Designing a notch filter, also known as a band-stop or band-reject filter, involves creating a filter that attenuates or removes a specific frequency or range of frequencies while allowing frequencies outside this range to pass through.
Design Methods Based on Subtraction
One fundamental approach to designing a notch filter, as indicated by the provided reference, relies on the principle of subtraction. This method involves taking a signal and removing the portion of it that lies within the frequency band you want to eliminate.
Method 1: Subtracting a Band-Pass Filter Output (1 – BP)
According to the reference, one way to build a notch filter is to construct it as a band-pass filter whose output is subtracted from the input (1 – BP).
Here's how this method works:
- Design a Band-Pass Filter (BP): First, you design a filter that passes the specific frequency band you want to remove from the original signal. The characteristics of this band-pass filter determine the characteristics of the resulting notch filter:
- Its center frequency will be the center frequency of the notch.
- Its bandwidth will determine how wide the notch is.
- Subtract from the Input: The output of this band-pass filter is then subtracted from the original input signal.
Think of the input signal as representing 100% of all frequencies. If a band-pass filter isolates and outputs, say, 100% of the frequencies between 50 Hz and 70 Hz (and 0% of others), subtracting this output from the original input signal (100% of all frequencies) effectively removes the 50 Hz to 70 Hz band, leaving everything else. This creates the "notch" at the targeted frequency range.
Method 2: Using Cascaded LP and HP Sections (Reference Variation)
The reference also mentions another way is with cascaded low-pass and high-pass sections, especially for the band-reject (wideband) case. It states: "In this case, the sections are in parallel, and the output is the difference."
This description suggests an architecture for achieving band-reject filtering, potentially wideband, that also involves a difference operation. While cascaded low-pass (LP) and high-pass (HP) filters typically form a band-pass filter (LP followed by HP or vice versa), the reference links this to a configuration where "sections are in parallel, and the output is the difference." This could imply a structure where the original signal path is split, perhaps with parallel LP and HP filters processing the signal, and their combined effect or outputs are used in a difference operation relative to the original input or another path to create the rejection band. This method is particularly noted for wideband band-reject applications.
Key Considerations in Notch Filter Design
When designing a notch filter, regardless of the specific implementation method (analog circuit, digital filter), several parameters are crucial:
- Center Frequency: The frequency at the very bottom of the notch, where maximum attenuation occurs.
- Bandwidth (or Q factor): This defines how wide the notch is. A narrow bandwidth creates a sharp notch suitable for removing a single tone (like 50 Hz or 60 Hz hum), while a wider bandwidth removes a broader range of frequencies (band-reject).
- Notch Depth: How much the frequencies within the notch are attenuated. A deep notch provides strong rejection.
- Transition Bands: The slopes on either side of the notch, indicating how quickly the filter transitions from passing frequencies to rejecting them.
By carefully designing the characteristics of the band-pass filter (for the 1-BP method) or the low-pass and high-pass sections (for the cascaded/parallel difference method), you can precisely control the frequency, width, and depth of the notch.
Narrowband vs. Wideband Notch
The reference highlights the use of cascaded LP/HP for the "band-reject (wideband) case." This points to a distinction often made in filter design:
| Characteristic | Narrowband Notch Filter | Wideband Band-Reject Filter |
|---|---|---|
| Primary Use | Removing specific tones (e.g., noise, hum) | Removing a range of frequencies |
| Bandwidth | Very narrow | Wider |
| Example Ref Method | Often achieved with the (1-BP) method | Reference mentions cascaded LP/HP in parallel difference configuration |
In summary, a primary way to design a notch filter is by strategically using a band-pass filter and subtracting its output from the input signal. Alternative configurations involving cascaded low-pass and high-pass sections used in a difference structure, as described in the reference, are also employed, particularly for achieving wideband band-reject characteristics.
