FT of DC; meaning of ESD and PSD

Fourier transform of a constant (DC) signal

Let $x(t) = A$ for all $t$. This signal is not absolutely integrable, so the integral $\int A e^{-j\omega t}dt$ does not converge in the ordinary sense. We find the transform using the impulse function and duality.

Step 1: Consider $X(j\omega) = 2\pi A\,\delta(\omega)$ and take its inverse FT:

$$\begin{aligned} x(t) &= \frac{1}{2\pi}\int_{-\infty}^{\infty} 2\pi A\,\delta(\omega)\,e^{j\omega t}\,d\omega \\ &= A\,e^{j0\cdot t} = A \end{aligned}$$

using the sifting property of $\delta(\omega)$.

Step 2: Since the inverse transform of $2\pi A\delta(\omega)$ is the constant $A$, the transform pair is

$$A \;\longleftrightarrow\; 2\pi A\,\delta(\omega)$$

(Equivalently, from duality: $\delta(t) \leftrightarrow 1$, so $1 \leftrightarrow 2\pi\delta(-\omega) = 2\pi\delta(\omega)$.) In terms of $f$ in Hz, $A \leftrightarrow A\,\delta(f)$.

Meaning: A DC signal has no variation, so all its content is at zero frequency. The spectrum is a single impulse at $\omega = 0$ with strength $2\pi A$.

 x(t)                    X(jw)
  |                        ^ 2*pi*A
 A|-----------------       |
  |                        |
--+-----------> t   -------+-------> w
                           0

Physical meaning of ESD and PSD

• Energy spectral density (ESD), $\Psi(\omega) = |X(j\omega)|^2$, applies to energy signals (finite energy, e.g. a pulse). It tells how the total energy of the signal is spread over frequency. The energy in a small band $d\omega$ around $\omega$ is $\frac{1}{2\pi}|X(j\omega)|^2d\omega$, and the total energy is $E = \frac{1}{2\pi}\int|X(j\omega)|^2d\omega$. Unit: joule per hertz.
• Power spectral density (PSD), $S(\omega) = \lim_{T\to\infty}\frac{|X_T(j\omega)|^2}{T}$, applies to power signals (periodic or random signals with finite average power). It tells how the average power is spread over frequency; total power $P = \frac{1}{2\pi}\int S(\omega)\,d\omega$. Unit: watt per hertz.

In practice, the ESD/PSD shows the bandwidth a signal needs, which frequency bands carry most energy or power, and how much noise power passes through a filter. Both are real, non-negative and even functions of $\omega$ for real signals.