2847:
523:
function (WDF) obtained for some signals is due to the auto-correlation function inherent in its formulation; however, the latter also causes the cross-term problem. Therefore, if we want to analyze a single-term signal, using the WDF may be the best approach; if the signal is composed of multiple components, some other methods like the Gabor transform, Gabor-Wigner distribution or
Modified B-Distribution functions may be better choices.
3407:
49:. Rather than viewing a 1-dimensional signal (a function, real or complex-valued, whose domain is the real line) and some transform (another function whose domain is the real line, obtained from the original via some transform), time–frequency analysis studies a two-dimensional signal – a function whose domain is the two-dimensional real plane, obtained from the signal via a time–frequency transform.
5037:(LCT) is really helpful. By LCTs, the shape and location on the time–frequency plane of a signal can be in the arbitrary form that we want it to be. For example, the LCTs can shift the time–frequency distribution to any location, dilate it in the horizontal and vertical direction without changing its area on the plane, shear (or twist) it, and rotate it (
300:
876:
5192:
not be possible in the case of non-stationary signals that are multicomponent as such components could overlap in both the time domain and also in the frequency domain; as a consequence, the only possible way to achieve component separation and therefore a signal decomposition is to implement a time–frequency filter.
118:
signal, this approach requires a complete description of the signal's behavior over all time. Indeed, one can think of points in the (spectral) frequency domain as smearing together information from across the entire time domain. While mathematically elegant, such a technique is not appropriate for
5191:
The concept of signal decomposition relates to the need to separate one component from the others in a signal; this can be achieved through a filtering operation which require a filter design stage. Such filtering is traditionally done in the time domain or in the frequency domain; however, this may
68:
assumes that signals are infinite in time or periodic, while many signals in practice are of short duration, and change substantially over their duration. For example, traditional musical instruments do not produce infinite duration sinusoids, but instead begin with an attack, then gradually decay.
5838:
As acoustic signals are used as speech in communication between the human-sender and -receiver, their undelayedly transmission in technical communication systems is crucial, which makes the use of simpler TFDs, such as the Gabor transform, suitable to analyze these signals in real-time by reducing
5842:
If frequency analysis speed is not a limitation, a detailed feature comparison with well defined criteria should be made before selecting a particular TFD. Another approach is to define a signal dependent TFD that is adapted to the data. In biomedicine, one can use time–frequency distribution to
522:
To analyze the signals well, choosing an appropriate time–frequency distribution function is important. Which time–frequency distribution function should be used depends on the application being considered, as shown by reviewing a list of applications. The high clarity of the Wigner distribution
122:
To harness the power of a frequency representation without the need of a complete characterization in the time domain, one first obtains a time–frequency distribution of the signal, which represents the signal in both the time and frequency domains simultaneously. In such a representation the
52:
The mathematical motivation for this study is that functions and their transform representation are tightly connected, and they can be understood better by studying them jointly, as a two-dimensional object, rather than separately. A simple example is that the 4-fold periodicity of the
5172:
The filtering methods mentioned above can’t work well for every signal which may overlap in the time domain or in the frequency domain. By using the time–frequency distribution function, we can filter in the
Euclidean time–frequency domain or in the fractional domain by employing the
5229:
Consequently, when the signal we tend to sample is composed of single component, we use the WDF; however, if the signal consists of more than one component, using the Gabor transform, Gabor-Wigner distribution function, or other reduced interference TFDs may achieve better results.
870:
697:
119:
analyzing a signal with indeterminate future behavior. For instance, one must presuppose some degree of indeterminate future behavior in any telecommunications systems to achieve non-zero entropy (if one already knows what the other person will say one cannot learn anything).
294:
5616:
5726:
5208:
is equivalent to the area of the time–frequency distribution of a signal. (This is actually just an approximation, because the TF area of any signal is infinite.) Below is an example before and after we combine the sampling theory with the time–frequency distribution:
57:– and the fact that two-fold Fourier transform reverses direction – can be interpreted by considering the Fourier transform as a 90° rotation in the associated time–frequency plane: 4 such rotations yield the identity, and 2 such rotations simply reverse direction (
107:, for the case when the signal frequency characteristics are varying with time. Since many signals of interest – such as speech, music, images, and medical signals – have changing frequency characteristics, time–frequency analysis has broad scope of applications.
5479:
5187:
Filter design in time–frequency analysis always deals with signals composed of multiple components, so one cannot use WDF due to cross-term. The Gabor transform, Gabor–Wigner distribution function, or Cohen's class distribution function may be better choices.
3640:
1372:
2670:
5253:
concentrates in time or in frequency, separately. By taking advantage of the time–frequency distribution, we can make it more efficient to modulate and multiplex. All we have to do is to fill up the time–frequency plane. We present an example as
5818:
306:
Once such a representation has been generated other techniques in time–frequency analysis may then be applied to the signal in order to extract information from the signal, to separate the signal from noise or interfering signals, etc.
5834:
Similarly, it is a characteristic of acoustic signals, that their frequency components undergo abrupt variations in time and would hence be not well represented by a single frequency component analysis covering their entire durations.
1931:
1635:
3078:
2777:
3168:
1824:
2267:
3710:
4969:
2396:
2166:
4038:
705:
532:
6002:
E. Sejdić, I. Djurović, J. Jiang, “Time-frequency feature representation using energy concentration: An overview of recent advances,” Digital Signal
Processing, vol. 19, no. 1, pp. 153-183, January 2009.
136:
5514:
5116:
5627:
3396:
1467:
5318:
5371:
5154:
of a signal. We can know the instantaneous frequency from the time–frequency plane directly if the image is clear enough. Because the high clarity is critical, we often use WDF to analyze it.
5024:
Decompose by the AF and the FRFT. Any non-stationary random process can be expressed as a summation of the fractional
Fourier transform (or chirp multiplication) of stationary random process.
5386:
123:
frequency domain will only reflect the behavior of a temporally localized version of the signal. This enables one to talk sensibly about signals whose component frequencies vary in time.
5162:
The goal of filter design is to remove the undesired component of a signal. Conventionally, we can just filter in the time domain or in the frequency domain individually as shown below.
3517:
3218:
4648:
4531:
1211:
3510:
2821:
1070:
2518:
2514:
954:
1975:
2928:
1513:
3314:
5741:
5506:
5148:
4133:
4075:
4709:
4409:
3866:
3244:
2841:
4377:
130:
to globally transform the following function into the frequency domain one could instead use these methods to describe it as a signal with a time varying frequency.
4173:
3443:
3895:
326:
There are several different ways to formulate a valid time–frequency distribution function, resulting in several well-known time–frequency distributions, such as:
5019:
4853:
1836:
1116:
5959:
4153:
4095:
3840:
3789:
3769:
3465:
2948:
2416:
1521:
1205:
2955:
2675:
5226:
causes an improvement in the clarity and readability of the representation, therefore improving its interpretation and application to practical problems.
5944:
3082:
1652:
96:, time–frequency analysis is a body of techniques and methods used for characterizing and manipulating signals whose statistics vary in time, such as
346:
5263:
As illustrated in the upper example, using the WDF is not smart since the serious cross-term problem make it difficult to multiplex and modulate.
2178:
3646:
356:
6117:
5201:
5831:
is an electromagnetic wave, so time–frequency analysis applies to optics in the same way as for general electromagnetic wave propagation.
