---
title: "Filter definition and denormalization from ωold to ωnew"
subject: "Filter Design"
chapter: 1
key: "semester_5/filter_design:filter-normalization-frequency-change"
source: https://www.bctengineer.com/q/semester_5/filter_design/filter-normalization-frequency-change
site: BCT Engineering
---

# Filter definition and denormalization from ωold to ωnew

Filter Design · Chapter 1: Introduction

- Asked in: 2081 Bhadra, 1+2+4 marks
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/filter_design/filter-normalization-frequency-change
- Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-1

## Question

What is Filter? Explain the significance of normalization and denormalization during filter design. Derive the expression to calculate the new values of elements that will change the operating frequency of a low pass filter from ωold to ωnew.

## Short answer

A filter is a frequency-selective two-port network. Normalization uses ωo=1 rad/s and R=1 Ω for universal tables and simpler design; denormalization scales to desired frequency and impedance. To change frequency from ωold to ωnew, set kf = ωnew/ωold; then Rnew = Rold, Lnew = Lold/kf, Cnew = Cold/kf. With impedance scaling km: Rnew = km·Rold, Lnew = (km/kf)·Lold, Cnew = Cold/(km·kf).

## Answer

#### Filter
A **filter** is a frequency-selective two-port network that passes signals in a chosen band of frequencies (passband) with little attenuation and attenuates signals at other frequencies (stopband).

#### Significance of normalization and denormalization
**Normalization** means designing the filter with convenient reference values, usually a cut-off (or half-power) frequency of $\omega_o = 1$ rad/s and a termination of $R = 1\ \Omega$. **Denormalization (scaling)** converts the normalized element values to the actual frequency and impedance level required.

Importance:
- Filter tables (Butterworth, Chebyshev, Bessel poles and ladder element values) are published only in normalized form; one table serves every frequency and impedance level.
- Calculations use simple numbers like 1, 1.414, 2 instead of values like $10^{-9}$ and $10^{4}$, so errors are fewer.
- Designs can be compared and reused: a 1 rad/s prototype becomes a 1 kHz or 1 MHz filter by simple scaling.
- Scaling lets the designer choose practical element values (e.g. capacitors in nF–μF, resistors in kΩ) without changing the shape of the response.

#### New element values for changing $\omega_{old}$ to $\omega_{new}$
Let $k_f = \omega_{new}/\omega_{old}$. For the same response at the new frequency, each element's reactance at $\omega_{new}$ must equal its old reactance at $\omega_{old}$:
$$
\begin{aligned}
\omega_{new} L_{new} = \omega_{old} L_{old} &\Rightarrow L_{new} = \frac{L_{old}}{k_f} \\
\frac{1}{\omega_{new} C_{new}} = \frac{1}{\omega_{old} C_{old}} &\Rightarrow C_{new} = \frac{C_{old}}{k_f} \\
R_{new} &= R_{old}
\end{aligned}
$$
The transfer function becomes $T_{new}(s) = T_{old}(s/k_f)$. Combined with impedance scaling by $k_m$: $R' = k_m R$, $L' = k_m L/k_f$, $C' = C/(k_m k_f)$.

**Example:** the normalized third-order Butterworth ladder (1 Ω, 1 H, 2 F, 1 H, 1 Ω, $\omega_o$ = 1 rad/s) scaled to $\omega_o = 10^4$ rad/s ($k_f = 10^4$) and 1 kΩ terminations ($k_m = 1000$) gives $R$ = 1 kΩ, $L = 1\times1000/10^4 = 0.1$ H and $C = 2/(1000\times10^4) = 0.2\ \mu$F.

## Check yourself

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

1. Why are filter tables published only in normalized form?
   A. Real element values cannot be tabulated because they are negative
   B. Normalized filters have a different response shape that is easier to build
   C. One table serves every frequency and impedance level after scaling
   D. Denormalization changes the response shape to the required one
   Answer: C. Scaling keeps the shape of the response, so a single normalized table can be scaled to any requirement. (Key point: Why normalize)

2. Which condition gives the new inductor value when frequency changes from wold to wnew?
   A. Lnew = kf Lold
   B. wnew Lnew = wold Lold, so Lnew = Lold/kf
   C. Lnew = Lold x wnew
   D. wnew Lnew = kf wold Lold
   Answer: B. The reactance at the new frequency must equal the old reactance at the old frequency. (Key point: Equal reactance condition)

3. For frequency scaling by kf = w_new/w_old, how do R, L and C change?
   A. R' = R, L' = L/kf, C' = C/kf
   B. R' = kf R, L' = L/kf, C' = C/kf
   C. R' = R, L' = kf L, C' = kf C
   D. R' = R, L' = L/kf, C' = C kf
   Answer: A. Equal impedance at kf*w requires kf*L' = L and kf*C' = C; R does not depend on frequency. (Key point: Frequency scaling)

4. What are the combined (km and kf) scaling formulas?
   A. R' = km R, L' = (kf/km) L, C' = C/(km kf)
   B. R' = km R, L' = km kf L, C' = C/(km kf)
   C. R' = R/km, L' = (km/kf) L, C' = C km kf
   D. R' = km R, L' = (km/kf) L, C' = C/(km kf)
   Answer: D. Applying kf then km multiplies L by km and divides by kf; C is divided by both. (Key point: Combined scaling)

## Related in Filter Design

- [Element values for changing half-power frequency](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Adenormalize-half-power-frequency-change) — Filter Design, chapter 1, asked 3×
- [Normalization, denormalization and element scaling equations](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Anormalization-element-scaling-equations) — Filter Design, chapter 1, asked 5×
- [Normalization and denormalization with example](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Anormalization-importance-example) — Filter Design, chapter 1
- [Need for all-pass filters; importance of normalization](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Aallpass-need-normalization) — Filter Design, chapter 2
- [Ideal filter and magnitude/frequency scaling](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Aideal-filter-magnitude-frequency-scaling) — Filter Design, chapter 1
- [Importance of frequency transformation; obtaining band pass](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Afrequency-transformation-importance-bandpass) — Filter Design, chapter 3, asked 3×

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- [Impulse response of DT ideal low pass filter](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aideal-lpf-discrete-time) — Signals and Systems, chapter 6
- [RC low pass filter operation](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Arc-low-pass-filter) — Basic Electronics Engineering, chapter 1, asked 5×
- [Transfer function of RC low-pass filter](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Arc-lowpass-transfer-function) — Signals and Systems, chapter 5, asked 2×