signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...

, a lapped transform is a type of

linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...

discrete block transformation where the

basis function In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be repres ...

s of the transformation overlap the block boundaries, yet the number of coefficients overall resulting from a series of overlapping block transforms remains the same as if a non-overlapping block transform had been used. Lapped transforms substantially reduce the blocking artifacts that otherwise occur with block transform coding techniques, in particular those using the discrete cosine transform. The best known example is the

modified discrete cosine transform The modified discrete cosine transform (MDCT) is a transform based on the type-IV discrete cosine transform (DCT-IV), with the additional property of being lapped: it is designed to be performed on consecutive blocks of a larger dataset, where ...

used in the

MP3 MP3 (formally MPEG-1 Audio Layer III or MPEG-2 Audio Layer III) is a coding format for digital audio developed largely by the Fraunhofer Society in Germany, with support from other digital scientists in the United States and elsewhere. Origin ...

Vorbis Vorbis is a free and open-source software project headed by the Xiph.Org Foundation. The project produces an audio coding format and software reference encoder/decoder ( codec) for lossy audio compression. Vorbis is most commonly used in con ...

, AAC, and

Opus ''Opus'' (pl. ''opera'') is a Latin word meaning "work". Italian equivalents are ''opera'' (singular) and ''opere'' (pl.). Opus or OPUS may refer to: Arts and entertainment Music * Opus number, (abbr. Op.) specifying order of (usually) publicatio ...

audio codec An audio codec is a device or computer program capable of encoding or decoding a digital data stream (a codec) that encodes or decodes audio. In software, an audio codec is a computer program implementing an algorithm that compresses and decompres ...

s. Although the best-known application of lapped transforms has been for audio coding, they have also been used for video and image coding and various other applications. They are used in video coding for coding I-frames in

VC-1 SMPTE 421, informally known as VC-1, is a video coding format. Most of it was initially developed as Microsoft's proprietary video format Windows Media Video 9 in 2003. With some enhancements including the development of a new Advanced Profile ...

and for image coding in the

JPEG XR JPEG XR (JPEG extended range) is an image compression standard for continuous tone photographic images, based on the HD Photo (formerly Windows Media Photo) specifications that Microsoft originally developed and patented. It supports both lossy a ...

format. More recently, a form of lapped transform has also been used in the development of the Daala video coding format.

References

Digital signal processing Linear algebra Discrete transforms {{electronics-stub