---
title: "Z-transform of ramp-and-hold continuous signal"
subject: "Digital Control System"
chapter: 2
key: "semester_6/digital_control_system:waveform-ramp-to-1-at-6"
source: https://www.bctengineer.com/q/semester_6/digital_control_system/waveform-ramp-to-1-at-6
site: BCT Engineering
---

# Z-transform of ramp-and-hold continuous signal

Digital Control System · Chapter 2: The Z-Transform

- Asked in: 2073 Magh, 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-1-at-6
- Whole chapter: https://www.bctengineer.com/md/semester_6/digital_control_system/chapter-2

## Question

Obtain the z-transform of x(t) for which time response is given by: [Figure: continuous time signal x(t) rising linearly from 0 at the origin to 1.0 at the tick labelled 6, then constant at 1.0; 0.5 marked on the x(t) axis; time-axis ticks labelled 1 (just left of the origin), 2 (at the origin), then 3 to 13]

## Short answer

Interpreted as ramp slope 1/4 reaching 1 at t=4 s: x(t)=(1/4)t – (1/4)(t–4)1(t–4). With T=1 s, X(z)=z^{-1}(1–z^{-4})/[4(1–z^{-1})^2]=(z^3+z^2+z+1)/[4z^3(z–1)].

## Answer

**Reading the figure:** the tick labels are offset. "2" sits at the origin, so the ramp starts at $t = 0$ and reaches 1.0 four divisions later (the tick labelled 6), then stays at 1.0. Each division is taken as one sampling period, $T = 1$ s. So $x(t)$ rises linearly from 0 to 1 in 4 s and then holds at 1. This is the standard textbook signal $x(t) = \tfrac14 t - \tfrac14(t-4)1(t-4)$.

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

The ramp has slope $\dfrac{1}{4}$ and reaches 1 at $t = 4$ s, after which it holds at 1:

$$
x(t) = \frac{1}{4}t\cdot1(t) - \frac{1}{4}(t-4)\cdot1(t-4)
$$

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

| $k$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | ... |
|---|---|---|---|---|---|---|---|---|
| $x(k)$ | 0 | 0.25 | 0.5 | 0.75 | 1 | 1 | 1 | ... |

#### Step 3: Z-transform

With $\mathcal{Z}[t] = \dfrac{Tz^{-1}}{(1-z^{-1})^2}$ ($T = 1$) and delay of 4 samples $\to z^{-4}$:

$$
\begin{aligned}
X(z) &= \frac{1}{4}\cdot\frac{z^{-1}}{(1-z^{-1})^2} - \frac{1}{4}z^{-4}\cdot\frac{z^{-1}}{(1-z^{-1})^2} \\
&= \frac{1}{4}\cdot\frac{z^{-1}(1 - z^{-4})}{(1-z^{-1})^2}
\end{aligned}
$$

Using $1 - z^{-4} = (1-z^{-1})(1+z^{-1}+z^{-2}+z^{-3})$:

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

#### Step 4: Check

Series: $X(z) = 0.25z^{-1} + 0.5z^{-2} + 0.75z^{-3} + z^{-4} + z^{-5} + \dots$, which matches the table.

Final value: $\lim_{z\to1}(1-z^{-1})X(z) = \dfrac{1\cdot 4}{4} = 1$, which is correct.

If the signal were instead read as reaching 1 after $n$ sampling periods, the same method gives $X(z) = \dfrac{z^{-1}(1-z^{-n})}{n(1-z^{-1})^2}$.

**Answer:** $X(z) = \dfrac{z^{-1}(1-z^{-4})}{4(1-z^{-1})^2} = \dfrac{z^3+z^2+z+1}{4z^3(z-1)}$ (with $T = 1$ s).

## Check yourself

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

1. x(t) =?
   A. (1/6)t 1(t) - (1/6)(t-6)1(t-6)
   B. t - (t-4)
   C. (1/4)t 1(t)
   D. (1/4)t 1(t) - (1/4)(t-4)1(t-4)
   Answer: D. Slope 1/4, delay 4. (Key point: Decomposition)

2. X(z)?
   A. (z^3+z^2+z+1)/(4z^3(z-1))
   B. (z^5+...+1)/(6z^5(z-1))
   C. 1/(z^4(z-1))
   D. z/(4(z-1)^2)
   Answer: A. Same as the 4-sample ramp. (Key point: Result)

3. For a ramp reaching 1 after n periods, X(z) = ?
   A. z^-1(1+z^-n)/(n(1-z^-1)^2)
   B. z^-1(1-z^-n)/(n(1-z^-1)^2)
   C. z^-n/(1-z^-1)
   D. n z^-1/(1-z^-1)^2
   Answer: B. Replace 4 by n. (Key point: General n)

## 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 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
- [Z-transform of a ramp-and-hold input (to 3)](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Awaveform-ramp-to-3) — Digital Control System, chapter 2

## Related in other subjects

- [Fourier transform of a ramp pulse](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aft-ramp-signal) — Data Communication, chapter 3
- [Thermocouple time constant and frequency response](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Athermocouple-first-order-response-numerical) — Instrumentation and Measurement, chapter 4