5182:
4857:
2295:
865:{\displaystyle x_{2}(t)={\begin{cases}\cos(\pi t);&t<10\\\cos(2\pi t);&10\leq t<20\\\cos(3\pi t);&t>20\end{cases}}}
692:{\displaystyle x_{1}(t)={\begin{cases}\cos(\pi t);&t<10\\\cos(3\pi t);&10\leq t<20\\\cos(2\pi t);&t>20\end{cases}}}
5991:
2000:
5508:
is the wavelength. When electromagnetic wave pass through a spherical lens or be reflected by a disk, the parameter matrix should be
3905:
373:
46:
28:
289:{\displaystyle x(t)={\begin{cases}\cos(\pi t);&t<10\\\cos(3\pi t);&10\leq t<20\\\cos(2\pi t);&t>20\end{cases}}}
5611:{\displaystyle {\begin{bmatrix}a&b\\c&d\end{bmatrix}}={\begin{bmatrix}1&0\\-{\frac {1}{\lambda f}}&1\end{bmatrix}}}
5912:(Wigner developed the Wigner–Ville distribution in 1932 in quantum mechanics, and Gabor was influenced by quantum mechanics – see
5721:{\displaystyle {\begin{bmatrix}a&b\\c&d\end{bmatrix}}={\begin{bmatrix}1&0\\{\frac {1}{\lambda R}}&1\end{bmatrix}}}
5167:
5917:
5214:
501:
5258:
5059:
316:
5377:
5033:
The following applications need not only the time–frequency distribution functions but also some operations to the signal. The
81:
372:
More information about the history and the motivation of development of time–frequency distribution can be found in the entry
5902:
3320:
1381:
58:
5277:
5898:
5333:
5174:
5038:
4662:
1642:
350:
330:
73:
5916:); this is reflected in the shared mathematics of the position-momentum plane and the time–frequency plane – as in the
409:
to ensure the time needed to represent and process a signal on a time–frequency plane allows real-time implementations.
366:
5954:
5474:{\displaystyle {\begin{bmatrix}a&b\\c&d\end{bmatrix}}={\begin{bmatrix}1&\lambda z\\0&1\end{bmatrix}},}
360:
3635:{\displaystyle SNR\approx 10\log _{10}{\frac {E_{x}}{\iint \limits _{(t,f)\in {\text{signal part}}}S_{x}(t,f)dtdf}}}
6122:
5034:
3172:
5222:
When we use the WDF, there might be the cross-term problem (also called interference). On the other hand, using
5949:
2846:
6087:
A. Papandreou-Suppappola, Applications in Time–Frequency Signal
Processing (CRC Press, Boca Raton, Fla., 2002)
5219:
It is noticeable that the number of sampling points decreases after we apply the time–frequency distribution.
5041:). This powerful operation, LCT, make it more flexible to analyze and apply the time–frequency distributions.
5881:, though these were not significantly applied to signal processing. More substantial work was undertaken by
5050:
5323:
which is similar to the time–frequency plane. When electromagnetic wave propagates through free-space, the
5234:
1367:{\displaystyle =\iint x(t+\tau /2,\xi _{1})x^{*}(t-\tau /2,\xi _{2})P(\xi _{1},\xi _{2})d\xi _{1}d\xi _{2}}
5848:
127:
5856:
4537:
4420:
2665:{\displaystyle =\iint x(\tau /2,\xi _{1})x^{*}(-\tau /2,\xi _{2})P(\xi _{1},\xi _{2})d\xi _{1}d\xi _{2}}
97:
3473:
2784:
962:
6042:
6101:. Taipei, Taiwan: Graduate Institute of Communication Engineering, National Taiwan University (NTU).
2421:
917:
736:
563:
160:
5925:
5868:
5324:
5151:
526:
As an illustration, magnitudes from non-localized
Fourier analysis cannot distinguish the signals:
5932:
5813:{\displaystyle {\begin{bmatrix}a&b\\c&d\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}.}
2286:
1990:
1935:
115:
3406:
2860:
1476:
17:
3262:
6070:
6031:"Techniques to Obtain Good Resolution and Concentrated Time-Frequency Distributions: A Review"
5987:
5909:
5852:
5491:
5237:
formalizes this, and provides a bound on the minimum number of time–frequency samples needed.
5124:
4102:
4044:
340:
111:
93:
54:
34:
5931:
An early practical motivation for time–frequency analysis was the development of radar – see
5908:
Particularly in the 1930s and 1940s, early time–frequency analysis developed in concert with
4670:
4388:
3845:
3223:
2826:
6060:
6050:
5964:
5844:
4187:
1926:{\displaystyle =\int _{-\infty }^{\infty }R_{x}(t,\tau )\cdot e^{-j2\pi f\tau }\cdot d\tau }
104:
69:
This is poorly represented by traditional methods, which motivates time–frequency analysis.
65:
4158:
3421:
5894:
5223:
3871:
1630:{\displaystyle S_{x}(t,f)=\int _{-\infty }^{\infty }R_{x}(t,\tau )e^{-j2\pi f\tau }d\tau }
334:
4974:
4714:
1077:
6046:
5878:
4138:
4080:
3794:
3774:
3723:
3450:
3073:{\displaystyle E=\int _{-\infty }^{\infty }R_{x}(\tau )\cdot e^{-j2\pi t\eta }\cdot dt}
2933:
2401:
1125:
2772:{\displaystyle S_{x}(f)=\int _{-\infty }^{\infty }R_{x}(\tau )e^{-j2\pi f\tau }d\tau }
41:
comprises those techniques that study a signal in both the time and frequency domains
6111:
413:
Below is a brief comparison of some selected time–frequency distribution functions.
5882:
5874:
5250:
5921:
5181:
3163:{\displaystyle =R_{x}(\tau )\int _{-\infty }^{\infty }e^{-j2\pi t\eta }\cdot dt}
2289:: the statistical properties do not change with t. Its auto-covariance function:
1819:{\displaystyle E=\int _{-\infty }^{\infty }E\cdot e^{-j2\pi f\tau }\cdot d\tau }
5913:
5886:
5246:
6074:
391:
in both time and frequency, to make it easier to be analyzed and interpreted.
384:
A time–frequency distribution function ideally has the following properties:
5905:(Ville 1948, in a signal processing context) was another foundational step.
5735:
is the radius of the disk. These corresponding results can be obtained from
2262:{\displaystyle =\int _{-\infty }^{\infty }R_{x}(t,\tau )e^{-j2\pi t\eta }dt}
6029:
Shafi, Imran; Ahmad, Jamil; Shah, Syed Ismail; Kashif, F. M. (2009-06-09).
3705:{\displaystyle SNR\approx 10\log _{10}{\frac {E_{x}}{\sigma \mathrm {A} }}}
6055:
6030:
5205:
6065:
5271:
We can represent an electromagnetic wave in the form of a 2 by 1 matrix
5890:
114:
can be extended to obtain the frequency spectrum of any slowly growing
77:
76:(STFT), but more sophisticated techniques have been developed, notably
64:
The practical motivation for time–frequency analysis is that classical
5166:
2845:
902:
For a random process x(t), we cannot find the explicit value of x(t).
