A Unit Impulse in Continuous-Time

Width of a sinc signal increases and approaches a constant, just like an eagle spreading its wings

This post treats the signals in continuous time which is different than the approach I adopted in my book. The book deals exclusively in discrete time.

A unit impulse is defined as

    \begin{equation*}     \delta (t) = \displaystyle{\lim_{\Delta \to 0}} \begin{cases}             \frac{1}{\Delta}   &  -\frac{\Delta}{2} < t < +\frac{\Delta}{2} \\             0  & \textmd{elsewhere}             \end{cases} \end{equation*}

The result is an impulse with zero width and infinite height, but a consequence of defining it in this way is that the area under the curve is unity.

    \begin{equation*}     \textmd{Area under a rectangle} = \Delta \cdot \frac{1}{\Delta} = 1 \end{equation*}

This is shown in Figure below.

A rectangle with unit area and how it transforms in a continuous-time unit impulse

Stated in another way,

    \begin{equation*}         \int \limits_{-\infty} ^{+\infty} \delta (t) dt = 1     \end{equation*}

A consequence of this property is that theoretically a particular value of a signal x(t) can be extracted in the following manner.

(1)   \begin{equation*}         \int \limits_{-\infty} ^{+\infty} x(t) \delta (t) dt = x(0), \qquad \int \limits_{-\infty} ^{+\infty} x(t) \delta (t-t_0) dt = x(t_0)     \end{equation*}

The Fourier Transform of a unit impulse can be derived through the help of Eq (1).

    \begin{equation*}         F\left\{\delta (t) \right\} = \int \limits_{-\infty} ^{+\infty} \delta (t) e^{-j 2\pi f t} dt = e^{-j 2\pi f\ \cdot \ 0} = 1     \end{equation*}

which is a constant for all frequencies from -\infty to +\infty. This is shown in Figure below.

Fourier transform of a unit impulse is a constant

Here, there is a chance of not emphasizing the underlying concept of a unit impulse — a rectangle with width approaching zero. An alternative and beautiful approach in this context is transforming a rectangular signal and taking the limit \delta \rightarrow 0. Since the Fourier Transform of a rectangle is a sinc function (\sin x / x),

    \begin{equation*}         F\left\{\delta (t) \right\} = \displaystyle{\lim_{\Delta \to 0}} \frac{\sin \pi f \delta}{\pi f \delta} \approx \frac{\pi f \delta}{\pi f \delta} = 1     \end{equation*}

where we used the approximation \sin \theta \approx \theta for small \theta. With \delta becoming small and approaching zero, the mainlobe of the sinc becomes wider and approaches \infty. Imagine an eagle spreading its wings. This illustrated in Figure below and perfectly demonstrates the concept that a signal narrow in time domain has a wide frequency domain representation and vice versa.

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