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## Differentiator

A particularly interesting factor is (1-Z), because the filter (1,-1) is like a time derivative. The time-derivative filter destroys zero frequency in the input signal. The zero frequency is with a Z-transform .To see that the filter (1-Z) destroys zero frequency, notice that .More formally, consider output Y(Z)=(1-Z)X(Z) made from the filter (1-Z) and any input X(Z). Since (1-Z) vanishes at Z=1, then likewise Y(Z) must vanish at Z=1. Vanishing at Z=1 is vanishing at frequency because from (20). Now we can recognize that multiplication of two functions of Z or of is the equivalent of convolving the associated time functions.

 Multiplication in the frequency domain is convolution in the time domain.

A popular mathematical abbreviation for the convolution operator is an asterisk: equation (9), for example, could be denoted by .I do not disagree with asterisk notation, but I prefer the equivalent expression Y(Z)=X(Z)B(Z), which simultaneously exhibits the time domain and the frequency domain.

The filter (1-Z) is often called a differentiator.'' It is displayed in Figure 7. ddt
Figure 7
A discrete representation of the first-derivative operator. The filter (1,-1) is plotted on the left, and on the right is an amplitude response, i.e., |1-Z| versus . (Press button to activate program Zplane. See appendix for details.)

The letter z'' plotted at the origin in Figure 7 denotes the root of 1-Z at Z=1, where .Another interesting filter is 1+Z, which destroys the highest possible frequency ,where .

A root is a numerical value for which a polynomial vanishes. For example, vanishes whenever Z=-2 or Z=1. Such a root is also called a zero.'' The fundamental theorem of algebra says that if the highest power of Z in a polynomial is ZN, then the polynomial has exactly N roots, not necessarily distinct. As N gets large, finding these roots requires a sophisticated computer program. Another complication is that complex numbers can arise. We will soon see that complex roots are exactly what we need to design filters that destroy any frequency.     Next: Gaussian examples Up: FOURIER AND Z-TRANSFORM Previous: Unit circle
Stanford Exploration Project
10/21/1998