996:
990:
986:
982:
979:
973:
966:
962:
959:
956:
953:
950:
947:
944:
924:
902:
880:
860:
857:
854:
851:
848:
845:
842:
839:
836:
833:
830:
806:
786:
774:
771:
758:
745:is called the
734:
714:
711:
708:
705:
702:
682:
662:
642:
610:
590:
570:
550:
527:
524:
521:
518:
495:
490:
486:
482:
479:
476:
473:
468:
464:
460:
457:
435:
431:
408:
404:
383:
363:
343:
323:
320:
317:
314:
279:
276:
273:
270:
250:
223:
203:
200:
164:
161:
158:
155:
131:
103:
79:
55:
15:
9:
6:
4:
3:
2:
4705:
4694:
4691:
4689:
4686:
4685:
4683:
4673:
4670:
4668:
4665:
4662:
4659:
4657:
4654:
4651:
4647:
4643:
4639:
4635:
4634:
4615:
4611:
4604:
4597:
4583:
4579:
4575:
4568:
4566:
4564:
4555:
4549:
4541:
4535:
4531:
4530:
4522:
4520:
4515:
4503:
4500:
4497:
4494:
4491:
4488:
4486:
4483:
4481:
4478:
4476:
4473:
4471:
4468:
4466:
4465:Phase problem
4463:
4461:
4458:
4456:
4453:
4451:
4448:
4446:
4443:
4441:
4438:
4436:
4433:
4432:
4422:
4418:
4412:
4398:
4395:, often just
4394:
4393:
4388:
4372:
4369:
4366:
4363:
4360:
4357:
4349:
4346:
4327:
4321:
4298:
4292:
4272:
4264:
4247:
4244:
4239:
4236:
4213:
4207:
4199:
4183:
4160:
4154:
4146:
4127:
4124:
4121:
4118:
4112:
4109:
4101:
4100:
4099:
4097:
4081:
4059:
4056:
4032:
4029:
4024:
4021:
4013:
4009:
4005:
3989:
3969:
3949:
3924:
3917:
3914:
3908:
3905:
3902:
3899:
3896:
3893:
3890:
3886:
3882:
3879:
3876:
3873:
3867:
3864:
3861:
3858:
3855:
3852:
3846:
3843:
3840:
3837:
3835:
3827:
3821:
3811:
3808:
3805:
3802:
3799:
3796:
3790:
3787:
3784:
3781:
3779:
3771:
3765:
3741:
3733:
3729:
3719:
3715:
3711:
3709:
3705:
3700:
3697:
3688:
3686:
3682:
3678:
3669:
3662:
3658:
3653:
3645:
3637:
3629:
3625:
3612:
3609:
3605:
3601:
3598:
3592:
3586:
3583:
3567:
3563:
3559:
3554:
3550:
3544:
3541:
3521:
3499:
3495:
3491:
3488:
3485:
3482:
3477:
3473:
3469:
3449:
3429:
3426:
3423:
3403:
3383:
3363:
3343:
3323:
3303:
3294:
3280:
3277:
3274:
3259:
3257:
3241:
3221:
3198:
3192:
3183:
3181:
3175:
3161:
3153:
3152:initial phase
3137:
3114:
3108:
3105:
3085:
3076:
3063:
3059:
3052:
3047:
3044:
3039:
3032:
3028:
3025:
3022:
3019:
3011:
2995:
2975:
2966:
2952:
2932:
2912:
2892:
2872:
2852:
2832:
2829:
2826:
2803:
2800:
2797:
2791:
2787:
2784:
2778:
2772:
2752:
2732:
2718:
2716:
2700:
2677:
2671:
2662:
2660:
2656:
2629:
2623:
2614:
2612:
2608:
2604:
2600:
2596:
2580:
2557:
2551:
2540:For sinusoids
2537:
2523:
2515:
2499:
2490:
2488:
2484:
2478:
2474:
2472:
2468:
2452:
2432:
2412:
2404:
2400:
2384:
2364:
2341:
2333:
2329:
2325:
2319:
2311:
2307:
2303:
2297:
2291:
2280:
2275:
2267:
2258:
2244:
2241:
2232:
2229:30 − 50 = −20
2205:
2198:
2193:
2189:
2186:
2183:
2177:
2170:
2165:
2152:
2145:
2140:
2136:
2133:
2130:
2124:
2117:
2112:
2097:
2083:
2063:
2041:
2037:
2016:
1996:
1970:
1964:
1958:
1954:
1951:
1945:
1939:
1931:
1915:
1906:
1892:
1872:
1852:
1826:
1820:
1814:
1811:
1805:
1799:
1779:
1759:
1739:
1717:
1713:
1703:
1690:
1687:
1677:
1674:
1668:
1665:
1662:
1657:
1653:
1646:
1637:
1624:
1621:
1608:
1602:
1599:
1593:
1590:
1587:
1581:
1561:
1538:
1532:
1518:
1515:
1501:
1495:
1492:
1487:
1483:
1476:
1470:
1467:
1462:
1457:
1451:
1447:
1443:
1440:
1433:
1426:
1421:
1417:
1414:
1411:
1405:
1399:
1379:
1376:
1356:
1353:
1333:
1330:
1321:
1307:
1287:
1265:
1261:
1251:
1249:
1233:
1210:
1204:
1197:. The phase
1182:
1178:
1157:
1149:
1144:
1128:
1124:
1100:
1097:
1094:
1090:
1087:
1084:
1074:
1037:
1020:
1006:
999:
994:
988:
984:
980:
977:
971:
964:
960:
957:
954:
948:
942:
922:
915:any argument
914:
900:
891:). Then the
878:
855:
849:
846:
840:
837:
834:
828:
820:
804:
784:
770:
756:
748:
