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 $(\cdots,1,1,1,\cdots )$with a Z-transform $(\cdots + Z^2 + Z^3 + Z^4 + \cdots )$.To see that the filter (1-Z) destroys zero frequency, notice that $(1-Z)(\cdots + Z^2 + Z^3 + Z^4 + \cdots )=0$.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 $\omega = 0$because $Z=\exp(i\omega \Delta t)$ from (20). Now we can recognize that multiplication of two functions of Z or of $\omega$ 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 $y_t = x_t {\rm * } b_t$.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 $\omega$. (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 $\omega = 0$.Another interesting filter is 1+Z, which destroys the highest possible frequency $(1,-1,1,-1,\cdots)$,where $\omega = \pi$.

A root is a numerical value for which a polynomial vanishes. For example, $2 - Z - Z^2 = (2 + Z)\, (1 - Z)$ 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 Z^N, 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.