Squelch
Squelch is a feature of a radio receiver or transceiver which mutes the output when a signal is not being received, such that it is not constantly producing static, because of this, the squelch can be seen as a kind of noise gate.
A traditional carrier squelch can be explained as a kind of threshold or gate that can be set above the noise floor, a knob on the radio controls the squelch, having no squelch is known as an "open squelch" where as increasing the threshold is known as "closing the squelch" only signals strong enough to "break through" this barrier will unmute the receiver, a rule of thumb for setting the squelch is open to the squelch fully and then close it until the static stops and close it a tad more, this gives the best sensitivity to weak signals while preventing static if the noise floor does not increase.
This basic form of squelch is included on nearly all radios along with more advanced kinds of squelch.
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Other types of Squelch
RF Squelch
RF Squelch is similar to a carrier squelch but instead of setting a threshold the operator instead sets a desired S-Meter value, only signals at or above this strength will unmute the receiver, the Yaesu FT-60 is one such radio which has this feature.
Continuous Tone Coded Squelch System
Sometimes called PL Tone or Private Line after the trademarked Motorola name, CTCSS embeds a subaudible low tone to your transmission, it is only subaudible because most receivers have a notch filter, the signal may be heard as a low hum on scanners and SDRs which lack this filter.
Only receiving radios set to the same tone will unmute upon receiving the transmission encoded with the tone, this allows mutiple users to share the same channel without confusion and thus it's used a lot in commerical PMR radios, where, e.g two shops next to each other can use the same channel but different tones.
In amateur radio, CTCSS is most commonly used to access repeaters, it prevents QRM from activating a repeater as the chances of QRM containing the exact tone needed is almost impossible.
CTCSS cannot prevent signal collisions when two users with two different tones are sharing the same channel, such is common on PMR.
Digitally Coded Squelch
DCS is similar to CTCSS but is a digital stream containing a digital code, only radios set to this code will unmute upon receiving the transmission, from the operators perspective it functions about the same as CTCSS but has a different sound to scanner listeners.
Unlike Carrier and RF Squelch, CTCSS and DCS cannot be falsed by an increasing noise floor, DCS and CTCSS are also examples of in-band signalling.
Other Systems
Other Systems known as Selcall, SELCAL, MDC, Quindar Tones (as used by NASA), but these are not used by amateur radio operators, a 1750 Hz tone may be required to trigger some repeaters but this is technically not a squelch.
Squelch Myths
Squelch and it's correct use has generated some myths and beliefs among the general public, commerical/PMR radio users, and new amateur radio operators due to confusion on how squelch systems actually work, they are as follows.
Squelch is the name of a sound a radio makes
This often refers to a burst of static or a data-like "sqwuak" sound after a radio is keyed down post transmission or a transmission it is receiving stops.
A burst of static is known as a squelch tail and is caused by the squelch being slow to activate, most modern radios have eliminated this.
The data sound, often heard on american police radios and scanners, is the sound of a system called MDC which appends data such as selective calling to the end of a radio transmission, due to squelch tail this signal can often come through the radio receiver, a similar thing can also happen with DCS.
Using a CTCSS or Private Line tone means scanner/SDR listeners can't listen to your conversations
This is a myth common among users of licence-by-rule radios such as PMR446, UHF CB, FRS, etc for both personal and business uses, they believe setting a CTCSS tone makes it so that external listeners cannot hear the conversations, this is not true, it only makes your conversations private to other users on the same channel with a different tone, external listeners can hear it all, CTCSS is not a scrambler or encryption, important to note also that scrambling nor encryption is allowed for use on the aforementioned bands, including amateur radio bands.