---
title: "Z-transform property proof"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:z-transform-property-proof-three"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/z-transform-property-proof-three
site: BCT Engineering
---

# Z-transform property proof

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2081 Chaitra (new course), 2 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/z-transform-property-proof-three
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

If Z[x(k)] = X(z) and y(k) = Σ_{m=0}^{k} x(m), then show that Y(z) = (z/(z − 1)) X(z).

## Short answer

From y(k)-y(k-1)=x(k), Z-transform gives Y(z)-z^{-1}Y(z)=X(z). Hence Y(z)= (z/(z-1))X(z).

## Answer

We use y(k) = Σ_{m=0}^{k} x(m) (the running sum of x).

**Proof.** Write the difference of consecutive terms, with y(−1) = 0:

```text
y(k) − y(k − 1) = x(k)
```

Take the Z-transform of both sides. By the right-shift property, Z[y(k − 1)] = z⁻¹ Y(z) (since y(−1) = 0):

```text
Y(z) − z⁻¹ Y(z) = X(z)
Y(z) (1 − 1/z) = X(z)
Y(z) = X(z) / (1 − z⁻¹) = (z/(z − 1)) X(z)
```

**Alternative (direct) proof:** Y(z) = Σₖ z^(−k) Σ_{m≤k} x(m). Interchanging the sums,

```text
Y(z) = Σ_m x(m) Σ_{k≥m} z^(−k) = Σ_m x(m) z^(−m) · 1/(1 − z⁻¹) = X(z) · z/(z − 1)
```

for |z| > 1. ∎

## Check yourself

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

1. y(k) = sum of x(m) for m up to k satisfies
   A. y(k) - y(k-1) = x(k)
   B. y(k) + y(k-1) = x(k)
   C. y(k) = x(k) y(k-1)
   D. y(k+1) = x(k)
   Answer: A. The new term x(k) is added. (Key point: Difference relation)

2. With y(-1) = 0, Z[y(k-1)] =
   A. z Y(z)
   B. z^-1 Y(z)
   C. Y(z) - 1
   D. z^-1 Y(z) - y(0)
   Answer: B. Right shift. (Key point: Shift)

3. Then Y(z) =
   A. (z-1)X(z)/z
   B. X(z)/(z-1)
   C. z X(z)/(z-1)
   D. z X(z)/(z+1)
   Answer: C. Y(1 - 1/z) = X. (Key point: Result)

## Related in Mathematics III

- [Z-transform property proof](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Az-transform-property-proof) — Mathematics III, chapter 3
- [Z-transform property proof](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Az-transform-property-proof-two) — Mathematics III, chapter 3
- [Laplace transform of derivative/shift property](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaplace-transform-of-derivative-shift-property) — Mathematics III, chapter 3
- [Laplace transform of derivative/shift property](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaplace-transform-of-derivative-shift-property-two) — Mathematics III, chapter 3

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- [Derive time shifting property of z-transform](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Az-time-shifting-derive) — Digital Signal Analysis and Processing, chapter 2
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