5204:, we can conclude that the minimum number of sampling points without
5213:
4964:{\displaystyle =\int _{t-B}^{t+B}Ew(t-\tau )e^{-j2\pi f\tau }d\tau }
5257:
5828:
3405:
2391:{\displaystyle R_{x}(t_{1},\tau )=R_{x}(t_{2},\tau )=R_{x}(\tau )}
874:
298:
2161:{\displaystyle E=\int _{-\infty }^{\infty }Ee^{-j2\pi t\eta }dt}
875:
299:
4033:{\displaystyle h(t)=x_{1}(t)+x_{2}(t)+x_{3}(t)+......+x_{k}(t)}
3467: : area of the time frequency distribution of the signal
72:
One of the most basic forms of time–frequency analysis is the
858:
685:
397:
to avoid confusing real components from artifacts or noise.
282:
5731:
respectively, where ƒ is the focal length of the lens and
5111:{\displaystyle {\frac {1}{2\pi }}{\frac {d}{dt}}\phi (t),}
905:
The value of x(t) is expressed as a probability function.
6099:
Time frequency analysis and wavelet transform class notes
5873:
Early work in time–frequency analysis can be seen in the
5924:(time–frequency analysis), ultimately both reflecting a
1376:(alternative definition of the auto-covariance function)
5786:
5750:
5675:
5636:
5562:
5523:
5434:
5395:
5342:
5286:
5744:
5630:
5517:
5494:
5389:
5336:
5280:
5127:
5062:
4977:
4860:
4717:
4673:
4540:
4423:
4391:
4190:
4161:
4141:
4105:
4083:
4047:
3908:
3874:
3848:
3797:
3777:
3726:
3649:
3520:
3476:
3453:
3424:
3391:{\displaystyle E=\sigma \delta (\tau )\delta (\eta )}
3323:
3265:
3226:
3175:
3085:
2958:
2936:
2863:
2829:
2787:
2678:
2521:
2424:
2404:
2298:
2181:
2003:
1938:
1839:
1655:
1524:
1479:
1462:{\displaystyle {\overset {\land }{R_{x}}}(t,\tau )=E}
1384:
1214:
1128:
1080:
965:
920:
708:
535:
403:
to ensure such methods benefit real-life application.
359:, Gabor–Wigner distribution function, and so on (see
139:
6011:P. Flandrin, "Time–frequency/Time–Scale Analysis,"
5313:{\displaystyle {\begin{bmatrix}x\\y\end{bmatrix}},}
5812:
5720:
5610:
5500:
5473:
5366:{\displaystyle {\begin{bmatrix}x\\y\end{bmatrix}}}
5365:
5312:
5142:
5110:
5013:
4963:
4847:
4703:
4642:
4525:
4403:
4371:
4167:
4147:
4127:
4089:
4069:
4032:
3889:
3860:
3834:
3783:
3763:
3704:
3634:
3504:
3459:
3437:
3402:Filter Design for a signal in additive white noise
3390:
3308:
3238:
3212:
3162:
3072:
2942:
2922:
2835:
2815:
2771:
2664:
2508:
2410:
2390:
2261:
2160:
1969:
1925:
1818:
1629:
1507:
1461:
1366:
1199:
1110:
1064:
948:
864:
691:
288:
6035:EURASIP Journal on Advances in Signal Processing
5327:occurs. We can operate with the 2 by 1 matrix
8:
3213:{\displaystyle =R_{x}(\tau )\delta (\eta )}
401:A list of desirable mathematical properties
5945:Motions in the time-frequency distribution
6064:
6054:
5960:Time–frequency analysis for music signals
5781:
5745:
5743:
5690:
5670:
5631:
5629:
5580:
5557:
5518:
5516:
5493:
5429:
5390:
5388:
5337:
5335:
5281:
5279:
5126:
5078:
5063:
5061:
4976:
4934:
4879:
4868:
4859:
4815:
4769:
4758:
4716:
4672:
4614:
4609:
4593:
4582:
4551:
4539:
4497:
4492:
4476:
4465:
4434:
4422:
4390:
4349:
4331:
4326:
4302:
4284:
4257:
4239:
4234:
4219:
4201:
4189:
4160:
4140:
4110:
4104:
4082:
4052:
4046:
4015:
3972:
3950:
3928:
3907:
3873:
3847:
3808:
3796:
3776:
3737:
3725:
3694:
3684:
3678:
3669:
3648:
3596:
3585:
3566:
3555:
3549:
3540:
3519:
3481:
3475:
3452:
3429:
3423:
3334:
3322:
3276:
3264:
3225:
3183:
3174:
3130:
3120:
3112:
3093:
3084:
3040:
3018:
3008:
3000:
2969:
2957:
2935:
2905:
2874:
2862:
2828:
2792:
2786:
2742:
2723:
2713:
2705:
2683:
2677:
2656:
2643:
2627:
2614:
2595:
2580:
2565:
2552:
2537:
2520:
2492:
2477:
2462:
2429:
2423:
2403:
2373:
2351:
2338:
2316:
2303:
2297:
2232:
2207:
2197:
2189:
2180:
2131:
2113:
2095:
2080:
2053:
2045:
2014:
2002:
1946:
1937:
1893:
1865:
1855:
1847:
1838:
1786:
1765:
1747:
1732:
1705:
1697:
1666:
1654:
1600:
1575:
1565:
1557:
1529:
1523:
1484:
1478:
1391:
1385:
1383:
1358:
1345:
1329:
1316:
1297:
1282:
1264:
1251:
1236:
1213:
1183:
1165:
1150:
1127:
1079:
1048:
1030:
1015:
970:
964:
925:
919:
731:
713:
707:
558:
540:
534:
155:
138:
103:It is a generalization and refinement of
5053:is the time rate of change of phase, or
415:
5975:
6013:Wavelet Analysis and its Applications
5982:L. Cohen, "Time–Frequency Analysis,"
5158:TF filtering and signal decomposition
357:Modified Wigner distribution function
311:Time–frequency distribution functions
7:
4135:'s are mutually independent for all
347:Bilinear time–frequency distribution
4663:STFT (Short Time Fourier Transform)
3897:is a non-stationary random process.
5824:Optics, acoustics, and biomedicine
5045:Instantaneous frequency estimation
4643:{\displaystyle E=\sum _{n=1}^{k}E}
4526:{\displaystyle E=\sum _{n=1}^{k}E}
3695:
3121:
3116:
3009:
3004:
2714:
2709:
2198:
2193:
2054:
2049:
1856:
1851:
1706:
1701:
1643:WDF (Wigner Distribution Function)
1566:
1561:
482:Gabor–Wigner distribution function
84:methods for unevenly spaced data.
25:
5245:Conventionally, the operation of
4711:should be satisfied. Otherwise,
883:But time–frequency analysis can.
