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Lecture #07

Lecture #07. Z-Transform. H. = The impulse response of system H. eigenfunction. eigenvalue. 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.

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Lecture #07

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  1. Lecture #07 Z-Transform signals & systems

  2. H = The impulse response of system H eigenfunction eigenvalue signals & systems

  3. signals & systems MIT signals & systems

  4. 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

  5. Let be a complex number The DTFT of a signal signals & systems

  6. Laplace/inverse laplace transfrom The z-transform of an arbitrary signal x[n] and the inverse z-transform Notation signals & systems

  7. 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

  8. signals & systems MIT signals & systems

  9. Example : Z-transform R.O.C signals & systems

  10. Properties of Z transform Linearity Right shift in time Left shift in time Time Multiplication Frequency Scaling Modulation signals & systems

  11. Convolution signals & systems

  12. Initial value theorem signals & systems

  13. Final value theorem signals & systems

  14. Some common Z transforms signals & systems

  15. Example : Inverse Z-transform signals & systems

  16. Example : the Z transform can be used to solve difference equations Taking the Z transform signals & systems

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