Transfer function of Butterworth LP
A complex transfer function can be expressed interns of real and imaginary terms.
Mathematically
\[
T(j\omega) = \text{Re}[T(j\omega)] + j\, \text{Im}[T(j\omega)] \tag{1}
\]
\[
\text{Here, the real part of } T(j\omega) \text{ is an even function:}
\]
\[
\text{Re}[T(-j\omega)] = \text{Re}[T(j\omega)]
\]
\[
\text{The imaginary part of } T(j\omega) \text{ is an odd function:}
\]
\[
\text{Im}[T(-j\omega)] = -\text{Im}[T(j\omega)]
\]
\[
\text{Since } T(-j\omega) \text{ is the complex conjugate of } T(j\omega), \text{ we have:}
\]
\[
T(-j\omega) = T^*(j\omega)
\]
\[
\Rightarrow T^*(j\omega) = \text{Re}[T(j\omega)] - j\, \text{Im}[T(j\omega)] \tag{2}
\]
\[
\text{Multiplying equations (1) and (2):}
\]
\[
T(j\omega) \cdot T^*(j\omega) =
\left( \text{Re}[T(j\omega)] + j\, \text{Im}[T(j\omega)] \right)
\left( \text{Re}[T(j\omega)] - j\, \text{Im}[T(j\omega)] \right)
\]
\[
= \left( \text{Re}[T(j\omega)] \right)^2 + \left( \text{Im}[T(j\omega)] \right)^2
\]
\[
\Rightarrow |T(j\omega)|^2 = \left( \text{Re}[T(j\omega)] \right)^2 + \left( \text{Im}[T(j\omega)] \right)^2
\]
\[
\text{This is called the magnitude squared function of } T(j\omega)
\]
\[
T(j\omega) \cdot T^*(j\omega) = T(s) \cdot T^*(s) = |T(s)|^2 = |T(j\omega)|^2
\]
\[
\text{Thus, the function } T(s)^2 \text{ or } |T(j\omega)|^2 \text{ is called the \textbf{magnitude squared function}.}
\]
Note:
\[
\text{Magnitude squared function:} \quad
\color{red}{
T(s) = \frac{s + 2}{s^3 + 2s^2 + 2s + 3}
}
\]
\[
\text{The complex conjugate of } T(s) \text{ is:} \quad
\color{red}{
T(-s) = \frac{-s + 2}{-s^3 + 2s^2 - 2s + 3}
}
\]
\[
\text{The magnitude squared function is given by:} \quad
\color{red}{
|T(s)|^2 = T(s) \cdot T^*(s) = T(s) \cdot T(-s)
}
\]
\[
\color{red}{
|T(s)|^2 =
\left( \frac{s + 2}{s^3 + 2s^2 + 2s + 3} \right) \cdot
\left( \frac{-s + 2}{-s^3 + 2s^2 - 2s + 3} \right)
}
\]
\[
|T(j\omega)|^2 \text{ is an even function which can be represented by using numerator and denominator which are both even functions.}
\]
\[
|T(j\omega)|^2 = |N(j\omega)|^2 \cdot |D(j\omega)|^2 = A(\omega^2) \, B(\omega^2)
\]
\[
\text{where } \omega \text{ is the normalized frequency}
\]
\[
|T_n(j\omega)|^2 = \frac{A_0 + A_2 \omega^2 + A_4 \omega^4 + \cdots + A_{2n} \omega^{2n}}{B_0 + B_2 \omega^2 + B_4 \omega^4 + \cdots + B_{2n} \omega^{2n}}
\]
\[
\text{where } T_n(j\omega) \text{ is of degree } n
\]
\[
\text{Let us assume:} \quad A_0 = B_0
\]
\[
\text{and } A_2 = A_4 = \cdots = A_{2n} = 0 \quad \Rightarrow \quad |T_n(0)|^2 = 1
\]
\[
\text{Then the expression becomes:}
\]
\[
|T_n(j\omega)|^2 = \frac{B_0}{B_0 + B_2 \omega^2 + B_4 \omega^4 + \cdots + B_{2n} \omega^{2n}}
\]
\[
\text{Again, let } B_2 = B_4 = \cdots = 0
\]
\[
\text{and } B_{2n} = B_0 \cdot \left(\frac{1}{\omega_0}\right)^{2n} \quad \text{[Madarain Expansion $\Rightarrow$ Flat Response]}
\]
\[
\text{Then the equation becomes:}
\]
\[
|T_n(j\omega)|^2 = \frac{B_0}{B_0 + B_0 \cdot \left(\frac{\omega}{\omega_0}\right)^{2n}} = \frac{B_0}{B_0 \left(1 + \left(\frac{\omega}{\omega_0}\right)^{2n}\right)} = \frac{1}{1 + \left(\frac{\omega}{\omega_0}\right)^{2n}}
\]
\[
\text{where } \omega_0 \text{ is the cutoff frequency}
\]
\[
\text{This is known as the Butterworth response for a low pass filter.}
\]
\[
\text{Normalizing the response at } \omega_0 = 1 \text{ rad/sec:}
\]
\[
|T_n(j\omega)|^2 = \frac{1}{1 + \omega^{2n}}
\]
\[
\Rightarrow |T_n(j\omega)| = \frac{1}{\sqrt{1 + \omega^{2n}}}
\]
What are the proporties of Butterworth Low pass Approximations?
From the above equation, the following properties can be considered
\begin{itemize}
\item It is an all-pole filter with zeros at infinity.
- At \( \omega = 0 \):
\[
|T_n(0)| = 1 \quad \text{for all values of } n \quad \text{(Result of normalization)}
\]
- At \( \omega = 1 \):
\[
|T_n(j)| = \frac{1}{\sqrt{2}} = 0.707 \quad \text{for all values of } n
\]
\[
\text{This is the half power point, gain will be } -3.2\, \text{dB or } 3\, \text{dB below}.
\]
- At \( \omega = \infty \):
\[
|T_n(\infty)| = 0 \quad \text{for all values of } n \quad \text{(Result of normalization)}
\]
- For large values of \( \omega \), \( T_n(j\omega) \) exhibits larger roll-off:
\[
|T_n(j\omega)| = \frac{1}{\sqrt{1 + \omega^{2n}}}
\]
\[
\text{For large } \omega, \quad 1 + \omega^{2n} \approx \omega^{2n}
\]
\[
\Rightarrow |T_n(j\omega)| \approx \frac{1}{\sqrt{\omega^{2n}}} = \frac{1}{\omega^n}
\]
- The attenuation \( \alpha \) is given by:
\[
\alpha = -20 \log \left( |T_n(j\omega)| \right)
\]
\[
\alpha = -20 \log \left( \frac{1}{\omega^n} \right) = 20 \log(\omega^n) = 20 n \log(\omega)
\]
- Butterworth response can be expanded in Taylor’s series form as:
\[
|T_n(j\omega)| = \frac{1}{\sqrt{1 + \omega^{2n}}} = \left( 1 + \omega^{2n} \right)^{-\frac{1}{2}}
\]
- Using Taylor expansion:
\[
|T_n(j\omega)| = 1 + \left(-\frac{1}{2}\right) \omega^{2n} + \frac{\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)}{2!} \omega^{4n} + \cdots
\]
- Approximating to first order:
\[
|T_n(j\omega)| \approx 1 - \frac{1}{2} \omega^{2n}
\]
- This response is maximally flat and approaches a brick-wall response for high order \( n \).
