--- title: "NBFM spectrum vs AM; Armstrong NBFM generation" subject: "Communication Systems" chapter: 3 key: "semester_6/communication_systems:nbfm-spectrum-vs-am-armstrong" source: https://www.bctengineer.com/q/semester_6/communication_systems/nbfm-spectrum-vs-am-armstrong site: BCT Engineering --- # NBFM spectrum vs AM; Armstrong NBFM generation Communication Systems · Chapter 3: Angle Modulation - Asked in: 2070 Bhadra (CS I), 2+5 marks - Video: [NarrowBand Frequency Modulation (NBFM) and Its Generation - Comparison of NBFM and AM](https://www.youtube.com/watch?v=E1nsrt34Y6Q) — Padmasri Naban - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/nbfm-spectrum-vs-am-armstrong - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-3 ## Question How is the spectrum of Narrow band FM similar to and different from the spectrum of conventional AM? Explain how NBFM is generated by using Armstrong's method. ## Short answer Both have a carrier and one pair of sidebands at fc ± fm and bandwidth 2fm. In NBFM the lower sideband is inverted, and the sidebands add at 90 degrees to the carrier, changing phase rather than amplitude. Armstrong's method integrates the message, modulates a 90-degree shifted carrier, and adds the carrier to produce NBFM. ## Answer #### NBFM spectrum compared with AM For single-tone modulation: $$ \begin{aligned} s_{AM}(t) &= A_c\cos\omega_c t + \frac{\mu A_c}{2}\cos(\omega_c+\omega_m)t + \frac{\mu A_c}{2}\cos(\omega_c-\omega_m)t \\ s_{NBFM}(t) &\approx A_c\cos\omega_c t + \frac{\beta A_c}{2}\cos(\omega_c+\omega_m)t - \frac{\beta A_c}{2}\cos(\omega_c-\omega_m)t \end{aligned} $$ **Similarities:** both have a carrier and one pair of sidebands at $f_c \pm f_m$; both have bandwidth $2f_m$; sideband amplitudes are proportional to the modulation index. **Differences:** in NBFM the **lower sideband is inverted** (180° phase shift). In AM the sideband phasors add to the carrier in phase, changing its **amplitude**; in NBFM they add at 90° to the carrier, changing mainly its **phase**. NBFM also has small amplitude variation and some higher-order terms that AM does not have. #### Generation of NBFM by Armstrong's method From the NBFM expression with general message: $$ s(t) \approx A_c\cos 2\pi f_c t - A_c\,\phi(t)\sin 2\pi f_c t, \qquad \phi(t) = 2\pi k_f\int_0^t m(\tau)\,d\tau $$ The second term is a DSB-SC wave of the integrated message on a carrier shifted by 90°. This leads to the block diagram: ```text m(t) +------------+ +----------+ -A.phi.sin wct ---->| Integrator |---->| Balanced |----------+ +------------+ | modulator| | +----------+ v ^ +---+ NBFM +----------+ | + |------> | -90 deg | +---+ | shift | ^ +----------+ | ^ | +-------------+ | Ac cos wct | | Crystal osc |---------+----------------+ | fc | +-------------+ ``` **Working:** 1. The message is integrated (so that the phase modulator produces FM rather than PM). 2. A crystal oscillator gives a very stable carrier $A_c\cos\omega_c t$. 3. The carrier is shifted by 90° and fed to a balanced modulator with the integrated message, producing $-A_c\phi(t)\sin\omega_c t$ (DSB-SC). 4. The carrier is added to this output, giving NBFM. The deviation is kept small ($\beta < 0.3$, about 25 Hz in practice) to limit distortion. Wideband FM is then obtained by frequency multipliers and a mixer. The main merit is crystal-controlled carrier stability. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which is a similarity between NBFM and AM spectra? A. Both have infinite sidebands B. Both have a carrier and one pair of sidebands, with bandwidth 2fm C. Both have no carrier D. Both have constant envelope Answer: B. For single tone, both have a carrier plus sidebands at fc +/- fm, giving 2fm bandwidth. (Key point: NBFM vs AM similarity) 2. What is the key difference in the sidebands of NBFM compared with AM? A. The upper sideband is missing B. NBFM has twice the bandwidth C. NBFM sidebands are at fc +/- 2fm D. The lower sideband is inverted (180 degrees), so the sidebands add at 90 degrees to the carrier and mainly change its phase Answer: D. In NBFM the lower sideband has opposite sign, making the sidebands combine in quadrature with the carrier (phase variation), unlike AM where they add in phase (amplitude variation). (Key point: NBFM vs AM difference) 3. In Armstrong's method the NBFM wave is made by adding the carrier to what? A. An AM wave of the message B. A DSB-SC wave of the integrated message on a 90-degree shifted carrier C. The differentiated message D. A SSB wave of the message Answer: B. s(t) ~ Ac cos wc t - Ac phi(t) sin wc t; the second term is the integrated message multiplied by a 90-degree shifted carrier in a balanced modulator. (Key point: Armstrong NBFM structure) 4. What is the main merit of the Armstrong method? A. Large deviation directly B. Crystal-controlled carrier stability C. No need for multipliers D. Very simple circuitry Answer: B. The carrier comes from a crystal oscillator; wideband FM is then obtained by multipliers and a mixer, keeping the stability. 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