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## Fourier transform of a sinusoid

We can use the phase shift formula above to find the Fourier transform of any complex sinusoid with frequency , simply by setting in the formula and using the Fourier transform for DC:

where is the Dirichlet kernel and is an ugly phase term:

Figure 9.3: Fourier transforms of complex sinusoids, with = 100: (a) with frequency ; (b) with frequency . (The effect of the phase winding term is not shown.)

If the sinusoid's frequency is an integer multiple of the fundamental frequency , the Dirichlet kernel is shifted to the left or right by an integer. In this case the zero crossings of the Dirichlet kernel line up with integer values of , so that only one partial is nonzero. This is pictured in Figure 9.3 (part a).

Figure 9.4: A complex sinusoid with frequency , forced to repeat every samples. ( is arbitrarily set to 100; only the real part is shown.)

Part (b) shows the result when the frequency falls halfway between two integers. The partials have amplitudes falling off roughly as in both directions, measured from the actual frequency . That the energy should be spread over many partials, when after all we started with a single sinusoid, might seem surprising at first. However, as shown in Figure 9.4, the signal repeats at a period which disagrees with the frequency of the sinusoid. As a result there is a discontinuity at the beginning of each period, and energy is flung over a wide range of frequencies.

Next: Fourier analysis of non-periodic Up: Properties of Fourier transforms Previous: Shifts and phase changes Contents Index