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31:
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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,
3724:
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
3728:
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
3277:
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
3768:
3709:
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.
3634:
2228:
2491:
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
2487:
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.
1524:
1712:
1646:
2115:
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
4434:
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.
2366:
3523:
1125:
34:
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.
3773:
1029:
3085:
515:
2828:
4271:
3548:
2119:
3674:
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.
1998:
1854:
4394:
4056:
2854:
3694:
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
4083:
2702:
2654:
2582:
1563:
1235:
880:
547:
343:
299:
4152:
3946:{\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}}}
3269:
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).
3159:
2914:
754:
630:
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1069:
734:
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123:
99:
75:
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3991:
3971:
4677:
3189:
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
3304:
is either identically zero, or is a sinusoidal signal with the same period and phase, whose amplitude is the difference of the original amplitudes.
4548:
2976:. In the clock analogy, this situation corresponds to the two hands turning at the same speed, so that the angle between them is constant.
3717:. 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
1528:
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".
949:
4455:
2669:
2289:
3026:
3196:
A well-known example of phase difference is the length of shadows seen at different points of Earth. To a first approximation, if
3193:
signal that reaches the receiving antenna in a straight line, and a copy of it that was reflected off a large building nearby.
2298:
549:
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.
2613:
2524:
3476:
2523:
when the phase difference is zero, the two signals will have the same sign and will be reinforcing each other. One says that
3307:
The phase shift of the co-sine function relative to the sine function is +90°. It follows that, for two sinusoidal signals
1946:
1074:
4671:
4703:
4503:, the quality of a wave to display a well defined phase relationship in different regions of its domain of definition
462:
2779:
4698:
4243:
2497:
2020:
is a "canonical" function of a phase angle in 0 to 2π, that describes just one cycle of that waveform; and
4512:
3738:
2280:
Illustration of phase shift. The horizontal axis represents an angle (phase) that is increasing with time.
1806:
4515:, a phase change that happens when a wave is reflected off of a boundary from fast medium to slow medium
3687:
187:
4660:
4495:
3654:
2284:
1154:
is an arbitrary "origin" value of the argument, that one considers to be the beginning of a cycle.
4500:
4460:
4402:
2833:
4364:
4028:
4656:
4470:
2678:
2630:
2558:
1539:
1211:
835:
523:
319:
275:
4061:
3629:{\displaystyle C={\sqrt {A^{2}+B^{2}}}\quad \quad {\text{ and }}\quad \quad \sin(\varphi )=B/C.}
2223:{\displaystyle 360\,\left\!\!\right]\quad \quad {\text{ and }}\quad \quad 360\,\left\!\!\right]}
1876:
is a function of an angle, defined only for a single full turn, that describes the variation of
4538:
4116:
4544:
4506:
3144:
2899:
739:
615:
55:
4666:
3112:
1036:
707:
4674:— Discusses the time-domain sources of phase shift in simple linear time-invariant circuits.
3646:
3020:
2725:
154:
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2248:
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205:
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130:
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3528:
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3002:
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2879:
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2707:
2530:
2506:
2493:
2459:
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2023:
2003:
1922:
1899:
1879:
1859:
1786:
1766:
1746:
1568:
1314:
1294:
1240:
1237:
is then the angle from the 12:00 position to the current position of the hand, at time
1164:
929:
907:
885:
811:
791:
763:
687:
667:
647:
595:
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388:
368:
348:
255:
228:
195:
136:
108:
84:
60:
51:
4328:
4299:
4214:
4161:
3638:
30:
4692:
4648:
4475:
313:
may be used instead of sine, depending on where one considers each period to start.)
126:
2724:. The periodic changes from reinforcement and opposition cause a phenomenon called
1071:
denotes the fractional part of a real number, discarding its integer part; that is,
4564:
4088:
3976:
3956:
3714:
191:
4584:
2040:
is a scaling factor for the amplitude. (This claim assumes that the starting time
3695:
829:
78:
43:
17:
3678:
3671:
2736:
The phase difference is especially important when comparing a periodic signal
2672:
of a composite signal or even different signals (e.g., voltage and current).
2547:
when the phases are different, the value of the sum depends on the waveform.
1519:{\displaystyle \varphi (t)=2\pi \left(\left\!\!\right]-{\frac {1}{2}}\right)}
3725:
motion of the test signal the offset between frequencies can be determined.
3718:
3691:
3667:
3266:
2238:, minus one full turn), and subtracting 50° from 30° gives a phase of 340° (
1707:{\displaystyle \varphi (t_{0}+kT)=0\quad \quad {\text{ for any integer }}k.}
1258:
306:
4450:
4431:
1940:
1641:{\displaystyle \varphi (t+T)=\varphi (t)\quad \quad {\text{ for all }}t.}
249:
2276:
4543:(3 ed.). Focal Press, Gulf Professional Publishing. p. 1499.
1334:
The formula above gives the phase as an angle in radians between 0 and
199:
39:
612:
is expressed as a fraction of the period, and then scaled to an angle
4682:
4490:
3721:, and the bottom sine signal represents a signal from the reference.
310:
316:
Usually, whole turns are ignored when expressing the phase; so that
105:-like quantity representing the fraction of the cycle covered up to
4620:
1161:
with a hand that turns at constant speed, making a full turn every
4361:
In the context of communication waveforms, the time-variant angle
3677:
3653:
3645:
3637:
3190:
2283:
2275:
1158:
102:
29:
552:
The term "phase" is also used when comparing a periodic function
3742:
302:
3686:
A real-world example of a sonic phase difference occurs in the
3453:
is a sinusoidal signal with the same frequency, with amplitude
517:) if the difference between them is a whole number of periods.
4605:
This content has been copied and pasted from an NIST web page
4565:"Federal Standard 1037C: Glossary of Telecommunications Terms"
3713:
A phase comparison can be made by connecting two signals to a
3109:
is a "canonical" representative for a class of signals, like
2476:
when the difference is zero, the two signals are said to be
2979:
In this case, the phase shift is simply the argument shift
2876:. Suppose also that the origin for computing the phase of
2361:{\displaystyle \varphi (t)=\varphi _{G}(t)-\varphi _{F}(t)}
1565:
of a periodic signal is periodic too, with the same period
1311:. For example, for a sinusoid, a convenient choice is any
2896:
has been shifted too. In that case, the phase difference
1331:
where the function's value changes from zero to positive.
