Sheraz Dip slides.pptx for assignments checking
- 1. Digital Signal Processing PRESENTER: SHERAZ HIADER NIM-BSEE-2020-33 INSTRUCTOR: ZULAIKHA KIRAN LTI System’s Z-transform (System Function) 1
- 2. What is LTI System? (linear time invariant system) ❑ Definition: A Linear Time Invariant (LTI) System is a system that possesses the property of superposition and has behavior and characteristics that are fixed over time. Examples: An electrical RLC circuit and a mechanical mass-spring-damper system are examples of Linear Time Invariant (LTI) systems, which can be described by linear differential equations with constant coefficients. Mathematically: 2
- 3. What is Z-Transform? 3 Definition: The z-transform is a mathematical transformation that converts discrete-time signals or systems from the time domain to the complex frequency domain, represented by the complex variable z. It is a fundamental tool for analyzing and designing discrete-time Linear Time Invariant (LTI) systems. Examples: The z-transform is used in the design of digital filters and digital control systems. Mathematically:
- 4. Analysis of LTI System using Z- Transform 4 The analysis of Linear Time-Invariant (LTI) Systems using the Z-Transform. It includes the convolution property equation is: Y(z) = H(z) X(z) And explanations of the variables involved in the equation: Y(z) represents the z-transform of the system output, H(z) is the z-transform of the impulse response, and X(z) is the z-transform of the system input.
- 5. Causality and Stability ❑Causality: A discrete-time LTI system is considered causal if and only if the region of convergence (ROC) of its system function lies outside the circle, including infinity. ❑Stability: The LTI system is stable if and only if the region of convergence (ROC) of the system function H(z) includes the unit circle, where the absolute value of z is equal to 1. 5
- 6. Causality and Stability for LTI System with Rational System Function ❑Causality: The Region of Convergence (ROC) is the exterior of the outermost pole. In the expression of H(z) as a ratio of polynomials in z, the order of the numerator cannot be greater than the order of the denominator. ❑Stability: For a causal system to be stable, all the poles of H(z) must lie inside the unit circle, meaning they must have a magnitude smaller than 1. It is noted that a system can be stable but not causal. 6
- 7. 7