The scaling property of the Dirac comb follows from the properties of the Dirac delta function.
Since  for positive real numbers , it follows that:
Note that requiring positive scaling numbers instead of negative ones is not a restriction because the negative sign would only reverse the order of the summation within , which does not affect the result.
It is clear that is periodic with period . That is,
for all t. The complex Fourier series for such a periodic function is
where the Fourier coefficients are (symbolically)
All Fourier coefficients are 1/T resulting in
When the period is one unit, this simplifies to
Remark: Most rigorously, Riemann or Lebesgue integration over any products including a Dirac delta function yields zero. For this reason, the integration above (Fourier series coefficients determination) must be understood "in the generalized functions sense". It means that, instead of using the characteristic function of an interval applied to the Dirac comb, one uses a so-called Lighthill unitary function as cutout function, see Lighthill 1958, p.62, Theorem 22 for details.
The Fourier transform of a Dirac comb is also a Dirac comb. For the Fourier transform expressed in frequency domain (Hz) the Dirac comb of period transforms into a rescaled Dirac comb of period i.e. for
is proportional to another Dirac comb, but with period in frequency domain (radian/s). The Dirac comb of unit period is thus an eigenfunction of to the eigenvalue
This result can be established (Bracewell 1986) by considering the respective Fourier transforms of the family of functions defined by
Since is a convergent series of Gaussian functions, and Gaussians transform into Gaussians, each of their respective Fourier transforms also results in a series of Gaussians, and explicit calculation establishes that
The functions and are thus each resembling a periodic function consisting of a series of equidistant Gaussian spikes and whose respective "heights" (pre-factors) are determined by slowly decreasing Gaussian envelope functions which drop to zero at infinity. Note that in the limit each Gaussian spike becomes an infinitely sharp Dirac impulse centered respectively at and for each respective and , and hence also all pre-factors in eventually become indistinguishable from . Therefore the functions and their respective Fourier transforms converge to the same function and this limit function is a series of infinite equidistant Gaussian spikes, each spike being multiplied by the same pre-factor of one, i.e. the Dirac comb for unit period:
Since , we obtain in this limit the result to be demonstrated:
Another manner to establish that the Dirac comb transforms into another Dirac comb starts by examining continuous Fourier transforms of periodic functions in general, and then specialises to the case of the Dirac comb. In order to also show that the specific rule depends on the convention for the Fourier transform, this will be shown using angular frequency with for any periodic function its Fourier transform
because Fourier transforming and leads to and This equation implies that nearly everywhere with the only possible exceptions lying at with and When evaluating the Fourier transform at the corresponding Fourier series expression times a corresponding delta function results. For the special case of the Fourier transform of the Dirac comb, the Fourier series integral over a single period covers only the Dirac function at the origin and thus gives for each This can be summarised by interpreting the Dirac comb as a limit of the Dirichlet kernel such that, at the positions all exponentials in the sum point into the same direction and add constructively. In other words, the continuous Fourier transform of periodic functions leads to
is another Dirac comb, but with period in angular frequency domain (radian/s).
As mentioned, the specific rule depends on the convention for the used Fourier transform. Indeed, when using the scaling property of the Dirac delta function, the above may be re-expressed in ordinary frequency domain (Hz) and one obtains again:
such that the unit period Dirac comb transforms to itself:
Finally, the Dirac comb is also an eigenfunction of the unitary continuous Fourier transform in angular frequency space to the eigenvalue 1 when because for the unitary Fourier transform
the above may be re-expressed as
Sampling and aliasingEdit
Multiplying any function by a Dirac comb transforms it into a train of impulses with integrals equal to the value of the function at the nodes of the comb. This operation is frequently used to represent sampling.
Since convolution with a delta function is equivalent to shifting the function by , convolution with the Dirac comb corresponds to replication or periodic summation:
This leads to a natural formulation of the Nyquist–Shannon sampling theorem. If the spectrum of the function contains no frequencies higher than B (i.e., its spectrum is nonzero only in the interval ) then samples of the original function at intervals are sufficient to reconstruct the original signal. It suffices to multiply the spectrum of the sampled function by a suitable rectangle function, which is equivalent to applying a brick-wall lowpass filter.
In time domain, this "multiplication with the rect function" is equivalent to "convolution with the sinc function" (Woodward 1953, p.33-34). Hence, it restores the original function from its samples. This is known as the Whittaker–Shannon interpolation formula.
Remark: Most rigorously, multiplication of the rect function with a generalized function, such as the Dirac comb, fails. This is due to undetermined outcomes of the multiplication product at the interval boundaries. As a workaround, one uses a Lighthill unitary function instead of the rect function. It is smooth at the interval boundaries, hence it yields determined multiplication products everywhere, see Lighthill 1958, p.62, Theorem 22 for details.
In linear statistics, the random variable is usually distributed over the real-number line, or some subset thereof, and the probability density of is a function whose domain is the set of real numbers, and whose integral from to is unity. In directional statistics, the random variable is distributed over the unit circle, and the probability density of is a function whose domain is some interval of the real numbers of length and whose integral over that interval is unity. Just as the integral of the product of a Dirac delta function with an arbitrary function over the real-number line yields the value of that function at zero, so the integral of the product of a Dirac comb of period with an arbitrary function of period over the unit circle yields the value of that function at zero.