4025:
of the sinusoid. These signals are periodic with period
2230:
respectively. Thus, for example, the sum of phase angles
405:
is said to be "at the same phase" at two argument values
4667:
Phase angle, phase difference, time delay, and frequency
3733:
Formula for phase of an oscillation or a periodic signal
3518:{\displaystyle -90^{\circ }<\varphi <+90^{\circ }}
2675:
If the frequencies are different, the phase difference
1650:
The phase is zero at the start of each period; that is
704:
is a "canonical" function for a class of signals, like
4367:
4331:
4302:
4282:
4246:
4217:
4193:
4164:
4119:
4091:
4064:
4058:, and they are identical except for a displacement of
4031:
3999:
3979:
3959:
3923:
3751:
1120:{\displaystyle \!]=x-\left\lfloor x\right\rfloor \!\,}
27:
The elapsed fraction of a cycle of a periodic function
3771:
3551:
3531:
3479:
3459:
3433:
3413:
3393:
3373:
3353:
3333:
3313:
3284:
3251:
3231:
3202:
3171:
3147:
3115:
3095:
3029:
3023:) of the two signals and then scaled to a full turn:
3005:
2985:
2962:
2942:
2922:
2902:
2882:
2862:
2836:
2782:
2762:
2742:
2710:
2681:
2633:
2590:
2561:
2533:
2509:
2462:
2442:
2422:
2394:
2374:
2301:
2251:
2122:
2093:
2073:
2046:
2026:
2006:
1949:
1925:
1902:
1882:
1862:
1809:
1789:
1769:
1749:
1722:
1656:
1591:
1571:
1542:
1409:
1386:
1363:
1340:
1317:
1297:
1270:
1243:
1214:
1187:
1167:
1133:
1077:
1039:
952:
932:
910:
888:
838:
814:
794:
766:
742:
710:
690:
670:
650:
618:
598:
578:
558:
526:
465:
438:
411:
391:
371:
351:
345:
is also a periodic function, with the same period as
322:
278:
258:
231:
208:
163:
139:
111:
87:
63:
3141:
is for all sinusoidal signals, then the phase shift
365:, that repeatedly scans the same range of angles as
4388:
4358:but are relative to their own specific references.
4346:
4317:
4288:
4265:
4232:
4199:
4179:
4146:
4097:
4077:
4050:
4005:
3985:
3965:
3945:
3757:
3628:
3537:
3517:
3465:
3445:
3419:
3399:
3379:
3359:
3339:
3319:
3296:
3257:
3237:
3217:
3177:
3153:
3133:
3101:
3079:
3011:
2991:
2968:
2948:
2928:
2908:
2888:
2868:
2848:
2822:
2768:
2748:
2716:
2696:
2664:), sinusoidal signals are sometimes said to be in
2648:
2596:
2576:
2555:For sinusoidal signals, when the phase difference
2539:
2515:
2468:
2448:
2428:
2400:
2380:
2360:
2260:
2222:
2099:
2079:
2059:
2032:
2012:
1992:
1931:
1908:
1888:
1868:
1848:
1795:
1775:
1755:
1735:
1706:
1640:
1577:
1557:
1518:
1395:
1372:
1349:
1323:
1303:
1283:
1249:
1229:
1200:
1173:
1146:
1119:
1063:
1023:
938:
916:
894:
874:
820:
800:
772:
748:
728:
696:
676:
656:
624:
604:
584:
564:
541:
509:
451:
424:
397:
377:
357:
337:
293:
264:
237:
217:
178:
145:
117:
93:
69:
3068:
3067:
3048:
3047:
2936:), called the 'phase shift' or 'phase offset' of
2214:
2213:
2186:
2185:
2161:
2160:
2133:
2132:
1492:
1491:
1442:
1441:
1264:The phase concept is most useful when the origin
1115:
1091:
1081:
1059:
1055:
1043:
1024:{\displaystyle \varphi (t)=2\pi \left\!\!\right]}
1015:
1014:
980:
979:
808:be a periodic function of one real variable, and
248:This convention is especially appropriate for a
4113:It can refer to a specified reference, such as
3080:{\displaystyle \varphi =2\pi \left\!\!\right].}
2999:, expressed as a fraction of the common period
510:{\displaystyle \varphi (t_{1})=\varphi (t_{2})}
4593:National Institute of Standards and Technology
1157:This concept can be visualized by imagining a
828:be its period (that is, the smallest positive
186:goes through each complete cycle). It may be
4583:Time and Frequency from A to Z (2010-05-12).
2612:. Then the signals have opposite signs, and
1716:Moreover, for any given choice of the origin
1536:With any of the above definitions, the phase
1181:seconds, and is pointing straight up at time
305:of the phase, multiplied by some factor (the
8:
4532:
4530:
2756:with a shifted and possibly scaled version
2368:between the phases of two periodic signals
252:function, since its value at any argument
4366:
4330:
4301:
4281:
4253:
4245:
4216:
4192:
4163:
4118:
4090:
4065:
4063:
4038:
4030:
3998:
3978:
3958:
3922:
3772:
3770:
3750:
3615:
3587:
3577:
3564:
3558:
3550:
3530:
3509:
3487:
3478:
3458:
3432:
3412:
3392:
3372:
3352:
3332:
3312:
3283:
3265:is the length seen at the same time at a
3250:
3230:
3201:
3170:
3146:
3114:
3094:
3053:
3028:
3004:
2984:
2961:
2941:
2921:
2901:
2881:
2861:
2835:
2801:
2781:
2761:
2741:
2709:
2680:
2632:
2589:
2560:
2532:
2508:
2461:
2441:
2421:
2393:
2373:
2343:
2321:
2300:
2250:
2191:
2179:
2169:
2138:
2126:
2121:
2092:
2072:
2051:
2045:
2025:
2005:
1968:
1948:
1924:
1901:
1881:
1861:
1808:
1788:
1768:
1748:
1727:
1721:
1693:
1667:
1655:
1627:
1590:
1570:
1541:
1501:
1476:
1461:
1448:
1408:
1385:
1362:
1339:
1316:
1296:
1275:
1269:
1242:
1213:
1192:
1186:
1166:
1138:
1132:
1116:
1076:
1060:
1051:
1047:
1038:
998:
985:
951:
931:
909:
887:
837:
813:
793:
765:
741:
709:
689:
669:
649:
617:
597:
577:
557:
525:
498:
476:
464:
443:
437:
416:
410:
390:
370:
350:
321:
277:
257:
230:
207:
162:
138:
110:
86:
62:
4578:
4576:
4574:
3661:
2823:{\displaystyle G(t)=\alpha \,F(t+\tau )}
2245:Similar formulas hold for radians, with
1357:. To get the phase as an angle between
4526:
4266:{\textstyle \varphi -{\frac {\pi }{2}}}
4109:can refer to several different things:
2604:radians), one says that the phases are
1993:{\displaystyle F(t)=A\,w(\varphi (t))}
2704:increases linearly with the argument
2656:is a quarter of turn (a right angle,
7:
4678:Open Source Physics JavaScript HTML5
736:is for all sinusoidal signals, then
632:spanning a whole turn, one gets the
4509:, a method of changing phase by 90°
4013:are constant parameters called the
3658:Representation of phase comparison.
