--- title: "Transfer function of zero order hold" subject: "Digital Control System" chapter: 1 key: "semester_6/digital_control_system:zoh-transfer-function" source: https://www.bctengineer.com/q/semester_6/digital_control_system/zoh-transfer-function site: BCT Engineering --- # Transfer function of zero order hold Digital Control System · Chapter 1: Introduction to Discrete Time Control System - Asked in: 2081 Chaitra, 4 marks; 2080 Chaitra, 6 marks; 2070 Magh, 8 marks; 2067 Mangsir, 6 marks (asked 4 times) - Video: [Zero Order Hold Transfer Function Proof. Based on IOE Syllabus . Digital Control System.....](https://www.youtube.com/watch?v=zAsuE-esMDU) — Aman Grid - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/digital_control_system/zoh-transfer-function - Whole chapter: https://www.bctengineer.com/md/semester_6/digital_control_system/chapter-1 ## Question Show that the transfer function of zero order hold circuit is given by Gₕ₀(s)=(1-e⁻ᵀˢ)/s. ## Short answer The ZOH holds each sample for one period T, giving impulse response g(t)=u(t)-u(t-T). Taking the Laplace transform of this rectangular pulse gives G_h0(s) = (1 - e^{-Ts})/s, the required transfer function. ## Answer A **zero order hold (ZOH)** holds each sample value constant until the next sample arrives. Its output is a staircase signal. Its transfer function is found from its impulse response. #### Impulse response Apply a unit impulse $\delta(t)$ (sample of strength 1 at $t=0$). The ZOH holds this value for one sampling period $T$ and then drops to zero: $$ g_{h0}(t) = \begin{cases} 1, & 0 \le t < T \\ 0, & \text{otherwise} \end{cases} $$ ```text g(t) 1 |------+ | | | | 0 +------+---------> t 0 T ``` This rectangular pulse is the difference of two unit steps: $$ g_{h0}(t) = u(t) - u(t-T) $$ #### Laplace transform Using $\mathcal{L}[u(t)] = \frac{1}{s}$ and the time-shift theorem $\mathcal{L}[f(t-T)u(t-T)] = e^{-Ts}F(s)$: $$ \begin{aligned} G_{h0}(s) &= \mathcal{L}[u(t)] - \mathcal{L}[u(t-T)] \\ &= \frac{1}{s} - \frac{e^{-Ts}}{s} \\ &= \frac{1-e^{-Ts}}{s} \end{aligned} $$ which is the required result. #### General input For an impulse-sampled input $x^*(t) = \sum x(kT)\delta(t-kT)$, the output is $h(t) = \sum_k x(kT)[u(t-kT) - u(t-(k+1)T)]$, whose Laplace transform is $X^*(s)\,\frac{1-e^{-Ts}}{s}$, confirming the same transfer function. #### Frequency response (for reference) Putting $s = j\omega$: $G_{h0}(j\omega) = T\,\frac{\sin(\omega T/2)}{\omega T/2}\,e^{-j\omega T/2}$, so the ZOH is a low-pass filter with a phase lag of $\omega T/2$ (a delay of half a sampling period). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the response of a zero order hold to a unit impulse? A. A unit pulse of height 1 lasting for 0 <= t < T, then zero B. A ramp rising from 1 to 2 over T C. An impulse of strength T D. A constant 1 for all t >= 0 Answer: A. The ZOH holds the sample value for one period T and then drops to zero, giving a rectangular pulse. (Key point: ZOH impulse response) 2. How is the rectangular pulse g(t) written using unit steps? A. u(t) + u(t-T) B. u(t) - u(t-T) C. u(t) * u(t-T) D. u(t-T) - u(t-2T) Answer: B. A step starting at 0 minus a delayed step starting at T leaves a pulse of height 1 between 0 and T. (Key point: Pulse as step difference) 3. Which property turns the delayed step u(t-T) into e^(-Ts)/s? A. Initial value theorem B. Convolution theorem C. Time-shift theorem of the Laplace transform D. Final value theorem Answer: C. L[f(t-T)u(t-T)] = e^(-Ts)F(s), and with F(s) = 1/s this gives e^(-Ts)/s. (Key point: Time-shift theorem) 4. What does putting s = jω show about the ZOH? A. It is a low-pass filter with phase lag ωT/2 (a delay of half a sampling period) B. It is a high-pass filter with phase lead C. It has unity gain and zero phase at all frequencies D. It amplifies high frequencies Answer: A. G(jω) = T sin(ωT/2)/(ωT/2) e^(-jωT/2): a sinc magnitude (low-pass) and phase lag ωT/2. (Key point: Frequency response of ZOH) ## Close variants set in other exams - [Transfer function of first order hold](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Afoh-transfer-function) — Digital Control System, chapter 1, asked 7× ## Related in Digital Control System - [Pulse transfer function with first order hold](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Afoh-system-pulse-tf) — Digital Control System, chapter 3 - [Data hold circuit and ZOH derivation](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Adata-hold-zoh-derivation) — Digital Control System, chapter 1 - [ZOH and FOH defined; ZOH transfer function](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Azoh-foh-definition) — Digital Control System, chapter 1 - [Series programming of a third-order transfer function](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Aseries-programming-z-plus-1) — Digital Control System, chapter 4, asked 2× - [Ladder programming of a third-order transfer function](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Aladder-z3-4z2) — Digital Control System, chapter 4, asked 2× - [Short note: Sample and hold circuit](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Ashort-note-sample-hold) — Digital Control System, chapter 1 ## Related in other subjects - [Butterworth order, poles and transfer function](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Abutterworth-order-0-5db-20db-2000) — Filter Design, chapter 2 - [3rd order Butterworth HPF from LPF transfer function](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Athird-order-hpf-from-lpf-tf) — Filter Design, chapter 5 - [Transfer function of op-amp phase-shift circuit](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-opamp-allpass-r1-c-r2) — Control Systems, chapter 2, asked 2× - [Transfer function of op-amp integrator-type circuit](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-opamp-r1-r2-c) — Control Systems, chapter 2