Fourier transform of the Dirac delta
/ dee-RAHK /
Strike a drum once, very sharply — an idealized instantaneous kick. To contain such a sudden spike, a signal must be built from every frequency at once, all in step. The Fourier transform of the Dirac delta makes this exact: the transform of an infinitely concentrated impulse is a perfectly flat spectrum, the same amount of every frequency. The sharpest possible thing in time is the broadest possible thing in frequency.
The Dirac delta, delta(x), is not an ordinary function — it is a distribution, defined by its sifting action: integral of delta(x) g(x) dx = g(0). Feeding it through the transform integral and using sifting gives FT{delta}(k) = integral of delta(x) e^{-i k x} dx = e^{0} = 1, a constant. By duality (and inversion), the reverse holds: the transform of the constant 1 is 2 pi delta(k). A shifted impulse delta(x minus a) transforms to e^{-i k a} (a pure phase, flat magnitude), and the transform of a pure wave e^{i a x} is 2 pi delta(k minus a) — a single spectral spike at the wave's frequency. This last pair is what lets a sinusoid, whose integral does not converge classically, have a Fourier transform at all.
These pairs are the connective tissue between the discrete world of Fourier series and the continuous world of the transform. Because a periodic signal is a sum of pure waves, its Fourier transform is a train of deltas — the continuous-spectrum picture of the original discrete spectral lines. The delta is also the identity element for convolution (convolving anything with delta returns it unchanged), which is why it represents an ideal, distortion-free channel. The one thing never to forget: delta is a distribution, a rule for acting on test functions, not a function with a value at each point, and treating its transform as an ordinary integral only works inside that distributional framework.
Since the transform of e^{i a x} is 2 pi delta(k minus a), and cos(a x) = (e^{i a x} + e^{-i a x})/2, the transform of cos(a x) is pi[delta(k minus a) + delta(k plus a)] — two spikes at plus and minus the frequency a, the spectrum of a pure tone.
A pure tone has a two-spike delta spectrum — the precise bridge from Fourier series lines to the continuous transform.
The delta is a distribution, not a function — it has no pointwise value, and its transform being a constant only makes rigorous sense as a statement about how it acts on test functions inside an integral.