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1193:{\displaystyle h_{\mathrm {ZOH} }(t)\,={\frac {1}{T}}\mathrm {rect} \left({\frac {t}{T}}-{\frac {1}{2}}\right)={\begin{cases}{\frac {1}{T}}&{\text{if }}0\leq t<T\\0&{\text{otherwise}}\end{cases}}}
456:
609:
794:{\displaystyle {\begin{aligned}x_{s}(t)&=\sum _{n=-\infty }^{\infty }x\cdot \delta \left({\frac {t-nT}{T}}\right)\\&{}=T\sum _{n=-\infty }^{\infty }x\cdot \delta (t-nT).\end{aligned}}}
1523:
1493:
834:) is equal to the mean value of the samples, so that the lowpass filter needed will have a DC gain of 1. Some authors use this scaling, while many others omit the time-scaling and the
384:
500:
1441:
1671:) (that, if ideally low-pass filtered, would result in the unique underlying bandlimited signal before sampling), but instead output a sequence of rectangular pulses,
821:
523:
522:
scaled to the sample values. The filter can then be analyzed in the frequency domain, for comparison with other reconstruction methods such as the
37:
1396:{\displaystyle H_{\mathrm {ZOH} }(f)={\mathcal {F}}\{h_{\mathrm {ZOH} }(t)\}={\frac {1-e^{-i2\pi fT}}{i2\pi fT}}=e^{-i\pi fT}\mathrm {sinc} (fT)}
1719:
527:
1815:
1020:{\displaystyle x_{\mathrm {ZOH} }(t)=\sum _{n=-\infty }^{\infty }x\cdot \mathrm {rect} \left({\frac {t-nT}{T}}-{\frac {1}{2}}\right)}
1793:
1768:
860:). It is identical to the rect function of Figure 1, except now scaled to have an area of 1 so the filter will have a DC gain of 1.
396:
1686:
function), means that there is an inherent effect of the ZOH on the effective frequency response of the DAC, resulting in a mild
601:
Begin by defining a continuous-time signal from the sample values, as above but using delta functions instead of rect functions:
1820:
95:
52:
339:{\displaystyle x_{\mathrm {ZOH} }(t)\,=\sum _{n=-\infty }^{\infty }x\cdot \mathrm {rect} \left({\frac {t-T/2-nT}{T}}\right)}
67:
1203:
74:
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129:
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579:
what a DAC does in reality, the DAC output can be modeled by applying the hypothetical sequence of dirac impulses,
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81:
140:
by holding each sample value for one sample interval. It has several applications in electrical communication.
349:
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63:
461:
561:
137:
1643:{\displaystyle H_{\mathrm {ZOH} }(s)={\mathcal {L}}\{h_{\mathrm {ZOH} }(t)\}\,={\frac {1-e^{-sT}}{sT}}\ }
1406:
125:
152:
Figure 1. The time-shifted and time-scaled rect function used in the time-domain analysis of the ZOH.
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133:
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823:, which arises naturally by time-scaling the delta function, has the result that the mean value of
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A zero-order hold reconstructs the following continuous-time waveform from a sample sequence
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The equation above for the output of the ZOH can also be modeled as the output of a
156:
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with impulse response equal to a rect function, and with input being a sequence of
598:) so that each input impulse results in the correct constant pulse in the output.
1694:, corresponding to a gain of sinc(1/2) = 2/Ď€). This drop is a consequence of the
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with such characteristics (which, for an LTI system, are fully described by the
26:
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175:
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44:
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Discretization of linear state space models (assuming zero-order hold)
18:
Model of signal reconstruction in digital-to-analog (DAC) converters
844:
174:
155:
147:
1690:
of gain at the higher frequencies (a 3.9224 dB loss at the
451:{\displaystyle \mathrm {rect} \left({\frac {t-T/2}{T}}\right)}
20:
1559:
1247:
1186:
842:, and hence dependent on the units of measurement of time.
