Group delay and phase delay

386:

amplitude modulation AM passband signal, where their sum exactly reconstructs the original angle-modulation (FM or PM) passband signal. While the (FM/PM) passband signal is not amplitude modulation, and therefore has no apparent outer envelope, the I and Q passband signals do indeed have amplitude modulation envelopes. (However, unlike with regular amplitude modulation, the I and Q envelopes do not resemble the wave shape of the baseband signals, even though 100 percent of the baseband signal is represented in a complex manner by their envelopes.) So, for each of the I and Q passband signals, a flat group delay ensures that neither the I pass band envelope nor the Q passband envelope will have wave shape distortion, so when the I passband signal and the Q passband signal is added back together, the sum is the original FM/PM passband signal, which will also be unaltered.

377:. If the inner red device group delay is completely flat in the frequency range of interest, the outer device will have the ideal of a phase delay that is also completely flat, where the contribution of distortion due to the outer LTI device's phase response—determined entirely by the inner device's possibly different phase response—is eliminated. In that case, the group delay of the inner red device and the phase delay of the outer device give the same time delay figure for the signal as a whole, from the baseband input to the baseband output. It is significant to note that it is possible for the inner (red) device to have a very non-flat phase delay (but flat group delay), while the outer device has the ideal of a perfectly flat phase delay. This is fortunate because in LTI device design, a flat group delay is easier to achieve than a flat phase delay. 6416: 3278: 5271: 3158: 6411:{\displaystyle {\begin{array}{lcl}Se=P(j\omega )_{even}&=&\sum _{k=0}^{N/2}P_{2k}(j\omega )^{2k}&=&\sum _{k=0}^{N/2}P_{2k}(-1)^{k}(\omega )^{2k}\\So=P(j\omega )_{odd}&=&-j\sum _{k=1}^{(N+1)/2}P_{2k-1}(j\omega )^{2k-1}&=&\sum _{k=1}^{(N+1)/2}P_{2k-1}(-1)^{k-1}(\omega )^{2k-1}\\De={\frac {d(P(j\omega )_{even})}{dx}}&=&-j\sum _{k=1}^{N/2}2kP_{2k}(j\omega )^{2k-1}&=&\sum _{k=1}^{N/2}2kP_{2k}(-1)^{k-1}(\omega )^{2k-1}\\Do={\frac {d(P(j\omega )_{odd})}{dx}}&=&\sum _{k=1}^{(N+1)/2}{(2k-1)}P_{2k-1}(j\omega )^{2k-2}&=&\sum _{k=1}^{(N+1)/2}{(2k-1)}P_{2k-1}(-1)^{k-1}(\omega )^{2k-2}\\\\{\frac {df(x)}{dx}}&=&{\frac {SeDo-SoDe}{Se^{2}}}\end{array}}} 5262: 343: 3313:). A filter whose group delay is negative over that signal's entire frequency range is able to use the signal's predictability to provide an illusion of a non-causal time advance. However, if the signal contains an unpredictable event (such as an abrupt change which makes the signal's spectrum exceed its band-limit), then the illusion breaks down. Circuits with negative group delay (e.g., Figure 2) are possible, though 4772: 7355: 2403: 8113: 8830: 7817: 3088: 713: 5257:{\displaystyle {\begin{aligned}&{\frac {dtan^{-1}(f(x))}{dx}}={\frac {df(x)/dx}{1+f(x)^{2}}}\\&f(x)={\frac {-jP(j\omega )_{odd}}{P(j\omega )_{even}}}\\&{\frac {df(x)}{dx}}={\frac {P(j\omega )_{even}{\frac {d(P(j\omega )_{odd})}{dx}}--jP(j\omega )_{odd}{\frac {d(-jP(j\omega )_{even})}{dx}}}{P(j\omega )_{even}^{2}}}\end{aligned}}} 7100: 2145: 7883: 3157: 3423:

Flanagan, Moore and Stone conclude that at 1, 2 and 4 kHz, a group delay of about 1.6 ms is audible with headphones in a non-reverberant condition. Other experimental results suggest that when the group delay in the frequency range from 300 Hz to 1 kHz is below 1.0 ms, it is

385:

In an angle-modulation system—such as with frequency modulation (FM) or phase modulation (PM)—the (FM or PM) passband signal applied to an LTI system input can be analyzed as two separate passband signals, an in-phase (I) amplitude modulation AM passband signal and a quadrature-phase (Q)

369:

creates the passband signal by shifting the baseband frequency components to a much higher frequency range. Although the frequencies are different, the passband signal carries the same information as the baseband signal. The demodulator does the inverse, shifting the passband frequencies back down

357:

system, which is itself an LTI system with a baseband output that is ideally an accurate copy of the baseband signal input. This system as a whole is referred to here as the outer LTI system/device, which contains an inner (red block) LTI system/device. As is often the case for a radio system, the

258:

as a function of frequency, from which group delay and phase delay can be calculated, typically occurs in devices such as microphones, amplifiers, loudspeakers, magnetic recorders, headphones, coaxial cables, and antialiasing filters. All frequency components of a signal are delayed when passed

7532: 3487:

It is possible to use digital signal processing techniques to correct the group delay distortion that arises due to the use of crossover networks in multi-way loudspeaker systems. This involves considerable computational modeling of loudspeaker systems in order to successfully apply delay

7538: 3427:

The waveform of any signal can be reproduced exactly by a system that has a flat frequency response and group delay over the bandwidth of the signal. Leach introduced the concept of differential time-delay distortion, defined as the difference between the phase delay and the group delay:

6629: 7091: 1469: 520: 2870: 2969: 7350:{\displaystyle {\begin{aligned}&Pe_{den}=2.4494897\omega ^{4}-4.6244874\omega ^{2}+1\\&Po_{den}=-3.8282201\omega ^{3}+3.0412127\omega \\&De_{den}=4(2.4494897)\omega ^{3}-2(4.6244874)\omega \\&Do_{den}=3(-3.8282201)\omega ^{2}+3.0412127\end{aligned}}} 306:. If the phase delay function at any given frequency—within a frequency range of interest—has the same constant of proportionality between the phase at a selected frequency and the selected frequency itself, the system/device will have the ideal of 250:

of a time varying physical quantity—for example a voltage signal—appears at the LTI system input, to the time when a copy of that same frequency component—perhaps of a different physical phenomenon—appears at the LTI system output.

2398:{\displaystyle {\begin{aligned}y(t)&=H(i\omega )\ e^{i\omega t}\ \\&=\left({\big |}H(i\omega ){\big |}e^{i\phi (\omega )}\right)\ e^{i\omega t}\ \\&={\big |}H(i\omega ){\big |}\ e^{i\left(\omega t+\phi (\omega )\right)}\ \\\end{aligned}}\ } 8108:{\displaystyle {\begin{aligned}&{\text{Group Delay}}=gd(T_{4}(j\omega ))=-{\frac {d\phi (\omega )}{d\omega }}\\&=-(\phi (1+1e-04)-\phi (1.-1e04))/2e-04\\&=5.1765432{\text{ sec}}\\&{\text{at }}\omega =1{\text{ rad/sec}}\end{aligned}}} 3277: 3087: 8117:

Since the group delay calculated by the digital derivative using a small delta is within 7 digits of accuracy when compared to the precise analytical calculation, the group delay calculation procedure and results are confirmed to be correct.

2525: 1312: 3069: 2747: 329:

The group delay is a convenient measure of the linearity of the phase with respect to frequency in a modulation system. For a modulation signal (passband signal), the information carried by the signal is carried exclusively in the

4644: 4422: 7812:{\displaystyle {\begin{aligned}&{\text{Group Delay}}=gd(T_{4}(j\omega ))=-{\frac {d\phi (\omega )}{d\omega }}\\&={\bigg }\\&=5.1765430{\text{ sec}}\\&{\text{at }}\omega =1{\text{ rad/sec}}\end{aligned}}} 1796: 310:, which results in a constant group delay. Since phase delay is a function of frequency giving time delay, a departure from the flatness of its function graph can reveal time delay differences among the various signal 7384: 2874:

For frequencies significantly lower than the cutoff frequency, the phase response is approximately linear (arctan for small inputs can be approximated as a line), so the group delay simplifies to a constant value of:

2661: 1889: 3891: 6931: 6454: 2776: 370:

to the original baseband frequency range. Ideally, the output (baseband) signal is a time delayed version of the input (baseband) signal where the waveform shape of the output is identical to that of the input.

2880: 6949: 358:

inner red LTI system in Fig 1 can represent two LTI systems in cascade, for example an amplifier driving a transmitting antenna at the sending end and the other an antenna and amplifier at the receiving end.

1153: 708:{\displaystyle y(t)=(h*x)(t)\mathrel {\stackrel {\text{def}}{=}} \int _{-\infty }^{\infty }h(t-\tau )\,x(\tau )\,\mathrm {d} \tau \mathrel {\stackrel {\text{def}}{=}} {\mathcal {L}}^{-1}\{H(s)\,X(s)\}\,,} 1323: 3614: 3340:, introduce group delay in the audio signal. It is therefore important to know the threshold of audibility of group delay with respect to frequency, especially if the audio chain is supposed to provide 3682: 8834: 3478: 1040: 7888: 7543: 7389: 7105: 4777: 2885: 2781: 2150: 2015: 9334: 3751:

states. If the state of polarization of the input signal is the linear state at 45° between the two eigenmodes, the input signal is divided equally into the two eigenmodes. The power of the

6809: 1194: 2591: 1722: 6726: 1954: 1687: 1584: 1530: 4234: 8665: 3265: 3230: 3196: 3127: 1925: 1654: 1613: 1501: 8726: 2037: 1976: 1555: 1216: 8583: 8509: 8459: 8417: 8313: 8170: 2134: 2428: 906: 8239: 2674: 2076: 1071: 972: 937: 864: 833: 798: 767: 508: 477: 439: 4765: 4720: 314:, in which case those differences will contribute to signal distortion, which is manifested as the output signal waveform shape being different from that of the input signal. 8375: 8346: 8268: 8199: 7876: 7853: 4517: 4266: 4151: 4004: 9195: 8615: 8541: 4298: 4183: 4122: 2436: 3709: 8767: 4488: 4090: 7377: 2769: 1224: 222: 4452: 3320:

Negative group delay filters can be made in both digital and analog domains. Applications include compensating for the inherent delay of low-pass filters, to create

298:

property and a phase delay property, where one can be calculated exactly from the other. Phase delay directly measures the device or system time delay of individual

6447: 4067: 2976: 4675: 2598: 736: 8419:

of an ideal linear phase response would be expected to have a value of 0 across the frequency range of interest (such as the pass band of a filter), while the

3554:

It is often desirable for the group delay to be constant across all frequencies; otherwise there is temporal smearing of the signal. Because group delay is

3332:

Group delay has some importance in the audio field and especially in the sound reproduction field. Many components of an audio reproduction chain, notably

3073:

As frequencies get even larger, the group delay decreases with the inverse square of the frequency and approaches zero as frequency approaches infinity.

9325: 8617:

converges on a finite quantity that may not be known ahead of time. Given this, a linear phase optimizing function may more easily be executed with a

4522: 9400: 4303: 7527:{\displaystyle {\begin{aligned}&Pe_{den}=-1.1749977\\&Po_{den}=-0.7870074\\&De_{den}=-0.548984\\&Do_{den}=-8.4434476\end{aligned}}} 1734: 6624:{\displaystyle {\text{Group Delay}}=gd(P(j\omega ))=-{\frac {d\phi (\omega )}{d\omega }}=-{\frac {(So*De+Se*Do)}{(Se^{2}+So^{2})}}{\text{ sec}}} 1819: 9107: 3778: 6816: 1557:). The phase of the sinusoid, as indicated by the positions of the zero crossings, is delayed in time by an amount equal to the phase delay, 7822:

The group delay calculation procedure and results may be confirmed to be correct by comparing them to the results derived from the digital

246:

The group delay and phase delay properties of a linear time-invariant (LTI) system are functions of frequency, giving the time from when a

7086:{\displaystyle T_{4}(j\omega )={\frac {1}{2.4494897(j\omega )^{4}+3.8282201(j\omega )^{3}+4.6244874(j\omega )^{2}+3.0412127(j\omega )+1}}} 6941: 511: 3489: 8461:

of a real-world approximately linear phase response may deviate from 0 by a small finite amount across the frequency range of interest.

9131:

Nakanishi, Toshihiro; Sugiyama, K.; Kitano, M. (2002-01-01). "Demonstration of negative group delays in a simple electronic circuit".

978:

defines the input-output characteristics of the LTI system. This convolution can be evaluated by using the integral expression in the

1464:{\displaystyle y(t)={\big |}H(i\omega ){\big |}\ A_{\text{env}}(t-\tau _{g})\cos {\big (}\omega (t-\tau _{\phi })+\theta {\big )}\;.} 1079: 9490: 282:

and equals the negative of the phase shift at each frequency divided by the value of that frequency. Group delay is the negative

9187: 3557: 322:

While phase delay describes the system's response to steady state sinusoidal components, group delay describes the response to

8839: 3627: 108:, a "pack" or "group" of oscillations centered around one frequency that travel together, formed for instance by multiplying ( 9614: 9333:. 144th Audio Engineering Society International Convention, Paper Number 10008. Audio Engineering Society. pp. 879–888. 8991: 8938: 3968: 2865:{\displaystyle {\begin{aligned}\tau _{g}(\omega )&={\frac {\omega _{o}}{\omega ^{2}+\omega _{o}^{2}}}\,.\\\end{aligned}}} 9056: 2964:{\displaystyle {\begin{aligned}\tau _{g}(\omega \ll \omega _{o})&\approx {\frac {1}{\omega _{o}}}=RC\,.\\\end{aligned}}} 9578:

McClellan, J.; Parks, T.; Rabiner, L. (1973). "A computer program for designing optimum FIR linear phase digital filters".

1317:

Applying the earlier convolution equation would reveal that the output of such an LTI system is very well approximated as:

9266:. ICASSP '80. IEEE International Conference on Acoustics, Speech, and Signal Processing. Vol. 5. pp. 1030–1033. 3711:

is a constant). The degree of nonlinearity of the phase indicates the deviation of the group delay from a constant value.

