DSP Lect_3_digital_signal_processing.pdf

Prepared by: Dr / Ahmed Nasser
Lecturer at Faculty of Engineering, Suez Canal University
(ahmed.nasser@eng.suez.edu.eg)
DSP
Lecture 4: The Z-Transform

九州大学UIプロジェクト Kyudai Taro,2007
• The z-transform is a very important tool in describing and analyzing digital systems. It also
offers techniques for digital filter design and frequency analysis of digital signals.
• The z-transform of a causal sequence x(n), designated by X(z) or Z(x(n)), is defined as:
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➢ The Z-Transform:
The Z-Transform
where z is the complex variable:
• We assume that the digital signal x(n) is a causal sequence, that is, x(n)=0 for n<0. Thus,
the above equation is referred to as one-sided z-transform or a unilateral transform

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✓ Example:
The Z-Transform
Solution

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✓ Example:
Solution
The Z-Transform

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The Z-Transform
✓ Example:
Solution

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The Z-Transform

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The Z-Transform

• The z-transform is a linear transformation, which implies:
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1- Linearity Property:
PROPERTIES of the Z-TRANSFORM

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Solution
✓ Example (Linearity Property):

• Given X(z), the z-transform of a sequence x(n), the z-transform of x(nm), the time shifted
sequence, is given by
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2- Shift Property:

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✓ Example (Linearity Property):
Solution

• Given two sequences x1(n) and x2(n), their convolution can be determined as follows:
• where ‘∗’ designates the linear convolution. In z-transform domain, we have
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3- Convolution Property:

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✓ Example (Convolution Property):
Solution

• Given the z-transfer function X(z), then initial value of the causal sequence x(n) can be
determined by
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4- Initial value theorem:
5- Final value theorem:
• Given the z-transfer function X(z), then final value of the causal sequence x(n) can be
determined by

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• The z-transform of a sequence x(n) and the inverse z-transform of a function X(z) are
defined as, respectively:
• The inverse of the z-transform may be obtained by at least three methods:
1. Partial fraction expansion and look-up table.
2. 2. Power series expansion.
3. 3. Inversion Formula Method.
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➢ INVERSE Z-TRANSFORM
INVERSE Z-TRANSFORM

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➢ Partial fraction expansion and look-up table:
INVERSE Z-TRANSFORM
• Example
Solution

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INVERSE Z-TRANSFORM
• Example
Solution

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INVERSE Z-TRANSFORM
• The general procedure is as follows:
1. Eliminate the negative powers of z for the z-transform function X(z).
2. Determine the rational function X(z)/z (assuming it is proper), and apply the partial fraction
expansion to the determined rational function X(z)/z using the formula in Table 5.3.
3. Multiply the expanded function X(z)/z by z on both sides of the equation to obtain X(z).
4. Apply the inverse z-transform using Table 5.1.

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INVERSE Z-TRANSFORM
• Example
Solution

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INVERSE Z-TRANSFORM
Solution

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DSP Lect_3_digital_signal_processing.pdf

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