3347:with same frequency and amplitudes
1849:{\displaystyle F(t)=f(\varphi (t))}
4619:Clint Goss; Barry Higgins (2013).
4456:In-phase and quadrature components
3698:of the sound of a warbling flute.
2670:in-phase and quadrature components
2624:implies a 180-degree phase shift.
25:
4154:, in which case we would say the
3688:warble of a Native American flute
3407:has phase shift +90° relative to
3273:For sinusoids with same frequency
385:goes through each period. Then,
4420:This section is an excerpt from
4672:ECE 209: Sources of Phase Shift
3593:
3592:
3586:
3585:
2175:
2174:
2168:
2167:
2067:chosen to compute the phase of
1919:In fact, every periodic signal
1916:ranges over a single period.
1692:
1691:
1626:
1625:
1291:is chosen based on features of
520:The numeric value of the phase
4341:
4335:
4312:
4306:
4227:
4221:
4174:
4168:
4141:
4126:
3881:
3860:
3841:
3835:
3825:
3804:
3785:
3779:
3606:
3600:
3212:
3206:
3128:
3122:
2916:is a constant (independent of
2817:
2805:
2792:
2786:
2691:
2685:
2643:
2637:
2571:
2565:
2355:
2349:
2333:
2327:
2311:
2305:
1987:
1984:
1978:
1972:
1959:
1953:
1843:
1840:
1834:
1828:
1819:
1813:
1682:
1660:
1622:
1616:
1607:
1595:
1552:
1546:
1419:
1413:
1224:
1218:
1092:
1088:
1082:
1078:
1056:
1052:
1044:
1040:
962:
956:
869:
863:
854:
842:
723:
717:
536:
530:
504:
491:
482:
469:
332:
326:
288:
282:
173:
167:
1:
4389:{\textstyle 2\pi ft+\varphi }
4296:, in which case we would say
4051:{\textstyle T={\frac {1}{f}}}
3690:. The amplitude of different
2849:{\displaystyle \alpha ,\tau }
2776:of it. That is, suppose that
2087:corresponds to argument 0 of
1783:depends only on its phase at
202:, thus increasing by 360° or
125:. It is expressed in such a
4540:Handbook for sound engineers
3765:in the following functions:
2527:is occurring. At arguments
4607:and is in the public domain
4078:{\textstyle {\frac {T}{4}}}
3739:simple harmonic oscillation
3225:is the length seen at time
2697:{\displaystyle \varphi (t)}
2649:{\displaystyle \varphi (t)}
2608:, and that the signals are
2577:{\displaystyle \varphi (t)}
2111:Adding and comparing phases
1695: for any integer
1558:{\displaystyle \varphi (t)}
1230:{\displaystyle \varphi (t)}
875:{\displaystyle F(t+T)=F(t)}
542:{\displaystyle \varphi (t)}
338:{\displaystyle \varphi (t)}
294:{\displaystyle \varphi (t)}
129:that it varies by one full
4720:
4419:
4147:{\textstyle \cos(2\pi ft)}
2627:When the phase difference
1743:, the value of the signal
2662:−90° = 270° = −π/2 = 3π/2
2525:constructive interference
272:then can be expressed as
245:completes a full period.
4651:". Prof. Jeffrey Hass. "
3715:two-channel oscilloscope
3154:{\displaystyle \varphi }
2909:{\displaystyle \varphi }
2616:occurs. Conversely, a
2614:destructive interference
1803:. Namely, one can write
749:{\displaystyle \varphi }
625:{\displaystyle \varphi }
592:of it. If the shift in
3134:{\displaystyle \sin(t)}
2498:superposition principle
2242:, plus one full turn).
1064:{\displaystyle \!]\!\,}
784:Mathematical definition
729:{\displaystyle \sin(t)}
572:with a shifted version
309:of the sinusoid). (The
4513:Reflection phase shift
4390:
4348:
4319:
4290:
4267:
4234:
4201:
4181:
4148:
4099:
4079:
4052:
4007:
3987:
3967:
3947:
3759:
3683:
3675:
3659:
3651:
3643:
3630:
3539:
3519:
3467:
3447:
3421:
3401:
3381:
3361:
3341:
3321:
3298:
3259:
3239:
3219:
3179:
3155:
3135:
3103:
3081:
3013:
2993:
2970:
2950:
2930:
2910:
2890:
2870:
2850:
2824:
2770:
2750:
2718:
2698:
2650:
2598:
2578:
2541:
2517:
2470:
2450:
2430:
2402:
2382:
2362:
2292:
2281:
2262:
2224:
2101:
2081:
2061:
2034:
2014:
1994:
1933:
1910:
1890:
1870:
1850:
1797:
1777:
1757:
1737:
1708:
1642:
1579:
1559:
1520:
1397:
1374:
1351:
1325:
1305:
1285:
1251:
1231:
1202:
1175:
1148:
1121:
1065:
1025:
940:
918:
896:
876:
822:
802:
774:
750:
730:
698:
678:
658:
626:
606:
586:
566:
543:
511:
453:
426:
399:
379:
359:
339:
295:
266:
239:
219:
180:
147:
119:
95:
71:
35:
4537:Ballou, Glen (2005).