3434: 1001: 9649: 1985: 9297: 6735: 147:

in a digital bit stream. However, for the ideal case of a constant group delay across the entire frequency range of a

9041: 9016: 8963: 8880: 4014:. Hence, any signal's propagation delay through the line simply equals the length of the line divided by this speed. 1161: 3539:

per unit length, which is the reciprocal of the group velocity of a particular mode. The measured group delay of a

2549: 1728:

at each frequency equals the negative of the phase shift at that frequency divided by the value of that frequency:

1696: 123:

dependent, which means that different frequency components experience different delays. As a result, the signal's

9747: 8845: 6641: 1930: 1663: 1560: 1506: 350:

A device's group delay can be exactly calculated from the device's phase response, but not the other way around.

4188: 9476:"The Differential Time-Delay Distortion and Differential Phase-Shift Distortion as Measures of Phase Linearity" 9085: 8620: 3957: 3241: 3206: 3172: 3103: 1901: 1630: 1589: 1477: 8670: 3303:

within some maximum frequency B, then it is predictable to a small degree (within time periods smaller than

2020: 1959: 1538: 1199: 227: 173: 51: 9630: 8546: 8472: 8422: 8380: 8276: 8133: 2088: 9718:"Introduction to Digital Filters with Audio Applications", Julius O. Smith III, (September 2007 Edition). 9431: 9133: 7095:

The numerator group delay by inspection is zero, so only the denominator group delay need be determined.

2411: 877: 17: 8208: 2050: 1045: 946: 911: 838: 807: 772: 741: 482: 451: 413: 9742: 4725: 4680: 871: 375:

For amplitude modulation, the inner red LTI device group delay becomes the outer LTI device phase delay

303: 93: 1532:

is the phase delay, and they are given by the expressions below (and potentially are functions of the

9670: 8778: 403: 9100: 8351: 8322: 8244: 8175: 7858: 7829: 4493: 4242: 4127: 3974: 8316: 3719: 3324:

filters, which can be used to quickly detect changes in the trends of sensor data or stock prices.

2520:{\displaystyle \phi (\omega )\ {\stackrel {\mathrm {def} }{=}}\ \arg \left\{H(i\omega )\right\}\;.} 407: 8172:, sometimes referred to as just, "phase deviation", is the difference between the phase response, 9717: 8588: 8514: 4654:

refer to the polynomials that contain only the even or odd order coefficients respectively. The

4271: 4156: 4095: 235: 3299:

group delay over frequency ranges where its phase response is positively-sloped. If a signal is

1307:{\displaystyle \left|{\frac {d}{dt}}\log {\big (}A_{\text{env}}(t){\big )}\right|\ll \omega \ .} 3687: 334:. Group delay therefore operates only with the frequency components derived from the envelope. 9450:"Discrimination of Group Delay in Clicklike Signals Presented via Headphones and Loudspeakers" 8808:— "The group velocity of light in a medium is the inverse of the group delay per unit length." 8731: 4457: 4072: 1978:) and equal, and their common value equals the overall delay of the system; and the unwrapped 7362: 4023: 3344:

reproduction. The best thresholds of audibility table has been provided by Blauert and Laws.

3064:{\displaystyle \tau _{g}(\omega {=}\omega _{o})={\frac {1}{2\omega _{o}}}={\frac {RC}{2}}\,.} 2754: 195: 71: 4427: 9415: 9225: 9152: 9068: 3737: 2742:{\displaystyle \phi (\omega )={\frac {\pi }{2}}-\arctan({\frac {\omega }{\omega _{o}}})\,.} 366: 259:

through such devices, or when propagating through space or a medium, such as air or water.

109: 9475: 6423: 4043: 353:

The simplest use case for group delay is illustrated in Figure 1 which shows a conceptual

8: 9655: 9216:

Plomp, R.; Steeneken, H. J. M. (1969). "Effect of Phase on the Timbre of Complex Tones".

9057:

https://www.tedpavlic.com/teaching/osu/ece209/lab3_opamp_FO/lab3_opamp_FO_phase_shift.pdf

6449:

in closed form, shown below after the equations have been reduced to a simplified form.

3548: 3532: 3166: 323: 189: 9419: 9229: 9156: 8901:"Phase Distortion and Phase Equalization in Audio Signal Processing — A Tutorial Review" 4657: 9554: 9381: 9168: 9142: 3964: 3283: 3093: 721: 3960:

with virtually no signal distortion such as pulse broadening during pulsed operation.

3484:

An ideal system should exhibit zero or negligible differential time-delay distortion.

9610: 9385: 9241: 9079: 9037: 9012: 8987: 8959: 8934: 8876: 5276: 4038: 4027: 3617: 3536: 1533: 940: 801: 395: 31: 9558: 9172: 9587: 9544: 9423: 9371: 9267: 9233: 9160: 8849: 2666: 2544: 1690: 446: 178: 169: 97: 9732: 9712: 9532: 9508: 9449: 8794: 6732:, may have the group delay determined by taking advantage of the phase relation, 3748: 3524: 3520: 3337: 2536: 131:

as it passes through the system. This distortion can cause problems such as poor

113: 9690: 9510:

Correction of Crossover Phase Distortion Using Reversed Time All-Pass IIR Filter

9271: 8900: 373:

In Figure 1, the outer system phase delay is the meaningful performance metric.

151:

signal and flat frequency response, the waveform will experience no distortion.

9737: 9376: 9359: 8811: 8805: 4639:{\displaystyle \phi (\omega )=tan^{-1}(-jP(j\omega )_{odd}/P(j\omega )_{even})} 3723: 3528: 983: 399: 295: 271: 255: 144: 79: 75: 4417:{\displaystyle \phi (\omega )=tan^{-1}(P(j\omega )_{imag}/P(j\omega )_{real})} 3963:

TTD is an important characteristic of lossless and low-loss, dispersion free,

2047:

More generally, it can be shown that for an LTI system with transfer function

9726: 9591: 8788: 6729: 3512: 3341: 2771:

for either this low-pass or high-pass filter yields the same group delay of:

1979: 342: 331: 279: 267: 132: 83: 59: 9549: 9286: 8348:

sampled over a frequency range of interest, and subtract it from the actual

1791:{\displaystyle \tau _{\phi }(\omega )=-{\frac {\phi (\omega )}{\omega }}\,.} 982:, or (according to the rightmost expression) by using multiplication in the 8783: 8127: 6420:

The equations above may be used to determine the group delay of polynomial

4031: 3333: 3300: 1895: 307: 140: 136: 9245: 3616:, it therefore follows that a constant group delay can be achieved if the 2656:{\displaystyle \phi (\omega )=-\arctan({\frac {\omega }{\omega _{o}}})\,.} 1884:{\displaystyle \tau _{g}(\omega )=-{\frac {d\phi (\omega )}{d\omega }}\,.} 9147: 8816: 8800: 3905:

is defined as the difference in propagation time between the eigenmodes:

3886:{\displaystyle E_{T}=(E_{i,x}\cdot t_{x})^{2}+(E_{i,y}\cdot t_{y})^{2}\,} 3752: 1618: 979: 442: 263: 148: 104: 67: 9707: 9287:"Loudspeaker phase measurements, transient response and audible quality" 6926:{\displaystyle {\text{Group Delay}}=gd(P2)=gd(P2_{num})-gd(P2_{den})sec} 7823: 4035: 3741: 3544: 2540: 1810: 354: 283: 186: 128: 55: 9237: 9164: 1586:. The envelope of the sinusoid is delayed in time by the group delay, 9427: 8202: 3956:

of the electrical signal. TTD allows for a wide instantaneous signal

3953: 3745: 3727: 3314: 1689:, are (potentially) frequency-dependent and can be computed from the 995: 231: 165: 120: 63: 3286:

simulation of an input (green) wave whose output (red) is ahead by

124: 87: 994:

Suppose that such a system is driven by a wave packet formed by a

986:

and then applying the inverse transform to return to time domain.

512:

Linear time-invariant system § Fourier and Laplace transforms

9188:"Five things to know about prediction and negative-delay filters" 4017: 3501: 3163: 2017:) is negative, with magnitude increasing linearly with frequency 275: 230:

process each sinusoidal component independently; the property of

182: 9448:

Flanagan, Sheila; Moore, Brian C. J.; Stone, Michael A. (2005).

2530: 9364:

IEEE/ACM Transactions on Audio, Speech, and Language Processing

3621: 3540: 3505: 2079: 1148:{\displaystyle x(t)=A_{\text{env}}(t)\cos(\omega t+\theta )\,.} 224:

with a fixed amplitude and phase and no beginning and no end.

8241:, and is a useful measurement to determine the linearity of 4185:, and then the group delay may be determined by solving for 9634: 9032:

Oppenheim, Alan V.; Willsky, Alan S.; Nawab, Hamid (1997).

8585:

always converges to 0 as the phase becomes linear, whereas

3609:{\textstyle \tau _{g}(\omega )=-{\frac {d\phi }{d\omega }}} 3516: 1813:

with respect to frequency) of the phase at that frequency:

47: 9264:

Group and phase delay requirements for loudspeaker systems

9124: 6940:

A four pole Legendre filter transfer function used in the

3971:

reveals that signals propagate through them at a speed of

2042: 9571: 27:

Delays experienced through a linear time-invariant system

9025: 8954:

Oppenheim, Alan V.; Schafer, R. W.; Buck, J. R. (1999).

3677:{\displaystyle \phi (\omega )=\phi (0)-\tau _{g}\omega } 3535:

measurement purposes, the quantity of interest is group

1196:

is slowly changing relative to the sinusoid's frequency

9668: 9130: 8947: 5266:

The above expressions contain four terms to calculate:

3290:, but with instability when the input turns on and off. 9530: 9441: 9401:"Group Delay Distortions in Electroacoustical Systems" 9357: 9323: 9036:. Upper Saddle River, New Jersey: Prentice-Hall, Inc. 8977: 8975: 8958:. Upper Saddle River, New Jersey: Prentice-Hall, Inc. 3764:

is the combination of the transmitted signals of both

3560: 3490:

Parks-McClellan FIR equiripple filter design algorithm

1619:

Mathematical definition of group delay and phase delay

9577: 9533:"Modeling and Delay-Equalizing Loudspeaker Responses" 9327:

Audibility of loudspeaker group-delay characteristics

9031: 8734: 8673: 8623: 8591: 8549: 8517: 8475: 8425: 8383: 8354: 8325: 8279: 8247: 8211: 8178: 8136: 7886: 7861: 7832: 7541: 7387: 7365: 7103: 6952: 6819: 6738: 6644: 6457: 6426: 5274: 4775: 4728: 4683: 4660: 4525: 4496: 4460: 4430: 4306: 4274: 4245: 4191: 4159: 4130: 4098: 4075: 4046: 3977: 3781: 3690: 3630: 3437: 3245: 3244: 3210: 3209: 3176: 3175: 3107: 3106: 2979: 2883: 2779: 2757: 2677: 2601: 2552: 2439: 2415: 2414: 2148: 2091: 2054: 2053: 2024: 2023: 1989: 1988: 1963: 1962: 1934: 1933: 1905: 1904: 1822: 1737: 1700: 1699: 1667: 1666: 1634: 1633: 1593: 1592: 1564: 1563: 1542: 1541: 1510: 1509: 1481: 1480: 1326: 1227: 1218:. This condition can be expressed mathematically as: 1203: 1202: 1165: 1164: 1082: 1049: 1048: 1005: 1004: 989: 950: 949: 915: 914: 880: 842: 841: 811: 810: 776: 775: 745: 744: 724: 523: 486: 485: 455: 454: 417: 416: 311: 247: 198: 185:

in time can alternatively be expressed as the sum of

9648:

Julius O. Smith III. "Phase Delay and Group Delay".

9598: 9531:

Mäkivirta, Aki; Liski, Juho; Välimäki, Vesa (2018).

9358:

Liski, Juho; Mäkivirta, Aki; Välimäki, Vesa (2021).

8953: 8875:. Englewood Cliffs, New Jersey: Prentice-Hall, Inc. 6728:, such as that typically found in the definition of 4034:

above may be solved analytically in closed form. A

3969:

Telegrapher's equations § Lossless transmission

3473:{\displaystyle \Delta \tau =\tau _{\phi }-\tau _{g}} 1035:{\displaystyle \displaystyle A_{\text{env}}(t)>0} 8972: 8873:

