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Chapter: Discrete Time Systems and Signal Processing - Discrete Time System Analysis

Plot and Properties of Z Transform

Z Transform Plot - Fig show the plot of z transforms. The z transform has real and imaginary parts. Thus a plot of imaginary part versus real part is called complex z-plane. The radius of circle is 1 called as unit circle.

Z TRANSFORM PLOT


Fig show the plot of z transforms. The z transform has real and imaginary parts. Thus a plot of imaginary part versus real part is called complex z-plane. The radius of circle is 1 called as unit circle.

This complex z plane is used to show ROC, poles and zeros. Complex variable z is also expressed in polar form as Z= rejω where r is radius of circle is given by |z| and ω is the frequency of the sequence in radians and given by z.




Tutorial problems:

Q) Determine z transform of following signals. Also draw ROC.

i)     x(n) = {1,2,3,4,5}

ii)    x(n)={1,2,3,4,5,0,7}

Q) Determine z transform and ROC for x(n) = (-1/3)n u(n) –(1/2)n u(-n-1).

Q) Determine z transform and ROC for x(n) = [ 3.(4n)–4(2n)] u(n).

Q) Determine z transform and ROC for x(n) = (1/2)n u(-n).

Q) Determine z transform and ROC for x(n) = (1/2)n {u(n) – u(n-10)}.

Q) Find linear convolution using z transform. X(n)={1,2,3} & h(n)={1,2}

 

 

 

PROPERTIES OF Z TRANSFORM (ZT)

1) Linearity

The linearity property states that if

 

z Transform of linear combination of two or more signals is equal to the same linear combination of z transform of individual signals.

 

2) Time shifting

 

The Time shifting property state that if


Thus shifting the sequence circularly by ‘k‘ samples is equivalent to multiplying its z transform by z –k

 

3) Scaling in z domain


Thus scaling in z transform is equivalent to multiplying by an  in time domain.


4) Time reversal Property


It means that if the sequence is folded it is equivalent to replacing z by z-1 in z domain.

 

5) Differentiation in z domain

The Differentiation property states that if



6) Convolution Theorem

The Circular property states that if


Convolution of two sequences in time domain corresponds to multiplication of its Z transform sequence in frequency domain.

 

7) Correlation Property

The Correlation of two sequences states that if



8) Initial value Theorem

Initial value theorem states that if



9) Final value Theorem

Final value theorem states that if




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