The Shah Function
This is going to be a fun page. We'll introduce the Shah Function, which is also known as the "bed of nails".
We'll then derive the Fourier Transform for this function, which gives a surprising result.
The Shah Function is defined as a train of
impulses,
equally spaced in time:
In equation [1], the period T is 1. The function of eq [1] is plotted in Figure 1:
Figure 1. Plot of Shah Function of Period T=1.
The Shah Function is regularly used in the mathematics of digital signal processing (DSP). The function
is used in the proof of the Sampling Theorem; notice that the Shah Function multiplied by any function g(t)
would only depend on g(t) at the locations of the impulses in the Shah Function - hence, g(t) is "sampled".
To determine the Fourier Transform of the Shah Function III(t), it is helpful to first find the Fourier Series of
III(t). You'll see why in the next section. Since the Shah Function is periodic, it can be represented
via a
Fourier Series:
Hence, we can find the
complex Fourier Series coefficients,
setting T=1 (note that the integration is from t=-0.5 to +0.5 - we can integrate over any period of the function
to calculate the coefficients, and in this case it makes more sense to do it this way than from t=0 to +1):
In equation [3], note that the Shah Function reduces from an infinite sum to a simple impulse,
because the only impulse that is within the integration region (t=-0.5 to +0.5) is the impulse at t=0. Also,
note that the final step used the
sifting property.
Equation [3] states that the Fourier Series of the Shah Function has complex coefficients c_n that are equal to 1,
for all n. This is a very neat property. Hence, we can rewrite the Shah Function, using the
Fourier Series representation, in equation [4]:
Now that we have the Fourier Series representation of the Shah Function in eq [4], the derivation for
the Fourier Transform is fairly straightforward. We simply make use of the change of summation and integration
property, and we're done:
The integral at the end of the derivation in equation [5] is found using the same math
as on the
complex exponential page. Hence,
we have a very surprising result:
Most people who know Fourier Transforms know that the
Gaussian Function has itself as its own
Fourier Transform. But most don't know that the Shah Function also possesses this property. Which makes it
even more awesome of a function.
Finally, we presents the Fourier Transform of the Shah Function for when the period is not T=1, but
rather for an arbitrary (positive) T:
Hence, if the Shah Function is sampled "slower" (that is, T>1), then the Fourier Transform
has impulses that occur more often (that is, at a frequency 1/T), and scaled by the factor 1/T.
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The Shah Function
[1]
The Fourier Series of the Shah Function
[2]
[3]
[4] The Fourier Transform of the Shah Function
[5]
The Fourier Transform of the Shah Function is the Shah Function.
[6]
Next: The Truncated Cosine