Z-Transform and Fourier transform

In digital signal processing, Z-transform is a very important analysis tool. But in common applications, we often only need to analyze the frequency response of a signal or system, that is to say, usually only the Fourier transform is required. So why introduce a Z-transform? What is the relationship between Z-transform and Fourier transform?

The physical meaning of the Fourier transform is very clear: the signal usually expressed in the time domain is decomposed into the superposition of multiple sinusoidal signals. Each sinusoidal signal can be fully characterized by amplitude, frequency, and phase. The signal after the Fourier transform is often called the spectrum, including the amplitude spectrum and the phase spectrum, respectively, the amplitude with frequency distribution and phase with frequency distribution. In nature, the frequency is clear physical significance, such as sound signal, the male voice deep and vigorous, this is mainly because the male speaker in the low-frequency component more, female compatriots more high-pitched crisp, this is mainly because female voice in higher-frequency components more. In terms of the amount of information contained in a signal, the signal of the time domain signal and its corresponding Fourier transform is exactly the same. What effect does the Fourier transform have? Because some of the signal mainly in the time domain performance of its characteristics, such as the process of capacitor charge and discharge, and some of the main signal in the frequency domain performance of its characteristics, such as mechanical vibration, human voice and so on. If the characteristics of the signal are mainly expressed in the frequency domain, then the corresponding time domain signal may appear disorganized, but it is very convenient to interpret in the frequency domain. In practice, when we collect a signal, in the absence of any prior information, intuition is to try to find some characteristics in the time domain, if the time domain is not found, it is natural to convert the signal to the frequency domain and then see what features. The time domain description of the signal and the frequency domain description, like the two sides of a coin, look different, but are actually the same thing. Because of this, in the usual signal and system analysis process, we are very concerned about the Fourier transform.

Since people only care about the frequency domain of the signal, then what is the Z-transform? To speak of the Z-transform, it is possible to go back to the Laplace transformation. The Laplace transform is a transformation method named after the French mathematician Laplace, mainly for the analysis of continuous signals. Both Laplace and Fourier were contemporaries, and their time in France was at the height of the Napoleonic era. In science also replaced the United Kingdom as the center of the world at that time, in the many scientific masters, Laplace, Lagrange, Fourier is among them the most brilliant three stars. Fourier's thesis that the signal can be decomposed into the superposition of sinusoidal signals includes Laplace and Lagrange.

Back to the point, although the Fourier transform is easy to use, and the physical meaning is clear, but one of the biggest problem is that its existence conditions are more stringent, such as the absolute integrable signal in the time domain may exist Fourier transform. The Laplace transform can be said to be a generalization of this concept. In nature, the exponential signal exp (-X) is one of the fastest decaying signals, which can easily satisfy the absolute integrable condition after the signal is multiplied by an exponential signal. Therefore, the original signal can be multiplied by the exponential signal to meet the conditions of the Fourier transform, the transformation is the Laplace transformation. This transformation transforms the differential equation into algebraic equations, which is very significant in the 18th century when the computer was far from being invented. As can be seen from the above analysis, the Fourier transform can be regarded as a special form of Laplace, that is, the exponential signal multiplied by exp (0). In other words, Laplace transform is a generalization of Fourier transform, and it is a more common form of expression. In the process of signal and system analysis, we can get the more common result of Laplace transform, and then get the special result of Fourier transform. This general-to-special solution has proved to be of great convenience in the analysis of continuous signals and systems.

The z-transform can be said to be a Laplace transform for discrete signals and systems, so it is easy to understand the importance of Z-transformations and to understand the relationship between Z-transformations and Fourier transforms. The z-plane in the Z-transform has a mapping relationship with the S-Plane in Laplace, Z=exp (Ts). In the Z-transform, the result of the unit circle corresponds to the result of the Fourier transform.

Z-Transform and Fourier transform