Lecture #07 Z-Transform meiling chen signals & systems

H = The impulse response of system H eigenvalue eigenfunction meiling chen signals & systems

meiling chen signals & systems MIT signals & systems

DTFT is a special case of Z transform 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 meiling chen signals & systems

Let be a complex number The DTFT of a signal meiling chen signals & systems

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

meiling chen signals & systems MIT signals & systems

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

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

Convolution meiling chen signals & systems

Initial value theorem meiling chen signals & systems

Final value theorem meiling chen signals & systems

Some common Z transforms meiling chen signals & systems

Example : Inverse Z-transform meiling chen signals & systems

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