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This lecture covers the Z-transform in the context of signals and systems, focusing on its definition, properties, and applications. Key concepts include the impulse response of a system, the relationship between the discrete-time Fourier transform (DTFT) and the Z-transform, and the Laplace transform. The region of convergence (ROC) is crucial to determining if the summation converges to a finite value. We explore various properties of the Z-transform, such as linearity, time shifts, and modulation, along with examples and theorems relevant to solving difference equations.
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Lecture #07 Z-Transform signals & systems
H = The impulse response of system H eigenfunction eigenvalue signals & systems
signals & systems MIT signals & systems
Discrete-time Fourier transform Where z is complex if DTFT is a special case of Z transform Same as FT is a special case of Laplace transform signals & systems
Let be a complex number The DTFT of a signal signals & systems
Laplace/inverse laplace transfrom The z-transform of an arbitrary signal x[n] and the inverse z-transform Notation signals & systems
Region of convergence (ROC) Critical question : Does the summation converge to a finite value In general that depends on the value of z Since Unique circle signals & systems
signals & systems MIT signals & systems
Example : Z-transform R.O.C signals & systems
Properties of Z transform Linearity Right shift in time Left shift in time Time Multiplication Frequency Scaling Modulation signals & systems
Convolution signals & systems
Initial value theorem signals & systems
Final value theorem signals & systems
Some common Z transforms signals & systems
Example : Inverse Z-transform signals & systems
Example : the Z transform can be used to solve difference equations Taking the Z transform signals & systems
