---
title: "Transfer function and magnitude for y[n] = 0.3y[n−1] + x[n]"
subject: "Signals and Systems"
chapter: 6
key: "semester_4/signals_and_systems:difference-equation-0-3-plot"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/difference-equation-0-3-plot
site: BCT Engineering
---

# Transfer function and magnitude for y[n] = 0.3y[n−1] + x[n]

Signals and Systems · Chapter 6: Discrete-Time Systems

- Asked in: 2074 Bhadra, 6 marks
- Video: [Q7. b. Frequency Response & Impulse Response | EnggClasses](https://www.youtube.com/watch?v=s9RZeAaRwFM) — EnggClasses
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/difference-equation-0-3-plot
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-6

## Question

For a system characterized by linear constant coefficient difference equation: y[n] = 0.3y[n−1] + x[n], find the transfer function, plot magnitude and find the impulse response of the system.

## Short answer

Take z-transforms to get H(z)=1/(1-0.3 z^-1). Magnitude is |H(e^jw)|=1/sqrt(1.09-0.6 cos w), maximum 1.4286 at w=0 and minimum 0.7692 at w=pi, so it is low-pass. Impulse response is h[n]=(0.3)^n u[n], causal and stable.

## Answer

System: $y[n]=0.3y[n-1]+x[n]$, initially at rest.

#### Transfer function
Take the z-transform:

$$
\begin{aligned}
Y(z)&=0.3z^{-1}Y(z)+X(z)\\
H(z)=\frac{Y(z)}{X(z)}&=\frac{1}{1-0.3z^{-1}}=\frac{z}{z-0.3}, \qquad |z|>0.3
\end{aligned}
$$

There is a pole at $z=0.3$ (inside the unit circle) and a zero at $z=0$. On the unit circle, $z=e^{j\omega}$:

$$
H(e^{j\omega})=\frac{1}{1-0.3e^{-j\omega}}
$$

#### Magnitude
$$
|H(e^{j\omega})|=\frac{1}{\sqrt{(1-0.3\cos\omega)^2+(0.3\sin\omega)^2}}=\frac{1}{\sqrt{1.09-0.6\cos\omega}}
$$

| $\omega$ | 0 | $\pi/4$ | $\pi/2$ | $3\pi/4$ | $\pi$ |
|---|---|---|---|---|---|
| $\lvert H\rvert$ | 1.4286 | 1.2256 | 0.9578 | 0.8126 | 0.7692 |

The magnitude is even in $\omega$ and periodic with period $2\pi$:

```text
 |H(e^jw)|
 1.43 |            *
      |         *     *
 1.0  |      *           *
      |   *                 *
 0.77 |*                       *
      +--+--------+--------+---> w
       -pi        0        pi
```

It is a **low pass** response: maximum $1/0.7$ at $\omega=0$, minimum $1/1.3$ at $\omega=\pm\pi$.

#### Impulse response
Using $a^n u[n]\leftrightarrow\frac{1}{1-az^{-1}}$ ($|z|>|a|$) with $a=0.3$:

$$
h[n]=(0.3)^n u[n]
$$

Recursion check with $x=\delta[n]$: $h[0]=1,\ h[1]=0.3,\ h[2]=0.09,\ h[3]=0.027,\dots$

**Answer: $H(z)=\dfrac{1}{1-0.3z^{-1}}$, $|H(e^{j\omega})|=\dfrac{1}{\sqrt{1.09-0.6\cos\omega}}$, $h[n]=(0.3)^n u[n]$** (causal and stable).

## Check yourself

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

1. What is H(z) for y[n]=0.3y[n-1]+x[n]?
   A. 1/(1-0.3z^(-1)) = z/(z-0.3), |z|>0.3
   B. 1/(1+0.3z^(-1))
   C. z/(z+0.3)
   D. (1-0.3z^(-1))
   Answer: A. Z-transform gives Y(1-0.3z^(-1))=X. (Key point: Transfer function)

2. Where are the pole and zero of H(z)?
   A. Pole at z=0, zero at z=0.3
   B. Pole at z=0.3 (inside unit circle), zero at z=0
   C. Pole at z=-0.3, zero at z=0
   D. Pole at z=0.3, zero at z=1
   Answer: B. H=z/(z-0.3) has numerator zero at 0 and denominator zero at 0.3. (Key point: Poles and zeros)

3. Which statement about |H(e^(jω))| is right?
   A. Odd in ω, with max at ω=π
   B. Not periodic, decreasing forever
   C. Even in ω, period 2π, max 1/0.7 at ω=0 and min 1/1.3 at ω=π
   D. Constant magnitude 1 (all-pass)
   Answer: C. Magnitude is 1/sqrt(1.09-0.6cosω), which is even and 2π-periodic. (Key point: Magnitude shape)

4. What is h[n], and what check confirms it?
   A. (0.3)^n u[n]; recursion gives 1, 0.6, 1.2...
   B. 0.3 u[n]; recursion gives 0.3 each time
   C. (0.3)^n u[-n]; recursion gives 1, 3.33, 11.1...
   D. (0.3)^n u[n]; recursion gives 1, 0.3, 0.09, 0.027...
   Answer: D. Feeding delta[n] into the recursion multiplies by 0.3 each step. (Key point: Impulse response)

## Close variants set in other exams

- [Transfer function of y[n] = 0.4y[n−1] + x[n]](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adifference-equation-0-4) — Signals and Systems, chapter 6, asked 9×
- [H(e^(jω)), magnitude plot and h[n] for y[n] = 0.5y[n−1] + x[n]](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Afirst-order-difference-plot) — Signals and Systems, chapter 6, asked 9×

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