Low Pass Filter
In most cases, filters extract the required frequency from a signal. Low-pass filters only permit frequencies falling below and attenuating frequencies above the cutoff frequency.
Low-pass filters generally have a form where the output decays at higher frequencies.
Low-Pass Filter Transfer Function
Let’s take the following simple transfer function:
\[ H(z) = \frac{1}{1 + 0.5z^{-1}} \]
Analysis of the Transfer Function
The denominator suggests it’s a low-pass filter because it has a single pole and the gain decreases as frequency increases (in the discrete-time case, this is for higher \(\omega\)).
At low frequencies, the magnitude will be close to 1, but as the frequency increases, the magnitude will drop.
A low pass filter's cutoff frequency is calculated as
Cut off frequency = 1 / 2*pi*R*C
High Pass Filter
All frequencies in a signal above the high pass filter's cutoff frequency can pass through the high pass filter. High-pass filters generally have a form where the output increases at higher frequencies.
High-Pass Filter Transfer Function
Now consider the following transfer function:
\[ H(z) = \frac{z^{-1}}{1 + 0.5z^{-1}} \]
Analysis of the Transfer Function
The numerator has z^{-1}, suggesting a high-pass filter because it includes a term that shifts the response at low frequencies, while passing higher frequencies more easily.
By examining the transfer function, you can classify the filter's behavior in terms of its frequency response.
A high pass filter's cutoff frequency is calculated as
Cut off frequency = 1 / 2*pi*R1*C1
Band Pass Filter
A band pass filter is a device that permits frequencies that fall within a specific frequency range. Both frequencies inside and outside of the field are attenuated. Band-pass filters will show a peak or resonance at a specific frequency.
Bandpass Filter Transfer Function
Now consider the following transfer function for a bandpass filter:
\[ H(z) = \frac{z^{-1} - z^{-2}}{1 + 0.5z^{-1} + 0.25z^{-2}} \]
Analysis of the Transfer Function
The numerator contains the terms z^{-1} and z^{-2}, suggesting that the filter allows a band of frequencies to pass while attenuating lower and higher frequencies.
The filter achieves this by having a zero at a specific frequency, which creates a notch at the desired frequency, thus passing a band of frequencies in the middle.
By examining the transfer function, you can classify the filter's behavior in terms of its frequency response. This includes its ability to pass signals within a specific frequency band and attenuate others outside that range.
