---
title: "Derive time shifting property of z-transform"
subject: "Digital Signal Analysis and Processing"
chapter: 2
key: "semester_7/digital_signal_analysis_and_processing:z-time-shifting-derive"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/z-time-shifting-derive
site: BCT Engineering
---

# Derive time shifting property of z-transform

Digital Signal Analysis and Processing · Chapter 2: Z-transform

- Asked in: 2069 Chaitra, 3 marks
- Video: [Time Shifting Property of Z Transform in Signals and Systems Explained](https://www.youtube.com/watch?v=1MebtI_yFVs) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/z-time-shifting-derive
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-2

## Question

Derive the time shifting property of z-transform.

## Short answer

x[n-n0]↔z^(-n0)X(z), with the same ROC except possibly at z=0 or ∞. Derivation: Σ_n x[n-n0] z^(-n) = Σ_m x[m] z^(-(m+n0)) = z^(-n0)X(z) by substitution m=n-n0.

## Answer

**Statement:** If x[n] ↔ X(z) with ROC R, then

x[n − n₀] ↔ z⁻ⁿ⁰ X(z), with the same ROC (except for the possible addition or removal of z = 0 or z = ∞).

**Derivation.** By definition, the z-transform of y[n] = x[n − n₀] is

Y(z) = Σ(n = −∞ to ∞) x[n − n₀] z⁻ⁿ

Let m = n − n₀, so n = m + n₀. As n runs over all integers, so does m:

Y(z) = Σ(m) x[m] z⁻⁽ᵐ⁺ⁿ⁰⁾ = z⁻ⁿ⁰ Σ(m) x[m] z⁻ᵐ = z⁻ⁿ⁰ X(z)

**Remarks:**
- A delay of one sample multiplies X(z) by z⁻¹; an advance of one sample multiplies it by z.
- Multiplying by z⁻ⁿ⁰ can add a pole (or zero) at z = 0 or ∞, so the ROC can change only at these two points.
- For the one-sided (unilateral) z-transform, the delay rule has extra initial-condition terms, e.g. x[n−1] ↔ z⁻¹X(z) + x[−1].

**Example:** δ[n − 3] ↔ z⁻³ (since δ[n] ↔ 1).

## Check yourself

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

1. x[n-n0] has z-transform
   A. z^n0 X(z)
   B. X(z) - n0
   C. X(z/n0)
   D. z^-n0 X(z)
   Answer: D. A delay multiplies by z^-n0. (Key point: Property)

2. Substituting m = n - n0 gives
   A. z^-(m-n0)
   B. z^-m only
   C. z^-(m+n0), pulling out z^-n0
   D. n0 z^-m
   Answer: C. The remaining sum is X(z). (Key point: Derivation step)

3. The ROC changes at most at
   A. the unit circle
   B. z = 0 or z = infinity
   C. every pole
   D. nowhere
   Answer: B. Multiplying by z^-n0 can add or remove z = 0 or infinity. (Key point: ROC)

## Related in Digital Signal Analysis and Processing

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

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