73:. Thus a non-time variable jumps from one value to another as time moves from one time period to the next. This view of time corresponds to a digital clock that gives a fixed reading of 10:37 for a while, and then jumps to a new fixed reading of 10:38, etc. In this framework, each variable of interest is measured once at each time period. The number of measurements between any two time periods is finite. Measurements are typically made at sequential
58:
1403:
of the same length as every other time period, and the measured variable is plotted as a height that stays constant throughout the region of the time period. In this graphical technique, the graph appears as a sequence of horizontal steps. Alternatively, each time period can be viewed as a detached
216:) will have some value at every instant of time. The electrical signals derived in proportion with the physical quantities such as temperature, pressure, sound etc. are generally continuous signals. Other examples of continuous signals are sine wave, cosine wave, triangular wave etc.
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Moreover, when a researcher attempts to develop a theory to explain what is observed in discrete time, often the theory itself is expressed in discrete time in order to facilitate the development of a time series or regression model.
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The signal is defined over a domain, which may or may not be finite, and there is a functional mapping from the domain to the value of the signal. The continuity of the time variable, in connection with the law of density of
68:
views values of variables as occurring at distinct, separate "points in time", or equivalently as being unchanged throughout each non-zero region of time ("time period")—that is, time is viewed as a
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actually occurs continuously, there being no moment when the economy is totally in a pause, it is only possible to measure economic activity discretely. For this reason, published data on, for example,
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point in time, usually at an integer value on the horizontal axis, and the measured variable is plotted as a height above that time-axis point. In this technique, the graph appears as a set of dots.
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Continuous signal may also be defined over an independent variable other than time. Another very common independent variable is space and is particularly useful in
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In many disciplines, the convention is that a continuous signal must always have a finite value, which makes more sense in the case of physical signals.
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from a continuous-time signal. When a discrete-time signal is obtained by sampling a sequence at uniformly spaced times, it has an associated
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Unlike a continuous-time signal, a discrete-time signal is not a function of a continuous argument; however, it may have been obtained by
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For some purposes, infinite singularities are acceptable as long as the signal is integrable over any finite interval (for example, the
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methods in which variables are indexed with a subscript indicating the time period in which the observation occurred. For example,
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When one attempts to empirically explain such variables in terms of other variables and/or their own prior values, one uses
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By observing an inherently discrete-time process, such as the weekly peak value of a particular economic indicator.
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an exact description requires the use of continuous time. In a continuous time context, the value of a variable
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are involved, because normally it is only possible to measure variables sequentially. For example, while
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Discrete-time signals may have several origins, but can usually be classified into one of two groups:
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The value of a finite (or infinite) duration signal may or may not be finite. For example,
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is the positive speed-of-adjustment parameter which is less than or equal to 1, and where
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in response to non-zero excess demand for a product can be modeled in continuous time as
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A signal of continuous amplitude and time is known as a continuous-time signal or an
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is the speed-of-adjustment parameter which can be any positive finite number, and
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of the price with respect to time (that is, the rate of change of the price),
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1490:"Digital Signal Processing: Instant access", Butterworth-Heinemann - page 8
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number of other points in time. The variable "time" ranges over the entire
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is a variable in the range from 0 to 1 inclusive whose value in period
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The values of a variable measured in continuous time are plotted as a
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short amount of time. Between any two points in time there are an
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to the value of income observed in the third time period, etc.
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is a finite duration signal but it takes an infinite value for
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A finite duration counterpart of the above signal could be:
30:"Discrete signal" redirects here. Not to be confused with
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A variable measured in discrete time can be plotted as a
138:
views variables as having a particular value only for an
1481:"Digital Signal Processing", Prentice Hall - pages 11–12
1333:{\displaystyle {\frac {dP}{dt}}=\lambda \cdot f(P,...)}
1197:{\displaystyle P_{t+1}=P_{t}+\delta \cdot f(P_{t},...)}
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Frameworks for modeling variables that evolve over time
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A typical example of an infinite duration signal is:
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at constant or variable rate. This process is called
285:{\displaystyle f(t)=\sin(t),\quad t\in \mathbb {R} }
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45:are two alternative frameworks within which
1104:Another example models the adjustment of a
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617:Any analog signal is continuous by nature.