4391:
4349:
4320:
4291:
4289:{\textstyle \varphi }
4268:
4235:
4202:
4200:{\textstyle \varphi }
4182:
4149:
4100:
4080:
4053:
4008:
4006:{\textstyle \varphi }
3988:
3968:
3948:
3760:
3758:{\textstyle \varphi }
3681:
3665:
3657:
3649:
3641:
3631:
3540:
3520:
3468:
3448:
3422:
3402:
3382:
3362:
3342:
3322:
3299:
3260:
3240:
3220:
3180:
3156:
3136:
3104:
3082:
3014:
2994:
2992:{\displaystyle \tau }
2971:
2951:
2931:
2911:
2891:
2871:
2851:
2825:
2771:
2751:
2719:
2699:
2651:
2599:
2579:
2542:
2518:
2471:
2451:
2431:
2403:
2383:
2363:
2287:
2279:
2263:
2261:{\displaystyle 2\pi }
2225:
2102:
2082:
2062:
2060:{\displaystyle t_{0}}
2035:
2015:
1995:
1943:can be expressed as
1934:
1911:
1891:
1871:
1851:
1798:
1778:
1758:
1738:
1736:{\displaystyle t_{0}}
1709:
1643:
1580:
1560:
1521:
1398:
1396:{\displaystyle +\pi }
1375:
1373:{\displaystyle -\pi }
1352:
1350:{\displaystyle 2\pi }
1326:
1306:
1286:
1284:{\displaystyle t_{0}}
1252:
1232:
1203:
1201:{\displaystyle t_{0}}
1176:
1149:
1147:{\displaystyle t_{0}}
1122:
1066:
1026:
941:
919:
897:
877:
823:
803:
775:
751:
731:
699:
679:
659:
627:
607:
587:
567:
544:
512:
454:
452:{\displaystyle t_{2}}
427:
425:{\displaystyle t_{1}}
400:
380:
360:
340:
296:
267:
240:
220:
218:{\displaystyle 2\pi }
181:
148:
120:
101:(such as time) is an
96:
72:
50:(symbol φ or ϕ) of a
33:
4496:Polarization (waves)
4400:, is referred to as
4365:
4329:
4300:
4280:
4244:
4215:
4191:
4162:
4117:
4089:
4062:
4029:
3997:
3977:
3957:
3769:
3749:
3729:than the reference.
3650:Out-of-phase signals
3549:
3529:
3477:
3457:
3431:
3411:
3391:
3371:
3351:
3331:
3311:
3282:
3249:
3229:
3218:{\displaystyle F(t)}
3200:
3169:
3145:
3113:
3093:
3027:
3003:
2983:
2960:
2940:
2920:
2900:
2880:
2860:
2834:
2780:
2760:
2740:
2708:
2679:
2631:
2597:{\displaystyle \pi }
2588:
2559:
2531:
2507:
2480:otherwise, they are
2460:
2440:
2420:
2392:
2372:
2299:
2288:Phase shifter using
2249:
2120:
2091:
2071:
2044:
2024:
2004:
1947:
1923:
1900:
1880:
1860:
1807:
1787:
1767:
1747:
1720:
1654:
1589:
1569:
1540:
1407:
1384:
1361:
1338:
1315:
1295:
1268:
1241:
1212:
1185:
1165:
1131:
1075:
1037:
950:
930:
908:
886:
836:
812:
792:
764:
740:
708:
688:
668:
648:
616:
596:
576:
556:
524:
463:
436:
409:
389:
369:
349:
320:
276:
256:
229:
206:
179:{\displaystyle F(t)}
161:
137:
109:
85:
61:
4704:Physical quantities
4659:, 2003. See also: (
4653:An Acoustics Primer
4501:Coherence (physics)
4461:Instantaneous phase
4403:instantaneous phase
3692:harmonic components
3446:{\displaystyle F+G}
3297:{\displaystyle F+G}
2830:for some constants
2732:For shifted signals
1629: for all
1403:, one uses instead
4657:Indiana University
4471:Phase cancellation
4430:is the phase of a
4386:
4344:
4315:
4286:
4263:
4230:
4197:
4177:
4144:
4095:
4075:
4048:
4003:
3983:
3963:
3943:
3941:
3932:
3755:
3684:
3676:
3660:
3652:
3644:
3626:
3535:
3515:
3463:
3443:
3417:
3397:
3377:
3357:
3337:
3317:
3294:
3255:
3235:
3215:
3175:
3161:called simply the
3151:
3131:
3099:
3077:
3009:
2989:
2966:
2946:
2926:
2906:
2886:
2866:
2846:
2820:
2766:
2746:
2714:
2694:
2646:
2594:
2574:
2537:
2513:
2496:systems, when the
2466:
2446:
2426:
2398:
2378:
2358:
2293:
2282:
2258:
2220:
2097:
2077:
2057:
2030:
2010:
1990:
1929:
1906:
1886:
1866:
1846:
1793:
1773:
1753:
1733:
1704:
1638:
1575:
1555:
1516:
1393:
1370:
1347:
1321:
1301:
1281:
1247:
1227:
1198:
1171:
1144:
1117:
1061:
1021:
936:
914:
892:
872:
818:
798:
770:
746:
726:
694:
674:
654:
622:
602:
582:
562:
539:
507:
449:
422:
395:
375:
355:
335:
291:
262:
235:
215:
176:
153:goes through each
143:
115:
91:
67:
36:
4550:978-0-240-80758-4
4507:Hilbert transform
4347:{\textstyle y(t)}
4318:{\textstyle x(t)}
4261:
4233:{\textstyle y(t)}
4180:{\textstyle x(t)}
4073:
4046:
3931:
3743:sinusoidal signal
3590:
3583:
3538:{\displaystyle F}
3466:{\displaystyle C}
3420:{\displaystyle F}
3400:{\displaystyle G}
3380:{\displaystyle B}
3360:{\displaystyle A}
3340:{\displaystyle G}
3320:{\displaystyle F}
3258:{\displaystyle G}
3245:at one spot, and
3238:{\displaystyle t}
3178:{\displaystyle G}
3102:{\displaystyle F}
3061:
3019:(in terms of the
3012:{\displaystyle T}
2969:{\displaystyle F}
2949:{\displaystyle G}
2929:{\displaystyle t}
2889:{\displaystyle G}
2869:{\displaystyle t}
2769:{\displaystyle G}
2749:{\displaystyle F}
2717:{\displaystyle t}
2540:{\displaystyle t}
2516:{\displaystyle t}
2484:with each other.
