Scarica TRANSFORM ANALYSIS OF LTI SYSTEM e più Formulari in PDF di Segnali E Trasmissioni solo su Docsity! TRANSFORM ANALYSIS OF LTI SYSTEM Reminder Example Pole-zero plot and region of possible convergence for the z transform in X ( z )= 1+z−1 (1−z−1)(1−0.5 z−1) System function of LTI systems Procedure for the analytical computation of the output of an LTI system using the convolution theorem of the z-transform: Equivalent system function of linear time-invariant systems combined in a) parallel connection and b) cascade connection: . Stability & Causality: -Casual system must be right sided: ROC is outside the outermost pole; - Stable system requires absolute summable impulse response ∑ k=−∞ +∞ |h [k ]|<∞: Absolute summability implies existence of DTFT; DTFT exist if unit circle is in the ROC; Therefore, stability implies that the ROC includes the unit circle. - Casual & stable system have all poles inside unit circle: -Casual hence the ROC is outside outermost pole; -Stable hence unit circle included in ROC; -This means outermost pole is inside unit circle; Hence all poles are inside unit circle. Linear constant-coefficient difference equations A class of LTI systems whose input and output sequences satisfy a linear constant-coefficient difference equation is written as: ∑ k=0 N ak y [n−k ]=∑ k=0 M bk x [n−k ] ∑ k=0 N ak z −kY (z)=∑ k=0 M bk z −k X (z ) H ( z )= ∑ k=0 M bk z −k ∑ k=0 N ak z −k Example LCCDE (causal): y [n ]=ay [n−1 ]+bx [n ] ,−1<a<1 System function: Y ( z )=a z−1Y ( z )+bX ( z )→H (z )= Y (z ) X (z) = b 1−a z−1 = bz z−a Since system is causal h [n ]=b anu [n] The system is stable because p=a and |a|<1 Example H ( z )= 1−z2 (1+0.9 z−1+0.6 z−2+0.05 z−3 ) Example y [n ]− 5 2 y [n−1 ]+ y [n−2]=x [n] H ( z )= 1 (1−12 z −1)(1−2 z−1) One-side z-Transform To solve LCCDEs with initial condition or to obtain output when an input is stepped into a system, we need the one-side z-transform Example Ideal and practical filters - Systems that are designed to pass some frequency components without significant distortion while severely or completely eliminating others are known as frequency selective filters. - By definition, an ideal frequency selective filter satisfies the requirements for distortionless response over one or more frequency bands. - Ideal filters are used in the early stages of a design process to specify modules in a signal processing systems. However they are not realizable in practice, they must be approximated by practical and non-ideal filters. - This is usually done by minimizing some approximation error between the non-ideal filter and a prototype ideal filter. Ideal filter: Typical filter requirements: - gain = 1 for wanted parts (pass band) - gain = 0 for unwanted parts (stop band) Ideal low pass filter h [n ]=IFTFT {H (e jωω)}= 1 2π ∫ −π +π H (e jωω)e jωωn dωω= 1 2π ∫ −ωc +ωc e jωωndωω→h [n ]= sinωcn πn Typical characteristics of a practical bandpass filter: Frequence response of Rational System Functions All LTI systems of practical interest are described by a difference equation of the form: ∑ k=0 N ak y [n−k ]=∑ k=0 M bk x [n−k ]