Theory and Application of Digital Signal Processing

2010:{\displaystyle \displaystyle -\omega \tau _{\phi }} 159: 9447: 9209: 8761: 8720: 8659: 8609: 8577: 8535: 8503: 8453: 8411: 8369: 8340: 8307: 8262: 8233: 8193: 8164: 8107: 7870: 7847: 7811: 7526: 7371: 7349: 7085: 6925: 6803: 6720: 6623: 6441: 6410: 5256: 4759: 4714: 4669: 4638: 4511: 4482: 4446: 4416: 4292: 4260: 4228: 4177: 4145: 4116: 4084: 4061: 3998: 3885: 3703: 3676: 3608: 3472: 3259: 3224: 3190: 3121: 3063: 2963: 2864: 2763: 2741: 2655: 2585: 2519: 2422: 2397: 2128: 2070: 2031: 2009: 1970: 1948: 1919: 1883: 1790: 1716: 1681: 1648: 1607: 1578: 1549: 1524: 1495: 1463: 1306: 1210: 1188: 1147: 1065: 1034: 966: 931: 900: 858: 827: 792: 761: 730: 707: 502: 471: 433: 216: 42:are two related ways of describing how a signal's 8933:(Second ed.). Oxford University Press, Inc. 8866: 8864: 8862: 8791:— low pass filter with maximally-flat group delay 7758: 7636: 943:of the LTI system and, like the impulse response 9724: 9604: 9218:The Journal of the Acoustical Society of America 8981: 6804:{\displaystyle \phi (P1/P2)=\phi (P1)-\phi (P2)} 3531:, to travel a given distance. For optical fiber 9647: 9605:Oppenheim, Alan V.; Schafer, Ronald W. (2010). 9580:IEEE Transactions on Audio and Electroacoustics 9392: 9324:Liski, J.; Mäkivirta, A.; Välimäki, V. (2018). 8982:Oppenheim, Alan V.; Schafer, Ronald W. (2014). 4677:in the numerator merely converts the imaginary 2751:Taking the negative derivative with respect to 1189:{\displaystyle \displaystyle A_{\text{env}}(t)} 294:A linear time-invariant system or device has a 96:). Group delay describes the time shift of the 9215: 8859: 8121: 4018:Group delay from transfer function polynomials 3952:(TTD) if the time delay is independent of the 9623: 9524: 9257: 9255: 9106:. Physical Meaning of Negative Group Delay?. 8464: 3082:Figure 2: Negative group delay filter circuit 2586:{\displaystyle \omega _{o}{=}{\frac {1}{RC}}} 2531:1st order low- or high-pass RC filter example 2336: 2314: 2248: 2226: 1805:at each frequency equals the negative of the 1452: 1414: 1366: 1344: 1282: 1256: 86:). Phase delay describes the time shift of a 9513:. 122nd Audio Engineering Society Convention 9469: 9467: 9408:Journal of the Acoustical Society of America 9319: 9317: 9185: 8871:Rabiner, Lawrence R.; Gold, Bernard (1975). 1717:{\displaystyle \displaystyle \phi (\omega )} 698: 670: 9398: 9179: 9000: 8894: 8892: 8870: 8511:over measuring or calculating group delay, 6721:{\displaystyle P2(S)=P_{num}(S)/P_{den}(S)} 1949:{\displaystyle \displaystyle \tau _{\phi }} 1682:{\displaystyle \displaystyle \tau _{\phi }} 1579:{\displaystyle \displaystyle \tau _{\phi }} 1525:{\displaystyle \displaystyle \tau _{\phi }} 192:, each based on the trigonometric function 9473: 9278: 9261: 9252: 4229:{\displaystyle -d\phi (\omega )|/d\omega } 2973:Similarly, right at the cutoff frequency, 2513: 1457: 337: 286:of phase shift with respect to frequency. 9713:Group Delay Explanations and Applications 9708:Discussion of Group Delay in Loudspeakers 9548: 9507:Adam, Veronique; Benz, Sebastien (2007). 9464: 9375: 9351: 9314: 9146: 8469:An advantage of measuring or calculating 3948:A transmitting apparatus is said to have 3882: 3057: 2953: 2854: 2735: 2649: 1877: 1784: 1141: 701: 685: 628: 615: 43: 9662: 9654:. Department of Electrical Engineering, 9537:Journal of the Audio Engineering Society 9506: 9500: 9483:Journal of the Audio Engineering Society 9454:Journal of the Audio Engineering Society 9360:"Audibility of Group-Delay Equalization" 9006: 8922: 8905:Journal of the Audio Engineering Society 8889: 8660:{\displaystyle \phi _{DLP}(\omega ){=}0} 6935: 3495: 2665:Similarly, the phase for a 1st-order RC 1073:can be expressed in the following form: 479:of the LTI system with the input signal 341: 112:) a sine wave by an envelope (such as a 9098: 9092: 4032:mathematical definition for group delay 3327: 3260:{\displaystyle \displaystyle \tau _{g}} 3225:{\displaystyle \displaystyle \tau _{g}} 3191:{\displaystyle \displaystyle \tau _{g}} 3122:{\displaystyle \displaystyle \tau _{g}} 3076: 2043:LTI system response to complex sinusoid 1920:{\displaystyle \displaystyle \tau _{g}} 1898:system (with non-inverting gain), both 1649:{\displaystyle \displaystyle \tau _{g}} 1608:{\displaystyle \displaystyle \tau _{g}} 1496:{\displaystyle \displaystyle \tau _{g}} 361: 14: 9725: 9609:. England: Pearson Education Limited. 9284: 8986:. England: Pearson Education Limited. 8721:{\displaystyle gd(\omega ){=}constant} 441:of an LTI system can be determined by 9641: 9061: 9011:(Second ed.). Cengage Learning. 8928: 8898: 2032:{\displaystyle \displaystyle \omega } 1971:{\displaystyle \displaystyle \omega } 1550:{\displaystyle \displaystyle \omega } 1211:{\displaystyle \displaystyle \omega } 346:Figure 1: Outer and Inner LTI Devices 9186:Castor-Perry, Kendall (2020-03-18). 9009:Analog and Digital Signal Processing 8578:{\displaystyle \phi _{DLP}(\omega )} 8504:{\displaystyle \phi _{DLP}(\omega )} 8454:{\displaystyle \phi _{DLP}(\omega )} 8412:{\displaystyle \phi _{DLP}(\omega )} 8308:{\displaystyle \phi _{DLP}(\omega )} 8165:{\displaystyle \phi _{DLP}(\omega )} 3543:through an optical fiber exhibits a 2129:{\displaystyle x(t)=e^{i\omega t}\ } 998:multiplied by an amplitude envelope 6633: 4022:If a transfer function or Sij of a 2423:{\displaystyle \displaystyle \phi } 1956:are constant (i.e., independent of 901:{\displaystyle {\mathcal {L}}^{-1}} 738:denotes the convolution operation, 380: 24: 9399:Blauert, J.; Laws, P. (May 1978). 8769:is not necessarily already known. 8234:{\displaystyle \phi _{L}(\omega )} 7862: 3943: 3438: 2471: 2468: 2465: 2071:{\displaystyle \displaystyle H(s)} 1066:{\displaystyle \displaystyle x(t)} 990:LTI system response to wave packet 967:{\displaystyle \displaystyle h(t)} 932:{\displaystyle \displaystyle H(s)} 908:is the inverse Laplace transform. 884: 859:{\displaystyle \displaystyle h(t)} 828:{\displaystyle \displaystyle x(t)} 793:{\displaystyle \displaystyle H(s)} 762:{\displaystyle \displaystyle X(s)} 656: 630: 592: 587: 503:{\displaystyle \displaystyle x(t)} 472:{\displaystyle \displaystyle h(t)} 434:{\displaystyle \displaystyle y(t)} 52:linear time-invariant (LTI) system 25: 9759: 9701: 4760:{\displaystyle P(j\omega )_{odd}} 4722:numerator to a real value, since 4715:{\displaystyle P(j\omega )_{odd}} 3551:mechanisms present in the fiber. 262:While a phase response describes 9496:from the original on 2022-10-09. 9474:Leach, Jr., W. Marshall (1989). 8833: This article incorporates 8828: 3276: 3156: 3137:for frequencies much lower than 3086: 514:expresses this relationship as: 160:Frequency components of a signal 9683: 9607:Discrete-Time Signal Processing 9340:from the original on 2022-10-09 9303:from the original on 2022-10-09 9198:from the original on 2022-06-28 9113:from the original on 2021-10-16 9069:"EELE503: Modern filter design" 8984:Discrete-Time Signal Processing 8956:Discrete-Time Signal Processing 8846:General Services Administration 6638:A polynomial ratio of the form 4767:by itself is purely imaginary. 1158:Also suppose that the envelope 241: 9050: 8686: 8680: 8646: 8640: 8604: 8598: 8572: 8566: 8530: 8524: 8498: 8492: 8448: 8442: 8406: 8400: 8370:{\displaystyle \phi (\omega )} 8364: 8358: 8341:{\displaystyle \phi (\omega )} 8335: 8329: 8302: 8296: 8273:A convenient means to measure 8263:{\displaystyle \phi (\omega )} 8257: 8251: 8228: 8222: 8205:portion of the phase response 8194:{\displaystyle \phi (\omega )} 8188: 8182: 8159: 8153: 8039: 8036: 8018: 8009: 7988: 7982: 7955: 7949: 7931: 7928: 7919: 7906: 7871:{\displaystyle \Delta \omega } 7848:{\displaystyle \phi (\omega )} 7842: 7836: 7750: 7741: 7731: 7719: 7709: 7706: 7701: 7698: 7680: 7674: 7656: 7653: 7610: 7604: 7586: 7583: 7574: 7561: 7324: 7315: 7279: 7273: 7254: 7248: 7071: 7062: 7047: 7037: 7022: 7012: 6997: 6987: 6972: 6963: 6911: 6889: 6877: 6855: 6843: 6834: 6798: 6789: 6780: 6771: 6762: 6742: 6715: 6709: 6685: 6679: 6657: 6651: 6610: 6578: 6573: 6537: 6514: 6508: 6490: 6487: 6478: 6472: 6436: 6430: 6335: 6329: 6295: 6288: 6273: 6263: 6240: 6225: 6211: 6199: 6158: 6148: 6125: 6110: 6096: 6084: 6047: 6032: 6022: 6016: 5979: 5972: 5957: 5947: 5874: 5864: 5789: 5771: 5761: 5755: 5718: 5711: 5696: 5686: 5654: 5642: 5601: 5591: 5559: 5547: 5503: 5493: 5465: 5458: 5449: 5439: 5378: 5368: 5301: 5291: 5224: 5214: 5195: 5177: 5167: 5155: 5134: 5124: 5098: 5083: 5073: 5067: 5043: 5033: 5010: 5004: 4969: 4959: 4939: 4929: 4911: 4905: 4885: 4878: 4853: 4847: 4821: 4818: 4812: 4806: 4742: 4732: 4697: 4687: 4633: 4615: 4605: 4582: 4572: 4560: 4535: 4529: 4512:{\displaystyle \phi (\omega )} 4506: 4500: 4411: 4393: 4383: 4357: 4347: 4341: 4316: 4310: 4287: 4278: 4261:{\displaystyle \phi (\omega )} 4255: 4249: 4211: 4207: 4201: 4172: 4163: 4146:{\displaystyle \phi (\omega )} 4140: 4134: 4111: 4102: 4056: 4050: 3999:{\displaystyle 1/{\sqrt {LC}}} 3873: 3840: 3828: 3795: 3655: 3649: 3640: 3634: 3620:of the device or medium has a 3577: 3571: 3547:dependence due to the various 3011: 2990: 2917: 2898: 2800: 2794: 2732: 2712: 2687: 2681: 2646: 2626: 2611: 2605: 2505: 2496: 2449: 2443: 2375: 2369: 2331: 2322: 2270: 2264: 2243: 2234: 2184: 2175: 2162: 2156: 2101: 2095: 2064: 2058: 1863: 1857: 1839: 1833: 1775: 1769: 1754: 1748: 1710: 1704: 1441: 1422: 1403: 1384: 1361: 1352: 1336: 1330: 1277: 1271: 1182: 1176: 1138: 1123: 1114: 1108: 1092: 1086: 1059: 1053: 1022: 1016: 960: 954: 925: 919: 852: 846: 821: 815: 786: 780: 755: 749: 695: 689: 682: 676: 625: 619: 612: 600: 560: 554: 551: 539: 533: 527: 496: 490: 465: 459: 427: 421: 317: 289: 211: 205: 13: 1: 9671:"Deviation from Linear Phase" 8822: 4006:for a distributed inductance 3896:The differential group delay 3515:, group delay is the transit 228:Linear time-invariant systems 154: 9669:Keysight Technologies, Inc. 3500:Group delay is important in 302:frequency components in the 174:Linear time-invariant system 7: 9651:Music 320 Background Reader 9296:(Application Note 17-198). 