173:) whose domain, which is often time, is a
584:signal is not integrable at infinity, but
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366:{\displaystyle f(t)=\sin(t),\quad t\in }
185:). That is, the function's domain is an
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1094:{\displaystyle x_{3}=4(8/9)(1/9)=32/81}
1520:Wagner, Thomas Charles Gordon (1959).
860:affects its value in the next period,
1387:is again the excess demand function.
1009:{\displaystyle x_{2}=4(1/3)(2/3)=8/9}
648:Discrete time is often employed when
189:. The function itself need not to be
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692:observed in unspecified time period
1501:The Nature of mathematical Modeling
49:that evolve over time are modeled.
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77:values of the variable "time".
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1343:where the left side is the
753:Discrete time makes use of
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114:By acquiring values of an
37:In mathematical dynamics,
29:
924:{\displaystyle x_{1}=1/3}
761:or logistic equation, is
623:digital signal processing
1360:{\displaystyle \lambda }
1111:in response to non-zero
664:will show a sequence of
1220:{\displaystyle \delta }
633:of continuous signals.
61:Discrete sampled signal
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607:{\displaystyle t^{-2}}
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86:discrete-time signal
1462:Time-scale calculus
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1391:Graphical depiction
883:{\displaystyle r=4}
152:continuous variable
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1437:Discrete calculus
1427:Bernoulli process
1380:{\displaystyle f}
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1115:for a product as
657:economic activity
644:Relevant contexts
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71:discrete variable
32:Discrete variable
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1020:=2 we have
935:=1 we have
931:, then for
858:nonlinearly
673:time series
90:time series
1533:Categories
1469:References
1016:, and for
677:regression
621:, used in
517:otherwise,
408:otherwise.
212:. This (a
191:continuous
1355:λ
1304:⋅
1301:λ
1215:δ
1161:⋅
1158:δ
847:parameter
841:in which
810:−
714:tractable
666:quarterly
650:empirical
597:−
567:−
455:∈
358:π
352:π
349:−
343:∈
324:
275:∈
256:
199:countable
179:connected
177:(e.g., a
175:continuum
47:variables
1524:. Wiley.
1422:Aliasing
1415:See also
668:values.
627:sampling
167:quantity
144:infinite
120:sampling
101:sampling
94:sequence
1247:is the
722:physics
75:integer
1507:
1207:where
690:income
214:signal
171:signal
1106:price
845:is a
614:is).
183:reals
161:or a
88:is a
1505:ISBN
890:and
629:and
482:and
373:and
41:and
675:or
548:.
321:sin
253:sin
169:(a
84:or
1535::
1251:.
1101:.
1089:81
1081:32
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696:,
205:.
157:A
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107:.
80:A
1513:.
1375:f
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1325:.
1322:.
1319:.
1316:,
1313:P
1310:(
1307:f
1298:=
1292:t
1289:d
1284:P
1281:d
1265:P
1235:f
1192:)
1189:.
1186:.
1183:.
1180:,
1175:t
1171:P
1167:(
1164:f
1155:+
1150:t
1146:P
1142:=
1137:1
1134:+
1131:t
1127:P
1109:P
1085:/
1078:=
1075:)
1072:9
1068:/
1064:1
1061:(
1058:)
1055:9
1051:/
1047:8
1044:(
1041:4
1038:=
1033:3
1029:x
1018:t
1004:9
1000:/
996:8
993:=
990:)
987:3
983:/
979:2
976:(
973:)
970:3
966:/
962:1
959:(
956:4
953:=
948:2
944:x
933:t
919:3
915:/
911:1
908:=
903:1
899:x
878:4
875:=
872:r
862:t
855:t
851:x
843:r
826:,
823:)
818:t
814:x
807:1
804:(
799:t
795:x
791:r
788:=
783:1
780:+
777:t
773:x
738:y
734:t
732:(
730:y
726:y
702:3
698:y
694:t
685:t
681:y
600:2
593:t
570:1
563:t
535:0
532:=
529:t
505:0
502:=
499:)
496:t
493:(
490:f
470:]
467:1
464:,
461:0
458:[
452:t
448:,
443:t
440:1
435:=
432:)
429:t
426:(
423:f
396:0
393:=
390:)
387:t
384:(
381:f
361:]
355:,
346:[
340:t
336:,
333:)
330:t
327:(
318:=
315:)
312:t
309:(
306:f
279:R
272:t
268:,
265:)
262:t
259:(
250:=
247:)
244:t
241:(
238:f
122:.
34:.
20:)
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