---
title: "PSD of a power signal and its interpretation"
subject: "Communication Systems"
chapter: 1
key: "semester_6/communication_systems:psdf-power-signal-interpretation"
source: https://www.bctengineer.com/q/semester_6/communication_systems/psdf-power-signal-interpretation
site: BCT Engineering
---

# PSD of a power signal and its interpretation

Communication Systems · Chapter 1: Introduction

- Asked in: 2067 Mangsir (CS I), 6+4 marks
- Video: [What is Power Spectral Density (PSD)?](https://www.youtube.com/watch?v=DoSLMEEo1Y0) — Iain Explains Signals, Systems, and Digital Comms
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/psdf-power-signal-interpretation
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1

## Question

Derive the expression for power spectral density function (psdf) for a power type signal. Give the interpretation of psdf.

## Short answer

For power signals with infinite energy, PSDF S_g(f) = lim_{T→∞} |G_T(f)|²/T (W/Hz) describes average power per unit bandwidth. It is derived by truncating the signal, applying Rayleigh's theorem, and taking the limit. S_g(f) is real, non-negative, even, and its integral gives total power. It carries no phase information and relates to autocorrelation via Wiener-Khinchin theorem.

## Answer

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 $S(f)=N_0/2$).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Why is a power signal truncated to g_T(t) when deriving the PSDF?
   A. The truncated signal has finite energy so its Fourier transform exists
   B. To make it periodic
   C. To remove noise
   D. To make it causal
   Answer: A. A power signal has infinite energy, so g_T(t) over a finite window is used to get G_T(f). (Key point: Why truncate the signal)

2. The PSDF is defined as
   A. lim |G_T(f)|^2 as T→∞
   B. lim |G_T(f)|^2 / T as T→∞
   C. lim T|G_T(f)|^2 as T→∞
   D. lim |G_T(f)| / T as T→∞
   Answer: B. Dividing the energy spectrum by T gives power per unit bandwidth, using Rayleigh's theorem. (Key point: PSDF definition)

3. By the Wiener-Khinchin theorem the PSD is
   A. The inverse transform of g(t)
   B. The Fourier transform of the time-averaged autocorrelation
   C. The square of g(t)
   D. The derivative of R_g(τ)
   Answer: B. S_g(f) and R_g(τ) are a Fourier pair; R_g(0) equals the total power. (Key point: Wiener-Khinchin)

4. Which is a correct property of the PSDF of a real signal?
   A. It carries phase information
   B. It is real, non-negative and even, with area equal to average power
   C. It can be negative
   D. It is always continuous
   Answer: B. S_g(f) >= 0, S_g(-f) = S_g(f), and its integral is P; it has no phase information. (Key point: Interpretation of PSDF)

## Close variants set in other exams

- [PSDF of an arbitrary signal](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Apsdf-arbitrary-signal) — Communication Systems, chapter 1, asked 3×

## Related in Communication Systems

- [PSD expression and relation with autocorrelation](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Apsd-expression-autocorrelation) — Communication Systems, chapter 1
- [ESD vs PSD; PSD of an arbitrary signal](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aesd-vs-psd-arbitrary-signal) — Communication Systems, chapter 1, asked 2×
- [Energy and power spectral density; spectrum analyzer](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aenergy-power-spectrum-analyzer) — Communication Systems, chapter 1, asked 2×
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- [White noise: relation between PSDF and AC function](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Awhite-noise-psdf-ac-relation) — Communication Systems, chapter 9
- [ESD, PSD and autocorrelation of white noise](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aesd-psd-white-noise-autocorrelation) — Communication Systems, chapter 9, asked 2×

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