2469:{\displaystyle t}
2449:{\displaystyle F}
2429:{\displaystyle G}
2401:{\displaystyle G}
2381:{\displaystyle F}
2207:
2172:
2154:
2100:{\displaystyle w}
2080:{\displaystyle F}
2033:{\displaystyle A}
2013:{\displaystyle w}
1932:{\displaystyle F}
1909:{\displaystyle t}
1889:{\displaystyle F}
1869:{\displaystyle f}
1796:{\displaystyle t}
1776:{\displaystyle t}
1763:for any argument
1756:{\displaystyle F}
1696:
1630:
1578:{\displaystyle T}
1509:
1484:
1471:
1324:{\displaystyle t}
1304:{\displaystyle F}
1250:{\displaystyle t}
1174:{\displaystyle T}
1008:
939:{\displaystyle t}
917:{\displaystyle F}
895:{\displaystyle t}
821:{\displaystyle T}
801:{\displaystyle F}
773:{\displaystyle G}
697:{\displaystyle F}
677:{\displaystyle F}
657:{\displaystyle G}
605:{\displaystyle t}
585:{\displaystyle G}
565:{\displaystyle F}
398:{\displaystyle F}
378:{\displaystyle t}
358:{\displaystyle F}
265:{\displaystyle t}
238:{\displaystyle t}
146:{\displaystyle t}
118:{\displaystyle t}
94:{\displaystyle t}
70:{\displaystyle F}
56:periodic function
16:(Redirected from
4711:
4683:Phase Difference
4649:What is a phase?
4635:
4634:
4632:
4631:
4616:
4610:
4604:
4602:
4600:
4580:
4569:
4568:
4561:
4555:
4554:
4534:
4395:
4393:
4392:
4387:
4353:
4351:
4350:
4345:
4324:
4322:
4321:
4316:
4295:
4293:
4292:
4287:
4276:It can refer to
4272:
4270:
4269:
4264:
4262:
4254:
4239:
4237:
4236:
4231:
4206:
4204:
4203:
4198:
4186:
4184:
4183:
4178:
4153:
4151:
4150:
4145:
4105:axis. The term
4104:
4102:
4101:
4096:
4084:
4082:
4081:
4076:
4074:
4066:
4057:
4055:
4054:
4049:
4047:
4039:
4012:
4010:
4009:
4004:
3992:
3990:
3989:
3984:
3972:
3970:
3969:
3964:
3952:
3950:
3949:
3944:
3942:
3938:
3934:
3933:
3924:
3764:
3762:
3761:
3756:
3745:is the value of
3707:Phase comparison
3702:Phase comparison
3642:In-phase signals
3635:
3633:
3632:
3627:
3619:
3591:
3588:
3584:
3582:
3581:
3569:
3568:
3559:
3544:
3542:
3541:
3536:
3524:
3522:
3521:
3516:
3514:
3513:
3492:
3491:
3473:and phase shift
3472:
3470:
3469:
3464:
3452:
3450:
3449:
3444:
3426:
3424:
3423:
3418:
3406:
3404:
3403:
3398:
3386:
3384:
3383:
3378:
3366:
3364:
3363:
3358:
3346:
3344:
3343:
3338:
3326:
3324:
3323:
3318:
3303:
3301:
3300:
3295:
3264:
3262:
3261:
3256:
3244:
3242:
3241:
3236:
3224:
3222:
3221:
3216:
3184:
3182:
3181:
3176:
3160:
3158:
3157:
3152:
3140:
3138:
3137:
3132:
3108:
3106:
3105:
3100:
3086:
3084:
3083:
3078:
3073:
3069:
3066:
3062:
3054:
3021:modulo operation
3018:
3016:
3015:
3010:
2998:
2996:
2995:
2990:
2975:
2973:
2972:
2967:
2955:
2953:
2952:
2947:
2935:
2933:
2932:
2927:
2915:
2913:
2912:
2907:
2895:
2893:
2892:
2887:
2875:
2873:
2872:
2867:
2855:
2853:
2852:
2847:
2829:
2827:
2826:
2821:
2775:
2773:
2772:
2767:
2755:
2753:
2752:
2747:
2723:
2721:
2720:
2715:
2703:
2701:
2700:
2695:
2663:
2659:
2655:
2653:
2652:
2647:
2603:
2601:
2600:
2595:
2583:
2581:
2580:
2575:
2546:
2544:
2543:
2538:
2522:
2520:
2519:
2514:
2475:
2473:
2472:
2467:
2456:. At values of
2455:
2453:
2452:
2447:
2435:
2433:
2432:
2427:
2410:phase difference
2407:
2405:
2404:
2399:
2387:
2385:
2384:
2379:
2367:
2365:
2364:
2359:
2348:
2347:
2326:
2325:
2268:instead of 360.
2267:
2265:
2264:
2259:
2241:
2237:
2233:
2229:
2227:
2226:
2221:
2219:
2215:
2212:
2208:
2203:
2192:
2173:
2170:
2166:
2162:
2159:
2155:
2150:
2139:
2106:
2104:
2103:
2098:
2086:
2084:
2083:
2078:
2066:
2064:
2063:
2058:
2056:
2055:
2039:
2037:
2036:
2031:
2019:
2017:
2016:
2011:
1999:
1997:
1996:
1991:
1939:with a specific
1938:
1936:
1935:
1930:
1915:
1913:
1912:
1907:
1895:
1893:
1892:
1887:
1875:
1873:
1872:
1867:
1855:
1853:
1852:
1847:
1802:
1800:
1799:
1794:
1782:
1780:
1779:
1774:
1762:
1760:
1759:
1754:
1742:
1740:
1739:
1734:
1732:
1731:
1713:
1711:
1710:
1705:
1697:
1694:
1672:
1671:
1647:
1645:
1644:
1639:
1631:
1628:
1584:
1582:
