Z-Transform and ROC of Finite Duration Sequences

The sequences having a finite number of samples are called the finite duration sequences. The finite duration sequences may be of following three types viz. −

• Right-Hand Sequences
• Left-Hand Sequences
• Two-Sided Sequences

Right-Hand Sequence

A sequence for which $\mathrm{\mathit{x\left ( n \right )}}$ = 0 for $\mathit{n}$ < $\mathit{n_{\mathrm{0}}}$ where $\mathit{n_{\mathrm{0}}}$ may be positive or negative but finite, is called the right hand sequence. If $\mathit{n_{\mathrm{0}}}$ ≥ 0, the resulting sequence is a causal sequence. The ROC of a causal sequence is the entire z-plane except at 𝑧 = 0.

Numerical Example (1)

Find the ROC and Z-Transform of the causal sequence.

$$\mathrm{\mathit{x\left ( n \right )}\mathrm{\, =\,} \begin{Bmatrix} 1, & 0,& -4,& 6,& 5,& 4 \ \uparrow & & & & & \ \end{Bmatrix} }$$

Solution

The given sequence is a right-hand sequence. The values of the given sequence are −

$$\mathrm{\mathit{x\left ( \mathrm{0} \right )\mathrm{\, =\,}\mathrm{1},x\left ( \mathrm{1} \right )\mathrm{\, =\,}\mathrm{0},x\left ( \mathrm{2} \right )\mathrm{\, =\,}\mathrm{-4},x\left ( \mathrm{3} \right )\mathrm{\, =\,}\mathrm{6},x\left ( \mathrm{4} \right )\mathrm{\, =\,}\mathrm{5},x\left ( \mathrm{5} \right )\mathrm{\, =\,}\mathrm{4}}}$$

The Z-transform of a sequence is given by,

$$\mathrm{\mathit{X\left ( z \right )\mathrm{\, =\,}\sum_{n\mathrm{\, =\,}-\infty }^{\infty }x\left ( n \right )z^{-n}}}$$

Thus, for the given values of the sequence, we get,

$$\mathrm{\mathit{Z\left [ x\left ( n \right ) \right ]\mathrm{\, =\,}X\left ( z \right )}}$$

$$\mathrm{\mathit{\mathrm{\, =\,}x\left ( \mathrm{0} \right )\mathrm{\, \mathrm{\, +\,}\,}x\left ( \mathrm{1} \right )z^{-\mathrm{1}}\mathrm{\, \mathrm{\, +\,}\,}x\left ( \mathrm{2} \right )z^{-\mathrm{2}}\mathrm{\, \mathrm{\, +\,}\,}x\left ( \mathrm{3} \right )z^{-\mathrm{3}}\mathrm{\, \mathrm{\, +\,}\,}x\left ( \mathrm{4} \right )z^{-\mathrm{4}}\mathrm{\, \mathrm{\, +\,}\,}x\left ( \mathrm{5} \right )z^{-\mathrm{5}}}}$$

$$\mathrm{\mathit{\therefore X\left ( z \right )\mathrm{\, =\,}\mathrm{1}-\mathrm{4}z^{\mathrm{-2}}\mathrm{\, \mathrm{\, +\,}\,}\mathrm{6}z^{\mathrm{-3}}\mathrm{\, \mathrm{\, +\,}\,}\mathrm{5}z^{\mathrm{-4}}\mathrm{\, \mathrm{\, +\,}\,}\mathrm{4}z^{\mathrm{-5}} }}$$

The given sequence is a causal sequence, thus $\mathrm{\mathit{X\left ( z \right )}}$ converges for all values of z except at 𝑧 = 0, i.e., the ROC is entire z-plane except at 𝑧 = 0.

Left-Hand Sequence

A sequence for which $\mathrm{\mathit{x\left ( n \right )}}$ = 0 for $\mathit{n}$ ≥ $\mathit{n_{\mathrm{0}}}$, where $\mathit{n_{\mathrm{0}}}$ is positive or negative but finite, is called the left-hand sequence. When $\mathit{n_{\mathrm{0}}}$ ≤ 0, then the resulting sequence is an anti-causal sequence. The ROC of an anti-causal sequence is the entire z-plane except at 𝑧 = ∞.

Numerical Example (2)

Find the Z-transform and ROC of the anti-causal sequence.

$$\mathrm{\mathit{x\left ( n \right )}\mathrm{\, =\,} \begin{Bmatrix} 1, & -2,& -1,& 2,& 3,& 4 \ & & & & & \uparrow\ \end{Bmatrix} }$$

Solution

The given sequence is a left-hand sequence. The values of the given sequence are −

$$\mathrm{\mathit{x\left (\mathrm{-5} \right )\mathrm{\, =\,}\mathrm{1},x\left (\mathrm{-4} \right )\mathrm{\, =\,}\mathrm{-2},x\left (\mathrm{-3} \right )\mathrm{\, =\,}\mathrm{-1},x\left (\mathrm{-2} \right )\mathrm{\, =\,}\mathrm{2},x\left (\mathrm{-1} \right )\mathrm{\, =\,}\mathrm{3},x\left (\mathrm{0} \right )\mathrm{\, =\,}\mathrm{4}}}$$

