Linear time-invariant (LTI) systems can be represented by the transfer function
It determines the output signal of an LTI system for a given input signal in the frequency domain:
This diagram illustrates how the output for a sine wave input signal is determined:
- Transform the input signal into the frequency domain.
- Weight the spectral components with the respective gain and phase of the transfer function.
- Transform the output signal's spectrum into the time domain.
LTI system's transfer function determines the output for a given input signal.
The response of a LTI system to a sine wave is a sine wave at the same frequency.
Amplitude and phase are determined by the transfer function's gain and phase at this frequency.
An RC low-pass filter is a LTI system. It shall be used to examine amplitude and phase of a complex valued frequency response.
Let us start deriving the RC low-pass filter's transfer function.
An RC low-pass filter is a potential divider circuit containing a resistor and a capacitor. It implements a first order low-pass.
Recalling the capacitor impedance the transfer function results to
is the cutoff frequency
The rearrangement in polar coordinates leads to:
Thus the absolute value yields the amplitude response, and the argument yields the phase response.
Amplitude response |H(f)| of RC low-pass with fc = 10MHz
Calculate the missing values in the table!
|f (frequency) [MHz]
(output amplitude) [V]
(phase shift) [
Example output amplitude and phase shift for
= 10 MHz and