Signals and Systems/Filter Terminology
When it comes to filters, there is a large amount of terminology that we need to discuss first, so the rest of the chapters in this section will make sense.
- Order (Filter Order)
- The order of a filter is an integer number, that defines how complex the filter is. In common filters, the order of the filter is the number of "stages" of the filter. Higher order filters perform better, but they have a higher delay, and they cost more.
- Pass Band
- In a general sense, the passband is the frequency range of the filter that allows information to pass. The passband is usually defined in the specifications of the filter. For instance, we could define that we want our passband to extend from 0 to 1000 Hz, and we want the amplitude in the entire passband to be higher than -1 db.
- Transition Band
- The transition band is the area of the filter between the passband and the stopband. Higher-order filters have a thinner transition band
- Stop Band
- The stop band of a filter is the frequency range where the signal is attenuated. Stop band performance is often defined in the specification for the filter. For instance, we might say that we want to attenuate all frequencies above 5000 Hz, and we want to attenuate them all by -40 db or more
- Cut-off Frequency
- The cut-off frequency of a filter is the frequency at which the filter "breaks", and changes (between pass band and transition band, or transition band and passband, for instance). The cut-off of a filter always has an attenuation of -3db. The -3 db point is the frequency that is cut in power by exactly 1/2.
Lowpass filters allow low frequency components to pass through, while attenuating high frequency components.
Lowpass filters are some of the most important and most common filters, and much of our analysis is going to be focused on them. Also, transformations exist that can be used to convert the mathematical model of a lowpass filter into a model of a highpass, bandpass, or bandstop filter. This means that we typically design lowpass filters and then transform them into the appropriate type.
Example: Telephone System
As an example of a lowpass filter, consider a typical telephone line. Telephone signals are bandlimited, which means that a filter is used to prevent certain frequency components from passing through the telephone network. Typically, the range for a phone conversation is 10Hz to 3˙000Hz. This means that the phone line will typically incorporate a lowpass filter that attenuates all frequency components above 3˙000Hz. This range has ben chosen because it includes all the information humans need for clearly understanding one another, so the effects of this filtering are not damaging to a conversation. Comparatively, CD recordings comprise most of the human hearing and their frequency components range up to 20˙000Hz or 20kHz.
Highpass filters allow high frequency components to pass through, while attenuating low frequency components.
Example: DSL Modems
Consider DSL modems, which are high-speed data communication devices that transmit over the existing telephone network. DSL signals operate in the high frequency ranges, above the 3000Hz limit for voice conversations. In order to separate the DSL data signal from the regular voice signal, the signal must be sent into two different filters: a lowpass filter to amplify the voice for the telephone signal, and a highpass filter to amplify the DSL data signal.
A bandpass filter allows a single band of frequency information to pass the filter, but will attenuate all frequencies above the band and below the band.
A good example of a bandpass filter is an FM radio tuner. In order to focus on one radio station, a filter must be used to attenuate the stations at both higher and lower frequencies.
A bandstop filter will allow high frequencies and low frequencies to pass through the filter, but will attenuate all frequencies that lay within a certain band.
Filters that cannot be classified into one of the above categories, are called gain or delay equalizers. They are mainly used to equalize the gain/phase in certain parts of the frequency spectrum as needed. More discussion on these kinds of advanced topics will be in Signal Processing.