---
title: "Z-transform of a ramp-and-hold input (to 3)"
subject: "Digital Control System"
chapter: 2
key: "semester_6/digital_control_system:waveform-ramp-to-3"
source: https://www.bctengineer.com/q/semester_6/digital_control_system/waveform-ramp-to-3
site: BCT Engineering
---

# Z-transform of a ramp-and-hold input (to 3)

Digital Control System · Chapter 2: The Z-Transform

- Asked in: 2080 Chaitra, 4 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/digital_control_system/waveform-ramp-to-3
- Whole chapter: https://www.bctengineer.com/md/semester_6/digital_control_system/chapter-2

## Question

The waveform below shows the input signal to a digital controller. Obtain the Z-transform of the input x(t). [Figure: x(t) rises linearly from 0 at t = 0 to 3 at t = 3 (passing 1 at t = 1 and 2 at t = 2), then stays constant at 3; dotted sample lines at t = 1, 2, …, 7]

## Short answer

Express x(t)=t·1(t)-(t-3)·1(t-3). Sampled at T=1 s, x(k)=0,1,2,3,3,3,… The z-transform is X(z)=z^{-1}(1-z^{-3})/(1-z^{-1})^2 = (z^2+z+1)/[z^2(z-1)].

## Answer

From the figure, $x(t)$ is a unit-slope ramp that stops rising at $t = 3$ and then holds at 3. Take the sampling period $T = 1$ s, from the sample lines at $t = 1, 2, \dots$.

#### Step 1: Write $x(t)$ as a ramp minus a delayed ramp

$$
x(t) = t\cdot 1(t) - (t-3)\cdot 1(t-3)
$$

For $t \le 3$ only the first term acts, giving $x = t$. For $t > 3$ the result is $t - (t-3) = 3$.

#### Step 2: Sampled values ($T = 1$)

| $k$ | 0 | 1 | 2 | 3 | 4 | 5 | ... |
|---|---|---|---|---|---|---|---|
| $x(k)$ | 0 | 1 | 2 | 3 | 3 | 3 | ... |

#### Step 3: Take the z-transform

Use $\mathcal{Z}[k] = \dfrac{z^{-1}}{(1-z^{-1})^2}$ (with $T = 1$) and the shift theorem $\mathcal{Z}[x(k-n)] = z^{-n}X(z)$:

$$
\begin{aligned}
X(z) &= \frac{z^{-1}}{(1-z^{-1})^2} - z^{-3}\frac{z^{-1}}{(1-z^{-1})^2} = \frac{z^{-1}(1 - z^{-3})}{(1-z^{-1})^2}
\end{aligned}
$$

Since $1 - z^{-3} = (1 - z^{-1})(1 + z^{-1} + z^{-2})$:

$$
X(z) = \frac{z^{-1}(1 + z^{-1} + z^{-2})}{1 - z^{-1}} = \frac{z^2 + z + 1}{z^2(z-1)}
$$

#### Step 4: Check by series

$$
X(z) = z^{-1} + 2z^{-2} + 3z^{-3}\left(1 + z^{-1} + z^{-2} + \dots\right) = z^{-1} + 2z^{-2} + 3z^{-3} + 3z^{-4} + \dots
$$

The coefficients $0, 1, 2, 3, 3, \dots$ match the samples.

Final value check: $\lim_{z\to1}(1-z^{-1})X(z) = 1\cdot(1+1+1) = 3$, which is correct.

**Answer:** $X(z) = \dfrac{z^{-1}(1 - z^{-3})}{(1 - z^{-1})^2} = \dfrac{z^2 + z + 1}{z^2(z-1)}$ (with $T = 1$ s).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How is the ramp that stops at 3 written?
   A. 3 t 1(t)
   B. t 1(t) - 3 1(t-3)
   C. t 1(t) - (t-3) 1(t-3)
   D. t 1(t) + (t-3) 1(t-3)
   Answer: C. Subtracting a ramp delayed by 3 s freezes the signal at 3. (Key point: Decomposition)

2. What does a 3-sample delay do to a z-transform?
   A. Multiplies by z^3
   B. Divides by 3
   C. Replaces z by z-3
   D. Multiplies by z^-3
   Answer: D. Z[x(k-n)] = z^-n X(z). (Key point: Delay theorem)

3. What is X(z) for T = 1?
   A. (z^2+z+1)/(z^2(z-1))
   B. z/(z-1)^2
   C. (z^2+z+1)/(z(z-1))
   D. 3z/(z-1)
   Answer: A. z^-1(1+z^-1+z^-2)/(1-z^-1) = (z^2+z+1)/(z^2(z-1)). (Key point: Result)

4. What does the final value theorem give?
   A. infinity
   B. 1
   C. 3
   D. 0
   Answer: C. (1 - z^-1) X(z) at z = 1 gives 1+1+1 = 3. (Key point: Final value)

## Related in Digital Control System

- [Z-transform of ramp-and-hold signal (T = 1 s)](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Awaveform-ramp-to-1-at-4) — Digital Control System, chapter 2
- [Z-transform of ramp-and-hold continuous signal](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Awaveform-ramp-to-1-at-6) — Digital Control System, chapter 2
- [Z-transform of ramp-and-hold curve (to 1 at t=2)](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Acurve-ramp-to-1-at-2) — Digital Control System, chapter 2
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- [Importance and operation of sample and hold](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Aimportance-of-sample-hold) — Digital Control System, chapter 1

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