--- title: "FS of unit-amplitude rectangular pulses" subject: "Signals and Systems" chapter: 2 key: "semester_4/signals_and_systems:rectangular-pulse-train-fs" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/rectangular-pulse-train-fs site: BCT Engineering --- # FS of unit-amplitude rectangular pulses Signals and Systems · Chapter 2: Fourier Series - Asked in: 2071 Magh, 5+1 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/rectangular-pulse-train-fs - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-2 ## Question Find Fourier series coefficients for periodic rectangular pulses with unity amplitude. Draw the magnitude spectrum. ## Short answer For a periodic rectangular pulse train of amplitude 1, period T, pulse width 2T1, FS coefficients are a0 = 2T1/T and a_k = sin(k ω0 T1)/(kπ) for k≠0. The magnitude spectrum is a sinc-shaped envelope; for T=4T1, a0=0.5, a1=1/π, a3=-1/(3π). ## Answer Take a periodic train of rectangular pulses with unity amplitude, period $T$ and pulse width $2T_1$, centred at $t = 0$: $$ x(t) = \begin{cases} 1, & |t| < T_1 \\ 0, & T_1 < |t| < T/2 \end{cases} \qquad x(t+T) = x(t), \quad \omega_0 = \frac{2\pi}{T} $$ ```text x(t) 1 ___ ___ ___ | | | | | | ______| |_____| |_____| |___ t -T1 T1 T-T1 T+T1 (width 2T1, period T) ``` #### Fourier series coefficients DC term (the average value): $$ a_0 = \frac{1}{T}\int_{-T_1}^{T_1} dt = \frac{2T_1}{T} $$ For $k \ne 0$: $$ \begin{aligned} a_k &= \frac{1}{T}\int_{-T_1}^{T_1} e^{-jk\omega_0 t}dt = \frac{1}{T}\cdot\frac{e^{-jk\omega_0 T_1} - e^{jk\omega_0 T_1}}{-jk\omega_0} \\ &= \frac{2}{k\omega_0 T}\sin(k\omega_0 T_1) = \frac{\sin(k\omega_0 T_1)}{k\pi} \end{aligned} $$ This can also be written $a_k = \frac{2T_1}{T}\,\mathrm{sinc}\!\left(\frac{2kT_1}{T}\right)$, with $\mathrm{sinc}(x) = \frac{\sin \pi x}{\pi x}$. #### Example: T = 4T₁ (duty cycle 50%) $a_k = \frac{\sin(k\pi/2)}{k\pi}$: $a_0 = 0.5$, $a_{\pm1} = 1/\pi \approx 0.318$, $a_{\pm2} = 0$, $a_{\pm3} = -1/(3\pi) \approx -0.106$, $a_{\pm5} \approx 0.064$. #### Magnitude spectrum ```text |a_k| 0.5 | 0.32 | | | 0.11 | | | | | 0.06 | | | | | | | --------+--+--+--+--+--+--+--- k -5 -3 -1 0 1 3 5 (|a_k| = 0 for even k != 0) ``` The lines sit at $k\omega_0$ under a sinc-shaped envelope. A wider pulse ($T_1$ larger) gives a narrower envelope. A longer period ($T$ larger) gives more closely spaced lines, and as $T \to \infty$ the scaled coefficients $Ta_k$ merge into the continuous Fourier transform $2\sin(\omega T_1)/\omega$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For a unity pulse train with width 2T1 and period T, a_0 is: A. T1/T B. 2T1/T C. 1 D. T/(2T1) Answer: B. a_0 is the average, 2T1/T. (Key point: DC coefficient) 2. For k not 0, a_k is: A. sin(k w0 T1)/(k pi) B. cos(k w0 T1)/(k pi) C. sin(k w0 T1)/k D. 2T1/T Answer: A. Integrating gives sin(k w0 T1)/(k pi) = (2T1/T) sinc(2kT1/T). (Key point: General coefficient) 3. With T = 4T1 (50% duty cycle), |a_k| for even k not 0 is: A. 1/pi B. 0 C. 0.5 D. 1/(2 pi) Answer: B. sin(k pi/2) = 0 for even k. (Key point: Even harmonics) 4. If the period T is made longer with the pulse width fixed, the spectral lines: A. Spread farther apart B. Disappear C. Become more closely spaced, approaching a continuous spectrum D. Move to zero Answer: C. Line spacing w0 = 2 pi/T shrinks; as T tends to infinity the lines merge into the Fourier transform. 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