838:, resulting in a low-pass filter model with a DC gain of
132:(DAC). That is, it describes the effect of converting a
883:)to the piecewise-constant signal (shown in Figure 2):
872:
that converts the sequence of modulated Dirac impulses
48:
1755:. Springer International Publishing AG. p. 459.
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534:or linear interpolation between sample values.
849:Figure 4. Impulse response of zero-order hold
8:
1594:
1564:
1495:commonly used in digital signal processing.
1488:{\displaystyle {\frac {\sin(\pi x)}{\pi x}}}
1282:
1252:
53:introducing citations to additional sources
124:) is a mathematical model of the practical
1752:Rudiments of Signal Processing and Systems
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1202:The effective frequency response is the
864:The zero-order hold is the hypothetical
198:, assuming one sample per time interval
43:Relevant discussion may be found on the
1741:
1698:property of a conventional DAC, and is
524:Whittaker–Shannon interpolation formula
379:{\displaystyle \mathrm {rect} (\cdot )}
552:), representing the discrete samples,
495:{\displaystyle x_{\mathrm {ZOH} }(t)}
160:Figure 2. Piecewise-constant signal
7:
1505:of the ZOH is found by substituting
1706:that might precede a conventional
1656:(DAC) do not output a sequence of
1579:
1576:
1573:
1539:
1536:
1533:
1436:{\displaystyle \mathrm {sinc} (x)}
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179:Figure 3. A modulated Dirac comb
14:
1720:Nyquist–Shannon sampling theorem
528:Nyquist–Shannon sampling theorem
36:relies largely or entirely on a
25:
1788:(fifth ed.). McGraw-Hill.
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628:
622:
537:In this method, a sequence of
489:
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273:
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227:
1:
592:linear, time-invariant filter
506:signal depicted in Figure 2.
458:is depicted in Figure 1, and
1654:digital-to-analog converters
1204:continuous Fourier transform
516:linear time-invariant filter
1786:Principles of Digital Audio
1708:analog-to-digital converter
130:digital-to-analog converter
1847:
1027:resulting in an effective
1816:Digital signal processing
1761:10.1007/978-3-030-76947-5
1206:of the impulse response.
1784:Ken C. Pohlmann (2000).
1652:The fact that practical
1031:(shown in Figure 4) of:
128:done by a conventional
1821:Electrical engineering
1644:
1489:
1437:
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1194:
1021:
943:
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562:continuous-time signal
510:Frequency-domain model
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138:continuous-time signal
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126:signal reconstruction
1749:Tom J. Moir (2022).
1524:
1450:
1443:is the (normalized)
1407:
1212:
1035:
887:
807:
605:
575:Even though this is
462:
397:
388:rectangular function
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206:
134:discrete-time signal
49:improve this article
1684:piecewise constant
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504:piecewise-constant
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1831:Signal processing
1692:Nyquist frequency
1639:
1635:
1503:transfer function
1500:Laplace transform
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1145:
1119:
1106:
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997:
816:{\displaystyle T}
703:
558:low-pass filtered
530:, or such as the
526:suggested by the
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330:
144:Time-domain model
114:
113:
99:
64:"Zero-order hold"
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1725:First-order hold
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1029:impulse response
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596:impulse response
532:first-order hold
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1704:sample and hold
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803:The scaling by
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118:zero-order hold
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30:
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5:
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1826:Control theory
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1658:dirac impulses
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539:Dirac impulses
520:dirac impulses
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47:. Please help
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1795:0-07-144156-5
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1770:9783030769475
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1445:sinc function
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560:to recover a
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393:The function
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66: –
65:
61:
60:Find sources:
54:
50:
46:
40:
39:
38:single source
34:This article
32:
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23:
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16:
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92:
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59:
35:
15:
1702:due to the
105:August 2021
1810:Categories
1736:References
870:LTI system
75:newspapers
1616:−
1608:−
1477:π
1466:π
1460:
1357:π
1351:−
1331:π
1312:π
1303:−
1295:−
1180:otherwise
1158:≤
1109:−
1000:−
985:−
957:⋅
940:∞
935:∞
932:−
922:∑
773:−
764:δ
761:⋅
744:∞
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