Kronecker delta

The Kronecker delta $\delta [k]$ is a discrete-index “indicator” function: $1$ at $k=0$ and $0$ everywhere else.

$\delta [k]=\{\begin{array}{ll}1,& k=0\\ 0,& k\ne 0.\end{array}$

This is the discrete analog of the continuous Dirac delta $\delta (t)$. The two are different objects: $\delta (t)$ is a distribution defined by its integral behavior, while $\delta [k]$ is an ordinary function taking values 0 or 1. Check the argument (continuous parenthesis $\delta (\cdot )$ vs discrete bracket $\delta [\cdot ]$) to disambiguate.

Stride-$m$ Kronecker delta

The course uses a generalization, the Kronecker delta with stride $m$:

${\delta }_m[k]=\{\begin{array}{ll}1,& k/m\text{ is an integer}\\ 0,& \text{otherwise}.\end{array}$

So ${\delta }_m[k]=1$ when $k$ is an integer multiple of $m$, and 0 otherwise.

• ${\delta }_1[k]=1$ for every integer $k$ — every integer is a multiple of $1$. So ${\delta }_1[k]$ is “always 1” and can be dropped from expressions.
• ${\delta }_{\infty }[k]=\delta [k]$ — only $k=0$ is a multiple of “infinity” (interpreted in the limit). This is just the ordinary discrete delta.
• ${\delta }_2[k]$ picks out even indices: $1$ at $k=0,\pm 2,\pm 4,\dots$, zero elsewhere.

Where this shows up

The stride-$m$ Kronecker delta appears throughout the Fourier series standard pairs. A signal that is the result of convolving a pulse with a periodic impulse train ${\delta }_{T_0}(t)$ has Fourier coefficients of the form

$c_x[k]=(\text{some sinc-like envelope})\cdot {\delta }_m[k],$

where $m$ is the integer such that the representation period $T=m{T}_0$. The ${\delta }_m[k]$ factor masks out the harmonic indices that aren’t multiples of $m$ — exactly the indices where the periodic-train spectrum has nonzero coefficients.

Why two notations matter

If you encounter $\delta (t)$ in a time-domain integral, it’s the continuous impulse — use the sifting property. If you encounter $\delta [k]$ in a discrete-index sum, it’s the Kronecker delta — just $1$ at $k=0$. Confusing them produces nonsense, because they have different defining properties (sifting for the continuous one, pointwise values for the discrete one).