The Shah Function

Previous: Decaying Exponential
Table of Fourier Transforms
Next: Truncated Cosine

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

The Shah Function is defined as a train of impulses, equally spaced in time:

the shah or sha function       [1]

In equation [1], the period T is 1. The function of eq [1] is plotted in Figure 1:

plot of shah function

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".

The Fourier Series of the Shah Function

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:

fourier series representation of shah function       [2]

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):

the fourier series coefficients for shah       [3]

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]:

series of sha function       [4]

The Fourier Transform of the Shah Function

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:

fourier transform of shah function       [5]

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:

The Fourier Transform of the Shah Function is the Shah Function.

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:

general Shah function transform       [6]

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.

Next: The Truncated Cosine

Previous: The Decaying Exponential

Fourier Transform Pairs List

The Fourier Transform (Home)

This page on the Fourier Transform of the shah function (bed of nails) is copyrighted. No portion can be reproduced except by permission from the author. Copyright, 2010-2011.