--- title: "Short note: Obtaining a pulse transfer function" subject: "Digital Control System" chapter: 3 key: "semester_6/digital_control_system:short-note-pulse-tf-procedure" source: https://www.bctengineer.com/q/semester_6/digital_control_system/short-note-pulse-tf-procedure site: BCT Engineering --- # Short note: Obtaining a pulse transfer function Digital Control System · Chapter 3: Analysis of Discrete Time Control System - Asked in: 2071 Magh, 4 marks - Video: [PULSE TRANSFER FUNCTION | BLOCK DIAGRAM | DIGITAL CONTROL SYSTEM](https://www.youtube.com/watch?v=7Z5X_paO3us) — VIEW EDUCATION CENTER - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/digital_control_system/short-note-pulse-tf-procedure - Whole chapter: https://www.bctengineer.com/md/semester_6/digital_control_system/chapter-3 ## Question Write a short note on general procedure for obtaining pulse transfer function. ## Short answer Draw the block diagram with all samplers and name signals. Write s‑domain equations, then take starred transforms noting that blocks not separated by a sampler must be combined as (G₁G₂)*. Obtain pulse transfer functions via partial fractions or the residue formula; with a ZOH use G(z) = (1−z⁻¹)Z[Gₚ/s]. Solve the algebraic equations for C(z)/R(z); if a continuous block precedes the first sampler, only C(z) can be found. ## Answer The **pulse transfer function** $G(z) = Y(z)/X(z)$ relates the z-transform of the sampled output to the z-transform of the sampled input, with zero initial conditions. The general procedure is: 1. **Draw the block diagram with all samplers.** Name the input of each sampler (for example $E$, $U$). Name the outputs as starred signals $E^*$, $U^*$. Add a fictitious sampler at the output if the output is continuous. 2. **Write the s-domain equations.** Express each sampler input and the output in terms of the starred signals and the input $R(s)$, e.g. $U(s) = G_1(s)E^*(s)$. 3. **Take the starred transform of each equation.** Use $[G(s)X^*(s)]^* = G^*(s)X^*(s)$. Elements not separated by a sampler must be combined first: $[G_1G_2X^*]^* = (G_1G_2)^*X^*$, and this is not equal to $G_1^*G_2^*X^*$. 4. **Find each pulse transfer function** using partial fractions or residues: $$ G(z) = \mathcal{Z}\big[G(s)\big] = \sum \text{Res}\left[G(s)\frac{z}{z-e^{Ts}}\right] $$ With a ZOH, $G(z) = (1-z^{-1})\,\mathcal{Z}\left[G_p(s)/s\right]$. 5. **Solve the algebraic equations** for $C(z)$ and form $C(z)/R(z)$. If $R(s)$ enters through a continuous block before the first sampler (as $GR(z)$), only $C(z)$ can be found. In that case no pulse transfer function exists. **Example:** Unity feedback with a sampler on the error, a ZOH and plant $G_p(s)$. Then $E^* = R^* - (G_hG_p)^*E^*$, which gives $\dfrac{C(z)}{R(z)} = \dfrac{G(z)}{1+G(z)}$ with $G(z) = (1-z^{-1})\mathcal{Z}[G_p/s]$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The pulse transfer function G(z) = Y(z)/X(z) relates A. Continuous input to continuous output B. The z-transform of the sampled output to that of the sampled input, with zero initial conditions C. Output to input at nonzero initial conditions D. Sampling period to gain Answer: B. It is defined for sampled signals with zero initial conditions. (Key point: Definition) 2. For blocks G1 and G2 not separated by a sampler, [G1 G2 X*]* equals A. G1*(s) G2*(s) X*(s) B. (G1G2)*(s) X*(s) C. G1(s) G2(s) X(s) D. (G1* + G2*) X* Answer: B. Elements not separated by a sampler must be combined first: (G1G2)* is not equal to G1*G2*. (Key point: Elements without a sampler) 3. What is added if the output is continuous? A. A fictitious sampler at the output B. An extra ZOH C. A differentiator D. A feedback path Answer: A. A fictitious sampler lets the sampled output be written as Y(z). (Key point: Output sampler) 4. When can only C(z) be found, with no pulse transfer function? A. When the sampling period is large B. When R(s) enters through a continuous block before the first sampler (giving GR(z)) C. When there is a ZOH D. When feedback is unity Answer: B. The input cannot then be separated out, so C(z) cannot be written as a ratio to R(z). 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