Signals and Systems. A signal is the representation of a physical wave

Transcription

1 Signals and Systems A signal is the representation of a physical wave Expressed as a variable in time-space, for instance x(t) Signals that might vary are the voltage or current of a circuit, the force in a mechanical circuit, heat flow in a thermal circuit, or hydraulic flow in a fluid circuit A system is a process that has as an input a signal and outputs a different signal (usually)

2 Signals and Systems A system is a process that has as an input a signal and outputs a different signal (usually) Expressed as a transfer function or an operator As an example an operator could be an integrator taking a signal and integrating over time, or it could be just as simple as multiplying by a constant

3 Signals and Systems The methods that are used in signals and systems are in general the same methods learned in linear algebra However this is only part of the story...the analytic part The other part of the story on how to actually model these and use them in the real world requires an introduction to numerical methods

4 Methods of approximating physical problems that are intractable using analytic techniques Physical problems are represented by mathematical equations A number of approximation techniques for the same physical expression are used depending on the situation Methods of assigning error to approximated solutions

5 While algorithms can be designed from scratch a number of packages exist along with a number of languages designed for numerical methods Languages/Packages (All platforms) MATLAB (main drawback: expensive) OCTAVE (very much like MATLAB: Free) Scilab (like MATLAB: Includes a Simulink like system) RLAB+ (kinda like MATLAB) Yorick (has heard of MATLAB) SciPy (not-like MATLAB) And more...

6 Continued Packages (almost always in Fortran though C translations do exist for most and some JAVA) LINPACK: Linear algebra pack LAPACK: Modern linear algebra pack MINPACK: non-linear problems ODEPACK: Differential equations QUADPACK: Integration, etc. FFTPACK: FFTs, of course SLAP: Sparse Linear Algebra Package BLAS: Basic Linear Algebra subprograms Many more (see list at NETLIB on web site)

7 Continued For Java numerical methods routines go to

8 Packages are compiled and made libraries EGR205% gcc -c -O2 test.f EGR205% gcc test.o -lm -lc -lslap -lblas -o test EGR205%./test Program gives output of some sort Example of subroutine used in say slap might be sgmres (see handout) which is a preconditioned GMRES (generalized minimum residual) iterative sparse Ax=b solver

9 Getting MATLAB Getting OCTAVE For windows (sourceforge is best) This gives step-by-step instructions on

10 Second law and the parachuting person F = ma F = F_d + F_u F_d = mg F_u = -cv 2 Therefore the analytic from is converted to digital form dv/dt = (mg cv 2 )/m = g cv 2 /m [v(i+1)-v(i)]/[t(i+1)-t(i)] = g cv(i)2 /m v(i+1)=v(i) + [g-cv(i)2 /m]*[t(i+1)-t(i)] Try programming and running this at home

11 Differential Equations First-order systems are divided into natural response and forced response parts Natural response = homogeneous (complementary) solution Forced response = particular solution dx t x t =K dt s f t

12 Natural Solution first dx n t x t =0 dt dx n t = x n t dt t / dx n t = e dx n t = dt dt x n t =e t / e c t / x n t = e

13 Forced solution Assume a constant force response (otherwise it is harder) dx f t dt x t =K s F dx f t =K s F for t 0

14 Full solution x t =x n t dx f t = e t / K s F

15 Writing good program Pseudocode Pretty much English in the manner you like If/then/else Case Do while Do until Do for i=a,b,step End etc Flowcharts

16 Flowchart Symbols

17 Programming example (C)

18 Programming example (Matlab)