The shah function is defined by
where 
is the delta function, so 
for 
(i.e., 
is not an integer). The shah function
is also called the sampling symbol or replicating symbol (Bracewell 1999, p. 77),
and is implemented in the Wolfram Language
as DiracComb[x].
It obeys the identities
The shah function is normalized so that
 |
(7) |
The "sampling property" is
 |
(8) |
and the "replicating property" is
 |
(9) |
where 
denotes convolution.
The two-dimensional sampling function, sometimes called the bed-of-nails function, is given by
 |
(10) |
which can be adjusted using a series of weights as
 |
(11) |
where 
is a reliability weight, 
is a density weight (weighting function), and 
is a taper. The two-dimensional shah function satisfies
 |
(12) |
(Bracewell 1999, p. 85).
See also
Convolution, Delta Function, Impulse Pair, Sinc Function
Explore with Wolfram|Alpha
References
Bracewell, R. "The Sampling or Replicating Symbol 
." In The
Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 77-79
and 85, 1999.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Shah Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ShahFunction.html