Cepstrum

The cepstrum was introduced in 1963 in a paper by Bogert, Healy and Tukey as a new transform a signal in the communication equipment. In that article, the cepstrum of a signal informally as the spectrum of the logarithm of the frequency spectrum (Fourier spectrum) is determined by this signal. The term cepstrum is derived from the term spectrum by the first four letters in the order were reversed.

The cepstrum is a function of so-called " quefrency " (English " quefrency " originated from the word "Frequency"). To explain:

• The spectrum of a signal is a function of the variable frequency.
• The cepstrum is ( as a spectrum of a function in the frequency domain ) the function of a new variable. For this variable, the term quefrency was introduced in the above-mentioned original article. The dimension of the quefrency is identical to the dimension of the independent variable of the considered original function, of which the spectrum is formed. For a time-dependent signal the quefrency therefore has the dimension of time.

Formulated exactly is the cepstrum is the inverse Fourier transform ( " Alanysis ") of the logarithmic, dimensionless made ​​by dividing by a reference value G0, one-sided auto power spectrum; as is the case for Cxx ( τ ), the cepstrum for the time signal x (t ) with the auto power spectrum Gxx ( f):

The advantage of the Cepstrumfunktion over the autocorrelation function, which can be calculated in a similar manner, lies in the frequency range in the logarithm. This go multiplications (eg, a spectrum with a frequency response ) in additions above. This remains valid because of the linear nature of the ( inverse ) Fourier transform in the cepstrum. Thus harmonic components in the signal can be clearly seen, even if they have relatively small amplitudes. The cepstrum is therefore particularly suitable for the separation of the effects of source and transmission characteristics, such as for the detection of echoes and possibly to suppress the echo effects by windowing ( as a derivative of " filtering" with " liftering ") of the cepstrum in not from echo signal affected area. By " liftering " even harmonic components in a signal from the other units can be separated, which is used in the analysis of language and in the early detection of damage to the machine, for example. In the latter case, the fact is exploited that often indicate damage to machines with rotating components by an increase of the harmonic components in the air - borne noise.