9272:10.1109/ICASSP.1980.1170852 9134:American Journal of Physics 8772: 8610:{\displaystyle gd(\omega )} 8536:{\displaystyle gd(\omega )} 8122:Deviation from Linear Phase 4293:{\displaystyle P(j\omega )} 4178:{\displaystyle P(j\omega )} 4117:{\displaystyle P(j\omega )} 10: 9764: 9377:10.1109/TASLP.2021.3087969 9084:: CS1 maint: url-status ( 8931:Linear Systems and Signals 8797:— from the example section 8465:Advantage over group delay 163: 92:component (a sine wave in 8779:Audio system measurements 3704:{\displaystyle \tau _{g}} 3336:and multiway loudspeaker 2535:The phase of a 1st-order 389: 278:), the phase delay is in 119:These delays are usually 9592:10.1109/TAU.1973.1162525 9285:Möller, Henning (1975). 9007:Ambardar, Ashok (1999). 8762:{\displaystyle constant} 8728:goal when the value for 8317:simple linear regression 4483:{\displaystyle j^{2N+1}} 4300:, use the definition of 4085:{\displaystyle j\omega } 3744:that are the 0° and 90° 3716:differential group delay 3488:equalization, using the 408:analog signal processing 9550:10.17743/jaes.2018.0053 9099:Bariska, Andor (2008). 7372:{\displaystyle \omega } 6942:Legendre filter example 4153:may be determined from 4069:may be taken along the 3523:, traveling at a given 3504:, and in particular in 2764:{\displaystyle \omega } 1503:is the group delay and 338:Basic modulation system 236:superposition principle 234:means they satisfy the 217:{\displaystyle \sin(x)} 50:when passing through a 9748:Electrical engineering 9101:"Negative Group Delay" 8841:Federal Standard 1037C 8835:public domain material 8763: 8722: 8661: 8611: 8579: 8537: 8505: 8455: 8413: 8371: 8342: 8309: 8264: 8235: 8195: 8166: 8109: 7878:of +/-1.e-04 rad/sec. 7872: 7855:, using a small delta 7849: 7813: 7528: 7373: 7351: 7087: 6927: 6805: 6722: 6625: 6443: 6412: 6223: 6108: 5927: 5844: 5666: 5571: 5425: 5354: 5258: 4761: 4716: 4671: 4640: 4513: 4484: 4448: 4447:{\displaystyle j^{2N}} 4418: 4294: 4262: 4230: 4179: 4147: 4118: 4086: 4063: 4000: 3887: 3705: 3684:where the group delay 3678: 3624:phase response (i.e., 3610: 3474: 3261: 3226: 3192: 3123: 3065: 2965: 2866: 2765: 2743: 2657: 2587: 2521: 2424: 2408:where the phase shift 2399: 2130: 2072: 2033: 2011: 1982:of the system (namely 1972: 1950: 1921: 1885: 1792: 1718: 1683: 1650: 1609: 1580: 1551: 1526: 1497: 1465: 1308: 1212: 1190: 1149: 1067: 1036: 968: 933: 902: 860: 829: 794: 763: 732: 709: 504: 473: 435: 347: 218: 8929:Lathi, B. P. (2005). 8764: 8723: 8662: 8612: 8580: 8538: 8506: 8456: 8414: 8372: 8343: 8310: 8265: 8236: 8196: 8167: 8110: 7873: 7850: 7814: 7529: 7374: 7352: 7088: 6936:Simple filter example 6928: 6806: 6723: 6626: 6444: 6413: 6183: 6068: 5899: 5816: 5626: 5531: 5397: 5326: 5259: 4762: 4717: 4672: 4641: 4514: 4490:is always imaginary, 4485: 4449: 4419: 4295: 4263: 4231: 4180: 4148: 4119: 4087: 4064: 4026:, is in a polynomial 4001: 3888: 3706: 3679: 3611: 3519:required for optical 3496:Group delay in optics 3475: 3262: 3227: 3193: 3124: 3066: 2966: 2867: 2766: 2744: 2658: 2588: 2522: 2425: 2400: 2131: 2073: 2034: 2012: 1973: 1951: 1922: 1886: 1793: 1719: 1684: 1651: 1610: 1581: 1552: 1527: 1498: 1466: 1309: 1213: 1191: 1150: 1068: 1037: 969: 934: 903: 861: 835:and impulse response 830: 795: 764: 733: 710: 505: 474: 436: 410:), the output signal 345: 219: 72:communications system 8732: 8671: 8621: 8589: 8547: 8515: 8473: 8423: 8381: 8352: 8323: 8277: 8245: 8209: 8176: 8134: 7884: 7859: 7830: 7826:of the phase angle, 7539: 7385: 7363: 7101: 6950: 6817: 6736: 6642: 6455: 6442:{\displaystyle P(S)} 6424: 5272: 4773: 4726: 4681: 4658: 4523: 4519:may be redefined as 4494: 4458: 4454:is always real, and 4428: 4304: 4272: 4243: 4189: 4157: 4128: 4096: 4092:axis and defined as 4073: 4062:{\displaystyle P(S)} 4044: 4024:scattering parameter 3975: 3779: 3688: 3628: 3558: 3435: 3328:Group delay in audio 3242: 3207: 3173: 3104: 3077:Negative group delay 2977: 2881: 2777: 2755: 2675: 2599: 2550: 2437: 2412: 2146: 2089: 2051: 2021: 1986: 1960: 1931: 1902: 1820: 1735: 1697: 1664: 1631: 1590: 1561: 1539: 1507: 1478: 1324: 1225: 1200: 1162: 1080: 1046: 1002: 947: 912: 878: 839: 808: 773: 742: 722: 521: 483: 452: 414: 367:Amplitude modulation 362:Amplitude Modulation 312:frequency components 196: 190:frequency components 110:amplitude modulation 44:frequency components 9656:Stanford University 9420:1978ASAJ...63.1478B 9262:Ashley, J. (1980). 9230:1969ASAJ...46..409P 9157:2002AmJPh..70.1117N 9074:. 2011. p. 11. 9034:Signals and Systems 5246: 3927: − 2850: 2082:of unit amplitude, 596: 324:amplitude modulated 248:frequency component 8899:Preis, D. (1982). 8759: 8718: 8657: 8607: 8575: 8533: 8501: 8451: 8409: 8367: 8338: 8305: 8260: 8231: 8191: 8162: 8105: 8103: 7868: 7845: 7809: 7807: 7524: 7522: 7369: 7347: 7345: 7083: 6923: 6801: 6718: 6621: 6439: 6408: 6406: 5254: 5252: 5223: 4757: 4712: 4670:{\displaystyle -j} 4667: 4636: 4509: 4480: 4444: 4414: 4290: 4258: 4226: 4175: 4143: 4114: 4082: 4059: 3996: 3965:transmission lines 3883: 3753:transmitted signal 3701: 3674: 3606: 3470: 3338:crossover networks 3295:Filters will have 3257: 3256: 3222: 3221: 3188: 3187: 3119: 3118: 3061: 2961: 2959: 2862: 2860: 2836: 2761: 2739: 2653: 2583: 2517: 2420: 2419: 2395: 2390: 2126: 2068: 2067: 2029: 2028: 2007: 2006: 1968: 1967: 1946: 1945: 1917: 1916: 1881: 1788: 1714: 1713: 1679: 1678: 1646: 1645: 1605: 1604: 1576: 1575: 1547: 1546: 1522: 1521: 1493: 1492: 1461: 1304: 1208: 1207: 1186: 1185: 1145: 1063: 1062: 1032: 1031: 964: 963: 929: 928: 898: 856: 855: 825: 824: 802:Laplace transforms 790: 789: 759: 758: 728: 705: 579: 500: 499: 469: 468: 431: 430: 348: 214: 9743:Signal processing 9633:. Microwaves101, 9631:"True Time Delay" 9616:978-0-13-198842-2 9294:Brüel & Kjaer 9238:10.1121/1.1911705 9165:10.1119/1.1503378 9141:(11): 1117–1121. 8993:978-1-292-02572-8 8940:978-0-19-515833-5 8667:goal than with a 8099: 8085: 8075: 7967: 7895: 7803: 7789: 7779: 7754: 7622: 7550: 7081: 6823: 6619: 6614: 6526: 6461: 6402: 6347: 6059: 5801: 5248: 5207: 5110: 5022: 4988: 4895: 4833: 4039:transfer function 4028:Laplace transform 3994: 3618:transfer function 3604: 3421: 3420: 3317:is not violated. 3055: 3037: 2942: 2852: 2730: 2701: 2644: 2581: 2481: 2476: 2454: 2394: 2387: 2343: 2301: 2282: 2208: 2189: 2125: 1875: 1782: 1534:angular frequency 1381: 1373: 1300: 1268: 1246: 1173: 1105: 1013: 941:transfer function 872:complex frequency 731:{\displaystyle *} 650: 648: 576: 574: 396:LTI system theory 114:tapering function 32:signal processing 16:(Redirected from 9755: 9695: 9694: 9687: 9681: 9680: 9678: 9677: 9666: 9660: 9659: 9645: 9639: 9638: 9627: 9621: 9620: 9602: 9596: 9595: 9575: 9569: 9568: 9566: 9565: 9552: 9528: 9522: 9521: 9519: 9518: 9504: 9498: 9497: 9495: 9480: 9471: 9462: 9461: 9445: 9439: 9438: 9436: 9430:. Archived from 9428:10.1121/1.381841 9414:(5): 1478–1483. 9405: 9396: 9390: 9389: 9379: 9355: 9349: 9348: 9346: 9345: 9339: 9332: 9321: 9312: 9311: 9309: 9308: 9302: 9291: 9282: 9276: 9275: 9259: 9250: 9249: 9213: 9207: 9206: 9204: 9203: 9192:planetanalog.com 9183: 9177: 9176: 9150: 9148:quant-ph/0201001 9128: 9122: 9121: 9119: 9118: 9112: 9105: 9096: 9090: 9089: 9083: 9075: 9073: 9065: 9059: 9054: 9048: 9047: 9029: 9023: 9022: 9004: 8998: 8997: 8979: 8970: 8969: 8951: 8945: 8944: 8926: 8920: 8919: 8917: 8916: 8896: 8887: 8886: 8868: 8854: 8853: 8848:. Archived from 8832: 8831: 8768: 8766: 8765: 8760: 8727: 8725: 8724: 8719: 8693: 8666: 8664: 8663: 8658: 8653: 8639: 8638: 8616: 8614: 8613: 8608: 8584: 8582: 8581: 8576: 8565: 8564: 8542: 8540: 8539: 8534: 8510: 8508: 8507: 8502: 8491: 8490: 8460: 8458: 8457: 8452: 8441: 8440: 8418: 8416: 8415: 8410: 8399: 8398: 8376: 8374: 8373: 8368: 8347: 8345: 8344: 8339: 8314: 8312: 8311: 8306: 8295: 8294: 8269: 8267: 8266: 8261: 8240: 8238: 8237: 8232: 8221: 8220: 8200: 8198: 8197: 8192: 8171: 8169: 8168: 8163: 8152: 8151: 8114: 8112: 8111: 8106: 8104: 8100: 8097: 8086: 8083: 8080: 8076: 8073: 8062: 8046: 7972: 7968: 7966: 7958: 7941: 7918: 7917: 7896: 7893: 7890: 7877: 7875: 7874: 7869: 7854: 7852: 7851: 7846: 7818: 7816: 7815: 7810: 7808: 7804: 7801: 7790: 7787: 7784: 7780: 7777: 7766: 7762: 7761: 7755: 7753: 7749: 7748: 7727: 7726: 7704: 7651: 7640: 7639: 7627: 7623: 7621: 7613: 7596: 7573: 7572: 7551: 7548: 7545: 7533: 7531: 7530: 7525: 7523: 7510: 7509: 7490: 7477: 7476: 7457: 7444: 7443: 7424: 7411: 7410: 7391: 7378: 7376: 7375: 7370: 7356: 7354: 7353: 7348: 7346: 7336: 7335: 7308: 7307: 7288: 7266: 7265: 7241: 7240: 7221: 7208: 7207: 7189: 7188: 7169: 7159: 7158: 7143: 7142: 7127: 7126: 7107: 7092: 7090: 7089: 7084: 7082: 7080: 7055: 7054: 7030: 7029: 7005: 7004: 6979: 6962: 6961: 6944:is shown below. 