EECE 301 Signals Systems Prof Mark Fowler Note Set 27 C T Systems Laplace Transform Power Tool for system analysis Reading Assignment Sections 6 1 6 3 of Kamen and Heck 1 18 Course Flow Diagram The arrows here show conceptual flow between ideas Note the parallel structure between the pink blocks C T Freq Analysis and the blue blocks D T Freq Analysis New Signal Models Ch 1 Intro C T Signal Model Functions on Real Line System Properties LTI Causal Etc D T Signal Model Functions on Integers New Signal Model Powerful Analysis Tool Ch 3 CT Fourier Signal Models Ch 5 CT Fourier System Models Ch 6 8 Laplace Models for CT Signals Systems Fourier Series Periodic Signals Fourier Transform CTFT Non Periodic Signals Frequency Response Based on Fourier Transform Transfer Function New System Model New System Model Ch 2 Diff Eqs C T System Model Differential Equations D T Signal Model Difference Equations Ch 2 Convolution Zero State Response C T System Model Convolution Integral Zero Input Response Characteristic Eq D T System Model Convolution Sum Ch 4 DT Fourier Signal Models DTFT for Hand Analysis DFT FFT for Computer Analysis Ch 5 DT Fourier System Models Freq Response for DT Based on DTFT New System Model New System Model Ch 7 Z Trans Models for DT Signals Systems Transfer Function New System Model2 18 What we have seen so far Diff Equations describe systems Differential Eq for CT Difference Eq for DT Convolution with the Impulse Response can be used to analyze the system An integral for CT A summation for DT Fourier Transform and Series describe what frequencies are in a signal CTFT for CT has an integral form DTFT for DT has a summation form There is a connection between them from the sampling theorem The Frequency Response of a system gives a multiplicative method of analysis Freq Response CTFT of impulse response for CT system Freq Response DTFT of impulse response for DT system We now look at two power tools for system analysis Laplace Transform for CT Systems Extension of CTFT Z Transform for DT Systems Extension of DTFT 3 18 Ch 6 Laplace Transform Transfer Function Back to C T signals and systems We ve seen that the FT is a useful tool for signal analysis understanding signal structure systems analysis design But only if Called Absolutely Integrable 1 System is in zero state 2 Impulse response satisfies 3 Input satisfies h t dt x t dt Well there are a few signals that we can handle with FT that do not satisfy this Sinusoids and unit step are two of them So frequency response is a tool that can only be used under these three conditions The Laplace Transform is a generalization of the CTFT it can handle cases when these three conditions are not met 4 18 There are two analysis methods that the Laplace Transform enables Zero state Sect 6 5 LT Transfer Function x t and h t may or may not be absolutely integrable So this just allows us to do the same thing that the FT does but for a larger class of signals systems Non zero state Sect 6 4 LT based solution of differential equations x t and h t may or may not be absolutely integrable This not only admits a larger class of signals systems it also gives a powerful tool for solving for both the zero state AND the zero input solutions ALL AT ONCE First we ll define the LT Next See some of its properties Then See how to use it in system analysis in these two ways 5 18 Section 6 1 Define the LT Two sided bilateral There are 2 types of LT One sided unilateral We ll only use the one sided LT Two Sided LT X 2 s x t e st dt with s j complex variable The book doesn t do this One Sided LT complex variable X 1 s x t e st dt 0 with s j One sided LT defined this way even if x t 0 t 0 But we will mostly focus on causal systems and causal inputs 6 18 One place the LT is most useful is when Causal signal 1 The system has Initial conditions at t 0 2 Input x t is applied at t 0 x t 0 t 0 This will be our focus in this course For this case X1 s X2 s Just use X s notation drop the 1 subscript a complex valued function Note that X s is of a complex variable s j X s Must plot on a plane the s plane j Im s j s plane Re s Similarly for X s 7 18 Example of Finding a LT Consider the signal x t e bt u t Back when we studied the FT we had to limit b to being b 0 with the LT we don t need to restrict that b This is a causal signal By definition of the LT 0 0 This is an easy integral to do X s e bt e st dt e s b t dt The limit is here by the definition of the integral 1 s b t X s e s b t t 0 1 limt e s b t 1 s b look at this If this limit does not converge then we say that the integral does not exist So we need to find out under what conditions this integral exists So let s look at the function inside this limit 8 18 e s b t e b j t if b 0 b if b 0 b t t Has Two Main Behaviors Thus lim t e s b t exists only for b So we can t find this X s for values of s such that Re s b But for s with Re s b we have no trouble For each X s we need to know at which s values things work This set of s is called the Region of Convergence ROC So for x t e bt u t We have X s 1 s b Don t worry too much about ROC at this level it kind of takes care of itself Re s b 9 18 This result and many others is on the Table of Laplace Transforms that is available on my web site Please use the tables from the website the ones in the book have some errors on them 10 18 e bt u t If b 0 then x t itself decays t For b 0 b is negative And we have on the s plane j This case can be handled by the FT and can also be handled by the LT ROC b e bt u t If b 0 then x t itself explodes t For b 0 b is positive And we have on the s plane j ROC b This case can t be handled by the FT but by restricting our focus to values of s in the …