Normalization and Denormalization

In most of the cases we consider the values of R, L S& C to be the order of unity. It is very difficult to build the capacitor of 1 f and inductor of 1 H. Besides this the practical values of capacitors available in the electronic circuit is of the order of microfarad or Pico farad. The circuit considered so far have normalized elemental values but practically these values are not realizable. So, we perform scaling to get the realizable components.

Normalization:

A normalized filter is one with passband cutoff at point \(\omega = 1\) rad/sec, which is \(\frac{1}{2\pi}\) hertz or 0.159 Hertz.

\[
\text{i.e., } f = \frac{1}{2\pi}
\]
Normalization is the process of scaling a filter's frequency specifications so that the cutoff frequency (or another reference frequency) is set to a standard value, typically 1 (or π radians/sample in digital filters). Working with normalized frequency units rather than absolute frequencies makes filter design and analysis easier.

\[
\omega_{\text{norm}} = \frac{\omega}{\omega_s / 2}
\]

Reason for Normalization

• Numerical computation become simple and it is easier to manipulate the numbers of the order of unity.
• it is easy to generate filters of similar characteristics of varying center frequency and impedance level.
• Filters work on various frequency ranges so, it’s better to standardize the design and implement it on various frequency ranges rather than designing a new one.
• It eliminates the need for extremely small or extremely large component values, such as MΩ (10⁶ Ω) resistors and pF (10⁻¹² F) capacitors. 
• It enables us to create filters with critical frequency specifications that fall within the range of ω = 1 rad/sec on the frequency axis. Without having to worry about units like Hz, Ω, F, or H, it enables us to work exclusively with dimensionless specifications and components.
• Filter designs are based on the use of design tables.

Here:

\[
W = 1 \text{ rad/sec}
\]

\[
2 \pi f = 1 \text{ rad/sec}
\]

\[
f = \frac{1}{2\pi} = 0.159 \text{ Hertz}
\]

\[
\text{New values of impedance of reactive components become:}
\]

\[
X_L = 2 \pi f L = 2 \pi \left(\frac{1}{2\pi}\right) L = L
\]

\[
X_C = \frac{1}{2 \pi f C} = \frac{1}{2 \pi \left(\frac{1}{2 \pi}\right) C} = \frac{1}{C}
\]

\[
\text{New values are } X_L(\text{new}) = L \quad \text{and} \quad X_C(\text{new}) = \frac{1}{C},
\]

\[
\text{which makes calculations much simpler.}
\]

IF Rc (an Arbitary REsistance is given then):

\[
L_n = \frac{\omega_s}{R_s} \, L
\]

\[
C_n = \omega_s \, R_s \, C
\]

\[
R_n = \frac{1}{R_s} \, R
\]

where \( R_s \) is an arbitrary scaling resistor (in ohms) that normalizes the impedance level and \( \omega_s \) is the radian frequency (in rad/sec) that normalizes and scales the frequency axis such that

\[
\frac{\omega}{\omega_s} = 1.
\]

Denomalaiztion

The opposite of normalization, denormalization involves converting the normalized frequency specifications back to their original analog or physical frequencies. When using the designed filter on actual signals with particular sampling rates or frequency bands, this step is required. The true frequency 𝜔, given a normalized cutoff frequency w(norm), is:

\[
\omega = \omega_{\text{norm}} \times \frac{\omega_s}{2}
\]

Anwer if  Q.N What is Normalization and Denormalization? Why normalization is important is filter Design. Derice formula to evaluate the new values of the resister, capacitor and inductor for w = 1rad/sec.