HOW TO DIVIDE NOISY SIGNALS

(This section is a heavy rewrite of classical material with a surprising twist near the end). Multiplication in the Fourier domain is convolution in the time domain. Division is deconvolution. We have already encountered the polynomial-division feedback operation X(Z)=Y(Z)/F(Z). The division is challenging when F has observational error. First by switching from the Z-domain to the $\omega$-domain we avoid needing to know if F is minimum phase. But the $\omega$-domain has pitfalls too. We may find for some real $\omega$ that $F(Z(\omega))$ vanishes so we cannot divide by that F. Failure erupts if zero-division occurs. More insidiously, when zero-division is avoided by a near miss, then results turn out poor.