PSD of a power signal and its interpretation

A power signal $g(t)$ has finite, non-zero average power and infinite energy, so its Fourier transform generally does not exist in the ordinary sense. Its frequency content is described by the power spectral density function (PSDF) $S_g(f)$, the average power per unit bandwidth.

Derivation

Step 1 – truncate the signal to a window of length $T$:

$$g_T(t)=g(t)\,\text{rect}\left(\frac{t}{T}\right)=\begin{cases}g(t), & \lvert t\rvert\le T/2\\0, & \text{otherwise}\end{cases}$$

$g_T(t)$ has finite energy, so its Fourier transform $G_T(f)$ exists.

Step 2 – average power of $g(t)$:

$$P=\lim_{T\to\infty}\frac{1}{T}\int_{-T/2}^{T/2}\lvert g(t)\rvert^2dt=\lim_{T\to\infty}\frac{1}{T}\int_{-\infty}^{\infty}\lvert g_T(t)\rvert^2dt$$

Step 3 – Rayleigh's energy theorem for $g_T(t)$:

$$\int_{-\infty}^{\infty}\lvert g_T(t)\rvert^2dt=\int_{-\infty}^{\infty}\lvert G_T(f)\rvert^2df$$

Step 4 – combine:

$$\begin{aligned} P&=\lim_{T\to\infty}\frac{1}{T}\int_{-\infty}^{\infty}\lvert G_T(f)\rvert^2df\\ &=\int_{-\infty}^{\infty}\lim_{T\to\infty}\frac{\lvert G_T(f)\rvert^2}{T}\,df \end{aligned}$$

Step 5 – define PSDF. Since total power must equal $\int S_g(f)df$:

$$S_g(f)=\lim_{T\to\infty}\frac{\lvert G_T(f)\rvert^2}{T}\quad(\text{W/Hz})$$

Alternative form (Wiener–Khinchin): the time-averaged autocorrelation $R_g(\tau)=\lim_{T\to\infty}\frac{1}{T}\int_{-T/2}^{T/2}g(t)g^*(t-\tau)dt$ and the PSD form a Fourier pair:

$$S_g(f)=\int_{-\infty}^{\infty}R_g(\tau)e^{-j2\pi f\tau}d\tau,\qquad R_g(0)=P=\int_{-\infty}^{\infty}S_g(f)df$$

Example: for $g(t)=A\cos2\pi f_ct$, $R_g(\tau)=\frac{A^2}{2}\cos2\pi f_c\tau$, so $S_g(f)=\frac{A^2}{4}[\delta(f-f_c)+\delta(f+f_c)]$ and $P=A^2/2$.

Interpretation of PSDF

1. $S_g(f)\,df$ is the average power contained in a narrow band $df$ at frequency $f$; the PSDF shows how the power is distributed over frequency.
2. Area = average power: $P=\int S_g(f)df$. Power in a band $f_1$ to $f_2$ (both sides) is $2\int_{f_1}^{f_2}S_g(f)df$ for real signals.
3. It is real, non-negative and even ($S_g(-f)=S_g(f)$) for real signals.
4. It carries no phase information: different signals can have the same PSDF.
5. Measured by a filter: if $g(t)$ is passed through an ideal narrow band-pass filter of bandwidth $\Delta f$ at $f$, the output power is about $2S_g(f)\Delta f$. This is how a spectrum analyzer measures PSD.
6. Through an LTI system: $S_y(f)=\lvert H(f)\rvert^2S_g(f)$, used to compute output noise power and filter design.
7. For periodic signals the PSDF is a set of impulses of weight $\lvert c_n\rvert^2$ at harmonics $nf_0$; for random noise it is continuous (e.g. white noise $S(f)=N_0/2$).