Signal filters
The impedance
Proposition: considering an ideal resistor \(R \in \mathbb{R}\). When a voltage difference \(v_i - v_o\) is applied to both ends a current \(I: \mathbb{R} \to \mathbb{R}\) will be induced given by
\[ v_i(t) - v_o(t) = R I(t), \]for all \(t \in \mathbb{R}\). By taking the Fourier transform of both sides of the expression we find
\[ V_i(\omega) - V_o(\omega) = R I(\omega), \]for all \(\omega \in \mathbb{R}\) with \(V_{i,o}: \mathbb{R} \to \mathbb{C}\) the fourier transforms of the voltage difference.
Proof:
Will be added later.
Proposition: considering a load coil with inductance \(L \in \mathbb{R}\). When a voltage difference \(v_i - v_o\) is applied to both ends a current \(I: \mathbb{R} \to \mathbb{R}\) will be induced given by
\[ v_i(t) - v_o(t) = L I'(t), \]for all \(t \in \mathbb{R}\). By taking the Fourier transform of both sides of the expression we find
\[ V_i(\omega) - V_o(\omega) = i \omega L I(\omega), \]for all \(\omega \in \mathbb{R}\) with \(V_{i,o}: \mathbb{R} \to \mathbb{C}\) the fourier transforms of the voltage difference.
Proof:
Will be added later.
Proposition: considering a capacitor with capacity \(C \in \mathbb{R}\). When a voltage difference \(v_i - v_o\) is applied to both ends a current \(I: \mathbb{R} \to \mathbb{R}\) will be induced given by
\[ v_i(t) - v_o(t) = \frac{1}{C} \int_{-\infty}^t I(t)dt, \]for all \(t \in \mathbb{R}\). By taking the Fourier transform of both sides of the expression we find
\[ V_i(\omega) - V_o(\omega) = \bigg(\frac{1}{i \omega C} + \frac{\pi \delta(\omega)}{C} \bigg) I(\omega), \]for all \(\omega \in \mathbb{R}\) with \(V_{i,o}: \mathbb{R} \to \mathbb{C}\) the fourier transforms of the voltage difference.
Proof:
Will be added later.
Definition: the complex impedance \(Z: \mathbb{R} \to \mathbb{C}\) is defined as
\[ V_i(\omega) - V_o(\omega) = Z(\omega) I(\omega) \]for all \(\omega \in \mathbb{R}\).
Therefore the complex impedance for the ideal resistor is given by \(Z(\omega) = R\) and for the load coil \(Z(\omega) = i \omega L\) for all \(\omega \in \mathbb{R}\).
Proposition: the impedance elements \(Z_i\) for \(i \in \{1, \dots, n\}\) with \(n \in \mathbb{N}\) in series can be summed to obtain \(Z\)
\[ Z = Z_1 + \dots + Z_n. \]
Proof:
Will be added later.
Proposition: the impedance elements \(Z_i\) for \(i \in \{1, \dots, n\}\) with \(n \in \mathbb{N}\) in parallel can be inversely summed to obtain \(Z\)
\[ \frac{1}{Z} = \frac{1}{Z_1} + \dots + \frac{1}{Z_n}. \]
Proof:
Will be added later.
The transfer function
Definition: the relation between the input and output voltage in the frequency domain \(V_{i,o}: \mathbb{R} \to \mathbb{C}\) can be written as
\[ V_o(\omega) = H(\omega) V_i(\omega), \]for all \(\omega \in \mathbb{R}\) with \(H: \mathbb{R} \to \mathbb{C}\) the transfer function.
The transfer function may be interpreted as a frequency filter of the signal.
Some ideal filters are given in the list below
- a low-pass filter removes all frequency components \(\omega > \omega_c\) with \(\omega_c \in \mathbb{R}\) the cut-off frequency,
- a high-pass filter removes all frequency components \(\omega < \omega_c\),
- a band-pass filter removes all frequency componets outside a particular frequency range,
- a band-stop filter removes all frequency compnents inside a particular frequency range.