1581:
1576:
1564:
1562:
1561:
1556:
1525:
1523:
1522:
1517:
1515:
1511:
1510:
1502:
1497:
1493:
1490:
1486:
1485:
1477:
1472:
1467:
1466:
1465:
1449:
1402:
1400:
1399:
1394:
1379:
1377:
1376:
1371:
1356:
1354:
1353:
1348:
1330:
1328:
1327:
1322:
1310:
1308:
1307:
1302:
1290:
1288:
1287:
1282:
1280:
1279:
1256:
1254:
1253:
1248:
1236:
1234:
1233:
1228:
1207:
1205:
1204:
1199:
1197:
1196:
1180:
1178:
1177:
1172:
1153:
1151:
1150:
1145:
1143:
1142:
1126:
1124:
1123:
1118:
1114:
1070:
1068:
1067:
1062:
1030:
1028:
1027:
1022:
1020:
1016:
1013:
1009:
1004:
1003:
1002:
986:
945:
943:
942:
937:
923:
921:
920:
915:
901:
899:
898:
893:
881:
879:
878:
873:
827:
825:
824:
819:
807:
805:
804:
799:
779:
777:
776:
771:
755:
753:
752:
747:
735:
733:
732:
727:
703:
701:
700:
695:
683:
681:
680:
675:
663:
661:
660:
655:
642:phase difference
631:
629:
628:
623:
611:
609:
608:
603:
591:
589:
588:
583:
571:
569:
568:
563:
548:
546:
545:
540:
516:
514:
513:
508:
503:
502:
481:
480:
458:
456:
455:
450:
448:
447:
431:
429:
428:
423:
421:
420:
404:
402:
401:
396:
384:
382:
381:
376:
364:
362:
361:
356:
344:
342:
341:
336:
300:
298:
297:
292:
271:
269:
268:
263:
244:
242:
241:
236:
225:as the variable
224:
222:
221:
216:
185:
183:
182:
177:
152:
150:
149:
144:
133:as the variable
124:
122:
121:
116:
100:
98:
97:
92:
76:
74:
73:
68:
21:
18:Phase difference
4719:
4718:
4714:
4713:
4712:
4710:
4709:
4708:
4689:
4688:
4644:
4639:
4638:
4629:
4627:
4618:
4617:
4613:
4598:
4596:
4582:
4581:
4572:
4563:
4562:
4558:
4551:
4536:
4535:
4528:
4523:
4518:
4466:Lissajous curve
4441:
4436:
4435:
4425:
4417:
4398:principal value
4363:
4362:
4327:
4326:
4298:
4297:
4278:
4277:
4242:
4241:
4213:
4212:
4189:
4188:
4160:
4159:
4115:
4114:
4087:
4086:
4060:
4059:
4027:
4026:
3995:
3994:
3975:
3974:
3955:
3954:
3940:
3939:
3900:
3896:
3844:
3829:
3828:
3788:
3767:
3766:
3747:
3746:
3737:The phase of a
3735:
3704:
3682:Out of phase AE
3589: and
3573:
3560:
3547:
3546:
3527:
3526:
3505:
3483:
3475:
3474:
3455:
3454:
3429:
3428:
3409:
3408:
3389:
3388:
3369:
3368:
3349:
3348:
3329:
3328:
3309:
3308:
3280:
3279:
3275:
3247:
3246:
3227:
3226:
3198:
3197:
3167:
3166:
3143:
3142:
3111:
3110:
3091:
3090:
3049:
3046:
3042:
3025:
3024:
3001:
3000:
2981:
2980:
2958:
2957:
2938:
2937:
2918:
2917:
2898:
2897:
2878:
2877:
2858:
2857:
2832:
2831:
2778:
2777:
2758:
2757:
2738:
2737:
2734:
2706:
2705:
2677:
2676:
2661:
2657:
2629:
2628:
2622:phase inversion
2586:
2585:
2557:
2556:
2553:
2529:
2528:
2505:
2504:
2458:
2457:
2438:
2437:
2418:
2417:
2390:
2389:
2370:
2369:
2339:
2317:
2297:
2296:
2295:The difference
2274:
2247:
2246:
2239:
2236:190 + 200 = 390
2235:
2231:
2193:
2187:
2184:
2180:
2171: and
2140:
2134:
2131:
2127:
2118:
2117:
2113:
2089:
2088:
2069:
2068:
2047:
2042:
2041:
2022:
2021:
2002:
2001:
1945:
1944:
1921:
1920:
1898:
1897:
1878:
1877:
1858:
1857:
1805:
1804:
1785:
1784:
1765:
1764:
1745:
1744:
1723:
1718:
1717:
1663:
1652:
1651:
1587:
1586:
1567:
1566:
1538:
1537:
1534:
1457:
1450:
1447:
1443:
1440:
1436:
1435:
1431:
1405:
1404:
1382:
1381:
1359:
1358:
1336:
1335:
1313:
1312:
1293:
1292:
1271:
1266:
1265:
1239:
1238:
1210:
1209:
1188:
1183:
1182:
1163:
1162:
1134:
1129:
1128:
1104:
1073:
1072:
1035:
1034:
994:
987:
981:
978:
974:
948:
947:
928:
927:
906:
905:
884:
883:
834:
833:
810:
809:
790:
789:
788:Let the signal
786:
762:
761:
738:
737:
706:
705:
686:
685:
666:
665:
646:
645:
614:
613:
594:
593:
574:
573:
554:
553:
522:
521:
494:
472:
461:
460:
439:
434:
433:
412:
407:
406:
387:
386:
367:
366:
347:
346:
318:
317:
274:
273:
254:
253:
227:
226:
204:
203:
159:
158:
135:
134:
107:
106:
83:
82:
59:
58:
28:
23:
22:
15:
12:
11:
5:
4717:
4715:
4707:
4706:
4701:
4699:Wave mechanics
4691:
4690:
4687:
4686:
4680:
4675:
4669:
4664:
4661:pages 1 thru 3
4655:", Section 8.
