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# m27 - The Shah Function

Module by: C. Sidney Burrus. E-mail the author

Summary: If distributions are allowed, the Fourier transform of an infinite string of delta functions is an infinite string of delta function. These are called shah functions because of the similarity to the Russion letter.

## Sampling Functions --- the Shah Function

Th preceding discussions used traditional Fourier techniques to develop sampling tools. If distributions or delta functions are allowed, the Fourier transform will exist for a much larger class of signals. One should take care when using distributions as if they were functions but it is a very powerful extension.

There are several functions which have equally spaced sequences of impulses that can be used as tools in deriving a sampling formula. These are called pitch fork" functions, picket fence functions, comb functions and shah functions. We start first with a finite length sequence to be used with the DFT. We define

⨿ M ( n ) = m = 0 L 1 δ ( n M m ) ⨿ M ( n ) = m = 0 L 1 δ ( n M m )
(1)
where N = L M N = L M . D F T { ⨿ M ( n ) } = n = 0 N 1 [ m = 0 L 1 δ ( n M m ) ] e j 2 π n k / N D F T { ⨿ M ( n ) } = n = 0 N 1 [ m = 0 L 1 δ ( n M m ) ] e j 2 π n k / N = m = 0 L 1 [ n = 0 N 1 δ ( n M m ) e j 2 π n k / N ] = m = 0 L 1 [ n = 0 N 1 δ ( n M m ) e j 2 π n k / N ] = m = 0 L 1 e j 2 π M m k / N = m = 0 L 1 e j 2 π m k / L = m = 0 L 1 e j 2 π M m k / N = m = 0 L 1 e j 2 π m k / L = { L < k > L = 0 0 otherwise = { L < k > L = 0 0 otherwise
= L l = 0 M 1 δ ( k L l ) = L ⨿ L ( k ) = L l = 0 M 1 δ ( k L l ) = L ⨿ L ( k )
(2)
For the DTFT we have a similar derivation: D T F T { ⨿ M ( n ) } = n = [ m = 0 L 1 δ ( n M m ) ] e j ω n D T F T { ⨿ M ( n ) } = n = [ m = 0 L 1 δ ( n M m ) ] e j ω n = m = 0 L 1 [ n = δ ( n M m ) e j ω n ] = m = 0 L 1 [ n = δ ( n M m ) e j ω n ] = m = 0 L 1 e j ω M m = m = 0 L 1 e j ω M m = { ω = k 2 π / M 0 otherwise = { ω = k 2 π / M 0 otherwise
= K l = 0 M 1 δ ( ω 2 π l / M l ) = K ⨿ 2 π / M ( ω ) = K l = 0 M 1 δ ( ω 2 π l / M l ) = K ⨿ 2 π / M ( ω )
(3)
where K K is a constant.

An alternate derivation for the DTFT uses the inverse DTFT. I D T F T { ⨿ 2 π / M ( ω ) } = 1 2 π π π ⨿ 2 π / M ( ω ) e j ω n ω I D T F T { ⨿ 2 π / M ( ω ) } = 1 2 π π π ⨿ 2 π / M ( ω ) e j ω n ω

= 1 2 π π π l δ ( ω 2 π l / M ) e j ω n ω = 1 2 π π π l δ ( ω 2 π l / M ) e j ω n ω
(4)
= 1 2 π l π π δ ( ω 2 π l / M ) e j ω n ω = 1 2 π l π π δ ( ω 2 π l / M ) e j ω n ω = 1 2 π l = 0 M 1 e 2 π l n / M = { M / 2 π n = M 0 otherwise = 1 2 π l = 0 M 1 e 2 π l n / M = { M / 2 π n = M 0 otherwise
= ( M 2 π ) ⨿ M ( n ) = ( M 2 π ) ⨿ M ( n )
(5)
Therefore,
⨿ M ( n ) ( 2 π M ) ⨿ 2 π / M ( ω ) ⨿ M ( n ) ( 2 π M ) ⨿ 2 π / M ( ω )
(6)

For regular Fourier transform, we have a string of impulse functions in both the time and frequency. This we see from:

F T { ⨿ T ( t ) } = n δ ( t n T ) e j ω t t = n δ ( t n T ) e j ω t t F T { ⨿ T ( t ) } = n δ ( t n T ) e j ω t t = n δ ( t n T ) e j ω t t
(7)
= n e j ω n T = { ω = 2 π / T 0 otherwise = n e j ω n T = { ω = 2 π / T 0 otherwise
(8)
= 2 π T ⨿ 2 π / T ( ω ) = 2 π T ⨿ 2 π / T ( ω )
(9)
The multiplicative constant is found from knowing the result for a single delta function.

These shah functions" will be useful in sampling signals in both the continuous time and discrete time cases.

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