---
title: "Z-transform of ramp-and-hold curve (to 1 at t=2)"
subject: "Digital Control System"
chapter: 2
key: "semester_6/digital_control_system:curve-ramp-to-1-at-2"
source: https://www.bctengineer.com/q/semester_6/digital_control_system/curve-ramp-to-1-at-2
site: BCT Engineering
---

# Z-transform of ramp-and-hold curve (to 1 at t=2)

Digital Control System · Chapter 2: The Z-Transform

- Asked in: 2068 Magh, 8 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/curve-ramp-to-1-at-2
- Whole chapter: https://www.bctengineer.com/md/semester_6/digital_control_system/chapter-2

## Question

Obtain Z transform of curve x(t) shown in figure below. [Figure: x(t) rises linearly from 0 at t = 0 through 0.5 at t = 1 to 1 at t = 2, then stays constant at 1; time axis marked 0 to 4]

## Short answer

x(t)=0.5t–0.5(t–2)1(t–2). With T=1 s: X(z)=0.5 z^{-1}(1–z^{-2})/(1–z^{-1})^2 = 0.5(z+1)/[z(z–1)].

## Answer

From the figure, $x(t)$ rises linearly with slope $0.5$ from 0 at $t = 0$ to 1 at $t = 2$, and then stays at 1. Take the sampling period $T = 1$ s, as marked on the axis. The general-$T$ form is also given.

#### Step 1: Express the curve with standard functions

$$
x(t) = 0.5\,t\cdot1(t) - 0.5\,(t-2)\cdot1(t-2)
$$

- For $0 \le t \le 2$: $x = 0.5t$
- For $t > 2$: $x = 0.5t - 0.5t + 1 = 1$

```text
 x(t)
  1 |        ________________
    |       /
0.5 |     /
    |   /
  0 +--/-----+-----+-----+----> t
    0     1     2     3     4
```

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

| $k$ | 0 | 1 | 2 | 3 | 4 | ... |
|---|---|---|---|---|---|---|
| $x(k)$ | 0 | 0.5 | 1 | 1 | 1 | ... |

#### Step 3: Z-transform

Use $\mathcal{Z}[t] = \dfrac{Tz^{-1}}{(1-z^{-1})^2}$ and the real translation theorem $\mathcal{Z}[x(t-nT)1(t-nT)] = z^{-n}X(z)$, with $n = 2/T$:

$$
X(z) = 0.5\cdot\frac{Tz^{-1}}{(1-z^{-1})^2}\left(1 - z^{-2/T}\right)
$$

With $T = 1$, $n = 2$:

$$
\begin{aligned}
X(z) &= \frac{0.5\,z^{-1}(1 - z^{-2})}{(1-z^{-1})^2} = \frac{0.5\,z^{-1}(1 + z^{-1})}{1 - z^{-1}} \\
&= \frac{0.5(z + 1)}{z(z-1)}
\end{aligned}
$$

#### Step 4: Verify by the definition

Directly from $X(z) = \sum x(k)z^{-k}$:

$$
\begin{aligned}
X(z) &= 0.5z^{-1} + z^{-2} + z^{-3} + z^{-4} + \dots \\
&= 0.5z^{-1} + \frac{z^{-2}}{1-z^{-1}} = \frac{0.5z^{-1}(1 - z^{-1}) + z^{-2}}{1 - z^{-1}} = \frac{0.5z^{-1} + 0.5z^{-2}}{1-z^{-1}}
\end{aligned}
$$

This is the same as Step 3.

#### Step 5: Checks with the limit theorems

- Initial value: $x(0) = \lim_{z\to\infty}X(z) = 0$, which is correct.
- Final value: $\lim_{z\to1}(1-z^{-1})X(z) = 0.5(1+1) = 1$, which is correct.

#### Note for other $T$

If $T = 0.5$ s, then $n = 4$ and $X(z) = \dfrac{0.25z^{-1}(1 - z^{-4})}{(1-z^{-1})^2}$, with samples $0, 0.25, 0.5, 0.75, 1, 1, \dots$

**Answer ($T = 1$ s):** $X(z) = \dfrac{0.5z^{-1}(1 - z^{-2})}{(1-z^{-1})^2} = \dfrac{0.5(z+1)}{z(z-1)}$.

## Check yourself

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

1. What is the slope of the rising part?
   A. 2
   B. 0.25
   C. 0.5
   D. 1
   Answer: C. Rises 1 over 2 s. (Key point: Slope)

2. x(t) =?
   A. 0.5t 1(t) - (t-2)1(t-2)
   B. 0.5t 1(t) + 0.5(t-2)1(t-2)
   C. t 1(t) - 0.5(t-2)
   D. 0.5t 1(t) - 0.5(t-2)1(t-2)
   Answer: D. Delayed ramp of the same slope. (Key point: Decomposition)

3. X(z) for T = 1?
   A. 0.5(z+1)/(z(z-1))
   B. 0.5/(z-1)^2
   C. (z+1)/(z-1)
   D. 0.5z/(z-1)
   Answer: A. 0.5z^-1(1+z^-1)/(1-z^-1). (Key point: Result)

4. Final value theorem result?
   A. 0
   B. 1
   C. 0.5
   D. 2
   Answer: B. 0.5(1+1) = 1. (Key point: Final value)

## Related in Digital Control System

- [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
- [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 minus delayed ramp](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Az-transform-ramp-minus-delayed-ramp) — Digital Control System, chapter 2

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