1. Operation in System Hany Ferdinando Dept. of Electrical Engineering Petra Christian University

2. Operation in system2 General Overview Convolution both in continuous- and discrete-time system De-convolution in discrete-time system

3. Operation in system3 Convolution (discrete) The output of a system can be analyzed with impulse response sequence Impulse response is response of a system due to an impulse sequence as input  n  h n means ‘input  n gives output h n ‘  c{  n }  c{h n }  c{  n±m }  c{h n±m }

4. Operation in system4 Convolution (discrete) For a signal u k, we can write u j {d k-j } gives output u j {h k-j } Applying the superposition property, the total response is merely the sum of the individual response of the u j {h k-j }

5. Operation in system5 Convolution (discrete) Thus the output sequence is If we let m = k-j, then

6. Operation in system6 Convolution Operation Calculate the sequence for every k To find y 1, we need h 1-n, then the shifted h 1-n is multiplied with u For others k, the procedures are similar

7. Operation in system7 Example and Exercise h(n) = (½) n for n ≥ 0 (even) and 0 for n is odd, u(n) = {1,2} do it with u*h and h*u h(n) = {1,2,1}, u(n) = {1,2,1} h(n) = (½) n for n ≥ 0, u(n) = (¼) n for n ≥ 0 … (get other exercises from the books)

8. Operation in system8 Convolution (discrete) If both signal is positive semi infinite, then we can use table matrix to calculate the convolution Place the values of one signal at the top row and the other at the left most column Be careful!! You have to verify the first result…!!

9. Operation in system9 Convolution (continuous) To derive procedure for convolution in continuous-time system is similar to that of discrete-time system The input is decomposed into a sum of impulse function, then express the output as a sum of the response resulting from individual impulses

10. Operation in system10 Convolution Operation The formula is Remark:  Inside the integral, ‘t’ is transformed into ‘  ’  h(t)  h(  ) and u(t)  u(  ) Get h(-  ) and shift it to the right to get h(t-  )

11. Operation in system11 Convolution Operation Always draw the signals before convolving them, this is important to get the limit for integration It is calculated based on the range, e.g. 0<t<1, 1<t<2, 2<t<3, etc.

12. Operation in system12 Examples and Exercises Convolve h(t) = 1 for 0≤t≤2 with u(t) = t for t≥0 h(t) = 1 for 0≤t≤1 and -1 for 1≤t≤2, u(t) = t for 0≤t≤2

13. Operation in system13 Deconvolution (discrete only) It is how to ‘undo’ convolution From the output y and input u relationship, one can derive the impulse response h, etc. Application:  To find transfer function of a system  To measure the linearity of unknown system

14. Operation in system14 Deconvolution formulation To find u: To find h:

15. Operation in system15 Exercise Input u = {1,½} and output y = (½) k for k ≥ 0. Find h!

16. Operation in system16 Next… The operations in system are discussed! Students have to exercise themselves in order to understand those operation well. The next topic is Fourier analysis. Read the Signals and Linear Systems by Robert A. Gabel (p. 239-255) or Modern Signals and System by Huibert Kwakernaak (p. 330-367) or Signals and System by Alan V. Oppenheim (p.161-179)