The Dirac-Delta Function - The Impulse

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The very useful Dirac-Delta Impulse functional has a simple Fourier Transform and derivation. Particularly, we will look at the shifted impulse:

shifted dirac delta impulse       [1]

Using the definition of the Fourier transform, and the sifting property of the dirac-delta, the Fourier Transform can be determined:

derivation of fourier transform for the dirac-delta impulse       [2]

So, the Fourier transform of the shifted impulse is a complex exponential.

Note that if the impulse is centered at t=0, then the Fourier transform is equal to 1 (i.e. a constant). This is a moment for reflection. The constant function, f(t)=1, is a function with no variation - there is an infinite amount of energy, but it is all contained within the d.c. term. Since the fourier transform evaluated at f=0, G(0), is the integral of the function. For f(t)=1, the integral is infinite, so it makes sense that the result should be infinite at f=0. And since the function f(t) has no variation, it should have no frequency components, so the fourier transform should be zero everywhere f does not equal 0. This last paragraph should be understood at an intuitive level.

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