--- title: "ZOH and FOH defined; ZOH transfer function" subject: "Digital Control System" chapter: 1 key: "semester_6/digital_control_system:zoh-foh-definition" source: https://www.bctengineer.com/q/semester_6/digital_control_system/zoh-foh-definition site: BCT Engineering --- # ZOH and FOH defined; ZOH transfer function Digital Control System · Chapter 1: Introduction to Discrete Time Control System - Asked in: 2078 Chaitra, 6 marks - Video: [Zero Order Hold & First Order Hold Sampling II Interpolation II Signals & Systems II KK Awasthi](https://www.youtube.com/watch?v=cyOK0bD5ojQ) — KK Awasthi EduHub - 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-foh-definition - Whole chapter: https://www.bctengineer.com/md/semester_6/digital_control_system/chapter-1 ## Question What do you mean by Zero Order Hold (ZOH) and First Order Hold? Determine the transfer function for ZOH. ## Short answer ZOH holds each sample constant for the full sampling period, producing a staircase output. FOH extrapolates a straight line using the two most recent samples. ZOH impulse response is a unit pulse of duration T; its transfer function is G_h0(s) = (1-e^(-Ts))/s. ## Answer #### Zero order hold and first order hold A **hold circuit** reconstructs a continuous signal from samples by extrapolating between sampling instants with a polynomial of order $n$. - **Zero order hold (ZOH):** holds the latest sample constant until the next one: $h(kT+\tau) = x(kT)$, $0 \le \tau < T$. Output is a staircase. It needs only the present sample. - **First order hold (FOH):** extrapolates a straight line using the slope of the last two samples: $h(kT+\tau) = x(kT) + \dfrac{x(kT)-x((k-1)T)}{T}\,\tau$. Transfer function $G_{h1}(s) = \frac{Ts+1}{T}\left(\frac{1-e^{-Ts}}{s}\right)^2$. ```text samples: . . . . ZOH: |‾‾‾|‾‾‾|‾‾‾|‾‾‾ FOH: / / / / ``` | ZOH | FOH | |---|---| | One sample used | Two samples used | | Staircase output | Ramp segments | | Simple, low cost | Needs memory of past sample | | Lag $\omega T/2$ | More lag at high frequency | #### Transfer function of ZOH Apply a unit impulse at $t=0$. The ZOH holds the value 1 for one sampling period and then drops to zero, so its impulse response is a rectangular pulse: $$ g_{h0}(t) = u(t) - u(t-T) $$ ```text g(t) 1 |------+ | | 0 +------+-------> t 0 T ``` Taking the Laplace transform and using the shift theorem $\mathcal{L}[u(t-T)] = e^{-Ts}/s$: $$ \begin{aligned} G_{h0}(s) &= \frac{1}{s} - \frac{e^{-Ts}}{s} \\ &= \frac{1-e^{-Ts}}{s} \end{aligned} $$ #### Frequency response With $s = j\omega$: $$ G_{h0}(j\omega) = T\,\frac{\sin(\omega T/2)}{\omega T/2}\,e^{-j\omega T/2} $$ so the ZOH acts as a low-pass filter with gain $T$ at low frequency, zero gain at multiples of $\omega_s$, and a phase lag of $\omega T/2$ (half-period delay). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. ZOH output between samples is: A. h(kT+t) = x(kT), a staircase B. A straight line with slope of last two samples C. Zero D. Equal to x((k+1)T) Answer: A. It holds the latest sample constant. (Key point: ZOH definition) 2. FOH extrapolates using: A. The present sample only B. The slope of the last two samples C. The next sample D. A constant Answer: B. h = x(kT) + [x(kT) - x((k-1)T)]/T t. (Key point: FOH definition) 3. ZOH has gain T at low frequency and phase: A. Lead of wT/2 B. Lag of wT C. Lag of wT/2 D. Zero phase Answer: C. G = T sinc(wT/2) e^(-jwT/2). (Key point: ZOH frequency response) 4. The ZOH gain is zero at: A. wm only B. w = 0 C. Odd multiples of ws/2 D. Multiples of ws Answer: D. sin(wT/2) = 0 at w = n ws. (Key point: ZOH zeros) ## Related in Digital Control System - [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× - [Transfer function of zero order hold](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Azoh-transfer-function) — Digital Control System, chapter 1, asked 7× - [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 transfer function and pulse transfer function](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Azoh-pulse-transfer-function) — 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× ## Related in other subjects - [First order filter: pole -100, zero -1000, gain 10](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Anoninverting-pole-100-zero-1000) — Filter Design, chapter 6 - [Models of zero, first and second order systems](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Azero-first-second-order-models) — Instrumentation and Measurement, chapter 4 - [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