5918:Heisenberg uncertainty principle
5488:is the propagation distance and
5267:Electromagnetic wave propagation
5256:
5212:
5202:Nyquist–Shannon sampling theorem
5180:
5165:
3505:{\displaystyle S_{n}(f)=\sigma }
2816:{\displaystyle S_{x}(f)=\sigma }
1065:{\displaystyle R_{x}(t,\tau )=E}
887:TF analysis and random processes
516:Medium (if recursively defined)
502:Cone-shape distribution function
18:Time–frequency signal processing
3715:Non-stationary random processes
3255:For additive white noise (AWN),
914:Auto-covariance function (ACF)
126:For instance rather than using
82:least-squares spectral analysis
5137:
5131:
5102:
5096:
5002:
4999:
4987:
4981:
4971:for zero-mean random process,
4927:
4915:
4909:
4906:
4900:
4894:
4842:
4808:
4796:
4790:
4784:
4751:
4742:
4739:
4727:
4721:
4692:
4689:
4683:
4677:
4637:
4634:
4622:
4602:
4572:
4569:
4557:
4544:
4520:
4517:
4505:
4485:
4455:
4452:
4440:
4427:
4360:
4357:
4337:
4319:
4313:
4310:
4290:
4277:
4268:
4265:
4245:
4227:
4207:
4194:
4122:
4116:
4064:
4058:
4027:
4021:
3984:
3978:
3962:
3956:
3940:
3934:
3918:
3912:
3884:
3878:
3829:
3826:
3814:
3801:
3758:
3755:
3743:
3730:
3614:
3602:
3579:
3567:
3493:
3487:
3470:The PSD of the white noise is
3385:
3379:
3373:
3367:
3355:
3352:
3340:
3327:
3297:
3294:
3282:
3269:
3207:
3201:
3195:
3189:
3105:
3099:
3030:
3024:
2990:
2987:
2975:
2962:
2917:
2911:
2895:
2892:
2880:
2867:
2804:
2798:
2735:
2729:
2695:
2689:
2633:
2607:
2601:
2571:
2558:
2531:
2509:{\displaystyle R_{x}(\tau )=E}
2503:
2500:
2483:
2470:
2456:
2450:
2441:
2435:
2385:
2379:
2363:
2344:
2328:
2309:
2225:
2213:
2124:
2121:
2101:
2088:
2068:
2062:
2035:
2032:
2020:
2007:
1964:
1952:
1883:
1871:
1776:
1773:
1753:
1740:
1720:
1714:
1687:
1684:
1672:
1659:
1593:
1581:
1547:
1535:
1502:
1490:
1456:
1453:
1441:
1435:
1429:
1423:
1414:
1402:
1335:
1309:
1303:
1270:
1257:
1224:
1194:
1191:
1171:
1158:
1138:
1132:
1099:
1096:
1090:
1084:
1059:
1056:
1036:
1023:
1003:
997:
988:
976:
949:{\displaystyle R_{x}(t,\tau )}
943:
931:
838:
826:
793:
781:
754:
745:
725:
719:
665:
653:
620:
608:
581:
572:
552:
546:
407:Lower computational complexity
380:Ideal TF distribution function
262:
250:
217:
205:
178:
169:
149:
143:
47:time–frequency representations
1:
5177:. An example is shown below.
1473:Power spectral density (PSD)
374:Time–frequency representation
110:Whereas the technique of the
59:reflection through the origin
29:Time–frequency representation
5920:(quantum mechanics) and the
5899:short-time Fourier transform
5175:fractional Fourier transform
5039:Fractional Fourier transform
4656:Short-time Fourier transform
463:Wigner distribution function
432:Good mathematical properties
351:Wigner distribution function
331:Short-time Fourier transform
74:short-time Fourier transform
5955:Spectral density estimation
5839:computational complexity.
5241:Modulation and multiplexing
2281:Stationary random processes
1970:{\displaystyle =S_{x}(t,f)}
317:Time–frequency distribution
6139:
5866:
5035:Linear canonical transform
4077:'s have zero mean for all
2923:{\displaystyle E=S_{x}(f)}
1508:{\displaystyle S_{x}(t,f)}
1074:In usual, we suppose that
314:
26:
5903:Wigner–Ville distribution
5889:(1947), an early form of
3309:{\displaystyle E=\sigma }
2287:Stationary random process
510:No (eliminated, in time)
6097:Ding, Jian-Jiun (2022).
5950:Multiresolution analysis
5501:{\displaystyle \lambda }
5143:{\displaystyle \phi (t)}
4128:{\displaystyle x_{n}(t)}
4070:{\displaystyle x_{n}(t)}
2854:When x(t) is stationary,
909:General random processes
437:Computational complexity
6118:Time–frequency analysis
5051:instantaneous frequency
4704:{\displaystyle E\neq 0}
4404:{\displaystyle m\neq n}
3861:{\displaystyle \eta =0}
3445:: energy of the signal
3239:{\displaystyle \eta =0}
2836:{\displaystyle \sigma }
367:Hilbert–Huang transform
39:time–frequency analysis
5849:electroencephalography
5814:
5722:
5612:
5502:
5475:
5380:with parameter matrix
5367:
5314:
5144:
5112:
5015:
4965:
4849:
4705:
4644:
4598:
4527:
4481:
4405:
4373:
4372:{\displaystyle E=EE=0}
4169:
4149:
4129:
4091:
4071:
4034:
3891:
3862:
3836:
3785:
3765:
3706:
3636:
3506:
3461:
3439:
3410:
3392:
3310:
3240:
3214:
3164:
3074:
2944:
2924:
2850:
2837:
2817:
2773:
2666:
2510:
2412:
2392:
2263:
2162:
1971:
1927:
1820:
1631:
1509:
1463:
1368:
1201:
1112:
1066:
950:
879:
866:
693:
361:Gabor–Wigner transform
303:
290:
128:tempered distributions
5857:otoacoustic emissions
5815:
5723:
5613:
5503:
5476:
5368:
5315:
5145:
5113:
5016:
4966:
4850:
4706:
4645:
4578:
4528:
4461:
4406:
4374:
4170:
4168:{\displaystyle \tau }
4150:
4130:
4092:
4072:
4035:
3892:
3863:
3837:
3786:
3766:
3707:
3637:
3507:
3462:
3440:
3438:{\displaystyle E_{x}}
3409:
3393:
3311:
3241:
3220:, (nonzero only when
3215:
3165:
3075:
2945:
2925:
2849:
2838:
2818:
2774:
2667:
2511:
2413:
2393:
2264:
2163:
1989:Relation between the
1972:
1928:
1821:
1641:Relation between the
1632:
1510:
1464:
1369:
1202:
1113:
1067:
951:
878:
867:
694:
302:
291:
5742:
5628:
5515:
5492:
5387:
5334:
5278:
5125:
5060:
4975:
4858:
4715:
4671:
4538:
4421:
4389:
4188:
4159:
4139:
4103:
4081:
4045:
3906:
3890:{\displaystyle x(t)}
3872:
3846:
3795:
3775:
3724:
3647:
3518:
3474:
3451:
3422:
3321:
3263:
3250:Additive white noise
3224:
3173:
3083:
2956:
2934:
2861:
2827:
2785:
2676:
2519:
2422:
2402:
2296:
2179:
2001:
1936:
1837:
1653:
1522:
1477:
1382:
1212:
1126:
1078:
963:
918:
706:
533:
137:
6056:10.1155/2009/673539
6047:2009EJASP2009..109S
5986:, New York, 1995.
5869:History of wavelets
5325:Fresnel diffraction
5152:instantaneous phase
5014:{\displaystyle E=0}
4890:
4848:{\displaystyle E=E}
4780:
4661:Random process for
4336:
4244:
3125:
3013:
2718:
2202:
2058:
1860:
1710:
1570:
1111:{\displaystyle E=0}
6019:, San Diego, 1999.
