NASA Technical Reports Server (NTRS) 20110024111: Analytic Method for Computing Instrument Pointing Jitter

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by  NASA Technical Reports Server (NTRS)

Quantum Search in Hilbert Space
A large database would be searched in one quan- NASA’s Jet Propulsion Laboratory,
tum computing operation. Pasadena, California
A proposed quantum-computing algo- unitary matrix, from which one would derive memory space exponentially large by use of
rithm would perform a search for an item the Hamiltonian operator (and hence the only linear resources. However, inasmuch as
of information in a database stored in a evolutionary operator) of a quantum sys- the necessary preprocessing (the mapping
Hilbert-space memory structure. The algo- tem. In other words, all the stored informa- of the stored information into a Hilbert space)
rithm is intended to make it possible to tion would be mapped onto a unitary oper- could be exponentially expensive, the pro-
search relatively quickly through a large ator acting on a quantum state that would posed algorithm would likely be most bene-
database under conditions in which avail- represent the item of information to be ficial in applications in which the resources
able computing resources would other- retrieved. Then one could exploit quantum available for preprocessing were much
wise be considered inadequate to perform parallelism: one could pose all search greater than those available for searching.
such a task. queries simultaneously by performing a This work was done by Michail Zak of
The algorithm would apply, more specifi- quantum measurement on the system. In Caltech for NASA’s Jet Propulsion
cally, to a relational database in which infor- so doing, one would effectively solve the Laboratory. For further information, access
mation would be stored in a set of Ncom- search problem in one computational step. the Technical Support Package (TSP) free
plex orthonormal vectors, each of Ndimen- One could exploit the direct- and inner- on-line at www.nasatech.com.
sions (where Ncan be exponentially large). product decomposability of the unitary NPO-30193
Each vector would constitute one row of a matrix to make the dimensionality of the
Analytic Method for Computing Instrument Pointing Jitter
Jitter can be computed more efficiently. NASA’s Jet Propulsion Laboratory,
Pasadena, California
A new method of calculating the root-
mean-square (rms) pointing jitter of a
er
scientific instrument (e.g., a camera, et
m
radar antenna, or telescope) is intro- a
ar
duced based on a state-space concept. P
In comparison with the prior method of ntrol
calculating the rms pointing jitter, the Co
present method involves significantly
less computation.
The rms pointing jitter of an instru-
e
ment (the square root of the jitter vari- u
al
ance shown in the figure) is an important V
or
physical quantity which impacts the s
n
design of the instrument, its actuators, Se
controls, sensory components, and sen- aw
R
sor-output-sampling circuitry. Using the
Sirlin, San Martin, and Lucke definition
of pointing jitter, the prior method of
computing the rms pointing jitter
on Correlated Uncorrelated
involves a frequency-domain integral of cti Events Event (Fault)
e
a rational polynomial multiplied by a et
D
transcendental weighting function, nt
necessitating the use of numerical-inte- ve
E
gration techniques. In practice, numeri-
cal integration complicates the problem
Time
of calculating the rms pointing error. In
contrast, the state-space method pro-
Instantaneous and Statistical Quantitiesare used to characterize the pointing of an instrument
vides exact analytic expressions that (that is, rotation of the instrument about an axis). The quantities shown here pertain to a pointing
can be evaluated without numerical inte- process y(t) at instant of time tduring an observation interval (window) of duration T that starts
gration. at time τ, E[] is an expectation operator denoting the ensemble average of the bracketed term, n(t)
The theoretical foundation of the is a zero-mean white-noise process, and Cov[] is an ensemble-average covariance operator.
state-space method includes a repre-
sentation of the pointing process as a state-space formulation results in the culation of a matrix exponential.
stationary process generated by a state- replacement of the aforementioned Additional simplifications may be possi-
space model driven by white noise. The weighted frequency integral with the cal- ble in certain applications by taking
56 NASA Tech Briefs, January 2003
advantage of well-known matrix expo- simplifying the calculations, the closed- Laboratory. For further information,
nential expressions and/or inverse form expressions provide insight into physi- access the Technical Support Package
Laplace transform relationships. Two cal mechanisms of jitter. (TSP) free on-line at www.nasatech.com.
useful examples of such simplifications This work was done by David Bayard NPO-30525
are given in the report. In addition to of Caltech for NASA’s Jet Propulsion
NASA Tech Briefs, January 2003 57