---
title: "z-transform of two given sequences"
subject: "Applied Mathematics"
chapter: 2
key: "semester_4/applied_mathematics:z-transform-sequences-15-10"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/z-transform-sequences-15-10
site: BCT Engineering
---

# z-transform of two given sequences

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2075 Baisakh, 2+2 marks
- Video: [Calculating Z transform of given discrete signals.](https://www.youtube.com/watch?v=3rHRevJhvIc) — Engg-Course-Made-Easy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/z-transform-sequences-15-10
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

Find the z-transform of the following sequences: (i) f(k) = {15, 10, 7, 4, 1, -1, 3, 6} [the origin (k = 0) is marked by an arrow under one of the terms, apparently 4]. (ii) f(k) = 5^k for k < 0; 2^k for k ≥ 0.

## Short answer

(i) F(z)=15z^3+10z^2+7z+4+z^{-1}-z^{-2}+3z^{-3}+6z^{-4}, ROC: entire z-plane except 0 and ∞. (ii) F(z)=3z/[(5-z)(z-2)], ROC: 2<|z|<5.

## Answer

Definition: F(z) = Σ f(k) z⁻ᵏ.

#### (i) f(k) = {15, 10, 7, 4, 1, −1, 3, 6}
The origin (k = 0) is at the fourth term (4), as read from the arrow. So the terms are:

```text
k    : -3  -2  -1   0   1   2   3   4
f(k) : 15  10   7   4   1  -1   3   6
```

F(z) = 15 z³ + 10 z² + 7 z + 4 + z⁻¹ − z⁻² + 3 z⁻³ + 6 z⁻⁴

The sequence is of finite length and two-sided, so the ROC is the whole z-plane except z = 0 and z = ∞.

#### (ii) f(k) = 5ᵏ for k < 0; 2ᵏ for k ≥ 0
Split into two parts.

**Part k < 0:** Σ (k = −∞ to −1) 5ᵏ z⁻ᵏ = Σ (m = 1 to ∞) (z/5)ᵐ (with m = −k)
= (z/5)/(1 − z/5) = z/(5 − z),  converges for |z| < 5.

**Part k ≥ 0:** Σ (k = 0 to ∞) (2/z)ᵏ = 1/(1 − 2/z) = z/(z − 2),  converges for |z| > 2.

Add:

F(z) = z/(5 − z) + z/(z − 2) = z [ (z − 2) + (5 − z) ] / [ (5 − z)(z − 2) ]

**F(z) = 3z / [ (5 − z)(z − 2) ],  ROC: 2 < |z| < 5**

## Check yourself

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

1. In sequence (i), which term is at k = 0 and what does that make the 15 term of F(z)?
   A. k = 0 is the term 4, so 15 sits at k = -3 and gives 15 z^3
   B. k = 0 is the first term 15, so F(z) starts with 15
   C. k = 0 is the term 4, so 15 gives 15 z^-3
   D. k = 0 is the last term 6, so 15 gives 15 z^7
   Answer: A. The arrow marks 4 as k = 0, so 15 is at k = -3 and f(k) z^-k gives 15 z^3. (Key point: Locating the origin)

2. For the sequence in (i), the terms to the left of the origin (k < 0) give
   A. negative powers of z
   B. positive powers of z (z^3, z^2, z)
   C. only constants
   D. terms that cancel with the right side
   Answer: B. z^-k with negative k is a positive power, so 7 -> 7z, 10 -> 10z^2, 15 -> 15z^3. (Key point: Sign of powers of z)

3. What is the ROC of the finite two-sided sequence in (i)?
   A. |z| > 1
   B. The whole plane including z = 0
   C. The whole z-plane except z = 0 and z = infinity
   D. |z| < 1
   Answer: C. It has both positive and negative powers of z, so z = 0 and z = infinity must be excluded; every other z gives a finite sum. (Key point: ROC of finite sequence)

4. For (ii), why is the ROC 2 < |z| < 5?
   A. Both parts need |z| > 5
   B. The k < 0 part needs |z| > 5 and the k >= 0 part |z| < 2
   C. The ROC is |z| > 5 because 5 is larger
   D. The k >= 0 part needs |z| > 2 and the k < 0 part needs |z| < 5; both must hold
   Answer: D. Sum of 2^k z^-k converges for |z| > 2; sum over k < 0 of (z/5)^m converges for |z| < 5. The ROC is the overlap. (Key point: Two-sided exponential ROC)

## Related in Applied Mathematics

- [Inverse z of z/(z²+7z+10)](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Ainverse-z-z-over-z2-7z-10) — Applied Mathematics, chapter 2
- [Heat equation, insulated ends, u(x,0)=lx-x²](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aheat-eq-insulated-ends-lx-x2) — Applied Mathematics, chapter 3, asked 2×
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## Related in other subjects

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