5933:ambiguity function
5810:
5801:
5775:
5718:
5712:
5661:
5608:
5602:
5548:
5498:
5471:
5462:
5420:
5363:
5357:
5310:
5301:
5235:Balian–Low theorem
5140:
5108:
5049:The definition of
5011:
4961:
4864:
4845:
4754:
4701:
4640:
4523:
4401:
4369:
4322:
4230:
4165:
4145:
4125:
4087:
4067:
4030:
3887:
3858:
3832:
3781:
3761:
3702:
3632:
3591:
3502:
3457:
3435:
3411:
3388:
3306:
3236:
3210:
3160:
3108:
3070:
2996:
2940:
2930:, (invariant with
2920:
2851:
2833:
2813:
2769:
2701:
2662:
2506:
2408:
2388:
2259:
2185:
2158:
2041:
1991:ambiguity function
1967:
1923:
1843:
1816:
1693:
1627:
1553:
1505:
1459:
1364:
1197:
1108:
1062:
946:
880:
862:
857:
689:
684:
489:Almost eliminated
304:
286:
281:
116:locally integrable
6123:Signal processing
5910:quantum mechanics
5853:electrocardiogram
5703:
5593:
5091:
5076:
4148:{\displaystyle t}
4090:{\displaystyle t}
3835:{\displaystyle E}
3784:{\displaystyle t}
3764:{\displaystyle E}
3700:
3630:
3588:
3562:
3460:{\displaystyle A}
2943:{\displaystyle t}
2843:is some constant.
2411:{\displaystyle t}
1400:
1200:{\displaystyle E}
520:
519:
341:Wavelet transform
112:Fourier transform
94:signal processing
55:Fourier transform
35:signal processing
16:(Redirected from
6130:
6103:
6102:
6094:
6088:
6085:
6079:
6078:
6068:
6058:
6026:
6020:
6009:
6003:
6000:
5994:
5980:
5965:Wavelet analysis
5845:electromyography
5819:
5817:
5816:
5811:
5806:
5805:
5780:
5779:
5727:
5725:
5724:
5719:
5717:
5716:
5704:
5702:
5691:
5666:
5665:
5617:
5615:
5614:
5609:
5607:
5606:
5594:
5592:
5581:
5553:
5552:
5507:
5505:
5504:
5499:
5480:
5478:
5477:
5472:
5467:
5466:
5425:
5424:
5372:
5370:
5369:
5364:
5362:
5361:
5319:
5317:
5316:
5311:
5306:
5305:
5260:
5216:
5184:
5169:
5149:
5147:
5146:
5141:
5117:
5115:
5114:
5109:
5092:
5090:
5079:
5077:
5075:
5064:
5020:
5018:
5017:
5012:
4970:
4968:
4967:
4962:
4954:
4953:
4889:
4878:
4854:
4852:
4851:
4846:
4835:
4834:
4779:
4768:
4710:
4708:
4707:
4702:
4649:
4647:
4646:
4641:
4621:
4620:
4619:
4618:
4597:
4592:
4556:
4555:
4532:
4530:
4529:
4524:
4504:
4503:
4502:
4501:
4480:
4475:
4439:
4438:
4410:
4408:
4407:
4402:
4378:
4376:
4375:
4370:
4353:
4335:
4330:
4306:
4289:
4288:
4261:
4243:
4238:
4223:
4206:
4205:
4174:
4172:
4171:
4166:
4154:
4152:
4151:
4146:
4134:
4132:
4131:
4126:
4115:
4114:
4096:
4094:
4093:
4088:
4076:
4074:
4073:
4068:
4057:
4056:
4039:
4037:
4036:
4031:
4020:
4019:
3977:
3976:
3955:
3954:
3933:
3932:
3896:
3894:
3893:
3888:
3867:
3865:
3864:
3859:
3842:is nonzero when
3841:
3839:
3838:
3833:
3813:
3812:
3790:
3788:
3787:
3782:
3770:
3768:
3767:
3762:
3742:
3741:
3711:
3709:
3708:
3703:
3701:
3699:
3698:
3689:
3688:
3679:
3674:
3673:
3641:
3639:
3638:
3633:
3631:
3629:
3601:
3600:
3590:
3589:
3586:
3560:
3559:
3550:
3545:
3544:
3511:
3509:
3508:
3503:
3486:
3485:
3466:
3464:
3463:
3458:
3444:
3442:
3441:
3436:
3434:
3433:
3397:
3395:
3394:
3389:
3339:
3338:
3315:
3313:
3312:
3307:
3281:
3280:
3245:
3243:
3242:
3237:
3219:
3217:
3216:
3211:
3188:
3187:
3169:
3167:
3166:
3161:
3150:
3149:
3124:
3119:
3098:
3097:
3079:
3077:
3076:
3071:
3060:
3059:
3023:
3022:
3012:
3007:
2974:
2973:
2949:
2947:
2946:
2941:
2929:
2927:
2926:
2921:
2910:
2909:
2879:
2878:
2842:
2840:
2839:
2834:
2822:
2820:
2819:
2814:
2797:
2796:
2778:
2776:
2775:
2770:
2762:
2761:
2728:
2727:
2717:
2712:
2688:
2687:
2671:
2669:
2668:
2663:
2661:
2660:
2648:
2647:
2632:
2631:
2619:
2618:
2600:
2599:
2584:
2570:
2569:
2557:
2556:
2541:
2515:
2513:
2512:
2507:
2496:
2482:
2481:
2466:
2434:
2433:
2417:
2415:
2414:
2409:
2397:
2395:
2394:
2389:
2378:
2377:
2356:
2355:
2343:
2342:
2321:
2320:
2308:
2307:
2268:
2266:
2265:
2260:
2252:
2251:
2212:
2211:
2201:
2196:
2167:
2165:
2164:
2159:
2151:
2150:
2117:
2100:
2099:
2084:
2057:
2052:
2019:
2018:
1976:
1974:
1973:
1968:
1951:
1950:
1932:
1930:
1929:
1924:
1913:
1912:
1870:
1869:
1859:
1854:
1825:
1823:
1822:
1817:
1806:
1805:
1769:
1752:
1751:
1736:
1709:
1704:
1671:
1670:
1636:
1634:
1633:
1628:
1620:
1619:
1580:
1579:
1569:
1564:
1534:
1533:
1514:
1512:
1511:
1506:
1489:
1488:
1468:
1466:
1465:
1460:
1401:
1396:
1395:
1386:
1373:
1371:
1370:
1365:
1363:
1362:
1350:
1349:
1334:
1333:
1321:
1320:
1302:
1301:
1286:
1269:
1268:
1256:
1255:
1240:
1206:
1204:
1203:
1198:
1187:
1170:
1169:
1154:
1117:
1115:
1114:
1109:
1071:
1069:
1068:
1063:
1052:
1035:
1034:
1019:
975:
974:
955:
953:
952:
947:
930:
929:
899:
898:
894:
871:
869:
868:
863:
861:
860:
718:
717:
698:
696:
695:
690:
688:
687:
545:
544:
416:
295:
293:
292:
287:
285:
284:
105:Fourier analysis
66:Fourier analysis
21:
6138:
6137:
6133:
6132:
6131:
6129:
6128:
6127:
6108:
6107:
6106:
6096:
6095:
6091:
6086:
6082:
6028:
6027:
6023:
6010:
6006:
6001:
5997:
5981:
5977:
5973:
5941:
5895:Gabor transform
5871:
5865:
5826:
5800:
5799:
5793:
5792:
5782:
5774:
5773:
5768:
5762:
5761:
5756:
5746:
5740:
5739:
5711:
5710:
5705:
5695:
5687:
5686:
5681:
5671:
5660:
5659:
5654:
5648:
5647:
5642:
5632:
5626:
5625:
5601:
5600:
5595:
5585:
5574:
5573:
5568:
5558:
5547:
5546:
5541:
5535:
5534:
5529:
5519:
5513:
5512:
5490:
5489:
5461:
5460:
5455:
5449:
5448:
5440:
5430:
5419:
5418:
5413:
5407:
5406:
5401:
5391:
5385:
5384:
5356:
5355:
5349:
5348:
5338:
5332:
5331:
5300:
5299:
5293:
5292:
5282:
