Code-excited linear prediction (CELP) is a linear predictive speech coding algorithm originally proposed by Manfred R. Schroeder and Bishnu S. Atal in 1985. At the time, it provided significantly better quality than existing low bit-rate algorithms, such as residual-excited linear prediction (RELP) and linear predictive coding (LPC) vocoders (e.g., FS-1015). Along with its variants, such as algebraic CELP, relaxed CELP, low-delay CELP and vector sum excited linear prediction, it is currently the most widely used speech coding algorithm. It is also used in MPEG-4 Audio speech coding. CELP is commonly used as a generic term for a class of algorithms and not for a particular codec. Background The CELP algorithm is based on four main ideas: Using the source-filter model of speech production through linear prediction (LP) (see the textbook "speech coding algorithm"); Using an adaptive and a fixed codebook as the input (excitation) of the LP model; Performing a search in closed-loop in a "perceptually weighted domain". Applying vector quantization (VQ) The original algorithm as simulated in 1983 by Schroeder and Atal required 150 seconds to encode 1 second of speech when run on a Cray-1 supercomputer. Since then, more efficient ways of implementing the codebooks and improvements in computing capabilities have made it possible to run the algorithm in embedded devices, such as mobile phones. CELP decoder Before exploring the complex encoding process of CELP we introduce the decoder here. Figure 1 describes a generic CELP decoder. The excitation is produced by summing the contributions from fixed (a.k.a. stochastic or innovation) and adaptive (a.k.a. pitch) codebooks: where is the fixed (a.k.a. stochastic or innovation) codebook contribution and is the adaptive (pitch) codebook contribution. The fixed codebook is a vector quantization dictionary that is (implicitly or explicitly) hard-coded into the codec. This codebook can be algebraic (ACELP) or be stored explicitly (e.g. Speex). The entries in the adaptive codebook consist of delayed versions of the excitation. This makes it possible to efficiently code periodic signals, such as voiced sounds. The filter that shapes the excitation has an all-pole model of the form , where is called the prediction filter and is obtained using linear prediction (Levinson–Durbin algorithm). An all-pole filter is used because it is a good representation of the human vocal tract and because it is easy to compute. CELP encoder The main principle behind CELP is called analysis-by-synthesis (AbS) and means that the encoding (analysis) is performed