Z-Transform and ROC of Finite Duration Sequences

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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 x(n) = 0 for n < n₀ where n₀ may be positive or negative but finite, is called the Right Hand Sequence. If n₀ ≥ 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

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

$$\mathrm{x(n) \:=\: \{1,\:0,\:−4,\:6,\:5,\:4\: \uparrow \}}$$

Solution

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

$$\mathrm{x(0) \:=\:1,\:x(1)\:=\:0,\:x(2)\:=\:−4,\:x(3)\:=\:6,\:x(4)\:=\:5,\:x(5)\:=\:4}$$

The Z-transform of a sequence is given by,

$$\mathrm{X(z) \:=\: \sum_{n=-\infty}^{\infty}\:x(n)z^{-n}}$$

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

$$\mathrm{Z[x(n)] \:=\: X(z)}$$

$$\mathrm{= \:x(0) \:+\:x(1)z^{−1}\:+\:x(2)z^{−2}\:+\:x(3)z^{−3}\:+\:x(4)z^{−4}\:+\:x(5)z^{−5}}$$

$$\mathrm{\therefore \:X(z)\:=\:1\:−\:4z^{−2}\:+\:6z^{−3}\:+\:5z^{−4}\:+\:4z^{−5}}$$

The given sequence is a causal sequence, thus X(z) 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 x(n) = 0 for n ≥ n₀, where n₀ is positive or negative but finite, is called the left-hand sequence. When n₀ ≤ 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

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

$$\mathrm{x(n) \:=\: \{1,\:−2,\:−1,\:2,\:3,\:4\: \uparrow \}}$$

Solution

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

$$\mathrm{x(−5)\:=\:1,\:x(−4)\:=\:−2,\:x(−3)\:=\:−1,\:x(−2)\:=\:2,\:x(−1)\:=\:3,\:x(0)\:=\:4}$$

As the Z-transform is given by,

$$\mathrm{X(z) \:=\: \sum_{n=-\infty}^{\infty}\:x(n)z^{-n}}$$

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

$$\mathrm{Z[x(n)]\:=\:X(z)}$$

$$\mathrm{=\:x(−5)z^5\:+\:x(−4)z^4\:+\:x(−3)z^3\:+\:x(−2)z^2\:+\:x(−1)z\:+\:x(0)}$$

$$\mathrm{\therefore\:X(z)\:=\:z^5\:−\:2z^4\:−\:z^3\:+\:2z^2\:+\:3z\:+\:4}$$

As the given sequence is an anti-causal sequence, therefore the X(z) 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

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

$$\mathrm{x(n) \:=\:\{5,\:1,\:2,\:3,\:4,\:0,\:5,\: \uparrow \}}$$

Solution

The values of the given two-sided sequence are −

$$\mathrm{x(−3) \:=\:5,\:x(−2)\:=\:1,\:x(−1)\:=\:2,\:x(0)\:=\:3,\:x(1)\:=\:4,\:x(2)\:=\:0,\:x(3)\:=\:5}$$

The Z-transform is given by,

$$\mathrm{X(z) \:=\: \sum_{n=-\infty}^{\infty}\:x(n)z^{-n}}$$

For the sequence values, the Z-transform is,

$$\mathrm{X(z) \:=\:x(−3)z^3\:+\:x(−2)z^2\:+\:x(−1)z\:+\:x(0)\:+\:x(1)z^{−1}\:+\:x(2)z^{−2}\:+\:x(3)z^{−3}}$$

$$\mathrm{\therefore\:\:X(z)\:=\:5z^3\:+\:z^2\:+\:2z\:+\:3\:+\:4z^{-1}\:+\:5z^{−3}}$$

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