5276:
5275:
5269:
5255:
5243:
5224:Gabor transform
5198:
5196:Sampling theory
5160:
5123:
5122:
5083:
5068:
5058:
5057:
5047:
5031:
4973:
4972:
4930:
4856:
4855:
4811:
4713:
4712:
4669:
4668:
4658:
4610:
4605:
4547:
4536:
4535:
4493:
4488:
4430:
4419:
4418:
4387:
4386:
4280:
4197:
4186:
4185:
4157:
4156:
4137:
4136:
4106:
4101:
4100:
4079:
4078:
4048:
4043:
4042:
4011:
3968:
3946:
3924:
3904:
3903:
3870:
3869:
3844:
3843:
3804:
3793:
3792:
3773:
3772:
3733:
3722:
3721:
3717:
3690:
3680:
3665:
3645:
3644:
3592:
3561:
3551:
3536:
3516:
3515:
3514:
3477:
3472:
3471:
3449:
3448:
3425:
3420:
3419:
3416:
3413:
3330:
3319:
3318:
3272:
3261:
3260:
3252:
3222:
3221:
3179:
3171:
3170:
3126:
3089:
3081:
3080:
3036:
3014:
2965:
2954:
2953:
2932:
2931:
2901:
2870:
2859:
2858:
2825:
2824:
2788:
2783:
2782:
2738:
2719:
2679:
2674:
2673:
2652:
2639:
2623:
2610:
2591:
2561:
2548:
2517:
2516:
2473:
2425:
2420:
2419:
2400:
2399:
2369:
2347:
2334:
2312:
2299:
2294:
2293:
2283:
2228:
2203:
2177:
2176:
2127:
2091:
2010:
1999:
1998:
1942:
1934:
1933:
1889:
1861:
1835:
1834:
1782:
1743:
1662:
1651:
1650:
1596:
1571:
1525:
1520:
1519:
1480:
1475:
1474:
1387:
1380:
1379:
1354:
1341:
1325:
1312:
1293:
1260:
1247:
1210:
1209:
1161:
1124:
1123:
1076:
1075:
1026:
966:
961:
960:
921:
916:
915:
911:
900:
896:
892:
890:
889:
882:
856:
855:
844:
817:
816:
799:
772:
771:
760:
732:
709:
704:
703:
683:
682:
671:
644:
643:
626:
599:
598:
587:
559:
536:
531:
530:
444:Gabor transform
389:High resolution
382:
335:Gabor transform
333:(including the
324:
319:
313:
280:
279:
268:
241:
240:
223:
196:
195:
184:
156:
135:
134:
90:
43:simultaneously,
31:
23:
22:
15:
12:
11:
5:
6136:
6134:
6126:
6125:
6120:
6110:
6109:
6105:
6104:
6089:
6080:
6021:
6017:Academic Press
6004:
5995:
5992:978-0135945322
5974:
5972:
5969:
5968:
5967:
5962:
5957:
5952:
5947:
5940:
5937:
5864:
5861:
5825:
5822:
5821:
5820:
5809:
5804:
5798:
5795:
5794:
5791:
5788:
5787:
5785:
5778:
5772:
5769:
5767:
5764:
5763:
5760:
5757:
5755:
5752:
5751:
5749:
5729:
5728:
5715:
5709:
5706:
5701:
5698:
5694:
5689:
5688:
5685:
5682:
5680:
5677:
5676:
5674:
5669:
5664:
5658:
5655:
5653:
5650:
5649:
5646:
5643:
5641:
5638:
5637:
5635:
5619:
5618:
5605:
5599:
5596:
5591:
5588:
5584:
5579:
5576:
5575:
5572:
5569:
5567:
5564:
5563:
5561:
5556:
5551:
5545:
5542:
5540:
5537:
5536:
5533:
5530:
5528:
5525:
5524:
5522:
5497:
5482:
5481:
5470:
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2022:
2017:
2013:
2009:
2006:
1995:
1994:
1986:
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1983:
1982:
1981:
1980:
1979:
1978:
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1966:
1963:
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1941:
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1911:
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987:
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429:
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419:
411:
410:
404:
398:
392:
381:
378:
370:
369:
364:
354:
344:
338:
323:
320:
315:Main article:
312:
309:
297:
296:
283:
278:
275:
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269:
267:
264:
261:
258:
255:
252:
249:
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174:
171:
168:
165:
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161:
159:
154:
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24:
14:
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10:
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5985:
5984:Prentice-Hall
5979:
5976:
5970:
5966:
5963:
5961:
5958:
5956:
5953:
5951:
5948:
5946:
5943:
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5938:
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5892:
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5875:Haar wavelets
5870:
5862:
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3662:
3659:
3656:
3653:
3650:
3642:
3626:
3623:
3620:
3617:
3611:
3608:
3605:
3597:
3593:
3582:
3576:
3573:
3570:
3563:
3556:
3552:
3546:
3541:
3537:
3533:
3530:
3527:
3524:
3521:
3512:
3499:
3496:
3490:
3482:
3478:
3468:
3454:
3446:
3430:
3426:
3417:
3414:
3408:
3401:
3400:
3382:
3376:
3370:
3364:
3361:
3358:
3349:
3346:
3343:
3335:
3331:
3324:
3317:
3303:
3300:
3291:
3288:
3285:
3277:
3273:
3266:
3259:
3258:
3254:
3253:
3249:
3247:
3233:
3230:
3227:
3204:
3198:
3192:
3184:
3180:
3176:
3157:
3154:
3151:
3146:
3143:
3140:
3137:
3134:
3131:
3127:
3113:
3109:
3102:
3094:
3090:
3086:
3067:
3064:
3061:
3056:
3053:
3050:
3047:
3044:
3041:
3037:
3033:
3027:
3019:
3015:
3001:
2997:
2993:
2984:
2981:
2978:
2970:
2966:
2959:
2951:
2937:
2914:
2906:
2902:
2898:
2889:
2886:
2883:
2875:
2871:
2864:
2853:
2852:
2848:
2844:
2830:
2810:
2807:
2801:
2793:
2789:
2780:
2779:White noise:
2766:
2763:
2758:
2755:
2752:
2749:
2746:
2743:
2739:
2732:
2724:
2720:
2706:
2702:
2698:
2692:
2684:
2680:
2657:
2653:
2649:
2644:
2640:
2636:
2628:
2624:
2620:
2615:
2611:
2604:
2596:
2592:
2588:
2585:
2581:
2577:
2574:
2566:
2562:
2553:
2549:
2545:
2542:
2538:
2534:
2528:
2525:
2522:
2497:
2493:
2489:
2486:
2478:
2474:
2467:
2463:
2459:
2453:
2447:
2444:
2438:
2430:
2426:
2418:, Therefore,
2405:
2382:
2374:
2370:
2366:
2360:
2357:
2352:
2348:
2339:
2335:
2331:
2325:
2322:
2317:
