# Cal3-Dinh_Hai (2017) The_Z_Transform SLIDES-GV.pdf -...

• 56

This preview shows page 1 - 9 out of 56 pages.

CALCULUS 3 Chapter 3 THE Z TRANSFORM Assoc. Profs. Nguyen Dinh & Nguyen Ngoc Hai INTERNATIONAL UNIVERSITY May 12, 2018 Assoc. Profs. Nguyen Dinh & Nguyen Ngoc Hai CALCULUS 3 Chapter 3 THE Z TRANSFORM
3 THE Z TRANSFORM 3.1 DEFINITION AND NOTATION A sequence of numbers may contain a lot of information. One concise way of storing this information is to wrap up the numbers together in a function. These functions are a very convenient tool for all sorts of computations, that would be difficult and tedious without them. z transforms are used to solve problems in discrete systems in a manner similar to the use of Laplace transforms for piecewise continuous systems. We take z transforms of sequences. Assoc. Profs. Nguyen Dinh & Nguyen Ngoc Hai CALCULUS 3 Chapter 3 THE Z TRANSFORM
3 THE Z TRANSFORM 3.1 DEFINITION AND NOTATION A sequence is a list of numbers. Sequences can be finite , like { x k } 5 0 = { 1 , 0 , - 8 , sin π 12 , - 0 . 35 } or infinite , like { x k } 0 = n 1 , 1 2 , 1 2 2 , · · · , 1 2 n , · · · o . Assoc. Profs. Nguyen Dinh & Nguyen Ngoc Hai CALCULUS 3 Chapter 3 THE Z TRANSFORM
3 THE Z TRANSFORM 3.1 DEFINITION AND NOTATION We shall allow some elements of the sequence with negative subscript : { y k } k = - 4 = { y - 4 , y - 3 , y - 2 , y - 1 , y 0 , y 1 , y 2 , · · · } { x k } k = -∞ = n 2 - k o k = -∞ = n · · · , 2 - k , · · · , 2 - 1 , 1 , 1 2 2 , · · · , 1 2 n , · · · o . Assoc. Profs. Nguyen Dinh & Nguyen Ngoc Hai CALCULUS 3 Chapter 3 THE Z TRANSFORM
3 THE Z TRANSFORM 3.1 DEFINITION AND NOTATION Definition (The transform) The z transform of a sequence { x k } -∞ is defined as Z { x k } -∞ = X ( z ) = X k = -∞ x k z k (1) whenever the sum exists and where z is a complex variable. The set of z for which it exists is called the region of existence of the z transform. Assoc. Profs. Nguyen Dinh & Nguyen Ngoc Hai CALCULUS 3 Chapter 3 THE Z TRANSFORM
3 THE Z TRANSFORM 3.1 DEFINITION AND NOTATION The process of taking the z transform of a sequence produces a function of a complex variable z . The symbol Z denotes the z transform operator . It is usual to refer to { x k } , X ( z ) as a z transform pair . Assoc. Profs. Nguyen Dinh & Nguyen Ngoc Hai CALCULUS 3 Chapter 3 THE Z TRANSFORM
3 THE Z TRANSFORM 3.1 DEFINITION AND NOTATION For sequences { x k } -∞ that are causal , that is x k = 0 for k < 0 , the z transform (1) reduces to Z { x k } -∞ = X ( z ) = X k =0 x k z k . A causal sequence is sometimes denoted by { x k } 0 . Assoc. Profs. Nguyen Dinh & Nguyen Ngoc Hai CALCULUS 3 Chapter 3 THE Z TRANSFORM
3 THE Z TRANSFORM 3.1 DEFINITION AND NOTATION
• • • 