Time-domain conjugate

A complex-valued signal

such as $e^{i\omega_0 t}$ can be imagined as a corkscrew, where the real and imaginary parts are plotted on the x- and y-axes, and time t runs down the axis of the screw. The complex conjugate of this signal reverses the y-axis and gives the screw an opposite handedness.  In Z-transform notation, the time-domain conjugate

is written  

$$\overline{B}(Z) \eq \overline{b_0} + \overline{b_1} e^{i\omega} + \overline{b_2} e^{i2\omega} + \cdots$$ (45)

Now consider the complex conjugate of a frequency function. In Z-transform notation this is written  

$$\overline{B(\omega )} \eq \overline{B} \left( {1\over Z} \ri... ...rline{b_1} e^{-i\omega} + \overline{b_2} e^{-i2\omega} + \cdots$$ (46)

To see that it makes a difference in which domain we take a conjugate, contrast the two equations (45) and (46). The function $\overline{B}({1\over Z}) B(Z)$ is a spectrum, whereas the function $\overline{b_t} \, b_t$is called an ``envelope function.''

For example, given complex-valued b_t vanishing for t<0, the composite filter $B(Z) \bar B(Z)$ is a causal filter with a real time function, whereas the filter $B(Z)\bar B(1/Z)$ is noncausal and also a real-valued function of time. (The latter filter would turn out to be symmetric in time only if all b_t were real.)

You might be tempted to think that $\overline{Z}=1/Z$,but that is true only if $\omega$ is real, and often it is not. Chapter is largely devoted to exploring the meaning of complex frequency.