4643:
4642:External links
4640:
4637:
4636:
4611:
4570:
4556:
4549:
4525:
4524:
4522:
4519:
4517:
4516:
4510:
4504:
4498:
4493:
4488:
4486:Phase velocity
4483:
4481:Phase spectrum
4478:
4473:
4468:
4463:
4458:
4453:
4448:
4446:Absolute phase
4442:
4440:
4437:
4428:Absolute phase
4426:
4422:Absolute phase
4418:
4416:
4415:Absolute phase
4413:
4412:
4411:
4385:
4382:
4379:
4376:
4373:
4370:
4359:
4354:have the same
4343:
4340:
4337:
4334:
4314:
4311:
4308:
4305:
4285:
4274:
4260:
4257:
4252:
4249:
4229:
4226:
4223:
4220:
4196:
4176:
4173:
4170:
4167:
4143:
4140:
4137:
4134:
4131:
4128:
4125:
4122:
4098:{\textstyle t}
4094:
4072:
4069:
4045:
4042:
4037:
4034:
4002:
3986:{\textstyle f}
3982:
3966:{\textstyle A}
3962:
3937:
3930:
3927:
3921:
3918:
3915:
3912:
3909:
3906:
3903:
3899:
3895:
3892:
3889:
3886:
3883:
3880:
3877:
3874:
3871:
3868:
3865:
3862:
3859:
3856:
3853:
3850:
3847:
3845:
3843:
3840:
3837:
3834:
3831:
3830:
3827:
3824:
3821:
3818:
3815:
3812:
3809:
3806:
3803:
3800:
3797:
3794:
3791:
3789:
3787:
3784:
3781:
3778:
3775:
3774:
3754:
3734:
3731:
3719:test frequency
3703:
3700:
3625:
3622:
3618:
3614:
3611:
3608:
3605:
3602:
3599:
3596:
3580:
3576:
3572:
3567:
3563:
3557:
3554:
3534:
3512:
3508:
3504:
3501:
3498:
3495:
3490:
3486:
3482:
3462:
3442:
3439:
3436:
3416:
3396:
3376:
3356:
3336:
3316:
3293:
3290:
3287:
3274:
3271:
3254:
3234:
3214:
3211:
3208:
3205:
3174:
3150:
3130:
3127:
3124:
3121:
3118:
3098:
3076:
3072:
3065:
3060:
3057:
3052:
3045:
3041:
3038:
3035:
3032:
3008:
2988:
2965:
2945:
2925:
2905:
2885:
2865:
2845:
2842:
2839:
2819:
2816:
2813:
2810:
2807:
2804:
2800:
2797:
2794:
2791:
2788:
2785:
2765:
2745:
2733:
2730:
2713:
2693:
2690:
2687:
2684:
2645:
2642:
2639:
2636:
2618:phase reversal
2593:
2573:
2570:
2567:
2564:
2552:
2549:
2536:
2512:
2503:For arguments
2465:
2445:
2425:
2408:is called the
2397:
2377:
2357:
2354:
2351:
2346:
2342:
2338:
2335:
2332:
2329:
2324:
2320:
2316:
2313:
2310:
2307:
2304:
2273:
2270:
2257:
2254:
2218:
2211:
2206:
2202:
2199:
2196:
2190:
2183:
2178:
2165:
2158:
2153:
2149:
2146:
2143:
2137:
2130:
2125:
2112:
2109:
2096:
2076:
2054:
2050:
2029:
2009:
1989:
1986:
1983:
1980:
1977:
1974:
1971:
1967:
1964:
1961:
1958:
1955:
1952:
1928:
1905:
1885:
1865:
1845:
1842:
1839:
1836:
1833:
1830:
1827:
1824:
1821:
1818:
1815:
1812:
1792:
1772:
1752:
1730:
1726:
1703:
1700:
1690:
1687:
1684:
1681:
1678:
1675:
1670:
1666:
1662:
1659:
1637:
1634:
1624:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1597:
1594:
1574:
1554:
1551:
1548:
1545:
1533:
1530:
1514:
1508:
1505:
1500:
1496:
1489:
1483:
1480:
1475:
1470:
1464:
1460:
1456:
1453:
1446:
1439:
1434:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1392:
1389:
1369:
1366:
1346:
1343:
1320:
1300:
1278:
1274:
1246:
1226:
1223:
1220:
1217:
1195:
1191:
1170:
1141:
1137:
1113:
1110:
1107:
1103:
1100:
1097:
1094:
1090:
1087:
1084:
1080:
1058:
1054:
1050:
1046:
1042:
1019:
1012:
1007:
1001:
997:
993:
990:
984:
977:
973:
970:
967:
964:
961:
958:
955:
935:
913:
891:
871:
868:
865:
862:
859:
856:
853:
850:
847:
844:
841:
817:
797:
785:
782:
769:
756:is called the
745:
725:
722:
719:
716:
713:
693:
673:
653:
621:
601:
581:
561:
538:
535:
532:
529:
506:
501:
497:
493:
490:
487:
484:
479:
475:
471:
468:
446:
442:
419:
415:
394:
374:
354:
334:
331:
328:
325:
290:
287:
284:
281:
261:
234:
214:
211:
175:
172:
169:
166:
142:
114:
90:
66:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4716:
4705:
4702:
4700:
4697:
4696:
4694:
4684:
4681:
4679:
4676:
4673:
4670:
4668:
4665:
4662:
4658:
4654:
4650:
4646:
4645:
4641:
4626:
4622:
4615:
4612:
4608:
4594:
4590:
4586:
4579:
4577:
4575:
4571:
4566:
4560:
4557:
4552:
4546:
4542:
4541:
4533:
4531:
4527:
4520:
4514:
4511:
4508:
4505:
4502:
4499:
4497:
4494:
4492:
4489:
4487:
4484:
4482:
4479:
4477:
4476:Phase problem
4474:
4472:
4469:
4467:
4464:
4462:
4459:
4457:
4454:
4452:
4449:
4447:
4444:
4443:
4438:
4433:
4429:
4423:
4414:
4409:
4406:, often just
4405:
4404:
4399:
4383:
4380:
4377:
4374:
4371:
4368:
4360:
4357:
4338:
4332:
4309:
4303:
4283:
4275:
4258:
4255:
4250:
4247:
4224:
4218:
4210:
4194:
4171:
4165:
4157:
4138:
4135:
4132:
4129:
4123:
4120:
4112:
4111:
4110:
4108:
4092:
4070:
4067:
4043:
4040:
4035:
4032:
4024:
4020:
4016:
4000:
3980:
3960:
3935:
3928:
3925:
3919:
3916:
3913:
3910:
3907:
3904:
3901:
3897:
3893:
3890:
3887:
3884:
3878:
3875:
3872:
3869:
3866:
3863:
3857:
3854:
3851:
3848:
3846:
3838:
3832:
3822:
3819:
3816:
3813:
3810:
3807:
3801:
3798:
3795:
3792:
3790:
3782:
3776:
3752:
3744:
3740:
3732:
3730:
3726:
3722:
3720:
3716:
3711:
3708:
3701:
3699:
3697:
3693:
3689:
3680:
3673:
3669:
3664:
3656:
3648:
3640:
3636:
3623:
3620:
3616:
3612:
3609:
3603:
3597:
3594:
3578:
3574:
3570:
3565:
3561:
3555:
3552:
3532:
3510:
3506:
3502:
3499:
3496:
3493:
3488:
3484:
3480:
3460:
3440:
3437:
3434:
