ECE04 Lesson 08 Z-Transform and the DT-LTI System (2) PDF

ECE04: SIGNALS, SPECTRA AND SIGNAL PROCESSING Z-TRANSFORM ENGR. ANTHONY RIEGO POINTS Z-TRANSFORM AND THE DT-LTI SYSTEM DIRECT Z-TRANSFORM REGION OF CONVERGENCE Z-TRANSFORM AS A COMPLEX VARIABLE CHARACTERISTIC FAMILIES OF SIGNALS WIT...

ECE04: SIGNALS, SPECTRA AND SIGNAL PROCESSING Z-TRANSFORM ENGR. ANTHONY RIEGO POINTS Z-TRANSFORM AND THE DT-LTI SYSTEM DIRECT Z-TRANSFORM REGION OF CONVERGENCE Z-TRANSFORM AS A COMPLEX VARIABLE CHARACTERISTIC FAMILIES OF SIGNALS WITH THEIR CORRESPONDING ROC PROPERTIES OF Z-TRANSFORM Z-TRANSFORMS OF COMMON FUNCTIONS Z-TRANSFORM Z-Transforms play an important role in the analysis of DT-LTI systems as the Laplace Transform does in the analysis of CT signals. Back to Agenda Page Z-TRANSFORM AND THE DT-LTI SYSTEM  THE DIRECT The z-transform of the discrete-time function x(n) is defined as: ∞ −𝒏 𝑿 𝒛 = ෍ 𝒙 𝒏 𝒛 𝒏=−∞ where 𝒛 is a complex variable. Z-TRANSFORM AND THE DT-LTI SYSTEM For convenience, the z-transform of a signal is denoted by: 𝑿 𝒛 ≡ 𝐙{𝐱 𝐧 } whereas the relationship is indicated by: 𝒛 𝒙 𝒏 ՞ 𝑿(𝒛)  REGION Since the z-transform is an infinite power series, it exists only for those values of 𝑧 for which the series converges. Z-TRANSFORM AND THE DT-LTI SYSTEM This 𝑟𝑒𝑔𝑖𝑜𝑛 𝑜𝑓 𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑐𝑒 (𝑅𝑂𝐶 )of 𝑋(𝑧) is the set of all values of 𝑧 for which 𝑋(𝑧) attains a finite value. Thus, at any time we cite a z-transform, we must also indicate its ROC. Determine the z-transform and region of convergence of the following finite duration signals: 1. x1 n = 1, 2, 5, 7, 0, 1 2. x2 n = 1, 2, 5, 7, 0, 1 ↑ 3. x 3 n = 0, 0, 1, 2, 5, 7, 0, 1 4. x4 n = 2, 4, 5, 7, 0, 1 ↑ 5. x5 n = δ 𝑛 6. x6 n = δ 𝑛 − 𝑘 ,k > 0 7. x7 n = δ 𝑛 + 𝑘 ,k > 0  It can be easily seen that the ROC of a finite duration signal is the entire z-plane, except possibly z = 0, and/or z = ∞. These points are excluded since 𝒛𝒌 (k > 0) becomes unbounded for z = ∞, and 