Unit Impulse and Unit Step in Continuous and Discrete Time

Relationship Between Impulse and Step Signals

In discrete time, the unit impulse is the first difference of the unit step:

$ \delta[n] = u[n] - u[n-1] $

In continuous time, the unit impulse is the derivative of the unit step:

$ \delta_{\Delta}(t) = \frac{d u_{\Delta}(t)}{dt} $

This describes how the discrete-time unit step is converted into the discrete-time unit impulse.

Conversely, the discrete-time unit step can be expressed as the accumulated sum of unit impulses:

$ u[n] = \sum_{m=-\infty}^{n} \delta[m] $

The continuous-time unit step can be expressed as the integral of the unit impulse:

$ u(t) = \int_{-\infty}^{t} \delta(\tau) , d\tau $

This is the integral of $\delta$ with respect to the dummy variable $\tau$, from negative infinity to $t$. It is a summation-like process with a limiting operation behind it.

We often replace the dummy variable $m$ or $\tau$ with another dummy variable such as $k$ or $\sigma$:

$ k = n - m $

$ \sigma = t - \tau $

With this substitution, the expressions can be rewritten as:

$ u[n] = \sum_{k = \infty}^{0} \delta[n - k] $

$ u(t) = \int_{-\infty}^{t} \delta(\tau)d\tau = \int_{\infty}^{0} \delta(t - \sigma)(-d\sigma) $

The negative sign in the second expression comes from the variable substitution:

$ \sigma = t - \tau $

$ d\sigma = -d\tau $

so:

$ d\tau = -d\sigma $

Reversing the Dummy Variable

For the discrete summation, changing the order from infinity-to-zero to zero-to-infinity does not introduce a sign change. It is still a summation over the same range of dummy values.

For the integral, reversing the integration limits requires changing the sign. Therefore the formulas can be rewritten as:

$ u[n] = \sum_{k = 0}^{\infty}\delta[n - k] $

$ u(t) = \int_{0}^{\infty} \delta(t - \tau)d\tau $

An integral can be understood as a limiting form of summation. Differentiation goes in the opposite direction of integration. Reversing summation order does not require a sign change, while reversing integration limits does.

The meaning of these transformations is that the impulse contributes only when the argument of $\delta$ is zero. In other words, $\delta[n-k]$ contributes only when $n = k$, and $\delta(t-\tau)$ contributes only when $t = \tau$.

By observing the range of the dummy variable, we can tell whether the right-hand side can produce a nonzero value:

$ \forall n < 0, u[n] = 0 $

$ \forall t < 0, u(t) = 0 $

Sampling with the Unit Impulse

Suppose we have a signal $x(t)$. A basic impulse property is:

$ x(t)\delta(t) = x(0)\delta(t) $

This means the impulse samples the signal at $t = 0$. This is already a simplified ideal statement.

In practice, the unit impulse is an idealized object. It can be thought of as the limiting case of $\delta_\Delta(t)$ as $\Delta \rightarrow 0$.

To sample $x(t)$ at $t_0$, we use a shifted impulse:

$ x(t)\delta(t - t_0) = x(t_0)\delta(t - t_0) $

The important point is that $\delta(t)$ is an ideal mathematical model. It is useful because it captures the sampling behavior cleanly, even though it is not an ordinary function in the usual sense.

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