6932: 6930: 6929: 6924: 6910: 6909: 6876: 6875: 6824: 6821: 6810: 6808: 6807: 6802: 6755: 6727: 6725: 6724: 6719: 6708: 6707: 6692: 6678: 6677: 6634:Polynomial ratio 6630: 6628: 6627: 6622: 6620: 6617: 6615: 6613: 6609: 6608: 6593: 6592: 6576: 6535: 6527: 6525: 6517: 6500: 6462: 6459: 6448: 6446: 6445: 6440: 6417: 6415: 6414: 6409: 6407: 6403: 6401: 6400: 6399: 6386: 6357: 6348: 6346: 6338: 6321: 6316: 6312: 6311: 6287: 6286: 6262: 6261: 6243: 6222: 6218: 6197: 6175: 6174: 6147: 6146: 6128: 6107: 6103: 6082: 6060: 6058: 6050: 6046: 6045: 6011: 5996: 5995: 5971: 5970: 5946: 5945: 5926: 5922: 5913: 5891: 5890: 5863: 5862: 5843: 5839: 5830: 5802: 5800: 5792: 5788: 5787: 5750: 5735: 5734: 5710: 5709: 5685: 5684: 5665: 5661: 5640: 5618: 5617: 5590: 5589: 5570: 5566: 5545: 5517: 5516: 5476: 5475: 5457: 5456: 5438: 5437: 5424: 5420: 5411: 5389: 5388: 5367: 5366: 5353: 5349: 5340: 5318: 5317: 5263: 5261: 5260: 5255: 5253: 5249: 5247: 5245: 5240: 5209: 5208: 5206: 5198: 5194: 5193: 5150: 5148: 5147: 5111: 5109: 5101: 5097: 5096: 5062: 5060: 5059: 5028: 5023: 5021: 5013: 4996: 4993: 4989: 4987: 4986: 4985: 4954: 4953: 4952: 4918: 4900: 4896: 4894: 4893: 4892: 4867: 4860: 4839: 4834: 4832: 4824: 4805: 4804: 4782: 4779: 4766: 4764: 4763: 4758: 4756: 4755: 4721: 4719: 4718: 4713: 4711: 4710: 4676: 4674: 4673: 4668: 4645: 4643: 4642: 4637: 4632: 4631: 4601: 4596: 4595: 4559: 4558: 4518: 4516: 4515: 4510: 4489: 4487: 4486: 4481: 4479: 4478: 4453: 4451: 4450: 4445: 4443: 4442: 4423: 4421: 4420: 4415: 4410: 4409: 4379: 4374: 4373: 4340: 4339: 4299: 4297: 4296: 4291: 4267: 4265: 4264: 4259: 4235: 4233: 4232: 4227: 4219: 4214: 4184: 4182: 4181: 4176: 4152: 4150: 4149: 4144: 4123: 4121: 4120: 4115: 4091: 4089: 4088: 4083: 4068: 4066: 4065: 4060: 4013: 4010:and capacitance 4009: 4005: 4003: 4002: 3997: 3995: 3987: 3985: 3892: 3890: 3889: 3884: 3881: 3880: 3871: 3870: 3858: 3857: 3836: 3835: 3826: 3825: 3813: 3812: 3791: 3790: 3740:. Consider two 3726:between the two 3724:propagation time 3710: 3708: 3707: 3702: 3700: 3699: 3683: 3681: 3680: 3675: 3670: 3669: 3615: 3613: 3612: 3607: 3605: 3603: 3595: 3587: 3570: 3569: 3479: 3477: 3476: 3471: 3469: 3468: 3456: 3455: 3347: 3346: 3312: 3311: 3307: 3289: 3280: 3270: 3266: 3264: 3263: 3258: 3255: 3254: 3237: 3233: 3231: 3229: 3228: 3223: 3220: 3219: 3201: 3197: 3195: 3194: 3189: 3186: 3185: 3160: 3150: 3146: 3145: 3141: 3136: 3132: 3128: 3126: 3125: 3120: 3117: 3116: 3090: 3070: 3068: 3067: 3062: 3056: 3051: 3043: 3038: 3036: 3035: 3034: 3018: 3010: 3009: 3000: 2989: 2988: 2970: 2968: 2967: 2962: 2960: 2943: 2941: 2940: 2928: 2916: 2915: 2897: 2896: 2871: 2869: 2868: 2863: 2861: 2853: 2851: 2849: 2844: 2832: 2831: 2821: 2820: 2811: 2793: 2792: 2770: 2768: 2767: 2762: 2748: 2746: 2745: 2740: 2731: 2729: 2728: 2716: 2702: 2694: 2667:high-pass filter 2662: 2660: 2659: 2654: 2645: 2643: 2642: 2630: 2592: 2590: 2589: 2584: 2582: 2580: 2569: 2567: 2562: 2561: 2545:cutoff frequency 2526: 2524: 2523: 2518: 2512: 2508: 2479: 2478: 2477: 2475: 2474: 2462: 2457: 2452: 2429: 2427: 2426: 2421: 2404: 2402: 2401: 2396: 2392: 2391: 2385: 2384: 2383: 2382: 2378: 2341: 2340: 2339: 2318: 2317: 2305: 2299: 2298: 2297: 2280: 2279: 2275: 2274: 2273: 2252: 2251: 2230: 2229: 2212: 2206: 2205: 2204: 2187: 2135: 2133: 2132: 2127: 2123: 2122: 2121: 2080:complex sinusoid 2077: 2075: 2074: 2069: 2038: 2036: 2035: 2030: 2016: 2014: 2013: 2008: 2005: 2004: 1977: 1975: 1974: 1969: 1955: 1953: 1952: 1947: 1944: 1943: 1926: 1924: 1923: 1918: 1915: 1914: 1890: 1888: 1887: 1882: 1876: 1874: 1866: 1849: 1832: 1831: 1797: 1795: 1794: 1789: 1783: 1778: 1764: 1747: 1746: 1723: 1721: 1720: 1715: 1688: 1686: 1685: 1680: 1677: 1676: 1655: 1653: 1652: 1647: 1644: 1643: 1614: 1612: 1611: 1606: 1603: 1602: 1585: 1583: 1582: 1577: 1574: 1573: 1556: 1554: 1553: 1548: 1531: 1529: 1528: 1523: 1520: 1519: 1502: 1500: 1499: 1494: 1491: 1490: 1470: 1468: 1467: 1462: 1456: 1455: 1440: 1439: 1418: 1417: 1402: 1401: 1383: 1382: 1379: 1371: 1370: 1369: 1348: 1347: 1313: 1311: 1310: 1305: 1298: 1291: 1287: 1286: 1285: 1270: 1269: 1266: 1260: 1259: 1247: 1245: 1234: 1217: 1215: 1214: 1209: 1195: 1193: 1192: 1187: 1175: 1174: 1171: 1154: 1152: 1151: 1146: 1107: 1106: 1103: 1072: 1070: 1069: 1064: 1041: 1039: 1038: 1033: 1015: 1014: 1011: 973: 971: 970: 965: 938: 936: 935: 930: 907: 905: 904: 899: 897: 896: 888: 887: 869: 866:, respectively, 865: 863: 862: 857: 834: 832: 831: 826: 799: 797: 796: 791: 768: 766: 765: 760: 737: 735: 734: 729: 714: 712: 711: 706: 669: 668: 660: 659: 652: 651: 649: 646: 644: 639: 633: 595: 590: 578: 577: 575: 572: 570: 565: 509: 507: 506: 501: 478: 476: 475: 470: 447:impulse response 445:the time-domain 440: 438: 437: 432: 381:Angle Modulation 223: 221: 220: 215: 179:Fourier analysis 170:Fourier analysis 21: 9763: 9762: 9758: 9757: 9756: 9754: 9753: 9752: 9723: 9722: 9704: 9699: 9698: 9689: 9688: 9684: 9675: 9673: 9667: 9663: 9646: 9642: 9629: 9628: 9624: 9617: 9603: 9599: 9576: 9572: 9563: 9561: 9543:(11): 922–934. 9529: 9525: 9516: 9514: 9505: 9501: 9493: 9478: 9472: 9465: 9460:(7/8): 593–611. 9446: 9442: 9434: 9403: 9397: 9393: 9356: 9352: 9343: 9341: 9337: 9330: 9322: 9315: 9306: 9304: 9300: 9289: 9283: 9279: 9260: 9253: 9224:(2B): 409–421. 9214: 9210: 9201: 9199: 9184: 9180: 9129: 9125: 9116: 9114: 9110: 9103: 9097: 9093: 9077: 9076: 9071: 9067: 9066: 9062: 9055: 9051: 9044: 9030: 9026: 9019: 9005: 9001: 8994: 8980: 8973: 8966: 8952: 8948: 8941: 8927: 8923: 8914: 8912: 8897: 8890: 8883: 8869: 8860: 8838: 8829: 8827: 8825: 8795:Legendre filter 8775: 8733: 8730: 8729: 8689: 8672: 8669: 8668: 8649: 8628: 8624: 8622: 8619: 8618: 8590: 8587: 8586: 8554: 8550: 8548: 8545: 8544: 8516: 8513: 8512: 8480: 8476: 8474: 8471: 8470: 8467: 8430: 8426: 8424: 8421: 8420: 8388: 8384: 8382: 8379: 8378: 8353: 8350: 8349: 8324: 8321: 8320: 8315:is to take the 8284: 8280: 8278: 8275: 8274: 8246: 8243: 8242: 8216: 8212: 8210: 8207: 8206: 8177: 8174: 8173: 8141: 8137: 8135: 8132: 8131: 8126:Deviation from 8124: 8102: 8101: 8096: 8082: 8078: 8077: 8072: 8060: 8059: 8042: 7970: 7969: 7959: 7942: 7940: 7913: 7909: 7892: 7887: 7885: 7882: 7881: 7860: 7857: 7856: 7831: 7828: 7827: 7821: 7806: 7805: 7800: 7786: 7782: 7781: 7776: 7764: 7763: 7757: 7756: 7744: 7740: 7722: 7718: 7705: 7652: 7650: 7635: 7634: 7625: 7624: 7614: 7597: 7595: 7568: 7564: 7547: 7542: 7540: 7537: 7536: 7521: 7520: 7499: 7495: 7488: 7487: 7466: 7462: 7455: 7454: 7433: 7429: 7422: 7421: 7400: 7396: 7388: 7386: 7383: 7382: 7364: 7361: 7360: 7344: 7343: 7331: 7327: 7297: 7293: 7286: 7285: 7261: 7257: 7230: 7226: 7219: 7218: 7203: 7199: 7178: 7174: 7167: 7166: 7154: 7150: 7138: 7134: 7116: 7112: 7104: 7102: 7099: 7098: 7050: 7046: 7025: 7021: 7000: 6996: 6983: 6978: 6957: 6953: 6951: 6948: 6947: 6938: 6899: 6895: 6865: 6861: 6820: 6818: 6815: 6814: 6751: 6737: 6734: 6733: 6697: 6693: 6688: 6667: 6663: 6643: 6640: 6639: 6636: 6616: 6604: 6600: 6588: 6584: 6577: 6536: 6534: 6518: 6501: 6499: 6458: 6456: 6453: 6452: 6425: 6422: 6421: 6405: 6404: 6395: 6391: 6387: 6358: 6356: 6354: 6349: 6339: 6322: 6320: 6317: 6314: 6313: 6298: 6294: 6276: 6272: 6248: 6244: 6224: 6214: 6198: 6187: 6181: 6176: 6161: 6157: 6133: 6129: 6109: 6099: 6083: 6072: 6066: 6061: 6051: 6035: 6031: 6012: 6010: 5998: 5997: 5982: 5978: 5960: 5956: 5938: 5934: 5918: 5914: 5903: 5897: 5892: 5877: 5873: 5855: 5851: 5835: 5831: 5820: 5808: 5803: 5793: 5774: 5770: 5751: 5749: 5737: 5736: 5721: 5717: 5699: 5695: 5671: 5667: 5657: 5641: 5630: 5624: 5619: 5604: 5600: 5576: 5572: 5562: 5546: 5535: 5523: 5518: 5506: 5502: 5478: 5477: 5468: 5464: 5452: 5448: 5430: 5426: 5416: 5412: 5401: 5395: 5390: 5381: 5377: 5359: 5355: 5345: 5341: 5330: 5324: 5319: 5304: 5300: 5275: 5273: 5270: 5269: 5251: 5250: 5241: 5227: 5210: 5199: 5180: 5176: 5151: 5149: 5137: 5133: 5102: 5086: 5082: 5063: 5061: 5046: 5042: 5029: 5027: 5014: 4997: 4995: 4991: 4990: 4972: 4968: 4955: 4942: 4938: 4919: 4917: 4898: 4897: 4888: 4884: 4868: 4856: 4840: 4838: 4825: 4797: 4793: 4783: 4781: 4776: 4774: 4771: 4770: 4745: 4741: 4727: 4724: 4723: 4700: 4696: 4682: 4679: 4678: 4659: 4656: 4655: 4618: 4614: 4597: 4585: 4581: 4551: 4547: 4524: 4521: 4520: 4495: 4492: 4491: 4465: 4461: 4459: 4456: 4455: 4435: 4431: 4429: 4426: 4425: 4396: 4392: 4375: 4360: 4356: 4332: 4328: 4305: 4302: 4301: 4273: 4270: 4269: 4244: 4241: 4240: 4215: 4210: 4190: 4187: 4186: 4158: 4155: 4154: 4129: 4126: 4125: 4097: 4094: 4093: 4074: 4071: 4070: 4045: 4042: 4041: 4030:form, then the 4020: 4011: 4007: 3986: 3981: 3976: 3973: 3972: 3950:true time delay 3946: 3944:True time delay 3939: 3926: 3913: 3904: 3876: 3872: 3866: 3862: 3847: 3843: 3831: 3827: 3821: 3817: 3802: 3798: 3786: 3782: 3780: 3777: 3776: 3763: 3695: 3691: 3689: 3686: 3685: 3665: 3661: 3629: 3626: 3625: 3596: 3588: 3586: 3565: 3561: 3559: 3556: 3555: 3498: 3464: 3460: 3451: 3447: 3436: 3433: 3432: 3361: 3356: 3351: 3330: 3309: 3305: 3304: 3291: 3287: 3281: 3272: 3268: 3250: 3246: 3243: 3240: 3239: 3235: 3215: 3211: 3208: 3205: 3204: 3202: 3199: 3181: 3177: 3174: 3171: 3170: 3161: 3152: 3148: 3143: 3139: 3138: 3134: 3130: 3112: 3108: 3105: 3102: 3101: 3100:group delay of 3091: 3079: 3044: 3042: 3030: 3026: 3022: 3017: 3005: 3001: 2996: 2984: 2980: 2978: 2975: 2974: 2958: 2957: 2936: 2932: 2927: 2920: 2911: 2907: 2892: 2888: 2884: 2882: 2879: 2878: 2859: 2858: 2845: 2840: 2827: 2823: 2822: 2816: 2812: 2810: 2803: 2788: 2784: 2780: 2778: 2775: 2774: 2756: 2753: 2752: 2724: 2720: 2715: 2693: 2676: 2673: 2672: 2638: 2634: 2629: 2600: 2597: 2596: 2573: 2568: 2563: 2557: 2553: 2551: 2548: 2547: 2537:low-pass filter 2533: 2492: 2488: 2464: 2463: 2458: 2456: 2455: 2438: 2435: 2434: 2413: 2410: 2409: 2389: 2388: 2356: 2352: 2348: 2344: 2335: 2334: 2313: 2312: 2303: 2302: 2287: 2283: 2257: 2253: 2247: 2246: 2225: 2224: 2223: 2219: 2210: 2209: 2194: 2190: 2165: 2149: 2147: 2144: 2143: 2111: 2107: 2090: 2087: 2086: 2052: 2049: 2048: 2045: 2022: 2019: 2018: 2000: 1996: 1987: 1984: 1983: 1961: 1958: 1957: 1939: 1935: 1932: 1929: 1928: 1910: 1906: 1903: 1900: 1899: 1867: 1850: 1848: 1827: 1823: 1821: 1818: 1817: 1765: 1763: 1742: 1738: 1736: 1733: 1732: 1698: 1695: 1694: 1672: 1668: 1665: 1662: 1661: 1639: 1635: 1632: 1629: 1628: 1621: 1598: 1594: 1591: 1588: 1587: 1569: 1565: 1562: 1559: 1558: 1540: 1537: 1536: 1515: 1511: 1508: 1505: 1504: 1486: 1482: 1479: 1476: 1475: 1451: 1450: 1435: 1431: 1413: 1412: 1397: 1393: 1378: 1374: 1365: 1364: 1343: 1342: 1325: 1322: 1321: 1281: 1280: 1265: 1261: 1255: 1254: 1238: 1233: 1232: 1228: 1226: 1223: 1222: 1201: 1198: 1197: 1170: 1166: 1163: 1160: 1159: 1102: 1098: 1081: 1078: 1077: 1047: 1044: 1043: 1042:, so the input 1010: 1006: 1003: 1000: 999: 992: 948: 945: 944: 913: 910: 909: 889: 883: 882: 881: 879: 876: 875: 867: 840: 837: 836: 809: 806: 805: 774: 771: 770: 743: 740: 739: 723: 720: 719: 661: 655: 654: 653: 645: 640: 638: 637: 629: 591: 583: 571: 566: 564: 563: 522: 519: 518: 484: 481: 480: 453: 450: 449: 415: 412: 411: 392: 383: 364: 340: 320: 292: 244: 197: 194: 193: 176: 164:Main articles: 162: 157: 46:are delayed in 28: 23: 22: 15: 12: 11: 5: 9761: 9751: 9750: 9745: 9740: 9735: 9721: 9720: 9715: 9710: 9703: 9702:External links 9700: 9697: 9696: 9682: 9661: 9640: 9622: 9615: 9597: 9586:(6): 506–526. 