3414:
3394:
3374:
3354:
3334:
3314:
3305:
3291:
3288:
3285:
3272:
3270:
3268:
3252:
3232:
3209:
3203:
3194:
3192:
3186:
3172:
3164:
3163:initial phase
3148:
3125:
3119:
3116:
3096:
3087:
3074:
3070:
3063:
3058:
3055:
3050:
3043:
3039:
3036:
3033:
3030:
3022:
3006:
2986:
2977:
2963:
2943:
2923:
2903:
2883:
2863:
2843:
2840:
2837:
2814:
2811:
2808:
2802:
2798:
2795:
2789:
2783:
2763:
2743:
2731:
2729:
2727:
2711:
2688:
2682:
2673:
2671:
2667:
2640:
2634:
2625:
2623:
2619:
2615:
2611:
2607:
2591:
2568:
2562:
2551:For sinusoids
2550:
2548:
2534:
2526:
2510:
2501:
2499:
2495:
2489:
2485:
2483:
2479:
2463:
2443:
2423:
2415:
2411:
2395:
2375:
2352:
2344:
2340:
2336:
2330:
2322:
2318:
2314:
2308:
2302:
2291:
2286:
2278:
2271:
2269:
2255:
2252:
2243:
2240:30 − 50 = −20
2216:
2209:
2204:
2200:
2197:
2194:
2188:
2181:
2176:
2163:
2156:
2151:
2147:
2144:
2141:
2135:
2128:
2123:
2110:
2108:
2094:
2074:
2052:
2048:
2027:
2007:
1981:
1975:
1969:
1965:
1962:
1956:
1950:
1942:
1926:
1917:
1903:
1883:
1863:
1837:
1831:
1825:
1822:
1816:
1810:
1790:
1770:
1750:
1728:
1724:
1714:
1701:
1698:
1688:
1685:
1679:
1676:
1673:
1668:
1664:
1657:
1648:
1635:
1632:
1619:
1613:
1610:
1604:
1601:
1598:
1592:
1572:
1549:
1543:
1531:
1529:
1526:
1512:
1506:
1503:
1498:
1494:
1487:
1481:
1478:
1473:
1468:
1462:
1458:
1454:
1451:
1444:
1437:
1432:
1428:
1425:
1422:
1416:
1410:
1390:
1387:
1367:
1364:
1344:
1341:
1332:
1318:
1298:
1276:
1272:
1262:
1260:
1244:
1221:
1215:
1208:. The phase
1193:
1189:
1168:
1160:
1155:
1139:
1135:
1111:
1108:
1105:
1101:
1098:
1095:
1085:
1048:
1031:
1017:
1010:
1005:
999:
995:
991:
988:
982:
975:
971:
968:
965:
959:
953:
933:
926:any argument
925:
911:
902:). Then the
889:
866:
860:
857:
851:
848:
845:
839:
831:
815:
795:
783:
781:
767:
759:
758:initial phase
743:
720:
714:
711:
691:
671:
651:
643:
639:
635:
619:
599:
579:
559:
550:
533:
527:
518:
499:
495:
488:
485:
477:
473:
466:
444:
440:
417:
413:
392:
372:
352:
329:
323:
314:
312:
308:
304:
285:
279:
259:
251:
246:
232:
212:
209:
201:
197:
193:
189:
170:
164:
156:
140:
132:
128:
112:
104:
88:
80:
64:
57:
53:
49:
45:
41:
32:
19:
4652:
4628:. Retrieved
4624:
4621:"The Warble"
4614:
4606:
4597:. Retrieved
4588:
4559:
4539:
4407:
4401:
4355:
4208:
4155:
4106:
4022:
4018:
4014:
3736:
3727:
3723:
3712:
3706:
3705:
3685:
3545:, such that
3306:
3276:
3195:
3187:
3162:
3088:
2978:
2956:relative to
2735:
2674:
2665:
2626:
2621:
2617:
2610:in antiphase
2609:
2605:
2554:
2502:
2490:
2486:
2482:out of phase
2481:
2477:
2436:relative to
2413:
2409:
2294:
2290:IQ modulator
2272:Phase shift
2244:
2114:
1918:
1715:
1649:
1535:
1532:Consequences
1527:
1333:
1263:
1156:
1032:
903:
787:
757:
664:relative to
641:
638:phase offset
637:
633:
551:
519:
315:
247:
192:angular unit
47:
37:
4685:Java Applet
3696:spectrogram
2414:phase shift
2232:190° + 200°
1257:, measured
830:real number
634:phase shift
44:mathematics
4693:Categories
4630:2013-03-06
4625:Flutopedia
4521:References
4207:, and the
4085:along the
3672:plane wave
3666:Left: the
3427:, the sum
2666:quadrature
2658:+90° = π/2
832:such that
459:(that is,
250:sinusoidal
4396:, or its
4384:φ
4372:π
4284:φ
4256:π
4251:−
4248:φ
4195:φ
4133:π
4124:
4019:frequency
4015:amplitude
4001:φ
3926:π
3920:−
3917:φ
3905:π
3894:
3879:φ
3867:π
3858:
3823:φ
3811:π
3802:
3753:φ
3668:real part
3604:φ
3598:
3511:∘
3497:φ
3489:∘
3481:−
3267:longitude
3149:φ
3120:
3056:τ
3040:π
3031:φ
2987:τ
2904:φ
2844:τ
2838:α
2815:τ
2799:α
2683:φ
2635:φ
2592:π
2584:is 180° (
2563:φ
2478:in phase;
2341:φ
2337:−
2319:φ
2303:φ
2256:π
2201:β
2198:−
2195:α
2148:β
2142:α
1976:φ
1832:φ
1658:φ
1614:φ
1593:φ
1544:φ
1499:−
1455:−
1429:π
1411:φ
1391:π
1368:π
1365:−
1345:π
1259:clockwise
1216:φ
1102:−
1049:⋅
992:−
972:π
954:φ
904:phase of
744:φ
715:
620:φ
528:φ
489:φ
467:φ
324:φ
307:amplitude
280:φ
213:π
81:variable
54:or other
4451:AC phase
4439:See also
4432:waveform
2856:and all
2668:, e.g.,
2606:opposite
2234:is 30° (
1941:waveform
1856:, where
1112:⌋
1106:⌊
882:for all
194:such as
188:measured
77:of some
4663:, 2013)
4599:12 June
4585:"Phase"
2726:beating
2500:holds.
200:radians
196:degrees
190:in any
40:physics
4595:(NIST)
4547:
4491:Phasor
4021:, and
3993:, and
3953:where
3387:, and
2494:linear
2000:where
1127:; and
684:. If
311:cosine
301:, the
155:period
46:, the
4408:phase
4356:phase
4209:phase
4156:phase
4107:phase
4023:phase
3670:of a
3525:from
3191:radio
1159:clock
1033:Here
640:, or
157:(and
127:scale
103:angle
48:phase
4601:2016
4589:Nist
4545:ISBN
4325:and
3500:<
3494:<
3367:and
3327:and
2388:and
1380:and
946:is
432:and
303:sine
131:turn
79:real
52:wave
42:and
4240:is
4211:of
4187:is
4158:of
4121:cos
3891:cos
3855:sin
3799:cos
3741:or
3595:sin
3165:of
3117:sin
3089:If
2660:or
2620:or
2416:of
2412:or
2205:360
2177:360
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