Definition and mathematical expression of 1.z transformation

Z transformation definition

Z-transform (ZT) is a mathematical transformation of discrete sequences, which is often used to solve linear time-invariant difference equations. Its position in discrete system is the same as that of Laplace transform in continuous system. Z-transform has become an important tool for analyzing linear time-invariant discrete systems, and is widely used in digital signal processing, computer control systems and other fields.

Mathematical representation of z-transform.

The relationship between DTFT and z-transform.

DTFT expression 4 of signal and system interpretation-discrete time Fourier transform;

The relationship between z transform and DTFT can be obtained, that is.

Therefore, DTFT is the Z transformation on the unit circle!

Second, the Z-transform of common signals.

① Pulse signal

Physical meaning: including all frequency components, such as thunder and other physical phenomena.

② Step signal

③ Unilateral exponential signal 1

④ Unilateral exponential signal 2

Thirdly, the convergence domain of Z-transform.

Convergence judgment of z-transform.

① Ratio discrimination method

② Root value discrimination method

Properties of convergence domain of z-transform

Fourthly, the nature of Z-transform.

① Linearity

② Time shift

③Z-domain scale transformation

④ Time domain inversion

⑤ Time domain expansion

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⑦ convolution property

⑧Z-domain difference/sequence linear weighting

Pet-name ruby initial value theorem

Attending the final value theorem

Verb (abbreviation of verb) uses Z-transform analysis to characterize LTI system.

① Causality

The essence of causal system:

1. For an LTI system, when the convergence region is an out-of-circle region and contains infinity points, the system is a causal system.

2. An LTI system with rational system function, if it satisfies ①: the convergence domain is outside the circle where the maximum pole of the module is located; ②: The order of the numerator polynomial is not greater than that of the denominator polynomial; This system is a causal system.

② Stability

Conclusion:

1. For LTI systems, if the convergence domain contains the unit circle, the system is stable.

2. For the causal LTI system, if the poles are in the unit circle, the system is stable.

Six, such as the application of Z transform.