9570: 9523: 9499: 9489:(9): 709–715. 9463: 9440: 9437:on 2015-09-30. 9391: 9350: 9313: 9277: 9251: 9208: 9178: 9123: 9091: 9060: 9049: 9042: 9024: 9017: 8999: 8992: 8971: 8964: 8946: 8939: 8921: 8888: 8881: 8857: 8856: 8852:on 2022-01-22. 8824: 8821: 8820: 8819: 8814: 8812:Phase velocity 8809: 8806:Group velocity 8803: 8798: 8792: 8786: 8781: 8774: 8771: 8758: 8755: 8752: 8749: 8746: 8743: 8740: 8737: 8717: 8714: 8711: 8708: 8705: 8702: 8699: 8696: 8692: 8688: 8685: 8682: 8679: 8676: 8656: 8652: 8648: 8645: 8642: 8637: 8634: 8631: 8627: 8606: 8603: 8600: 8597: 8594: 8574: 8571: 8568: 8563: 8560: 8557: 8553: 8532: 8529: 8526: 8523: 8520: 8500: 8497: 8494: 8489: 8486: 8483: 8479: 8466: 8463: 8450: 8447: 8444: 8439: 8436: 8433: 8429: 8408: 8405: 8402: 8397: 8394: 8391: 8387: 8366: 8363: 8360: 8357: 8337: 8334: 8331: 8328: 8304: 8301: 8298: 8293: 8290: 8287: 8283: 8259: 8256: 8253: 8250: 8230: 8227: 8224: 8219: 8215: 8190: 8187: 8184: 8181: 8161: 8158: 8155: 8150: 8147: 8144: 8140: 8123: 8120: 8095: 8092: 8089: 8081: 8079: 8071: 8068: 8065: 8063: 8061: 8058: 8055: 8052: 8049: 8045: 8041: 8038: 8035: 8032: 8029: 8026: 8023: 8020: 8017: 8014: 8011: 8008: 8005: 8002: 7999: 7996: 7993: 7990: 7987: 7984: 7981: 7978: 7975: 7973: 7971: 7965: 7962: 7957: 7954: 7951: 7948: 7945: 7939: 7936: 7933: 7930: 7927: 7924: 7921: 7916: 7912: 7908: 7905: 7902: 7899: 7891: 7889: 7867: 7864: 7844: 7841: 7838: 7835: 7799: 7796: 7793: 7785: 7783: 7775: 7772: 7769: 7767: 7765: 7760: 7752: 7747: 7743: 7739: 7736: 7733: 7730: 7725: 7721: 7717: 7714: 7711: 7708: 7703: 7700: 7697: 7694: 7691: 7688: 7685: 7682: 7679: 7676: 7673: 7670: 7667: 7664: 7661: 7658: 7655: 7649: 7646: 7643: 7638: 7633: 7630: 7628: 7626: 7620: 7617: 7612: 7609: 7606: 7603: 7600: 7594: 7591: 7588: 7585: 7582: 7579: 7576: 7571: 7567: 7563: 7560: 7557: 7554: 7546: 7544: 7519: 7516: 7513: 7508: 7505: 7502: 7498: 7494: 7491: 7489: 7486: 7483: 7480: 7475: 7472: 7469: 7465: 7461: 7458: 7456: 7453: 7450: 7447: 7442: 7439: 7436: 7432: 7428: 7425: 7423: 7420: 7417: 7414: 7409: 7406: 7403: 7399: 7395: 7392: 7390: 7368: 7359:Evaluating at 7342: 7339: 7334: 7330: 7326: 7323: 7320: 7317: 7314: 7311: 7306: 7303: 7300: 7296: 7292: 7289: 7287: 7284: 7281: 7278: 7275: 7272: 7269: 7264: 7260: 7256: 7253: 7250: 7247: 7244: 7239: 7236: 7233: 7229: 7225: 7222: 7220: 7217: 7214: 7211: 7206: 7202: 7198: 7195: 7192: 7187: 7184: 7181: 7177: 7173: 7170: 7168: 7165: 7162: 7157: 7153: 7149: 7146: 7141: 7137: 7133: 7130: 7125: 7122: 7119: 7115: 7111: 7108: 7106: 7079: 7076: 7073: 7070: 7067: 7064: 7061: 7058: 7053: 7049: 7045: 7042: 7039: 7036: 7033: 7028: 7024: 7020: 7017: 7014: 7011: 7008: 7003: 6999: 6995: 6992: 6989: 6986: 6982: 6977: 6974: 6971: 6968: 6965: 6960: 6956: 6937: 6934: 6922: 6919: 6916: 6913: 6908: 6905: 6902: 6898: 6894: 6891: 6888: 6885: 6882: 6879: 6874: 6871: 6868: 6864: 6860: 6857: 6854: 6851: 6848: 6845: 6842: 6839: 6836: 6833: 6830: 6827: 6800: 6797: 6794: 6791: 6788: 6785: 6782: 6779: 6776: 6773: 6770: 6767: 6764: 6761: 6758: 6754: 6750: 6747: 6744: 6741: 6730:filter designs 6717: 6714: 6711: 6706: 6703: 6700: 6696: 6691: 6687: 6684: 6681: 6676: 6673: 6670: 6666: 6662: 6659: 6656: 6653: 6650: 6647: 6635: 6632: 6612: 6607: 6603: 6599: 6596: 6591: 6587: 6583: 6580: 6575: 6572: 6569: 6566: 6563: 6560: 6557: 6554: 6551: 6548: 6545: 6542: 6539: 6533: 6530: 6524: 6521: 6516: 6513: 6510: 6507: 6504: 6498: 6495: 6492: 6489: 6486: 6483: 6480: 6477: 6474: 6471: 6468: 6465: 6438: 6435: 6432: 6429: 6398: 6394: 6390: 6385: 6382: 6379: 6376: 6373: 6370: 6367: 6364: 6361: 6355: 6353: 6350: 6345: 6342: 6337: 6334: 6331: 6328: 6325: 6319: 6318: 6315: 6310: 6307: 6304: 6301: 6297: 6293: 6290: 6285: 6282: 6279: 6275: 6271: 6268: 6265: 6260: 6257: 6254: 6251: 6247: 6242: 6239: 6236: 6233: 6230: 6227: 6221: 6217: 6213: 6210: 6207: 6204: 6201: 6196: 6193: 6190: 6186: 6182: 6180: 6177: 6173: 6170: 6167: 6164: 6160: 6156: 6153: 6150: 6145: 6142: 6139: 6136: 6132: 6127: 6124: 6121: 6118: 6115: 6112: 6106: 6102: 6098: 6095: 6092: 6089: 6086: 6081: 6078: 6075: 6071: 6067: 6065: 6062: 6057: 6054: 6049: 6044: 6041: 6038: 6034: 6030: 6027: 6024: 6021: 6018: 6015: 6009: 6006: 6003: 6000: 5999: 5994: 5991: 5988: 5985: 5981: 5977: 5974: 5969: 5966: 5963: 5959: 5955: 5952: 5949: 5944: 5941: 5937: 5933: 5930: 5925: 5921: 5917: 5912: 5909: 5906: 5902: 5898: 5896: 5893: 5889: 5886: 5883: 5880: 5876: 5872: 5869: 5866: 5861: 5858: 5854: 5850: 5847: 5842: 5838: 5834: 5829: 5826: 5823: 5819: 5815: 5812: 5809: 5807: 5804: 5799: 5796: 5791: 5786: 5783: 5780: 5777: 5773: 5769: 5766: 5763: 5760: 5757: 5754: 5748: 5745: 5742: 5739: 5738: 5733: 5730: 5727: 5724: 5720: 5716: 5713: 5708: 5705: 5702: 5698: 5694: 5691: 5688: 5683: 5680: 5677: 5674: 5670: 5664: 5660: 5656: 5653: 5650: 5647: 5644: 5639: 5636: 5633: 5629: 5625: 5623: 5620: 5616: 5613: 5610: 5607: 5603: 5599: 5596: 5593: 5588: 5585: 5582: 5579: 5575: 5569: 5565: 5561: 5558: 5555: 5552: 5549: 5544: 5541: 5538: 5534: 5530: 5527: 5524: 5522: 5519: 5515: 5512: 5509: 5505: 5501: 5498: 5495: 5492: 5489: 5486: 5483: 5480: 5479: 5474: 5471: 5467: 5463: 5460: 5455: 5451: 5447: 5444: 5441: 5436: 5433: 5429: 5423: 5419: 5415: 5410: 5407: 5404: 5400: 5396: 5394: 5391: 5387: 5384: 5380: 5376: 5373: 5370: 5365: 5362: 5358: 5352: 5348: 5344: 5339: 5336: 5333: 5329: 5325: 5323: 5320: 5316: 5313: 5310: 5307: 5303: 5299: 5296: 5293: 5290: 5287: 5284: 5281: 5278: 5277: 5244: 5239: 5236: 5233: 5230: 5226: 5222: 5219: 5216: 5213: 5205: 5202: 5197: 5192: 5189: 5186: 5183: 5179: 5175: 5172: 5169: 5166: 5163: 5160: 5157: 5154: 5146: 5143: 5140: 5136: 5132: 5129: 5126: 5123: 5120: 5117: 5114: 5108: 5105: 5100: 5095: 5092: 5089: 5085: 5081: 5078: 5075: 5072: 5069: 5066: 5058: 5055: 5052: 5049: 5045: 5041: 5038: 5035: 5032: 5026: 5020: 5017: 5012: 5009: 5006: 5003: 5000: 4994: 4992: 4984: 4981: 4978: 4975: 4971: 4967: 4964: 4961: 4958: 4951: 4948: 4945: 4941: 4937: 4934: 4931: 4928: 4925: 4922: 4916: 4913: 4910: 4907: 4904: 4901: 4899: 4891: 4887: 4883: 4880: 4877: 4874: 4871: 4866: 4863: 4859: 4855: 4852: 4849: 4846: 4843: 4837: 4831: 4828: 4823: 4820: 4817: 4814: 4811: 4808: 4803: 4800: 4796: 4792: 4789: 4786: 4780: 4778: 4754: 4751: 4748: 4744: 4740: 4737: 4734: 4731: 4709: 4706: 4703: 4699: 4695: 4692: 4689: 4686: 4666: 4663: 4635: 4630: 4627: 4624: 4621: 4617: 4613: 4610: 4607: 4604: 4600: 4594: 4591: 4588: 4584: 4580: 4577: 4574: 4571: 4568: 4565: 4562: 4557: 4554: 4550: 4546: 4543: 4540: 4537: 4534: 4531: 4528: 4508: 4505: 4502: 4499: 4477: 4474: 4471: 4468: 4464: 4441: 4438: 4434: 4424:. Given that 4413: 4408: 4405: 4402: 4399: 4395: 4391: 4388: 4385: 4382: 4378: 4372: 4369: 4366: 4363: 4359: 4355: 4352: 4349: 4346: 4343: 4338: 4335: 4331: 4327: 4324: 4321: 4318: 4315: 4312: 4309: 4289: 4286: 4283: 4280: 4277: 4257: 4254: 4251: 4248: 4225: 4222: 4218: 4213: 4209: 4206: 4203: 4200: 4197: 4194: 4174: 4171: 4168: 4165: 4162: 4142: 4139: 4136: 4133: 4113: 4110: 4107: 4104: 4101: 4081: 4078: 4058: 4055: 4052: 4049: 4019: 4016: 3993: 3990: 3984: 3980: 3945: 3942: 3931: 3918: 3914: = | 3909: 3900: 3894: 3893: 3879: 3875: 3869: 3865: 3861: 3856: 3853: 3850: 3846: 3842: 3839: 3834: 3830: 3824: 3820: 3816: 3811: 3808: 3805: 3801: 3797: 3794: 3789: 3785: 3758: 3698: 3694: 3673: 3668: 3664: 3660: 3657: 3654: 3651: 3648: 3645: 3642: 3639: 3636: 3633: 3602: 3599: 3594: 3591: 3585: 3582: 3579: 3576: 3573: 3568: 3564: 3529:group velocity 3497: 3494: 3482: 3481: 3467: 3463: 3459: 3454: 3450: 3446: 3443: 3440: 3419: 3418: 3415: 3412: 3408: 3407: 3404: 3401: 3397: 3396: 3393: 3390: 3386: 3385: 3382: 3379: 3375: 3374: 3371: 3368: 3364: 3363: 3358: 3353: 3329: 3326: 3293: 3292: 3282: 3275: 3273: 3253: 3249: 3218: 3214: 3184: 3180: 3169:simulation of 3162: 3155: 3153: 3115: 3111: 3092: 3085: 3083: 3078: 3075: 3060: 3054: 3050: 3047: 3041: 3033: 3029: 3025: 3021: 3016: 3013: 3008: 3004: 2999: 2995: 2992: 2987: 2983: 2956: 2952: 2949: 2946: 2939: 2935: 2931: 2926: 2923: 2921: 2919: 2914: 2910: 2906: 2903: 2900: 2895: 2891: 2887: 2886: 2857: 2848: 2843: 2839: 2835: 2830: 2826: 2819: 2815: 2809: 2806: 2804: 2802: 2799: 2796: 2791: 2787: 2783: 2782: 2760: 2738: 2734: 2727: 2723: 2719: 2714: 2711: 2708: 2705: 2700: 2697: 2692: 2689: 2686: 2683: 2680: 2652: 2648: 2641: 2637: 2633: 2628: 2625: 2622: 2619: 2616: 2613: 2610: 2607: 2604: 2579: 2576: 2572: 2566: 2560: 2556: 2532: 2529: 2528: 2527: 2516: 2511: 2507: 2504: 2501: 2498: 2495: 2491: 2487: 2484: 2473: 2470: 2467: 2461: 2451: 2448: 2445: 2442: 2418: 2406: 2405: 2381: 2377: 2374: 2371: 2368: 2365: 2362: 2359: 2355: 2351: 2347: 2338: 2333: 2330: 2327: 2324: 2321: 2316: 2311: 2308: 2306: 2304: 2296: 2293: 2290: 2286: 2278: 2272: 2269: 2266: 2263: 2260: 2256: 2250: 2245: 2242: 2239: 2236: 2233: 2228: 2222: 2218: 2215: 2213: 2211: 2203: 2200: 2197: 2193: 2186: 2183: 2180: 2177: 2174: 2171: 2168: 2166: 2164: 2161: 2158: 2155: 2152: 2151: 2139:the output is 2137: 2136: 2120: 2117: 2114: 2110: 2106: 2103: 2100: 2097: 2094: 2066: 2063: 2060: 2057: 2044: 2041: 2027: 2003: 1999: 1995: 1992: 1966: 1942: 1938: 1913: 1909: 1892: 1891: 1880: 1873: 1870: 1865: 1862: 1859: 1856: 1853: 1847: 1844: 