Limit of the Fourier Series

ECE 210: As Period (T) → ∞, Fundamental Frequency (ω₀) → 0
Period (T) = 4.00 s Fundamental (ω₀) = 1.571 rad/s
−10−5051000.511.5
Time Domain: f​T​(t)Time (t)Amplitude
−20−15−10−505101520−0.500.511.52
Continuous F(ω)Discrete F(nω₀)Frequency Domain: SpectrumFrequency ω (rad/s)F(ω)
Pedagogical Note for Students:
1. Time Domain: As \(T\) increases, the neighboring pulses move further away. In the limit \(T \to \infty\), only the central pulse remains (it becomes aperiodic).
2. Frequency Domain: The orange bars represent the discrete Fourier Series coefficients (scaled by \(T\), exactly as \(F(n\omega_0)\) in lecture). Notice their width is exactly \(\omega_0\).
3. The Calculus Limit: As \(\omega_0 \to 0\), the discrete orange bars pack together to perfectly fill the continuous blue envelope \(F(\omega)\). The sum of their areas \(\sum F(n\omega_0)\omega_0\) becomes the Riemann integral \(\int F(\omega)d\omega\).