---
title: "Why spectral transformation is required"
subject: "Digital Signal Analysis and Processing"
chapter: 6
key: "semester_7/digital_signal_analysis_and_processing:why-spectral-transformation"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/why-spectral-transformation
site: BCT Engineering
---

# Why spectral transformation is required

Digital Signal Analysis and Processing · Chapter 6: IIR Filter Design

- Asked in: 2070 Asar, 2 marks
- 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/why-spectral-transformation
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-6

## Question

Why is spectral transformation required?

## Short answer

Spectral transformation converts a prototype low-pass filter into other types (low-pass, high-pass, band-pass, band-stop) or adjusts cutoff frequency. It lets a single normalized design derive any filter via simple s-domain or z-domain mappings, saving effort and enabling high-pass/band-stop designs without aliasing in impulse-invariance methods.

## Answer

Spectral (frequency) transformation converts a filter of one type or cut-off into another. It is required because:

- **One prototype, many filters.** Design theory (Butterworth, Chebyshev) is worked out for a low-pass prototype with a normalised cut-off. Transformations turn this into a low-pass filter with another cut-off, or a high-pass, band-pass or band-stop filter, so no separate design procedure is needed for each type.
- **Saves design effort.** An existing, well-tested low-pass design can be reused by changing the cut-off or type.
- **Simple, systematic method.** In the analog domain we replace s; in the digital domain we replace z⁻¹ by an all-pass function, for example z⁻¹ → −(z⁻¹ + α)/(1 + αz⁻¹) for low-pass to high-pass.
- **Needed with impulse invariance.** High-pass and band-stop analog filters are not band-limited, so impulse invariance would alias them badly. Instead, a digital low-pass filter is designed first and then converted to the required type by a digital-domain transformation.

## Check yourself

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

1. Why is spectral transformation needed?
   A. It reduces the filter order
   B. One low-pass prototype can be converted into many filter types and cut-offs
   C. It removes aliasing
   D. It makes filters FIR
   Answer: B. Design theory is done once for a normalised prototype. (Key point: Main reason)

2. In the digital domain what is replaced?
   A. z by s
   B. z by e^jw
   C. z^-1 by an all-pass function
   D. Nothing
   Answer: C. An all-pass mapping preserves stability and magnitude shape. (Key point: Digital domain)

3. What is saved?
   A. Memory only
   B. Hardware
   C. Sampling rate
   D. Design effort, since an existing design is reused
   Answer: D. No separate procedure is needed per type. (Key point: Benefit)

## Related in other subjects

- [How transformation reduces design steps; band stop design](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Abandstop-design-steps-reduction) — Filter Design, chapter 3