1841: 1838: 1835: 1830: 1826: 1799: 1798: 1787: 1781: 1777: 1774: 1771: 1768: 1762: 1759: 1756: 1753: 1750: 1745: 1741: 1712: 1709: 1706: 1703: 1675: 1671: 1642: 1638: 1620: 1617: 1601: 1597: 1572: 1568: 1545: 1518: 1514: 1489: 1485: 1472: 1471: 1460: 1454: 1449: 1446: 1443: 1438: 1434: 1430: 1427: 1424: 1421: 1416: 1411: 1408: 1405: 1400: 1396: 1392: 1389: 1386: 1377: 1368: 1363: 1360: 1357: 1354: 1351: 1346: 1341: 1338: 1335: 1332: 1329: 1315: 1314: 1303: 1297: 1294: 1290: 1284: 1279: 1276: 1273: 1264: 1258: 1253: 1250: 1244: 1241: 1237: 1231: 1206: 1184: 1181: 1178: 1169: 1156: 1155: 1144: 1140: 1137: 1134: 1131: 1128: 1125: 1122: 1119: 1116: 1113: 1110: 1101: 1097: 1094: 1091: 1088: 1085: 1061: 1058: 1055: 1052: 1030: 1027: 1024: 1021: 1018: 1009: 991: 988: 984:Laplace domain 962: 959: 956: 953: 939:is called the 927: 924: 921: 918: 895: 892: 886: 854: 851: 848: 845: 823: 820: 817: 814: 788: 785: 782: 779: 757: 754: 751: 748: 727: 716: 715: 704: 700: 697: 694: 691: 688: 684: 681: 678: 675: 672: 667: 664: 658: 643: 636: 632: 627: 624: 621: 618: 614: 611: 608: 605: 602: 599: 594: 589: 586: 582: 569: 562: 559: 556: 553: 550: 547: 544: 541: 538: 535: 532: 529: 526: 498: 495: 492: 489: 467: 464: 461: 458: 429: 426: 423: 420: 400:control theory 391: 388: 382: 379: 363: 360: 339: 336: 319: 316: 296:phase response 291: 288: 256:phase response 243: 240: 213: 210: 207: 204: 201: 161: 158: 156: 153: 145:bit-error rate 80:digital filter 76:ethernet cable 26: 9: 6: 4: 3: 2: 9760: 9749: 9746: 9744: 9741: 9739: 9736: 9734: 9731: 9730: 9728: 9719: 9716: 9714: 9711: 9709: 9706: 9705: 9692: 9691:"Group Delay" 9686: 9672: 9665: 9657: 9653: 9652: 9644: 9636: 9632: 9626: 9618: 9612: 9608: 9601: 9593: 9589: 9585: 9581: 9574: 9560: 9556: 9551: 9546: 9542: 9538: 9534: 9527: 9512: 9511: 9503: 9492: 9488: 9484: 9477: 9470: 9468: 9459: 9455: 9451: 9444: 9433: 9429: 9425: 9421: 9417: 9413: 9409: 9402: 9395: 9387: 9383: 9378: 9373: 9370:: 2189–2201. 9369: 9365: 9361: 9354: 9336: 9329: 9328: 9320: 9318: 9299: 9295: 9288: 9281: 9273: 9269: 9265: 9258: 9256: 9247: 9243: 9239: 9235: 9231: 9227: 9223: 9219: 9212: 9197: 9193: 9189: 9182: 9174: 9170: 9166: 9162: 9158: 9154: 9149: 9144: 9140: 9136: 9135: 9127: 9109: 9102: 9095: 9087: 9081: 9070: 9064: 9058: 9053: 9045: 9043:0-13-814757-4 9039: 9035: 9028: 9020: 9018:9780534954093 9014: 9010: 9003: 8995: 8989: 8985: 8978: 8976: 8967: 8965:0-13-754920-2 8961: 8957: 8950: 8942: 8936: 8932: 8925: 8911:(11): 774–794 8910: 8906: 8902: 8895: 8893: 8884: 8882:0-13-914101-4 8878: 8874: 8867: 8865: 8863: 8858: 8855: 8851: 8847: 8843: 8842: 8836: 8818: 8815: 8813: 8810: 8807: 8804: 8802: 8799: 8796: 8793: 8790: 8789:Bessel filter 8787: 8785: 8782: 8780: 8777: 8776: 8770: 8756: 8753: 8750: 8747: 8744: 8741: 8738: 8735: 8715: 8712: 8709: 8706: 8703: 8700: 8697: 8694: 8690: 8683: 8677: 8674: 8654: 8650: 8643: 8635: 8632: 8629: 8625: 8601: 8595: 8592: 8569: 8561: 8558: 8555: 8551: 8527: 8521: 8518: 8495: 8487: 8484: 8481: 8477: 8462: 8445: 8437: 8434: 8431: 8427: 8403: 8395: 8392: 8389: 8385: 8361: 8355: 8332: 8326: 8318: 8299: 8291: 8288: 8285: 8281: 8271: 8254: 8248: 8225: 8217: 8213: 8204: 8185: 8179: 8156: 8148: 8145: 8142: 8138: 8129: 8119: 8115: 8098: rad/sec 8093: 8090: 8087: 8069: 8066: 8064: 8056: 8053: 8050: 8047: 8043: 8033: 8030: 8027: 8024: 8021: 8015: 8012: 8006: 8003: 8000: 7997: 7994: 7991: 7985: 7979: 7976: 7974: 7963: 7960: 7952: 7946: 7943: 7937: 7934: 7925: 7922: 7914: 7910: 7903: 7900: 7897: 7879: 7865: 7839: 7833: 7825: 7819: 7802: rad/sec 7797: 7794: 7791: 7773: 7770: 7768: 7745: 7737: 7734: 7728: 7723: 7715: 7712: 7695: 7692: 7689: 7686: 7683: 7677: 7671: 7668: 7665: 7662: 7659: 7647: 7644: 7641: 7631: 7629: 7618: 7615: 7607: 7601: 7598: 7592: 7589: 7580: 7577: 7569: 7565: 7558: 7555: 7552: 7534: 7517: 7514: 7511: 7506: 7503: 7500: 7496: 7492: 7484: 7481: 7478: 7473: 7470: 7467: 7463: 7459: 7451: 7448: 7445: 7440: 7437: 7434: 7430: 7426: 7418: 7415: 7412: 7407: 7404: 7401: 7397: 7393: 7380: 7379:= 1 rad/sec: 7366: 7357: 7340: 7337: 7332: 7328: 7321: 7318: 7312: 7309: 7304: 7301: 7298: 7294: 7290: 7282: 7276: 7270: 7267: 7262: 7258: 7251: 7245: 7242: 7237: 7234: 7231: 7227: 7223: 7215: 7212: 7209: 7204: 7200: 7196: 7193: 7190: 7185: 7182: 7179: 7175: 7171: 7163: 7160: 7155: 7151: 7147: 7144: 7139: 7135: 7131: 7128: 7123: 7120: 7117: 7113: 7109: 7096: 7093: 7077: 7074: 7068: 7065: 7059: 7056: 7051: 7043: 7040: 7034: 7031: 7026: 7018: 7015: 7009: 7006: 7001: 6993: 6990: 6984: 6980: 6975: 6969: 6966: 6958: 6954: 6945: 6943: 6933: 6920: 6917: 6914: 6906: 6903: 6900: 6896: 6892: 6886: 6883: 6880: 6872: 6869: 6866: 6862: 6858: 6852: 6849: 6846: 6840: 6837: 6831: 6828: 6825: 6812: 6795: 6792: 6786: 6783: 6777: 6774: 6768: 6765: 6759: 6756: 6752: 6748: 6745: 6739: 6731: 6712: 6704: 6701: 6698: 6694: 6689: 6682: 6674: 6671: 6668: 6664: 6660: 6654: 6648: 6645: 6631: 6605: 6601: 6597: 6594: 6589: 6585: 6581: 6570: 6567: 6564: 6561: 6558: 6555: 6552: 6549: 6546: 6543: 6540: 6531: 6528: 6522: 6519: 6511: 6505: 6502: 6496: 6493: 6484: 6481: 6475: 6469: 6466: 6463: 6450: 6433: 6427: 6418: 6396: 6392: 6388: 6383: 6380: 6377: 6374: 6371: 6368: 6365: 6362: 6359: 6351: 6343: 6340: 6332: 6326: 6323: 6308: 6305: 6302: 6299: 6291: 6283: 6280: 6277: 6269: 6266: 6258: 6255: 6252: 6249: 6245: 6237: 6234: 6231: 6228: 6219: 6215: 6208: 6205: 6202: 6194: 6191: 6188: 6184: 6178: 6171: 6168: 6165: 6162: 6154: 6151: 6143: 6140: 6137: 6134: 6130: 6122: 6119: 6116: 6113: 6104: 6100: 6093: 6090: 6087: 6079: 6076: 6073: 6069: 6063: 6055: 6052: 6042: 6039: 6036: 6028: 6025: 6019: 6013: 6007: 6004: 6001: 5992: 5989: 5986: 5983: 5975: 5967: 5964: 5961: 5953: 5950: 5942: 5939: 5935: 5931: 5928: 5923: 5919: 5915: 5910: 5907: 5904: 5900: 5894: 5887: 5884: 5881: 5878: 5870: 5867: 5859: 5856: 5852: 5848: 5845: 5840: 5836: 5832: 5827: 5824: 5821: 5817: 5813: 5810: 5805: 5797: 5794: 5784: 5781: 5778: 5775: 5767: 5764: 5758: 5752: 5746: 5743: 5740: 5731: 5728: 5725: 5722: 5714: 5706: 5703: 5700: 5692: 5689: 5681: 5678: 5675: 5672: 5668: 5662: 5658: 5651: 5648: 5645: 5637: 5634: 5631: 5627: 5621: 5614: 5611: 5608: 5605: 5597: 5594: 5586: 5583: 5580: 5577: 5573: 5567: 5563: 5556: 5553: 5550: 5542: 5539: 5536: 5532: 5528: 5525: 5520: 5513: 5510: 5507: 5499: 5496: 5490: 5487: 5484: 5481: 5472: 5469: 5461: 5453: 5445: 5442: 5434: 5431: 5427: 5421: 5417: 5413: 5408: 5405: 5402: 5398: 5392: 5385: 5382: 5374: 5371: 5363: 5360: 5356: 5350: 5346: 5342: 5337: 5334: 5331: 5327: 5321: 5314: 5311: 5308: 5305: 5297: 5294: 5288: 5285: 5282: 5279: 5267: 5264: 5242: 5237: 5234: 5231: 5228: 5220: 5217: 5211: 5203: 5200: 5190: 5187: 5184: 5181: 5173: 5170: 5164: 5161: 5158: 5152: 5144: 5141: 5138: 5130: 5127: 5121: 5118: 5115: 5112: 5106: 5103: 5093: 5090: 5087: 5079: 5076: 5070: 5064: 5056: 5053: 5050: 5047: 5039: 5036: 5030: 5024: 5018: 5015: 5007: 5001: 4998: 4982: 4979: 4976: 4973: 4965: 4962: 4956: 4949: 4946: 4943: 4935: 4932: 4926: 4923: 4920: 4914: 4908: 4902: 4889: 4881: 4875: 4872: 4869: 4864: 4861: 4857: 4850: 4844: 4841: 4835: 4829: 4826: 4815: 4809: 4801: 4798: 4794: 4790: 4787: 4784: 4768: 4752: 4749: 4746: 4738: 4735: 4729: 4707: 4704: 4701: 4693: 4690: 4684: 4664: 4661: 4653: 4649: 4628: 4625: 4622: 4619: 4611: 4608: 4602: 4598: 4592: 4589: 4586: 4578: 4575: 4569: 4566: 4563: 4555: 4552: 4548: 4544: 4541: 4538: 4532: 4526: 4503: 4497: 4475: 4472: 4469: 4466: 4462: 4439: 4436: 4432: 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Retrieved 9664: 9650: 9643: 9625: 9606: 9600: 9583: 9579: 9573: 9562:. Retrieved 9540: 9536: 9526: 9515:. Retrieved 9509: 9502: 9486: 9482: 9457: 9453: 9443: 9432:the original 9411: 9407: 9394: 9367: 9363: 9353: 9342:. Retrieved 9326: 9305:. Retrieved 9293: 9280: 9263: 9221: 9217: 9211: 9200:. Retrieved 9191: 9181: 9138: 9132: 9126: 9115:. Retrieved 9094: 9063: 9052: 9033: 9027: 9008: 9002: 8983: 8955: 8949: 8930: 8924: 8913:. Retrieved 8908: 8904: 8872: 8850:the original 8840: 8826: 8784:Linear phase 8468: 8272: 8128:Linear Phase 8125: 8116: 7880: 7820: 7535: 7381: 7358: 7097: 7094: 6946: 6939: 6813: 6637: 6451: 6419: 5268: 5265: 4769: 4651: 4647: 4238: 4021: 3962: 3949: 3947: 3936: 3932: 3928: 3923: 3919: 3915: 3910: 3906: 3901: 3897: 3895: 3769: 3765: 3759: 3755: 3749:polarization 3734: 3730: 3715: 3713: 3553: 3510: 3499: 3486: 3483: 3426: 3422: 3334:loudspeakers 3331: 3321: 3319: 3301:band-limited 3296: 3294: 3097: 3072: 2972: 2877: 2873: 2773: 2750: 2671: 2664: 2595: 2539:formed by a 2534: 2407: 2138: 2078:driven by a 2046: 1896:linear phase 1893: 1806: 1802: 1800: 1725: 1693:phase shift 1657: 1624: 1622: 1473: 1316: 1157: 993: 975: 717: 393: 384: 374: 372: 365: 352: 349: 328: 321: 308:linear phase 304:steady state 299: 293: 261: 253: 245: 242:Introduction 226: 181:reveals how 177: 143:, or a high 141:analog audio 137:analog video 127:experiences 118: 103: 98: 94:steady state 88: 39: 35: 29: 8817:Wave packet 8801:Eye pattern 7894:Group Delay 7549:Group Delay 6822:Group Delay 6460:Group Delay 3424:inaudible. 1980:phase shift 1803:group delay 1726:phase delay 1658:phase delay 1625:group delay 980:time domain 443:convoluting 326:sinusoids. 