As the Z-transform is given by,

$$\mathrm{\mathit{X\left ( z \right )\mathrm{\, =\,}\sum_{n\mathrm{\, =\,}-\infty }^{\infty }x\left ( n \right )z^{-n}}}$$

Hence, for the given sequence values, the Z-transform is,

$$\mathrm{\mathit{Z\left [ x\left ( n \right ) \right ]\mathrm{\, =\,}X\left ( z \right )}}$$

$$\mathrm{\mathit{\mathrm{\, =\,}x\left ( \mathrm{-5} \right )z^{\mathrm{5}}\mathrm{\, \mathrm{\, +\,}\,}x\left ( \mathrm{-4} \right )z^{\mathrm{4}}\mathrm{\, \mathrm{\, +\,}\,}x\left ( \mathrm{-3} \right )z^{\mathrm{3}}\mathrm{\, \mathrm{\, +\,}\,}x\left ( \mathrm{-2} \right )z^{\mathrm{2}}\mathrm{\, \mathrm{\, +\,}\,}x\left ( \mathrm{-1} \right )z\mathrm{\, \mathrm{\, +\,}\,}x\left ( \mathrm{0} \right )}}$$

$$\mathrm{\mathit{\therefore X\left ( z \right )\mathrm{\, =\,}z^{\mathrm{5}}-\mathrm{2}z^{\mathrm{4}}-z^{\mathrm{3}}\mathrm{\, \mathrm{\, +\,}\,}\mathrm{2}z^{\mathrm{2}}\mathrm{\, \mathrm{\, +\,}\,}\mathrm{3}z\mathrm{\, \mathrm{\, +\,}\,}\mathrm{4} }}$$

As the given sequence is an anti-causal sequence, therefore the $\mathrm{\mathit{X\left ( z \right )}}$ converses for all values of z except at 𝑧 = ∞, i.e., the ROC is the entire z-plane except at 𝑧 = ∞.

Two-Sided Sequence

A two-sided sequence is the one which exists on both the left and right sides. The ROC of a two-sided sequence is the entire z-plane except at 𝑧 = 0 and 𝑧 = ∞.

Numerical Example (3)

Find the Z-transform and ROC of the two-sided sequence.

$$\mathrm{\mathit{x\left ( n \right )}\mathrm{\, =\,} \begin{Bmatrix} 5, & 1,& 2,& 3,& 4,& 0,& 5,& \ & & & \uparrow& & \ \end{Bmatrix} }$$

Solution

The values of the given two-sided sequence are −

$$\mathrm{\mathit{x\left (\mathrm{-3} \right )\mathrm{\, =\,}\mathrm{5},x\left (\mathrm{-2} \right )\mathrm{\, =\,}\mathrm{1},x\left (\mathrm{-1} \right )\mathrm{\, =\,}\mathrm{2},x\left (\mathrm{0} \right )\mathrm{\, =\,}\mathrm{3},x\left (\mathrm{1} \right )\mathrm{\, =\,}\mathrm{4},x\left (\mathrm{2} \right )\mathrm{\, =\,}\mathrm{0},x\left (\mathrm{3} \right )\mathrm{\, =\,}\mathrm{5}}}$$

The Z-transform is given by,

$$\mathrm{\mathit{X\left ( z \right )\mathrm{\, =\,}\sum_{n\mathrm{\, =\,}-\infty }^{\infty }x\left ( n \right )z^{-n}}}$$

For the sequence values, the Z-transform is,

$$\mathrm{\mathit{X\left ( z \right )\mathrm{\, =\,}x\left ( \mathrm{-3} \right )z^{\mathrm{3}}\mathrm{\, +\,}x\left ( \mathrm{-2} \right )z^{\mathrm{2}}\mathrm{\, +\,}x\left ( \mathrm{-1} \right )z\mathrm{\, +\,}x\left ( \mathrm{0} \right )\mathrm{\, +\,}x\left ( \mathrm{1} \right )z^{\mathrm{-1}}\mathrm{\, +\,}x\left ( \mathrm{2} \right )z^{\mathrm{-2}}\mathrm{\, +\,}x\left ( \mathrm{3} \right )z^{\mathrm{-3}}}}$$

$$\mathrm{\mathit{\therefore X\left ( z \right )\mathrm{\, =\,}\mathrm{5}z^{\mathrm{3}}\mathrm{\, +\,}z^{\mathrm{2}}\mathrm{\, +\,}\mathrm{2}z\mathrm{\, +\,}\mathrm{3}\mathrm{\, +\,}\mathrm{4}z^{\mathrm{-1}}\mathrm{\, +\,}\mathrm{5}z^{\mathrm{-3}}}}$$

The ROC is the entire z-plane except at 𝑧 = 0 and 𝑧 = ∞.