Digital Signal Processing Lecture Notes 2024-2025 PDF

Year: 2024-2025 Fall Semester Digital Signal Processing Dr. Manal Shaban Dr. Nabil Sabour Dr. Adel Fathy Khalifa Z-Transform  Z-Transform Z-transform, like the Laplace transform, is a mathematical tool for the design, analysis and monitoring...

Year: 2024-2025 Fall Semester Digital Signal Processing Dr. Manal Shaban Dr. Nabil Sabour Dr. Adel Fathy Khalifa Z-Transform  Z-Transform Z-transform, like the Laplace transform, is a mathematical tool for the design, analysis and monitoring of systems. The z-transform is the discrete-time part of the Laplace transform and a generalization of the Fourier transform of a sampled signal. A working knowledge of the z-transform is essential to the study of digital filters and systems.  Z-Transform Like Laplace transform, the z-transform allows insight into the behavior and the stability of discrete-time systems. Z-transform enables us to obtain a broader characterization of discrete time LTI systems and their interaction with signals than is possible with the DTFT. For example: - DTFT can be applied to stable LTI systems since it exists only if the sum of the impulse response is finite. In contrast z-transform of the impulse response exists for unstable LTI systems.  Z-Transform Z-transform simplify the analysis of the LTIS as follows: - It is easier to test stability and causality using z-transform (Like Laplace transform Continuous time signals). - Convolution of two time domain signals is equivalent to multiplication of their z-transforms. - Finding the output of a LTIS using z-transform. Derivation of Z-transform Definition of Z-transform  Geometric Representation of Z-transform The variable z in z-transform is complex and can be represented as a circle with radius r where r represents magnitude of z. B(z) X (z) = A(z) B(z) is the numerator polynomial A(z) is the denominator polynomial The roots of B(z) is called the zeros of X(z) The roots of A(z) is called the poles of X(z) Geometric Representation of Z-transform  Remember: Discrete –Time Fourier Transform (DTFT) The discrete-time Fourier Transform is given by:   x(e )=  x(n)e   x(n)   n=− j − j n n=− Where: x(e j ) is a complex valued continuous function w = 2π f [rad/sec] f is the digital frequency measured in [ C/S] Relation between Z-Transform and DTFT: since z = re j  r =1   − j n x(z) =  x(n) z −n z=e j x(e j )=  x(n) e n=− n=− Z-transform DTFT  Region of Convergence (ROC) Properties of ROC:  Examples 1 2  3 1 z 4 = = 1− z −1 z −1 ROC consists of Remember: when a  1 5 6 7 a