Model fitting by least squares

HOW TO DIVIDE NOISY SIGNALS

If "inversion" is dividing by a matrix, then the place to begin is dividing one number by another, say one function of frequency by another function of frequency. A single parameter fitting problem arises in Fourier analysis, where we seek a ``best answer'' at each frequency, then combine all the frequencies to get a best signal. Thus emerges a wide family of interesting and useful applications. However, Fourier analysis first requires us to introduce complex numbers into statistical estimation.

Multiplication in the Fourier domain is convolution in the time domain. Fourier-domain division is time-domain deconvolution. This division is challenging when the divisor has observational error. Failure erupts if zero division occurs. More insidious are the poor results we obtain when zero division is avoided by a near miss.