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This paper examines how people communicate with computers using speech. Automatic speech recognition (ASR) transforms speech into text, while automatic speech synthesis [or text-to-speech (TTS)] performs the reverse task. ASR has been largely developed based on speech coding theory, while simulating certain spectral analyses performed by the ear. Typically, a Fourier transform is employed, but following the auditory Bark scale and simplifying the spectral representation with a decorrelation into cepstral coefficients. Current ASR provides good accuracy and performance on limited practical tasks, but exploits only the most rudimentary knowledge about human production and perception phenomena. The popular mathematical model called the hidden Markov model (HMM) is examined; first-order HMMs are efficient but ignore long-range correlations in actual speech. Common language models use a time window of three successive words in their syntactic-semantic analysis. Speech synthesis is the automatic generation of a speech waveform, typically from an input text. As with ASR, TTS starts from a database of information previously established by analysis of much training data, both speech and text. Previously analyzed speech is stored in small units in the database, for concatenation in the proper sequence at runtime. TTS systems first perform text processing, including "letter-to-sound" conversion, to generate the phonetic transcription. Intonation must be properly specified to approximate the naturalness of human speech. Modern synthesizers using large databases of stored spectral patterns or waveforms output highly intelligible synthetic speech, but naturalness remains to be improved.