---
title: "Characteristics of different window functions"
subject: "Digital Signal Analysis and Processing"
chapter: 5
key: "semester_7/digital_signal_analysis_and_processing:window-characteristics"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/window-characteristics
site: BCT Engineering
---

# Characteristics of different window functions

Digital Signal Analysis and Processing · Chapter 5: FIR Filter Design

- Asked in: 2070 Chaitra, 4 marks
- Video: [Types of Window in DSP - Its Comparison - DTSP](https://www.youtube.com/watch?v=-uZDx4PzRWo) — Padmasri Naban
- 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/window-characteristics
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-5

## Question

Discuss the characteristics of different types of window functions.

## Short answer

Windows trade off main-lobe width (transition width) and side-lobe level (ripple/attenuation). Rectangular: 4π/M width, −13 dB side lobe, 21 dB attenuation. Bartlett, Hanning, Hamming, Blackman provide increasing attenuation (25, 44, 53, 74 dB) and wider main lobes (8π/M or 12π/M). All are symmetric for linear phase. Kaiser allows adjustable β for any attenuation.

## Answer

A window w(n) of length M truncates the ideal impulse response. In frequency the result is the ideal response convolved with W(e^jω), so two features of W matter: the **main-lobe width** (sets the transition width) and the **side-lobe level** (sets the ripple/attenuation).

| Window | w(n), 0 ≤ n ≤ M−1 | Main-lobe width | Peak side lobe | Min. stop-band attenuation |
|---|---|---|---|---|
| Rectangular | 1 | 4π/M | −13 dB | 21 dB |
| Bartlett (triangular) | 1 − 2\|n − α\|/(M−1) | 8π/M | −25 dB | 25 dB |
| Hanning | 0.5 − 0.5 cos(2πn/(M−1)) | 8π/M | −31 dB | 44 dB |
| Hamming | 0.54 − 0.46 cos(2πn/(M−1)) | 8π/M | −41 dB | 53 dB |
| Blackman | 0.42 − 0.5 cos(2πn/(M−1)) + 0.08 cos(4πn/(M−1)) | 12π/M | −57 dB | 74 dB |
| Kaiser | I₀(β√(1−((n−α)/α)²)) / I₀(β) | depends on β | adjustable | adjustable |

(α = (M−1)/2 in all windows.)

#### General characteristics
- All are symmetric about α, so the filter has **linear phase**.
- Tapered windows (Bartlett, Hanning, Hamming, Blackman) fall smoothly towards the ends, which lowers the side lobes (less Gibbs ripple) but widens the main lobe (wider transition band). The rectangular window is the opposite.
- The ripple in the pass band and stop band is almost equal and set by the window, not by M. Increasing M only narrows the transition band.
- Hamming has the lowest peak side lobe among the 8π/M windows; Blackman has the best attenuation but needs the longest filter.
- The Kaiser window gives continuous control of the trade-off through β.

## Check yourself

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

1. Which two window features matter, and what does each set?
   A. Main-lobe width sets transition width; side-lobe level sets ripple/attenuation
   B. Main-lobe width sets ripple; side-lobe level sets transition width
   C. Length sets ripple; phase sets attenuation
   D. Peak value sets transition; mean sets ripple
   Answer: A. A wide main lobe gives a wider transition band; high side lobes mean more ripple. (Key point: Two key features)

2. What is the trade-off of tapered windows compared with rectangular?
   A. Lower side lobes but wider main lobe
   B. Higher side lobes and narrower main lobe
   C. Lower side lobes and narrower main lobe
   D. No change
   Answer: A. Tapering reduces Gibbs ripple at the cost of a wider transition band. (Key point: Tapered vs rectangular)

3. Which window has the best attenuation but needs the longest filter?
   A. Hamming
   B. Blackman
   C. Hanning
   D. Bartlett
   Answer: B. Blackman gives 74 dB with a 12pi/M main lobe. (Key point: Best attenuation)

4. What does increasing M do for a given window?
   A. Narrows the transition band; ripple is set by the window
   B. Reduces ripple to zero
   C. Increases ripple
   D. Changes linear phase to nonlinear
   Answer: A. Ripple is almost independent of M; only the transition width shrinks. (Key point: Effect of M)

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