--- title: "Energy and power spectral density; spectrum analyzer" subject: "Communication Systems" chapter: 1 key: "semester_6/communication_systems:energy-power-spectrum-analyzer" source: https://www.bctengineer.com/q/semester_6/communication_systems/energy-power-spectrum-analyzer site: BCT Engineering --- # Energy and power spectral density; spectrum analyzer Communication Systems · Chapter 1: Introduction - Asked in: 2079 Chaitra (CS I), 4+6 marks; 2067 Shrawan (CS I), 4+6 marks (asked 2 times) - 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/energy-power-spectrum-analyzer - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1 ## Question Define energy spectrum and power spectrum density functions. Briefly explain the operation of analog spectrum analyzer. ## Short answer Energy spectral density for an energy signal is |X(f)|² (J/Hz); power spectral density for a power signal is lim_{T→∞} |X_T(f)|²/T (W/Hz). An analog spectrum analyzer mixes the input with a swept local oscillator, passes the product through a narrow IF filter, envelope-detects it, and displays amplitude versus frequency using a synchronized ramp. ## Answer #### Energy spectral density (ESD) For an **energy signal** $x(t)$ (finite energy, zero average power) with Fourier transform $X(f)$, the energy spectral density shows how the energy is spread over frequency: $$ \Psi_x(f) = |X(f)|^2 \quad (\text{J/Hz}) $$ The total energy is the area under the ESD (Rayleigh's theorem): $$ E = \int_{-\infty}^{\infty} |x(t)|^2\,dt = \int_{-\infty}^{\infty} |X(f)|^2\,df $$ ESD is real, non-negative and even (for real $x(t)$), and it is the Fourier transform of the autocorrelation function $R_x(\tau)$. #### Power spectral density (PSD) For a **power signal** (finite non-zero average power, infinite energy) the Fourier transform may not exist, so we truncate the signal to $x_T(t)$ over $-T/2 < t < T/2$ and define $$ S_x(f) = \lim_{T\to\infty} \frac{|X_T(f)|^2}{T} \quad (\text{W/Hz}) $$ The average power is $P = \int_{-\infty}^{\infty} S_x(f)\,df$. PSD is real, non-negative, even for real signals, and $S_x(f)$ and $R_x(\tau)$ form a Fourier transform pair (Wiener–Khinchine theorem). #### Analog (swept-tuned) spectrum analyzer A spectrum analyzer displays signal amplitude (or power) versus frequency on a screen. The common analog type is the **swept-tuned superheterodyne** analyzer. ```text Input +-------+ +-----+ +--------+ +-------+ ------->| Atten |-->| LPF |-->| Mixer |-->| IF | +-------+ +-----+ +--------+ | filter| ^ +-------+ | | +---------+ +--------+ | Swept LO| |Log amp | | (VCO) | |Env det | +---------+ +--------+ ^ | +---------+ +--------+ | Ramp | | Video | |generator| | filter | +---------+ +--------+ | X | Y +-->[ DISPLAY ]<+ ``` Operation: 1. **Attenuator and low-pass filter (preselector):** set a safe input level and remove image frequencies. 2. **Ramp (sweep) generator:** produces a sawtooth voltage. It tunes the local oscillator and also moves the display trace horizontally, so the X-axis becomes frequency. 3. **Mixer and swept LO:** the LO frequency is swept; a component of the input at $f_{in}$ is converted to the fixed IF when $|f_{LO} - f_{in}| = f_{IF}$. So, at each instant, only one input frequency reaches the IF stage. 4. **IF filter:** a narrow band-pass filter; its bandwidth is the **resolution bandwidth (RBW)**, which decides how closely two spectral lines can be separated. 5. **Log amplifier and envelope detector:** convert the IF signal amplitude to a DC level, usually on a dB scale. 6. **Video filter:** smooths the detected output to reduce noise on the display. 7. **Display:** the vertical (Y) deflection is the detected amplitude and the horizontal (X) deflection is the ramp, so the screen shows amplitude versus frequency. As the LO sweeps once over its range, each spectral component of the input appears as a peak at its frequency position, giving the amplitude spectrum (or power spectrum) of the signal. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For an energy signal x(t) with Fourier transform X(f), the energy spectral density is: A. |X(f)|^2, in J/Hz B. |X(f)|, in V/Hz C. |X(f)|^2 / T, in W/Hz D. X(f) itself, in J/Hz Answer: A. ESD is the squared magnitude of the Fourier transform, and its area gives total energy (Rayleigh's theorem). (Key point: ESD definition) 2. Why is a power signal truncated to x_T(t) before defining its PSD? A. Because power signals have zero energy B. Because the Fourier transform of a power signal may not exist C. Because truncation removes noise from the signal D. Because the PSD is defined only for periodic signals Answer: B. A power signal has infinite energy, so its Fourier transform may not exist; the truncated signal has a transform, and dividing |X_T|^2 by T and letting T go to infinity gives the PSD. (Key point: Why truncate for PSD) 3. In the swept-tuned spectrum analyzer, what does the ramp (sawtooth) generator do? A. It sets the IF filter bandwidth B. It smooths the detected output C. It removes image frequencies D. It tunes the local oscillator and also drives the horizontal display axis Answer: D. The same sawtooth sweeps the LO and moves the trace along X, so the horizontal axis represents frequency. (Key point: Role of the ramp generator) 4. What decides how closely two spectral lines can be separated on the analyzer display? A. The bandwidth of the narrow IF filter (RBW) B. The bandwidth of the video filter C. The gain of the log amplifier D. The attenuator setting at the input Answer: A. The IF band-pass filter bandwidth is the resolution bandwidth; lines closer than this merge on the display. (Key point: Resolution bandwidth) ## Related in Communication Systems - [Short note: Analog spectrum analyzer](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ashort-note-spectrum-analyzer) — Communication Systems, chapter 1 - [Working principle of analog spectrum analyzer](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aanalog-spectrum-analyzer) — Communication Systems, chapter 1 - [Two-tone AM: net modulation index, power, spectrum](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aam-two-tone-50-20-12) — Communication Systems, chapter 2, asked 2× - [Two-tone AM: spectrum, power, net index (50, 15, 20)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aam-two-tone-50-15-20) — Communication Systems, chapter 2, asked 2× - [Two-tone AM: equation, index, power, spectrum](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aam-two-tone-10cos-15cos-20cos) — Communication Systems, chapter 2 - [Two-tone AM: spectrum, net index, power, efficiency](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aam-two-tone-100-30-40) — Communication Systems, chapter 2 ## Related in other subjects - [Energy vs power spectral density](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-vs-power-spectral-density) — Signals and Systems, chapter 3 - [Energy density spectrum of CT signals](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-density-spectrum) — Signals and Systems, chapter 3, asked 3× - [Short note: Spectrum analyzer](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Aspectrum-analyzer-note) — RF and Microwave Engineering, chapter 8, asked 2× - [Digital vs analog spread spectrum](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aspread-spectrum-digital-vs-analog) — Data Communication, chapter 6