𝒛−𝒌 (k > 0) becomes unbounded for z = 0. In many cases, we can express the sum of the finite or infinite series for the z-transform in a closed expression. SAMPLE PROBLEM SAMPLE PROBLEM Determine the z- S A ofM P L transform and region E PROBLEM convergence of the signal: SAMPLE 𝒙 𝒏 = 𝒂 𝒏 𝒖(𝒏) PROBLEM SAMPLE PROBLEM SAMPLE SAMPLE PROBLEM SAMPLE PROBLEM Determine the z- S A ofM P L transform and region E PROBLEM convergence of the signal: SAMPLE 𝒙 𝒏 = −𝒂𝒏 𝒖(−𝒏 − 𝟏) PROBLEM SAMPLE PROBLEM SAMPLE  THE Z- TRANSFORM AS A Since 𝒛 is a complex variable, let us express 𝑧 in polar form 𝒋𝜽 𝒛 = 𝒓𝒆 where 𝑟 = 𝑧 and 𝜃 ≤ 𝑧 Z-TRANSFORM AND THE DT-LTI SYSTEM Then, 𝑿(𝒛) can be expressed as: ∞ 𝒋𝜽 −𝒏 𝑿(𝒛)ȁ𝒛=𝒓𝒆𝒋𝜽 = σ𝒏=−∞ 𝒙 𝒏 (𝒓𝒆 ) ∞ −𝒏 −𝒋𝜽𝒏 = σ𝒏=−∞ 𝒙 𝒏 𝒓 𝒆  THE Z- TRANSFORM AS A In the ROC of 𝑿(𝒛), ȁ X (z) ȁ < ∞, since the z-transform must have a finite value. ∞ −𝒏 −𝒋𝜽𝒏 𝑿 𝒛 = ෍ 𝒙 𝒏 𝒓 𝒆 𝒏=−∞ Z-TRANSFORM AND THE DT-LTI SYSTEM ∞ ≤ ෍ 𝒙 𝒏 −𝒏 𝒓 𝒆 −𝒋𝜽𝒏 𝒏=−∞ ∞ −𝒏 ≤ ෍ 𝒙 𝒏 𝒓 𝒏=−∞ −𝒏 Hence | X (z) | is finite if 𝒙 𝒏 𝒓 is absolutely summable.  THE Z- TRANSFORM AS A Our problem is finding the values for which will make 𝑥 𝑛 𝑟 −𝑛 absolutely summable, so −1 ∞ −𝑛 𝑥 𝑛 𝑋(𝑧) ≤ ෍ 𝑥 𝑛 𝑟 +෍ 𝑛 𝑟 Z-TRANSFORM AND THE DT-LTI SYSTEM 𝑛=−∞ 𝑛=0 ∞ ∞ 𝑛 𝑥 𝑛 ≤ ෍ 𝑥 −𝑛 𝑟 +෍ 𝑟𝑛 𝑛=1 𝑛=0 In which case 𝑟 in the first term must be small enough for the first term to be finite, but big enough for the second term to prevent it from vanishing.  THE Z- TRANSFORM AS A The ROC for the first term is The ROC for the second term is a circle with some radius 𝑟1. outside the circle of radius 𝑟2. Z-TRANSFORM AND THE DT-LTI SYSTEM  REGION Therefore, the ROC for X(z) is the area common among the two terms, where 𝑟2 < 𝑟 < 𝑟1. Z-TRANSFORM AND THE DT-LTI SYSTEM  CHARACTERISTIC FAMILIES OF SIGNALS FINITE-DURATION Unilateral, Causal 𝑥 𝑛 ,0 ≤ 𝑛 ≤ 𝑁 𝑅𝑂𝐶: 𝑧 ≠ 0 (positive side of the timeline) Unilateral,Anti- 𝑥 𝑛 , −𝑁 ≤ 𝑛 ≤ 0 𝑅𝑂𝐶: 𝑧 ≠ ∞ Causal (negative side of the timeline) Z-TRANSFORM AND THE DT-LTI SYSTEM Bilateral 𝑥 𝑛 , −𝑁 ≤ 𝑛 ≤ 𝑁 𝑅𝑂𝐶: 𝑧 ≠ 0, 𝑧 ≠ ∞ (both sides of the timeline) INFINITE-DURATION Unilateral, Causal 𝑥 𝑛 ,0 ≤ 𝑛 ≤ ∞ 𝑅𝑂𝐶: 𝑧 > 𝑟 (positive side of the timeline) (outside of the circle) Unilateral, Anti- 𝑥 𝑛 , −∞ ≤ 𝑛 ≤ 0 𝑅𝑂𝐶: 𝑧 < 𝑟 Causal (negative side of the timeline) (inside of the circle) Bilateral 𝑥 𝑛 , −∞ ≤ 𝑛 ≤ ∞ 𝑅𝑂𝐶: 𝑟2 < 𝑧 < 𝑟1 (both sides of the timeline) (annular region)  PROPERTIES OF LINEARITY IF: 𝒛 𝒛 𝒙𝟏 𝒏 ՞ 𝑿𝟏 (𝒛) and 𝒙𝟐 𝒏 ՞ 𝑿𝟐 (𝒛) THEN: 𝒛 Z-TRANSFORM AND THE DT-LTI SYSTEM 𝒙 𝒏 = 𝒂𝟏 𝒙𝟏 𝒏 + 𝒂𝟐 𝒙𝟐 𝒏 ՞ 𝑿 𝒛 = 𝒂𝟏 𝑿𝟏 𝒛 + 𝒂𝟐 𝑿𝟐 𝒛 TIME-SHIFTING IF: 𝒛 𝒙 𝒏 ՞ 𝑿(𝒛) THEN: 𝒛 −𝒌 𝒙 𝒏 − 𝒌 ՞𝒛 𝑿(𝒛)  PROPERTIES OF TIME REVERSAL (FOLDING) IF: 𝒛 𝒙 𝒏 ՞ 𝑿(𝒛) 𝑅𝑂𝐶: 𝑟1 < 𝑧 < 𝑟2 THEN: 𝒛 −𝟏 1 1 Z-TRANSFORM AND THE DT-LTI SYSTEM 𝒙 −𝒏 ՞ 𝑿 𝒛 𝑅𝑂𝐶: > 𝑧 > 𝑟1 𝑟2 AMPLITUDE SCALING IN THE Z-DOMAIN IF: 𝒛 𝒙 𝒏 ՞ 𝑿(𝒛) 𝑅𝑂𝐶: 𝑟1 < 𝑧 < 𝑟2 THEN: 𝒛 𝒏 −𝟏 𝒂 𝒙 𝒏 ՞ 𝑿(𝒂 𝒛) 𝑅𝑂𝐶: 𝑎 𝑟1 < 𝑧 < 𝑎 𝑟2  PROPERTIES OF DIFFERENTIATION IF: 𝒛 𝒙 𝒏 ՞ 𝑿(𝒛) THEN: 𝒛 𝒅𝑿(𝒛) Z-TRANSFORM AND THE DT-LTI SYSTEM 𝒏𝒙 𝒏 ՞ − 𝒛 𝒅𝒛 CONVOLUTION IF: 𝒛 𝒛 𝒙𝟏 𝒏 ՞ 𝑿𝟏 (𝒛) and 𝒙𝟐 𝒏 ՞ 𝑿𝟐 (𝒛) THEN: 𝒛 𝒙 𝒏 = 𝒙𝟏 𝒏 ∗ 𝒙𝟐 𝒏 ՞ 𝑿 𝒛 = 𝑿𝟏 𝒛 ∙ 𝑿𝟐 𝒛  Z-TRANSFORM SIGNAL Z-TRANSFORM SIGNAL Z-TRANSFORM x(n) X(z) x(n) X(z) 𝟏 − 𝒛−𝟏 𝒄𝒐𝒔𝝎𝒐 𝜹(𝒏) 𝟏 (𝒄𝒐𝒔𝝎𝒐 𝒏)𝒖(𝒏) 𝟏 − 𝟐𝒛−𝟏 𝒄𝒐𝒔𝝎𝒐 + 𝒛−𝟐 Z-TRANSFORM AND THE DT-LTI SYSTEM 𝟏 𝒛−𝟏 𝒔𝒊𝒏𝝎𝒐 𝒖(𝒏) (𝒔𝒊𝒏𝝎𝒐 𝒏)𝒖(𝒏) 𝟏 − 𝟐𝒛−𝟏 𝒄𝒐𝒔𝝎𝒐 + 𝒛−𝟐 𝟏 − 𝒛−𝟏 𝟏 𝟏 − 𝒂𝒛−𝟏 𝒄𝒐𝒔𝝎𝒐 𝒂𝒏 𝒖(𝒏) (𝒂𝒏 𝒄𝒐𝒔𝝎𝒐 𝒏)𝒖(𝒏) 𝟏 − 𝟐𝒂𝒛−𝟏 𝒄𝒐𝒔𝝎𝒐 + 𝒂𝟐 𝒛−𝟐 𝟏 − 𝒂𝒛−𝟏 −𝟏 𝒂𝒛 𝒂𝒛−𝟏 𝒔𝒊𝒏𝝎𝒐 𝒏𝒂𝒏 𝒖(𝒏) (𝒂𝒏 𝒔𝒊𝒏𝝎𝒐 𝒏)𝒖(𝒏) 𝟏 − 𝟐𝒂𝒛−𝟏 𝒄𝒐𝒔𝝎𝒐 + 𝒂𝟐 𝒛−𝟐 (𝟏 − 𝒂𝒛−𝟏 )𝟐 SAMPLE PROBLEM SAMPLE PROBLEM Convolve the two S A properties M P ofL E sequences using PROBLEM z- transform: S A 𝒙M 𝒙 𝒏 P = {𝟏, L −𝟐, 𝟏} E PROBLEM 𝟏 𝒏 = {𝟏, 𝟏, 𝟏, 𝟏, 𝟏, 𝟏} 𝟐 SAMPLE PROBLEM SAMPLE Determine the z-transform and the ROC of the signals: A. 𝒙𝟏 𝒏 = 𝟏, 𝟐, 𝟓, 𝟕, 𝟎, 𝟏 B. 𝒙𝟐 𝒏 = 𝟏, 𝟐, 𝟓, 𝟕, 𝟎, 𝟏 C. 𝒙𝟑 𝒏 = {𝟎, 𝟎, 𝟏, 𝟐, 𝟓, 𝟕, 𝟎, 𝟏} lISTENING PREPARE FOR YOUR TAKE-HOME QUIZ 02 lISTENING ABOUT THE TOPIC. lISTENING lISTENING lISTENING QUIZ WILL BE UPLOADED ON YOUR lISTENING GOOGLE CLASSROOM THIS JANUARY 20. lISTENING lISTENING