---
title: "H(e^(jω)) and h[n] 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"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/difference-equation-0-3
site: BCT Engineering
---

# H(e^(jω)) and h[n] for y[n] = 0.3y[n−1] + x[n]

Signals and Systems · Chapter 6: Discrete-Time Systems

- Asked in: 2076 Bhadra, 7 marks
- 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
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-6

## Question

Find the frequency response H(e^(jω)) and impulse response h[n] for a system characterized by linear constant coefficient difference equation y[n] = 0.3y[n−1] + x[n].

## Short answer

Frequency response: H(e^jω) = 1/(1 - 0.3 e^(-jω)). Impulse response: h[n] = (0.3)^n u[n], a causal, stable lowpass system.

## Answer

Given $y[n]=0.3y[n-1]+x[n]$, i.e. $y[n]-0.3y[n-1]=x[n]$ (system initially at rest).

#### Frequency response
Take the DTFT of both sides, using the time-shift property $y[n-1]\leftrightarrow e^{-j\omega}Y(e^{j\omega})$:

$$
\begin{aligned}
Y(e^{j\omega})-0.3e^{-j\omega}Y(e^{j\omega}) &= X(e^{j\omega})\\
H(e^{j\omega})=\frac{Y(e^{j\omega})}{X(e^{j\omega})} &= \frac{1}{1-0.3e^{-j\omega}}
\end{aligned}
$$

Magnitude and phase (using $e^{-j\omega}=\cos\omega-j\sin\omega$):

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

$$
\angle H(e^{j\omega})=-\tan^{-1}\!\left(\frac{0.3\sin\omega}{1-0.3\cos\omega}\right)
$$

| $\omega$ | 0 | $\pi/4$ | $\pi/2$ | $3\pi/4$ | $\pi$ |
|---|---|---|---|---|---|
| $\lvert H\rvert$ | 1.4286 | 1.2256 | 0.9578 | 0.8126 | 0.7692 |
| $\angle H$ (deg) | 0 | −15.07 | −16.70 | −9.93 | 0 |

The gain falls from $1/0.7$ at $\omega=0$ to $1/1.3$ at $\omega=\pi$, so it is a **low pass** system.

#### Impulse response
Use the standard DTFT pair $a^n u[n]\leftrightarrow\frac{1}{1-ae^{-j\omega}}$ for $|a|<1$. With $a=0.3$:

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

Check by recursion with $x[n]=\delta[n]$, $h[-1]=0$: $h[0]=1$, $h[1]=0.3$, $h[2]=0.09$, $h[3]=0.027,\dots$ which matches $(0.3)^n$.

**Answer: $H(e^{j\omega})=\dfrac{1}{1-0.3e^{-j\omega}}$ and $h[n]=(0.3)^n u[n]$.** The system is causal and stable ($\sum h=1/0.7\approx1.43$).

## Check yourself

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

1. Which DTFT property is used on y[n-1] when solving y[n]=0.3y[n-1]+x[n]?
   A. y[n-1] becomes e^(-jω)Y(e^(jω))
   B. y[n-1] becomes e^(jω)Y(e^(jω))
   C. y[n-1] becomes 0.3Y(e^(jω))
   D. y[n-1] becomes Y(e^(jω))/jω
   Answer: A. A delay by one sample multiplies the transform by e^(-jω). (Key point: DTFT of delay)

2. What is H(e^(jω)) for y[n]=0.3y[n-1]+x[n]?
   A. 1/(1+0.3e^(-jω))
   B. 1/(1-0.3e^(-jω))
   C. 1-0.3e^(-jω)
   D. 0.3/(1-e^(-jω))
   Answer: B. Y(1-0.3e^(-jω))=X, so H=1/(1-0.3e^(-jω)). (Key point: Frequency response)

3. What is h[n]?
   A. (0.3)^n u[-n]
   B. 0.3 delta[n-1]
   C. (0.3)^n u[n]
   D. (1/0.3)^n u[n]
   Answer: C. Using the pair a^n u[n] <-> 1/(1-a e^(-jω)) with a=0.3. (Key point: Impulse response)

4. Why is this a low pass system?
   A. Gain is 0 at ω=0 and rises to a maximum at ω=π
   B. Gain is the same at all frequencies
   C. Gain is infinite at ω=0
   D. Gain is 1/0.7 at ω=0 and falls to 1/1.3 at ω=π
   Answer: D. The magnitude 1/sqrt(1.09-0.6cosω) is largest at ω=0 and smallest at ω=π. (Key point: Low pass nature)

## Close variants set in other exams

- [H(e^(jω)) 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-equation) — Signals and Systems, chapter 6, asked 8×
- [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×
- [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×

## Related in Signals and Systems

- [Impulse and frequency response of y[n] + 0.55y[n−1] = x[n]](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adifference-equation-0-55) — Signals and Systems, chapter 6
- [Impulse response of y[n] − (3/4)y[n−1] + (1/8)y[n−2] = 2x[n]](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Asecond-order-difference-equation) — Signals and Systems, chapter 6, asked 2×
- [Transfer function and magnitude for y[n] = 0.3y[n−1] + x[n]](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adifference-equation-0-3-plot) — Signals and Systems, chapter 6, asked 4×
- [Frequency response; ideal LPF impulse response](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aideal-lpf-cutoff-impulse) — Signals and Systems, chapter 5, asked 2×
- [Impulse and frequency response of y[n] = 0.5y[n−1] + x[n] + x[n+1]](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adifference-equation-noncausal) — Signals and Systems, chapter 6
- [First-order system impulse and step response](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Afirst-order-system-response) — Signals and Systems, chapter 5

## Related in other subjects

- [Need for difference equation; LCCDE and system function](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Awhy-difference-equation-lccde) — Digital Signal Analysis and Processing, chapter 1, asked 2×
- [Difference equation need; pole-zero, magnitude response](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Awhy-difference-eq-magnitude-response) — Digital Signal Analysis and Processing, chapter 3, asked 3×
- [Difference equation example; poles/zeros response](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Apole-zero-define-difference-eq) — Digital Signal Analysis and Processing, chapter 3
- [Linear momentum and impulse-momentum equation](https://www.bctengineer.com/question.md?key=semester_5%2Ffluid_machines%3Alinear-momentum-impulse-equation) — Fluid Machines, chapter 1