Home | 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= re^{j}^{ω} 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.(4^{n})–4(2^{n})]
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 a^{n} 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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