747:initial phase
732:
709:
703:
700:
680:
660:
640:
632:
628:
624:
608:
588:
568:
548:
539:
522:
516:
507:
488:
484:
477:
474:
466:
462:
455:
433:
429:
406:
402:
381:
361:
341:
318:
312:
303:
301:
297:
293:
274:
268:
248:
240:
235:
221:
201:
198:
190:
186:
182:
178:
159:
153:
145:
129:
121:
117:
101:
93:
77:
69:
53:
46:
42:
38:
34:
30:
21:
4641:
4617:. Retrieved
4613:
4610:"The Warble"
4603:
4595:
4586:. Retrieved
4577:
4548:
4528:
4396:
4390:
4344:
4197:
4144:
4095:
4011:
4007:
4003:
3725:
3716:
3712:
3701:
3695:
3694:
3674:
3534:, such that
3295:
3265:
3184:
3176:
3151:
3077:
2967:
2945:relative to
2724:
2663:
2654:
2615:
2610:
2606:
2599:in antiphase
2598:
2594:
2543:
2491:
2479:
2475:
2471:out of phase
2470:
2466:
2425:relative to
2402:
2398:
2283:
2279:IQ modulator
2261:Phase shift
2233:
2103:
1907:
1704:
1638:
1524:
1521:Consequences
1516:
1322:
1252:
1145:
1021:
892:
776:
746:
653:relative to
630:
627:phase offset
626:
622:
540:
508:
304:
236:
181:angular unit
36:
26:
4674:Java Applet
3685:spectrogram
2403:phase shift
2221:190° + 200°
1246:, measured
819:real number
623:phase shift
33:mathematics
4682:Categories
4619:2013-03-06
4614:Flutopedia
4510:References
4196:, and the
4074:along the
3661:plane wave
3655:Left: the
3416:, the sum
2655:quadrature
2647:+90° = π/2
821:such that
448:(that is,
239:sinusoidal
4385:, or its
4373:φ
4361:π
4273:φ
4245:π
4240:−
4237:φ
4184:φ
4122:π
4113:
4008:frequency
4004:amplitude
3990:φ
3915:π
3909:−
3906:φ
3894:π
3883:
3868:φ
3856:π
3847:
3812:φ
3800:π
3791:
3742:φ
3657:real part
3593:φ
3587:
3500:∘
3486:φ
3478:∘
3470:−
3256:longitude
3138:φ
3109:
3045:τ
3029:π
3020:φ
2976:τ
2893:φ
2833:τ
2827:α
2804:τ
2788:α
2672:φ
2624:φ
2581:π
2573:is 180° (
2552:φ
2467:in phase;
2330:φ
2326:−
2308:φ
2292:φ
2245:π
2190:β
2187:−
2184:α
2137:β
2131:α
1965:φ
1821:φ
1647:φ
1603:φ
1582:φ
1533:φ
1488:−
1444:−
1418:π
1400:φ
1380:π
1357:π
1354:−
1334:π
1248:clockwise
1205:φ
1091:−
1038:⋅
981:−
961:π
943:φ
893:phase of
733:φ
704:
609:φ
517:φ
478:φ
456:φ
313:φ
296:amplitude
269:φ
202:π
70:variable
43:or other
4440:AC phase
4428:See also
4421:waveform
2845:and all
2657:, e.g.,
2595:opposite
2223:is 30° (
1930:waveform
1845:, where
1101:⌋
1095:⌊
871:for all
183:such as
177:measured
66:of some
4652:, 2013)
4588:12 June
4574:"Phase"
2715:beating
2489:holds.
189:radians
185:degrees
179:in any
29:physics
4584:(NIST)
4536:
4480:Phasor
4010:, and
3982:, and
3942:where
3376:, and
2483:linear
1989:where
1116:; and
673:. If
300:cosine
290:, the
144:period
35:, the
4397:phase
4345:phase
4198:phase
4145:phase
4096:phase
4012:phase
3659:of a
3514:from
3180:radio
1148:clock
1022:Here
629:, or
146:(and
116:scale
92:angle
37:phase
4590:2016
4578:Nist
4534:ISBN
4314:and
3489:<
3483:<
3356:and
3316:and
2377:and
1369:and
935:is
421:and
292:sine
120:turn
68:real
41:wave
31:and
4229:is
4200:of
4176:is
4147:of
4110:cos
3880:cos
3844:sin
3788:cos
3730:or
3584:sin
3154:of
3106:sin
3078:If
2649:or
2609:or
2405:of
2401:or
2194:360
2166:360
2141:360
2113:360
2096:.)
1885:as
1250:.
749:of
701:sin
633:of
187:or
27:In
4684::
4612:.
4580:.
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1234:t
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1183:0
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1085:=
1082:]
1078:]
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1072:[
1068:[
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1030:[
1007:]
1000:]
995:T
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829:F
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