2313:
2304:
2300:
2288:
2285:
2284:
2280:
2256:
2253:
2248:
2245:
2242:
2239:
2236:
2233:
2229:
2222:
2219:
2216:
2208:
2204:
2190:
2186:
2182:
2175:
2174:
2173:
2172:
2171:
2170:
2169:
2168:
2155:
2152:
2147:
2144:
2141:
2138:
2135:
2132:
2128:
2118:
2114:
2110:
2107:
2104:
2096:
2092:
2085:
2081:
2077:
2074:
2071:
2065:
2059:
2046:
2042:
2038:
2029:
2026:
2023:
2015:
2011:
2004:
1997:
1996:
1992:
1988:
1987:
1961:
1958:
1955:
1947:
1943:
1939:
1920:
1917:
1914:
1909:
1906:
1903:
1900:
1897:
1894:
1890:
1886:
1880:
1877:
1874:
1866:
1862:
1848:
1844:
1840:
1833:
1832:
1831:
1830:
1829:
1828:
1827:
1826:
1813:
1810:
1807:
1802:
1799:
1796:
1793:
1790:
1787:
1783:
1779:
1770:
1766:
1762:
1759:
1756:
1748:
1744:
1737:
1733:
1729:
1726:
1723:
1717:
1711:
1698:
1694:
1690:
1681:
1678:
1675:
1667:
1663:
1656:
1649:
1648:
1644:
1640:
1639:
1624:
1621:
1616:
1613:
1610:
1607:
1604:
1601:
1597:
1590:
1587:
1584:
1576:
1572:
1558:
1554:
1550:
1544:
1541:
1538:
1530:
1526:
1518:
1517:
1499:
1496:
1493:
1485:
1481:
1472:
1471:
1450:
1447:
1444:
1438:
1432:
1426:
1420:
1417:
1411:
1408:
1405:
1397:
1392:
1388:
1378:
1375:
1359:
1355:
1351:
1346:
1342:
1338:
1330:
1326:
1322:
1317:
1313:
1306:
1298:
1294:
1290:
1287:
1283:
1279:
1276:
1273:
1265:
1261:
1252:
1248:
1244:
1241:
1237:
1233:
1230:
1227:
1221:
1218:
1215:
1208:
1188:
1184:
1180:
1177:
1174:
1166:
1162:
1155:
1151:
1147:
1144:
1141:
1135:
1129:
1122:
1121:
1105:
1102:
1093:
1087:
1081:
1073:
1053:
1049:
1045:
1042:
1039:
1031:
1027:
1020:
1016:
1012:
1009:
1006:
1000:
994:
991:
985:
982:
979:
971:
967:
959:
958:
940:
937:
934:
926:
922:
913:
912:
908:
906:
903:
895:
886:
884:
877:
852:
849:
846:
841:
835:
832:
829:
823:
820:
813:
810:
807:
804:
801:
796:
790:
787:
784:
778:
775:
768:
765:
762:
757:
751:
748:
742:
739:
733:
728:
722:
714:
710:
702:
701:
679:
676:
673:
668:
662:
659:
656:
650:
647:
640:
637:
634:
631:
628:
623:
617:
614:
611:
605:
602:
595:
592:
589:
584:
578:
575:
569:
566:
560:
555:
549:
541:
537:
529:
528:
527:
524:
515:
512:
509:
506:
504:
503:
499:
498:
494:
491:
488:
485:
483:
480:
479:
475:
472:
469:
466:
464:
461:
460:
456:
453:
450:
447:
445:
442:
441:
438:
435:
433:
430:
428:
425:
423:
420:
418:
417:
414:
408:
405:
402:
399:
396:
395:No cross-term
393:
390:
387:
386:
385:
379:
377:
375:
368:
365:
362:
358:
355:
352:
348:
345:
342:
339:
336:
332:
329:
328:
327:
321:
318:
310:
308:
301:
276:
273:
270:
265:
259:
256:
253:
247:
244:
237:
234:
231:
228:
225:
220:
214:
211:
208:
202:
199:
192:
189:
186:
181:
175:
172:
166:
163:
157:
152:
146:
140:
133:
132:
131:
129:
124:
120:
117:
113:
108:
106:
101:
99:
95:
87:
85:
83:
79:
75:
70:
67:
62:
60:
56:
50:
48:
44:
40:
36:
30:
19:
6098:
6092:
6083:
6066:1721.1/50243
6038:
6034:
6024:
6016:
6012:
6007:
5998:
5983:
5978:
5930:
5907:
5883:Dennis Gabor
5872:
5843:analyze the
5841:
5837:
5833:
5827:
5732:
5730:
5620:
5485:
5483:
5375:
5322:
5270:
5262:
5251:multiplexing
5244:
5232:
5228:
5221:
5218:
5211:
5199:
5190:
5186:
5179:
5171:
5164:
5161:
5120:
5048:
5032:
5029:Applications
4667:
3771:varies with
3643:
3513:
3469:
3447:
3418:
3415:
3412:
2952:
2857:
2781:
2292:
904:
901:
881:
525:
521:
500:
481:
462:
443:
436:
431:
426:
421:
412:
406:
400:
394:
388:
383:
371:
325:
322:Formulations
305:
125:
121:
109:
102:
91:
71:
63:
51:
42:
38:
32:
5928:structure.
5922:Gabor limit
5887:Gabor atoms
5879:Alfréd Haar
3587:signal part
1993:and the ACF
1645:and the PSD
6112:Categories
6015:, Vol. 10
5971:References
5926:symplectic
5914:Gabor atom
5893:, and the
5885:, such as
5877:(1909) of
5867:See also:
5247:modulation
1118:for any t,
427:Cross-term
353:, or WDF),
349:function (
88:Motivation
27:See also:
6075:1687-6180
5855:(ECG) or
5697:λ
5587:λ
5578:−
5496:λ
5443:λ
5129:ϕ
5094:ϕ
5073:π
4959:τ
4951:τ
4945:π
4936:−
4925:τ
4922:−
4904:τ
4873:−
4866:∫
4840:τ
4832:τ
4826:π
4817:−
4806:τ
4803:−
4788:τ
4763:−
4756:∫
4696:≠
4632:τ
4626:η
4580:∑
4567:τ
4561:η
4463:∑
4396:≠
4347:τ
4344:−
4333:∗
4300:τ
4255:τ
4252:−
4241:∗
4217:τ
4163:τ
3850:η
3824:τ
3818:η
3692:σ
3676:
3660:≈
3583:∈
3564:∬
3547:
3531:≈
3500:σ
3383:η
3377:δ
3371:τ
3365:δ
3362:σ
3350:τ
3344:η
3304:σ
3228:η
3205:η
3199:δ
3193:τ
3152:⋅
3147:η
3141:π
3132:−
3122:∞
3117:∞
3114:−
3110:∫
3103:τ
3062:⋅
3057:η
3051:π
3042:−
3034:⋅
3028:τ
3010:∞
3005:∞
3002:−
2998:∫
2985:τ
2979:η
2831:σ
2811:σ
2767:τ
2759:τ
2753:π
2744:−
2733:τ
2715:∞
2710:∞
2707:−
2703:∫
2654:ξ
2641:ξ
2625:ξ
2612:ξ
2593:ξ
2578:τ
2575:−
2567:∗
2550:ξ
2535:τ
2526:∬
2490:τ
2487:−
2479:∗
2460:τ
2439:τ
2383:τ
2361:τ
2326:τ
2249:η
2243:π
2234:−
2223:τ
2199:∞
2194:∞
2191:−
2187:∫
2148:η
2142:π
2133:−
2111:τ
2108:−
2097:∗
2078:τ
2055:∞
2050:∞
2047:−
2043:∫
2030:τ
2024:η
1921:τ
1915:⋅
1910:τ
1904:π
1895:−
1887:⋅
1881:τ
1857:∞
1852:∞
1849:−
1845:∫
1814:τ
1808:⋅
1803:τ
1797:π
1788:−
1780:⋅
1763:τ
1760:−
1749:∗
1730:τ
1707:∞
1702:∞
1699:−
1695:∫
1625:τ
1617:τ
1611:π
1602:−
1591:τ
1567:∞
1562:∞
1559:−
1555:∫
1451:τ
1412:τ
1398:∧
1356:ξ
1343:ξ
1327:ξ
1314:ξ
1295:ξ
1280:τ
1277:−
1266:∗
1249:ξ
1234:τ
1219:∬
1181:τ
1178:−
1167:∗
1148:τ
1046:τ
1043:−
1032:∗
1013:τ
986:τ
941:τ
833:π
824:
805:≤
788:π
779:
749:π
743:
660:π
651:
632:≤
615:π
606:
576:π
570:
257:π
248:
229:≤
212:π
203:
173:π
167:
100:signals.