318:Group delay 290:Phase delay 264:phase shift 149:bandlimited 105:wave packet 68:loudspeaker 54:(such as a 40:phase delay 36:group delay 18:Group delay 9727:Categories 9676:2024-09-09 9564:2022-05-22 9517:2022-05-22 9344:2022-05-21 9307:2022-05-22 9202:2023-06-13 9117:2022-10-28 8915:2022-05-22 8823:References 8201:, and the 7824:derivative 4036:polynomial 3742:eigenmodes 3728:eigenmodes 3720:difference 3549:dispersion 3545:wavelength 3533:dispersion 3322:zero phase 2541:RC circuit 1811:derivative 1809:(i.e. the 355:modulation 300:sinusoidal 284:derivative 254:A varying 187:sinusoidal 155:Background 129:distortion 89:sinusoidal 56:microphone 9386:236192266 8684:ω 8644:ω 8626:ϕ 8602:ω 8570:ω 8552:ϕ 8528:ω 8496:ω 8478:ϕ 8446:ω 8428:ϕ 8404:ω 8386:ϕ 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6141:− 6120:− 6070:∑ 6029:ω 5990:− 5976:ω 5965:− 5951:− 5901:∑ 5885:− 5871:ω 5818:∑ 5811:− 5768:ω 5729:− 5715:ω 5704:− 5690:− 5679:− 5628:∑ 5612:− 5598:ω 5584:− 5533:∑ 5526:− 5500:ω 5462:ω 5443:− 5399:∑ 5375:ω 5328:∑ 5298:ω 5221:ω 5174:ω 5159:− 5131:ω 5116:− 5113:− 5080:ω 5040:ω 4966:ω 4936:ω 4921:− 4799:− 4739:ω 4694:ω 4662:− 4612:ω 4579:ω 4564:− 4553:− 4533:ω 4527:ϕ 4504:ω 4498:ϕ 4390:ω 4354:ω 4334:− 4314:ω 4308:ϕ 4285:ω 4253:ω 4247:ϕ 4224:ω 4205:ω 4199:ϕ 4193:− 4170:ω 4138:ω 4132:ϕ 4109:ω 4080:ω 3958:bandwidth 3954:frequency 3860:⋅ 3815:⋅ 3693:τ 3672:ω 3663:τ 3659:− 3647:ϕ 3638:ω 3632:ϕ 3601:ω 3593:ϕ 3584:− 3575:ω 3563:τ 3462:τ 3458:− 3453:ϕ 3449:τ 3442:τ 3439:Δ 3362:(Cycles) 3355:Threshold 3350:Frequency 3315:causality 3284:Transient 3248:τ 3213:τ 3179:τ 3110:τ 3028:ω 3003:ω 2994:ω 2982:τ 2934:ω 2925:≈ 2909:ω 2905:≪ 2902:ω 2890:τ 2838:ω 2825:ω 2814:ω 2798:ω 2786:τ 2759:ω 2722:ω 2718:ω 2710:⁡ 2704:− 2696:π 2685:ω 2679:ϕ 2636:ω 2632:ω 2624:⁡ 2618:− 2609:ω 2603:ϕ 2555:ω 2503:ω 2486:⁡ 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−1 ms 3164:LTspice 3142:⁄ 3094:Circuit 870:is the 404:digital 276:radians 272:degrees 183:signals 9733:Optics 9613: 9557: 9384: 9244: 9171: 9040: 9015: 8990: 8962: 8937: 8879: 8377:. The 8203:linear 4646:where 3762:,total 3746:linear 3622:linear 3541:signal 3511:In an 3506:optics 3352:(kHz) 3236:10 kHz 2707:arctan 2621:arctan 2480: 2453: 2393: 2386: 2342: 2300: 2281: 2207: 2188: 2124: 1724:. The 1656:, and 1372: 1299: 874:, and 718:where 390:Theory 172:, and 9738:Waves 9555:S2CID 9494:(PDF) 9479:(PDF) 9435:(PDF) 9404:(PDF) 9382:S2CID 9338:(PDF) 9331:(PDF) 9301:(PDF) 9290:(PDF) 9169:S2CID 9143:arXiv 9111:(PDF) 9104:(PDF) 9072:(PDF) 8837:from 8543:, is 4268:from 3537:delay 3521:power 3357:(ms) 3234:) to 3198:from 3149:1 kHz 3135:−1 ms 3096:with 2543:with 1894:In a 1807:slope 1474:Here 976:fully 102:of a 82:, or 9635:IEEE 9611:ISBN 9242:PMID 9086:link 9038:ISBN 9013:ISBN 8988:ISBN 8960:ISBN 8935:ISBN 8877:ISBN 4650:and 4648:even 3768:and 3733:and 3714:The 3525:mode 3517:time 3373:1.6 3288:1 ms 3269:0 ms 3200:1 Hz 2669:is: 2593:is: 1927:and 1801:The 1623:The 1026:> 769:and 402:and 139:and 48:time 38:and 9588:doi 9545:doi 9424:doi 9372:doi 9268:doi 9234:doi 9161:doi 8319:of 4652:odd 4124:. 3940:|. 3722:in 3527:'s 3417:16 3403:1.5 3370:3.2 3367:0.5 3131:−RC 2483:arg 2430:is 1407:cos 1380:env 1267:env 1249:log 1172:env 1118:cos 1104:env 1012:env 647:def 573:def 406:or 274:or 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7438:e 7435:d 7431:o 7427:P 7413:= 7408:n 7405:e 7402:d 7398:e 7394:P 7338:+ 7333:2 7325:) 7316:( 7313:3 7310:= 7305:n 7302:e 7299:d 7295:o 7291:D 7280:) 7274:( 7271:2 7263:3 7255:) 7249:( 7246:4 7243:= 7238:n 7235:e 7232:d 7228:e 7224:D 7210:+ 7205:3 7191:= 7186:n 7183:e 7180:d 7176:o 7172:P 7164:1 7161:+ 7156:2 7140:4 7129:= 7124:n 7121:e 7118:d 7114:e 7110:P 7078:1 7075:+ 7072:) 7066:j 7063:( 7057:+ 7052:2 7048:) 7041:j 7038:( 7032:+ 7027:3 7023:) 7016:j 7013:( 7007:+ 7002:4 6998:) 6991:j 6988:( 6981:1 6976:= 6973:) 6967:j 6964:( 6959:4 6955:T 6921:c 6918:e 6915:s 6912:) 6907:n 6904:e 6901:d 6897:2 6893:P 6890:( 6887:d 6884:g 6878:) 6873:m 6870:u 6867:n 6863:2 6859:P 6856:( 6853:d 6850:g 6847:= 6844:) 6841:2 6838:P 6835:( 6832:d 6829:g 6826:= 6799:) 6796:2 6793:P 6790:( 6781:) 6778:1 6775:P 6772:( 6766:= 6763:) 6760:2 6757:P 6753:/ 6749:1 6746:P 6743:( 6716:) 6713:S 6710:( 6705:n 6702:e 6699:d 6695:P 6690:/ 6686:) 6683:S 6680:( 6675:m 6672:u 6669:n 6665:P 6661:= 6658:) 6655:S 6652:( 6649:2 6646:P 6611:) 6606:2 6602:o 6598:S 6595:+ 6590:2 6586:e 6582:S 6579:( 6574:) 6571:o 6568:D 6562:e 6559:S 6556:+ 6553:e 6550:D 6544:o 6541:S 6538:( 6529:= 6520:d 6515:) 6509:( 6503:d 6494:= 6491:) 6488:) 6482:j 6479:( 6476:P 6473:( 6470:d 6467:g 6464:= 6437:) 6434:S 6431:( 6428:P 6397:2 6393:e 6389:S 6384:e 6381:D 6378:o 6375:S 6369:o 6366:D 6363:e 6360:S 6352:= 6344:x 6341:d 6336:) 6333:x 6330:( 6327:f 6324:d 6309:2 6303:k 6300:2 6296:) 6289:( 6284:1 6278:k 6274:) 6270:1 6264:( 6259:1 6253:k 6250:2 6246:P 6241:) 6238:1 6232:k 6229:2 6226:( 6220:2 6216:/ 6212:) 6209:1 6206:+ 6203:N 6200:( 6195:1 6192:= 6189:k 6179:= 6172:2 6166:k 6163:2 6159:) 6152:j 6149:( 6144:1 6138:k 6135:2 6131:P 6126:) 6123:1 6117:k 6114:2 6111:( 6105:2 6101:/ 6097:) 6094:1 6091:+ 6088:N 6085:( 6080:1 6077:= 6074:k 6064:= 6056:x 6053:d 6048:) 6043:d 6040:d 6037:o 6033:) 6026:j 6023:( 6020:P 6017:( 6014:d 6008:= 6005:o 6002:D 5993:1 5987:k 5984:2 5980:) 5973:( 5968:1 5962:k 5958:) 5954:1 5948:( 5943:k 5940:2 5936:P 5932:k 5929:2 5924:2 5920:/ 5916:N 5911:1 5908:= 5905:k 5895:= 5888:1 5882:k 5879:2 5875:) 5868:j 5865:( 5860:k 5857:2 5853:P 5849:k 5846:2 5841:2 5837:/ 5833:N 5828:1 5825:= 5822:k 5814:j 5806:= 5798:x 5795:d 5790:) 5785:n 5782:e 5779:v 5776:e 5772:) 5765:j 5762:( 5759:P 5756:( 5753:d 5747:= 5744:e 5741:D 5732:1 5726:k 5723:2 5719:) 5712:( 5707:1 5701:k 5697:) 5693:1 5687:( 5682:1 5676:k 5673:2 5669:P 5663:2 5659:/ 5655:) 5652:1 5649:+ 5646:N 5643:( 5638:1 5635:= 5632:k 5622:= 5615:1 5609:k 5606:2 5602:) 5595:j 5592:( 5587:1 5581:k 5578:2 5574:P 5568:2 5564:/ 5560:) 5557:1 5554:+ 5551:N 5548:( 5543:1 5540:= 5537:k 5529:j 5521:= 5514:d 5511:d 5508:o 5504:) 5497:j 5494:( 5491:P 5488:= 5485:o 5482:S 5473:k 5470:2 5466:) 5459:( 5454:k 5450:) 5446:1 5440:( 5435:k 5432:2 5428:P 5422:2 5418:/ 5414:N 5409:0 5406:= 5403:k 5393:= 5386:k 5383:2 5379:) 5372:j 5369:( 5364:k 5361:2 5357:P 5351:2 5347:/ 5343:N 5338:0 5335:= 5332:k 5322:= 5315:n 5312:e 5309:v 5306:e 5302:) 5295:j 5292:( 5289:P 5286:= 5283:e 5280:S 5243:2 5238:n 5235:e 5232:v 5229:e 5225:) 5218:j 5215:( 5212:P 5204:x 5201:d 5196:) 5191:n 5188:e 5185:v 5182:e 5178:) 5171:j 5168:( 5165:P 5162:j 5156:( 5153:d 5145:d 5142:d 5139:o 5135:) 5128:j 5125:( 5122:P 5119:j 5107:x 5104:d 5099:) 5094:d 5091:d 5088:o 5084:) 5077:j 5074:( 5071:P 5068:( 5065:d 5057:n 5054:e 5051:v 5048:e 5044:) 5037:j 5034:( 5031:P 5025:= 5019:x 5016:d 5011:) 5008:x 5005:( 5002:f 4999:d 4983:n 4980:e 4977:v 4974:e 4970:) 4963:j 4960:( 4957:P 4950:d 4947:d 4944:o 4940:) 4933:j 4930:( 4927:P 4924:j 4915:= 4912:) 4909:x 4906:( 4903:f 4890:2 4886:) 4882:x 4879:( 4876:f 4873:+ 4870:1 4865:x 4862:d 4858:/ 4854:) 4851:x 4848:( 4845:f 4842:d 4836:= 4830:x 4827:d 4822:) 4819:) 4816:x 4813:( 4810:f 4807:( 4802:1 4795:n 4791:a 4788:t 4785:d 4753:d 4750:d 4747:o 4743:) 4736:j 4733:( 4730:P 4708:d 4705:d 4702:o 4698:) 4691:j 4688:( 4685:P 4665:j 4634:) 4629:n 4626:e 4623:v 4620:e 4616:) 4609:j 4606:( 4603:P 4599:/ 4593:d 4590:d 4587:o 4583:) 4576:j 4573:( 4570:P 4567:j 4561:( 4556:1 4549:n 4545:a 4542:t 4539:= 4536:) 4530:( 4507:) 4501:( 4476:1 4473:+ 4470:N 4467:2 4463:j 4440:N 4437:2 4433:j 4412:) 4407:l 4404:a 4401:e 4398:r 4394:) 4387:j 4384:( 4381:P 4377:/ 4371:g 4368:a 4365:m 4362:i 4358:) 4351:j 4348:( 4345:P 4342:( 4337:1 4330:n 4326:a 4323:t 4320:= 4317:) 4311:( 4288:) 4282:j 4279:( 4276:P 4256:) 4250:( 4221:d 4217:/ 4212:| 4208:) 4202:( 4196:d 4173:) 4167:j 4164:( 4161:P 4141:) 4135:( 4112:) 4106:j 4103:( 4100:P 4077:j 4057:) 4054:S 4051:( 4048:P 4012:C 4008:L 3992:C 3989:L 3983:/ 3979:1 3937:y 3935:, 3933:t 3929:t 3924:x 3922:, 3920:t 3916:t 3911:t 3907:D 3902:t 3898:D 3878:2 3874:) 3868:y 3864:t 3855:y 3852:, 3849:i 3845:E 3841:( 3838:+ 3833:2 3829:) 3823:x 3819:t 3810:x 3807:, 3804:i 3800:E 3796:( 3793:= 3788:T 3784:E 3770:y 3766:x 3760:T 3756:E 3735:Y 3731:X 3697:g 3667:g 3656:) 3653:0 3650:( 3644:= 3641:) 3635:( 3598:d 3590:d 3581:= 3578:) 3572:( 3567:g 3480:. 3466:g 3445:= 3414:2 3411:8 3400:4 3392:1 3389:2 3381:2 3378:1 3310:B 3306:1 3252:g 3238:( 3217:g 3203:( 3183:g 3151:. 3140:1 3114:g 3059:. 3053:2 3049:C 3046:R 3040:= 3032:o 3024:2 3020:1 3015:= 3012:) 3007:o 2998:= 2991:( 2986:g 2955:. 2951:C 2948:R 2945:= 2938:o 2930:1 2918:) 2913:o 2899:( 2894:g 2856:. 2847:2 2842:o 2834:+ 2829:2 2818:o 2808:= 2801:) 2795:( 2790:g 2737:. 2733:) 2726:o 2713:( 2699:2 2691:= 2688:) 2682:( 2651:. 2647:) 2640:o 2627:( 2615:= 2612:) 2606:( 2578:C 2575:R 2571:1 2565:= 2559:o 2515:. 2510:} 2506:) 2500:i 2497:( 2494:H 2490:{ 2472:f 2469:e 2466:d 2460:= 2450:) 2444:( 2380:) 2376:) 2370:( 2364:+ 2361:t 2354:( 2350:i 2346:e 2337:| 2332:) 2326:i 2323:( 2320:H 2315:| 2310:= 2295:t 2289:i 2285:e 2277:) 2271:) 2265:( 2259:i 2255:e 2249:| 2244:) 2238:i 2235:( 2232:H 2227:| 2221:( 2217:= 2202:t 2196:i 2192:e 2185:) 2179:i 2176:( 2173:H 2170:= 2163:) 2160:t 2157:( 2154:y 2119:t 2113:i 2109:e 2105:= 2102:) 2099:t 2096:( 2093:x 2065:) 2062:s 2059:( 2056:H 1912:g 1879:. 1869:d 1864:) 1858:( 1852:d 1843:= 1840:) 1834:( 1829:g 1786:. 1776:) 1770:( 1758:= 1755:) 1749:( 1711:) 1705:( 1641:g 1600:g 1488:g 1459:. 1453:) 1445:+ 1442:) 1426:t 1423:( 1415:( 1404:) 1399:g 1388:t 1385:( 1376:A 1367:| 1362:) 1356:i 1353:( 1350:H 1345:| 1340:= 1337:) 1334:t 1331:( 1328:y 1302:. 1289:| 1283:) 1278:) 1275:t 1272:( 1263:A 1257:( 1243:t 1240:d 1236:d 1230:| 1183:) 1180:t 1177:( 1168:A 1143:. 1139:) 1133:+ 1130:t 1124:( 1115:) 1112:t 1109:( 1100:A 1096:= 1093:) 1090:t 1087:( 1084:x 1060:) 1057:t 1054:( 1051:x 1029:0 1023:) 1020:t 1017:( 1008:A 961:) 958:t 955:( 952:h 926:) 923:s 920:( 917:H 894:1 885:L 868:s 853:) 850:t 847:( 844:h 822:) 819:t 816:( 813:x 787:) 784:s 781:( 778:H 756:) 753:s 750:( 747:X 703:, 699:} 696:) 693:s 690:( 687:X 683:) 680:s 677:( 674:H 671:{ 666:1 657:L 642:= 631:d 626:) 620:( 617:x 613:) 604:t 601:( 598:h 568:= 561:) 558:t 555:( 552:) 549:x 543:h 540:( 537:= 534:) 531:t 528:( 525:y 497:) 494:t 491:( 488:x 466:) 463:t 460:( 457:h 428:) 425:t 422:( 419:y 212:) 209:x 206:( 20:)

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Knowledge

Group delay and phase delay

Index