EEL4514C Communication Systems and Components, Fall 2026, Lecture 20

Overview

Mingyue Ji develops the properties and bandwidth analysis of angle modulation, contrasting PM’s phase sensitivity with FM’s frequency sensitivity and showing why FM tolerates a cubic nonlinearity better than DSB-SC. A Taylor expansion reveals FM’s theoretically infinite bandwidth, motivates narrowband FM when the modulation index is small, and leads to a sampling-based method for analyzing wideband FM.

Key takeaways

PM changes phase directly according to m(t), while FM changes instantaneous frequency according to m(t); consequently, FM phase deviation depends on the integral of the message.
A cubic nonlinearity can widen DSB-SC’s carrier-band message term from bandwidth B to roughly 3B, whereas FM’s corresponding cubic contribution mainly changes amplitude and phase deviation.
The Taylor expansion of FM contains powers of the integrated message at every order, so its strict mathematical bandwidth is infinite even though higher-order sidebands may carry little energy.
Narrowband FM is accurate when kfA(t) ≪ 1 and can be generated by adding a carrier to a DSB-SC-like term based on the integrated message; its approximate bandwidth is 2B.
For a message bandlimited to B, piecewise-constant samples spaced 1/(2B) apart provide a Nyquist-rate representation that makes FM’s phase integral easier to evaluate interval by interval.

Chapters

0:00 Midterm Scope, Date, and Lecture Schedule

The midterm is planned for Wednesday, October 21, at about 7:30 PM, with the room booking to be confirmed.
Exam coverage runs through angle modulation, the final topic in analog communication before digital communication begins.
Mingyue Ji plans to upload a prior-year practice exam and prerecorded lectures during the travel week; there will be no Friday lecture.

1:10 PM and FM: Phase, Frequency, and the General Angle-Modulation Model

Instantaneous angular frequency is the time derivative of phase; PM sets phase to ωct + kp m(t), where kp is measured in radians per volt.
FM sets instantaneous angular frequency to ωc + kf m(t), where kf is frequency sensitivity in radians per second per volt; integrating it gives phase ωct + kf∫m(τ)dτ.
A general angle modulator applies an LTI system to m(t) to produce phase deviation; PM and FM are special cases, with phase sensitivity and frequency sensitivity respectively.

6:34 Angle-Modulation Properties and DSB-SC’s Cubic-Distortion Problem

When carrier frequency greatly exceeds message bandwidth, angle-modulated signals have nearly constant power over a carrier-period-scale observation window.
Angle modulation is nonlinear in the message, unlike the linear DSB-SC modulation operator discussed in class.
For a DSB-SC input passed through y = a s + b s³, the cubic term creates an in-band carrier-frequency component proportional to m³(t), whose bandwidth can grow to about 3B.

10:21 Why FM Is More Robust to a Cubic Nonlinearity

For an FM input Ac cos(ωct + φ(t)), cubic expansion produces a carrier-frequency component with a changed amplitude and phase deviation scaled by a constant.
The additional cubic harmonic appears around three times the carrier frequency and can be removed with filtering.
Unlike DSB-SC’s m³(t) term, FM’s carrier-band term does not broaden the message bandwidth, making FM more robust to this nonlinearity.

16:43 Why FM Bandwidth Is Hard to Determine Directly

Bandwidth is assessed from a signal’s frequency-domain energy distribution, not just from its time-domain waveform.
FM contains a cosine whose phase includes an integral of the message, so directly Fourier-transforming the waveform is difficult.
Mingyue Ji introduces complex exponentials as a more tractable way to expose FM’s spectral components.

20:26 Taylor Expansion Reveals FM Sidebands at Every Order

Writing A(t) = ∫m(τ)dτ expresses FM as the real part of Ac exp(jωct) exp(jkfA(t)).
Expanding exp(jkfA(t)) produces terms involving A(t), A²(t), A³(t), and higher powers, each modulated around the carrier.
If A(t) has bandwidth B, its nth power can have bandwidth up to nB, so the sequence of FM sidebands extends to arbitrarily high frequencies.

31:30 FM’s Infinite Theoretical Bandwidth and the Small-Index Condition

Because higher powers of A(t) generate progressively wider sidebands, the Taylor-series analysis gives FM infinite bandwidth in the strict mathematical sense.
When kfA(t) is much smaller than 1, higher-order terms become small and contribute little energy despite the infinite theoretical extent.
The size of kfA(t) provides a practical condition for simplifying bandwidth analysis; later analysis will offer another bandwidth measure.

35:30 Narrowband FM as a Carrier Plus a DSB-SC-Like Term

For kfA(t) ≪ 1, FM is approximated by Ac cos(ωct) − Ac kfA(t)sin(ωct), retaining only the carrier and first-order term.
The message-bearing term resembles DSB-SC with A(t), the integral of m(t), as its message and a 90-degree phase shift from the usual cosine form.
A practical narrowband-FM modulator integrates m(t), scales it by kf, multiplies it by a sine carrier in a DSB-SC modulator, then adds the unmodulated carrier; the approximate bandwidth is 2B.

40:49 Wideband FM Analysis via Nyquist-Rate Message Sampling

When kfA(t) is not small, the narrowband Taylor approximation is inadequate; wideband FM requires a different way to analyze the phase integral.
The method approximates m(t) with piecewise-constant samples m(tk), using sample spacing 1/(2B) for a message bandlimited to B.
Replacing the message on each interval by a constant makes the phase integral explicit over that interval, allowing the resulting FM spectrum to be analyzed piece by piece.
Mingyue Ji sets the message amplitude range from −Mp to Mp before continuing the wideband analysis in the prerecorded follow-up.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Mingyue Ji.