---
title: "NBFM time domain expression; NBFM to WBFM"
subject: "Communication Systems"
chapter: 3
key: "semester_6/communication_systems:nbfm-time-domain-wbfm"
source: https://www.bctengineer.com/q/semester_6/communication_systems/nbfm-time-domain-wbfm
site: BCT Engineering
---

# NBFM time domain expression; NBFM to WBFM

Communication Systems · Chapter 3: Angle Modulation

- Asked in: 2073 Bhadra (CS I), 6+4 marks
- Video: [Narrow Band Frequency Modulation NBFM (Basics, Derivation, Bandwidth, Formula & Importance)](https://www.youtube.com/watch?v=ZyiHw6z2weY) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/nbfm-time-domain-wbfm
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-3

## Question

Find the time domain expression for Narrowband FM signal. How NBFM can be used to generate Wideband FM signal?

## Short answer

NBFM (β<<1): s(t) ≈ A_c cos 2πf_ct + (βA_c/2)cos 2π(f_c+f_m)t – (βA_c/2)cos 2π(f_c–f_m)t. To generate WBFM, multiply NBFM's frequency deviation and carrier via cascade of multipliers and shift carrier with a mixer; e.g., NBFM → ×n_1 → mixer → ×n_2 gives high deviation at desired carrier.

## Answer

#### Time-domain expression of NBFM

An FM wave modulated by a single tone $m(t) = A_m \cos 2\pi f_m t$ is

$$
s(t) = A_c \cos\left[2\pi f_c t + \beta \sin 2\pi f_m t\right], \qquad \beta = \frac{k_f A_m}{f_m} = \frac{\Delta f}{f_m}
$$

Expanding with $\cos(A+B) = \cos A\cos B - \sin A \sin B$:

$$
s(t) = A_c \cos 2\pi f_c t \,\cos(\beta \sin 2\pi f_m t) - A_c \sin 2\pi f_c t\, \sin(\beta \sin 2\pi f_m t)
$$

For **narrowband FM**, $\beta \ll 1$ (in practice $\beta < 0.3$). Then

$$
\cos(\beta \sin 2\pi f_m t) \approx 1, \qquad \sin(\beta \sin 2\pi f_m t) \approx \beta \sin 2\pi f_m t
$$

so

$$
s(t) \approx A_c \cos 2\pi f_c t - \beta A_c \sin 2\pi f_c t \,\sin 2\pi f_m t
$$

Using $\sin A \sin B = \tfrac12[\cos(A-B) - \cos(A+B)]$:

$$
s_{NBFM}(t) \approx A_c \cos 2\pi f_c t + \frac{\beta A_c}{2}\cos 2\pi (f_c+f_m)t - \frac{\beta A_c}{2}\cos 2\pi (f_c-f_m)t
$$

So NBFM has a carrier and **one pair of side frequencies** at $f_c \pm f_m$, like AM, but the lower side frequency has a **negative sign** (180° phase shift). Its bandwidth is $2f_m$, the same as AM.

#### Generating WBFM from NBFM (Armstrong / indirect method)

WBFM is obtained from NBFM using **frequency multipliers** and a **mixer**:

```text
m(t) -> [Integrator] -> [Balanced ] -> (+) -> NBFM
                        [modulator]     ^     f1, df1 (b1 < 0.3)
                             ^          |
                         [-90 deg]      |
                             ^          |
         crystal osc f1 -----+----------+

NBFM -> [x n1] -> [Mixer] -> [x n2] -> WBFM
                     ^                fc, df
                  [LO fLO]
```

1. A stable crystal oscillator, a −90° phase shifter, a balanced modulator and an adder (which adds the carrier back) produce NBFM with small deviation $\Delta f_1$ at carrier $f_1$.
2. A frequency multiplier (a non-linear device + band-pass filter) multiplies the **instantaneous frequency** by $n$. Both carrier and deviation are multiplied: $f_c \to n f_1$, $\Delta f \to n \Delta f_1$, so $\beta \to n\beta_1$.
3. Because the required deviation needs a large $n$, the carrier would become too high. A **mixer** with a local oscillator shifts the carrier down ($f_{LO} - n_1 f_1$) **without changing the deviation**.
4. A second multiplier $n_2$ raises the deviation (and carrier) to the final values:

$$
\Delta f = n_1 n_2 \Delta f_1, \qquad f_c = n_2\,|f_{LO} - n_1 f_1|
$$

Example: $\Delta f_1 = 25$ Hz needs $n_1 n_2 = 3000$ to give 75 kHz deviation for broadcast FM. The method gives a very stable carrier (crystal controlled) and is used in FM broadcast transmitters.

## Check yourself

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

1. What is the NBFM time-domain form?
   A. Ac cos wc t - beta Ac sin wc t sin wm t
   B. Ac cos wc t + beta Ac cos wm t
   C. Ac(1 + beta cos wm t) cos wc t
   D. Ac cos(beta wc t)
   Answer: A. From cos(A+B) with small beta approximations. (Key point: NBFM expression)

2. How does NBFM differ from AM in its sidebands?
   A. Both sidebands are absent
   B. The lower side frequency has a negative sign (180 degree phase shift)
   C. NBFM has two pairs
   D. Carrier is absent
   Answer: B. Spectrum: carrier + (beta Ac/2)cos(wc+wm)t - (beta Ac/2)cos(wc-wm)t. (Key point: Sideband sign)

3. What does an n-times multiplier do to beta?
   A. Leaves it unchanged
   B. Divides it by n
   C. Multiplies it by n
   D. Adds n to it
   Answer: C. Delta f becomes n Delta f1 while fm is unchanged. (Key point: Multiplier effect on beta)

4. Why use a mixer?
   A. To raise deviation
   B. To remove noise
   C. To integrate the signal
   D. To shift the carrier down without changing the deviation
   Answer: D. Otherwise the carrier would become too high. (Key point: Mixer role)

## Related in Communication Systems

- [Generate FM (50 MHz, β = 1) from NBFM](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anbfm-to-fm-50mhz) — Communication Systems, chapter 3
- [Narrowband vs wideband FM; stereo FM broadcasting](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anbfm-wbfm-stereo-fm) — Communication Systems, chapter 3
- [Angle modulation; single tone FM time and frequency domain](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aangle-mod-single-tone-fm-time-freq) — Communication Systems, chapter 3
- [FM expression after change of modulating signal](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Afm-new-waveform-500hz-2khz) — Communication Systems, chapter 3
- [NBFM and WBFM; Armstrong method](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anbfm-wbfm-armstrong) — Communication Systems, chapter 3
- [General FM expression; FM using phase modulator](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Afm-general-expression-phase-modulator) — Communication Systems, chapter 3

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- [Time-domain specifications of 25/(s²+8s+25)](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atr-tp-ts-mp-25-s2) — Control Systems, chapter 3