EEL4514C Communication Systems and Components, Fall 2026, Lecture 08

Overview

Mingyue Ji connects the Fourier transform of a single-period pulse to the Fourier-series coefficients of the periodic waveform formed by repeating it: the coefficients are samples of the pulse transform at harmonics, scaled by 1/T₀. A rectangular pulse produces a sinc-shaped spectrum, whose regularly spaced zeros explain how frequency-shifted, orthogonal subcarriers can be packed in OFDM, a principle used in 4G, 5G, 6G, and Wi-Fi.

Key takeaways

For a periodic repetition of a single-period waveform g(t), the Fourier-series coefficient at harmonic n is cₙ = G(n/T₀)/T₀, so a continuous Fourier transform can directly yield the discrete series coefficients.
A unit-height rectangular pulse of width τ has transform G(f) = τ sinc(τf), with zeros at f = k/τ for nonzero integers k.
The spectrum of a periodic signal is formed by sampling the single-period waveform’s transform at multiples of 1/T₀; the pulse shape determines the sample amplitudes.
Multiplication by a cosine shifts a real signal’s spectrum into positive- and negative-frequency copies, enabling signals to be placed in a channel’s usable frequency band.
OFDM uses frequency-shifted subcarriers aligned with one another’s spectral zeros to achieve orthogonality and pack multiple signals closely in frequency; Ji cites 4G, 5G, 6G, and Wi-Fi as applications.

Chapters

0:00 Fourier Transform Review: Continuous Spectra and Real-Signal Symmetry

Unlike a periodic signal’s discrete harmonics, a nonperiodic signal generally requires a continuous range of frequency components.
For real-valued signals, the negative-frequency spectrum is the complex conjugate of the positive-frequency spectrum, so it adds no independent information.
Mingyue Ji reviews the Fourier transform as a linear operator and notes that frequency-domain multiplication replaces time-domain convolution in LTI-system analysis.

4:43 Impulse and Cosine Transforms Lead to Frequency-Domain Modulation

The Dirac delta has a Fourier transform equal to 1 at every frequency; an LTI system with impulse response δ(t) therefore leaves its input unchanged.
A shifted delta in frequency corresponds to a complex exponential in time; combining positive and negative shifts produces a cosine.
Multiplying a real signal by a cosine shifts its spectrum in both directions, a practical way to move signal content into a channel’s passband.

8:28 Representing a Periodic Waveform as Repeated Copies of One Pulse

Ji introduces the link between Fourier transforms and Fourier series, with the goal of calculating series coefficients from a transform table.
A periodic signal s(t) with period T₀ is represented as repeated, shifted copies of a single-period waveform g(t): s(t) = Σₙ g(t − nT₀).
The pulse width τ describes the nonzero portion of each period; this representation lets the transform of g(t) stand in for a direct Fourier-series calculation.

13:39 Deriving Fourier-Series Coefficients from the Pulse Transform

The Fourier-series coefficient is defined by integrating the periodic signal over one fundamental period against the complex exponential basis.
Within that period, the signal equals g(t), so the coefficient integral can use the single-period pulse instead.
Comparing the coefficient integral with the Fourier transform gives cₙ = (1/T₀)G(n/T₀): evaluate the continuous transform at harmonic frequencies and scale by 1/T₀.

18:14 Rectangular Pulses Produce a Sinc Spectrum

For a unit-height rectangular pulse of width τ, direct integration of e^(−j2πft) over −τ/2 to τ/2 gives G(f) = τ sinc(τf).
Using sinc(x) = sin(πx)/(πx), the spectrum has its peak value τ at f = 0 and zeros at nonzero integer multiples of 1/τ.
The Fourier-series coefficients of the repeated pulse are consequently cₙ = (τ/T₀)sinc(nτ/T₀), samples of the sinc envelope at f = n/T₀.

28:47 Harmonic Sampling and Sinc Zeros Determine Which Coefficients Vanish

The periodic signal’s spectrum consists of discrete samples spaced by the fundamental frequency 1/T₀, with sample heights set by the sinc envelope.
A coefficient is zero when a sampled harmonic lands on a sinc zero, which occurs when n/T₀ is a nonzero integer multiple of 1/τ.
Changing the relationship between τ and T₀ changes which harmonic samples coincide with spectral zeros and therefore which Fourier-series coefficients disappear.

34:12 Sinc-Zero Orthogonality Explains OFDM Subcarrier Packing

Frequency-shifting copies of the sinc spectrum by integer multiples of 1/τ places each copy’s center at the other copies’ zero crossings.
In the time domain, those frequency shifts correspond to multiplying the signal by complex exponentials; receiver-side frequency sampling can separate the orthogonal components without interference.
Ji connects this subcarrier-packing principle to OFDM, naming 4G, 5G, 6G, and Wi-Fi as technologies that use it.

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.