1. Week 4-5 1.Real time state machines 2.LTI systems– the [A,B,C,D] representation 3.Differential equation-approximation by discrete- time system 4.Response– total response = zero-input response + zero-state response 5.Convolution-zero-state response = convolution of impulse response and input signal 6.Impulse response

2. Recall function-and-set description of state machines States, Inputs, Outputs, initialState, update function s(0) = initialState s(n+1) = nextState(s(n), x(n)) y(n) = output(s(n), x(n)) } (s(n+1), y(n)) = update(s(n),x(n))

3. Real time state machines Above (Ch 4) n represents step We now consider machines in which n represents real time, eg. seconds, micro-seconds, etc. The only difference this makes is that we cannot have the absent input We consider machines in which inputs, outputs and states are represented as tuples of real numbers

4. Delay3 D DD x(n)x(n-1)x(n-2)x(n-3) s 1 (n)s 2 (n)s 3 (n) y(n) nextState s 1 (n+1) = x(n) s 2 (n+1) = s 1 (n) s 3 (n+1) = s 2 (n) y(n) = s 3 (n) output

5. 4pt Moving average D DD x(n)x(n-1)x(n-2)x(n-3) s 1 (n)s 2 (n)s 3 (n) y(n) 1/4 + nextState s 1 (n+1) = x(n) s 2 (n+1) = s 1 (n) s 3 (n+1) = s 2 (n) y(n) = s 1 (n) + s 2 (n) + s 3 (n) + x(n) output 1 4 1 4 1 4 1 4

6. x 1 (n) x M (n) s 1 (n) s N (n) y 1 (n) y K (n)............ x(n)  R M s(n)  R N y(n)  R K MIMO system SISO system if M = K =1 LTI systems

7. Infinite state systems with linear update function System = (R N, R M, R K, update, initialState) States = R N, Inputs = R M, Outputs = R K, initialState = s(0) update: R N  R M  R N  R K is a linear function so there are matrices A (N  N), B (N  M), C (K  N), D(K  M) such that s(n+1) = A s(n) + B x(n) y(n) = C s(n) + Dx(n)

8. v(t) i(t) q(t) R C Differential equations

9. Response to input signal x = (x(0), x(1), … )  [Ints 0  R] State response is s = (s(0), s(1), … )  [Ints 0  R] s(0) = initialState s(n+1) = a s(n) + b x(n), n  0 s(1) = as(0) + bx(0) s(2) = as(1) + bx(1) = a 2 s(0) + abx(0) + bx(1) … s(n) = a n s(0) + a n-1 bx(0) + a n-2 bx(1) + … + bx(n-1)

10. Output response y = (y(0), y(1), … )  [Ints 0  R] is obtained from state response and y(n) = cs(n) + dx(n) zero-input response zero-state response (total) response =+

11. Zero-state response is Define Then the zero-state response is the convolution sum h: Integers 0  R is the (zero-state) impulse response

12. The Kronecker delta or impulse at time k is the input signal n k graph of impulse at k

13. Notes/Responses/Echo

14. v(t) i 1 (t) q 1 (t) R1R1 C1C1 R2R2 C2C2 q 2 (t) i 2 (t)

15. Define a state machine by Substitute from differential equation above to get Note: A is 2  2, b is 2  1, c T is 1  2, d is 1  1

16. Recall Zero-state response is Define Then the zero-state response is the convolution sum h: Integers 0  R is the (zero-state) impulse response

17. The Kronecker delta or impulse at time k is the input signal n k graph of impulse at k The impulse signal 1 If k = 0, write  instead of  0, and call it the impulse

18. Recall the general formula for the zero-state response to any input signal x: So the response to the impulse is obtained by setting x = , which gives That is why h is called the (zero-state) impulse response.

19. Suppose the input signal is  k, impulse at k. Substitution in the general formula gives its (zero-state) response as which is the impulse response delayed by k 0-state response 0 n 0 k k+n k Time- invariance

20. 5 0 1 graph of m  h(m) m 0-5 m 10-4 m graph of m  h(4- m) 40 m graph of m  h(-m) flip graph of m  h(1- m) flip & drag

21. Convolution mechanics by flip and drag 01 23 x(m) m 01 2 h(m) m 1 2 1/3 2/3 1 0 -2 m h(0-m) y(0) = 1/3 x 1 = 1/3 10 m h(1-m) y(1) = 2/3 x 1 + 1/3 x 2 = 4/3 21 0 m h(2-m) y(2) = 1 x 1 + 2/3 x 2 + 1/3 x 1= 8/3

22. Convolution, time-invariance and linearity