HOPE THE FOLLOWING ABOUT THE PROPERTIES OF THE TWO TRANSFORMS WILL MAKE THINGS CLEAR:
FOURIER TRANSFORM:
The time and frequency domains are alternative ways of representing signals. The Fourier transform is the mathematical relationship between these two representations. If a signal is modified in one domain, it will also be changed in the other domain, although usually not in the same way. For example, it was shown in the last chapter that convolving time domain signals results in their frequency spectra being multiplied. Other mathematical operations, such as addition, scaling and shifting, also have a matching operation in the opposite domain. These relationships are called properties of the Fourier Transform, how a mathematical change in one domain results in a mathematical change in the other domain.
Z TRANSFORM:
Just as analog filters are designed using the Laplace transform, recursive digital filters are developed with a parallel technique called the z-transform. The overall strategy of these two transforms is the same: probe the impulse response with sinusoids and exponentials to find the system's poles and zeros. The Laplace transform deals with differential equations, the s-domain, and the s-plane. Correspondingly, the z-transform deals with difference equations, the z-domain, and the z-plane. However, the two techniques are not a mirror image of each other; the s-plane is arranged in a rectangular coordinate system, while the z-plane uses a polar format. Recursive digital filters are often designed by starting with one of the classic analog filters, such as the Butterworth, Chebyshev, or elliptic. A series of mathematical conversions are then used to obtain the desired digital filter. The z-transform provides the framework for this mathematics. The Chebyshev filter design program presented in Chapter 20 uses this approach, and is discussed in detail in this chapter.