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In principle a compact subwoofer is simply a small subwoofer, which is the loudspeaker that handles bass in music and rumbling effects in soundtracks. There are several ways to make the subwoofer smaller, such as making it passive rather than active. Exactly what counts as compact depends very much on the context.
A subwoofer is a specific type of loudspeaker, designed to carry low frequency audio, usually referred to in musical terms as the bass. It can be used alongside a pair of stereo speakers or as part of a surround sound system. The frequencies carried by a subwoofer are usually so low that humans cannot detect their location. This means that unlike normal speakers, it does not matter where a subwoofer is placed, and there is no need to make sure it is the same size as other speakers.
A compact subwoofer can be either active or passive. Active means that the subwoofer has its own amplification system within its casing, requiring a separate power source. Passive means that the amplification is taken care of by the sound source such as a hi-fi or surround sound receiver. Generally, surround systems use an active subwoofer, and the receiver may not be able to provide the amplification for a passive subwoofer. That said, a passive subwoofer can be much smaller because it doesn't need the amplifier, which means a compact subwoofer is more likely to be passive.
There is no clearly defined size that qualifies as a compact subwoofer. It can sometimes be a marketing term and may simply mean the smallest unit in a particular range. What count as compact can also depend on the context.
For home cinema speakers, compact may refer to the subwoofer's width and depth rather than its height. This is because floorspace is often more likely to be a problem. For car stereo systems, compact may refer mainly to depth, though it depends on the orientation of the subwoofer within the vehicle. For computer speaker systems, the definition of compact is usually much smaller than in other contexts, and may indicate the subwoofer will fit easily on a desktop or shelf.
There are several ways to save space when making a compact subwoofer. In some cases it is simply a case of compromising on quality and performance. In others, computer techniques can be used to adjust the sound to make the most efficient use of the space within the subwoofer.