98:transient
5939:See also
5891:wavelets
5859:(OAEs).
5206:aliasing
3868:, then
2823:, where
2398:for any
78:wavelets
6043:Bibcode
5863:History
5851:(EEG),
5847:(EMG),
5200:By the
5150:is the
4155:'s and
422:Clarity
6073:
5990:
5901:. The
5484:where
5254:below.
5121:where
4411:, then
4182:then:
891:": -->
454:Worst
448:Worst
5829:Light
2672:PSD,
513:Good
507:Good
495:High
492:Good
486:Good
476:High
473:Best
467:Best
6071:ISSN
6039:2009
5988:ISBN
5621:and
5249:and
5233:The
3791:and
893:edit
850:>
811:<
766:<
677:>
638:<
593:<
470:Yes
457:Low
274:>
235:<
190:<
80:and
6061:hdl
6051:doi
5378:LCT
5376:by
4385:if
3900:If
3720:If
3667:log
3538:log
3246:)
821:cos
776:cos
740:cos
648:cos
603:cos
567:cos
451:No
245:cos
200:cos
164:cos
92:In
61:).
33:In
6114::
6069:.
6059:.
6049:.
6037:.
6033:.
5935:.
4175:'s
4097:'s
3671:10
3663:10
3542:10
3534:10
2950:)
853:20
814:20
802:10
769:10
680:20
641:20
629:10
596:10
376:.
363:).
337:),
277:20
238:20
226:10
193:10
37:,
6077:.
6063::
6053::
6045::
5808:.
5803:]
5797:y
5790:x
5784:[
5777:]
5771:d
5766:c
5759:b
5754:a
5748:[
5733:R
5714:]
5708:1
5700:R
5693:1
5684:0
5679:1
5673:[
5668:=
5663:]
5657:d
5652:c
5645:b
5640:a
5634:[
5604:]
5598:1
5590:f
5583:1
5571:0
5566:1
5560:[
5555:=
5550:]
5544:d
5539:c
5532:b
5527:a
5521:[
5486:z
5469:,
5464:]
5458:1
5453:0
5446:z
5438:1
5432:[
5427:=
5422:]
5416:d
5411:c
5404:b
5399:a
5393:[
5359:]
5353:y
5346:x
5340:[
5308:,
5303:]
5297:y
5290:x
5284:[
5138:)
5135:t
5132:(
5106:,
5103:)
5100:t
5097:(
5088:t
5085:d
5081:d
5070:2
5066:1
5009:0
5006:=
5003:]
5000:)
4997:f
4994:,
4991:t
4988:(
4985:X
4982:[
4979:E
4956:d
4948:f
4942:2
4939:j
4932:e
4928:)
4919:t
4916:(
4913:w
4910:]
4907:)
4901:(
4898:x
4895:[
4892:E
4887:B
4884:+
4881:t
4876:B
4870:t
4862:=
4843:]
4837:d
4829:f
4823:2
4820:j
4813:e
4809:)
4800:t
4797:(
4794:w
4791:)
4785:(
4782:x
4777:B
4774:+
4771:t
4766:B
4760:t
4752:[
4749:E
4746:=
4743:]
4740:)
4737:f
4734:,
4731:t
4728:(
4725:X
4722:[
4719:E
4699:0
4693:]
4690:)
4687:t
4684:(
4681:x
4678:[
4675:E
4638:]
4635:)
4629:,
4623:(
4616:n
4612:x
4607:A
4603:[
4600:E
4595:k
4590:1
4587:=
4584:n
4576:=
4573:]
4570:)
4564:,
4558:(
4553:h
4549:A
4545:[
4542:E
4521:]
4518:)
4515:f
4512:,
4509:t
4506:(
4499:n
4495:x
4490:W
4486:[
4483:E
4478:k
4473:1
4470:=
4467:n
4459:=
4456:]
4453:)
4450:f
4447:,
4444:t
4441:(
4436:h
4432:W
4428:[
4425:E
4399:n
4393:m
4367:0
4364:=
4361:]
4358:)
4355:2
4351:/
4341:t
4338:(
4328:n
4324:x
4320:[
4317:E
4314:]
4311:)
4308:2
4304:/
4297:+
4294:t
4291:(
4286:m
4282:x
4278:[
4275:E
4272:=
4269:]
4266:)
4263:2
4259:/
4249:t
4246:(
4236:n
4232:x
4228:)
4225:2
4221:/
4214:+
4211:t
4208:(
4203:m
4199:x
4195:[
4192:E
4143:t
4123:)
4120:t
4117:(
4112:n
4108:x
4085:t
4065:)
4062:t
4059:(
4054:n
4050:x
4028:)
4025:t
4022:(
4017:k
4013:x
4009:+
4006:.
4003:.
4000:.
3997:.
3994:.
3991:.
3988:+
3985:)
3982:t
3979:(
3974:3
3970:x
3966:+
3963:)
3960:t
3957:(
3952:2
3948:x
3944:+
3941:)
3938:t
3935:(
3930:1
3926:x
3922:=
3919:)
3916:t
3913:(
3910:h
3885:)
3882:t
3879:(
3876:x
3856:0
3853:=
3830:]
3827:)
3821:,
3815:(
3810:x
3806:A
3802:[
3799:E
3779:t
3759:]
3756:)
3753:f
3750:,
3747:t
3744:(
3739:x
3735:W
3731:[
3728:E
3696:A
3686:x
3682:E
3657:R
3654:N
3651:S
3627:f
3624:d
3621:t
3618:d
3615:)
3612:f
3609:,
3606:t
3603:(
3598:x
3594:S
3580:)
3577:f
3574:,
3571:t
3568:(
3557:x
3553:E
3528:R
3525:N
3522:S
3497:=
3494:)
3491:f
3488:(
3483:n
3479:S
3455:A
3431:x
3427:E
3386:)
3380:(
3374:)
3368:(
3359:=
3356:]
3353:)
3347:,
3341:(
3336:x
3332:A
3328:[
3325:E
3301:=
3298:]
3295:)
3292:f
3289:,
3286:t
3283:(
3278:g
3274:W
3270:[
3267:E
3234:0
3231:=
3208:)
3202:(
3196:)
3190:(
3185:x
3181:R
3177:=
3158:t
3155:d
3144:t
3138:2
3135:j
3128:e
3106:)
3100:(
3095:x
3091:R
3087:=
3068:t
3065:d
3054:t
3048:2
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