Week 4-5
1. Real time state machines2. LTI systems– the [A,B,C,D] representation3. 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 signal6. Impulse response
Recall function-and-set description of state machines
States, Inputs, Outputs, initialState, update function
s(0) = initialStates(n+1) = nextState(s(n), x(n))y(n) = output(s(n), x(n)) }
(s(n+1), y(n)) = update(s(n),x(n))
Real time state machines
Above (Ch 4) n represents step
We now consider machines in which n representsreal 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 andstates are represented as tuples of real numbers
Delay3
D D Dx(n) x(n-1) x(n-2) x(n-3)
s1(n) s2(n) s3(n)
y(n)
nextState s1(n+1) = x(n)
s2(n+1) = s1(n)
s3(n+1) = s2(n)
y(n) = s3(n)output
4pt Moving average
D D Dx(n) x(n-1) x(n-2) x(n-3)
s1(n) s2(n) s3(n)
y(n)1/4 +1/4
1/41/4
nextState s1(n+1) = x(n)
s2(n+1) = s1(n)
s3(n+1) = s2(n)
y(n) = s1(n) + s2(n) + s3(n) + x(n) output 14
14
14
14
x1(n)
xM(n)
s1(n)
sN(n)
y1(n)
yK(n)
.
...
.
.
x(n)RM s(n)RN y(n)RK
MIMO system
SISO system if M = K =1
LTI systems
Infinite state systems with linear update function
System = (RN, RM, RK, update, initialState)
States = RN, Inputs = RM, Outputs = RK, initialState = s(0)
update: RN RM RN RK 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)
Response
to input signalx = (x(0), x(1), …) [Ints0 R]
State response is
s = (s(0), s(1), …) [Ints0 R]
s(0) = initialStates(n+1) = a s(n) + b x(n), n 0
s(1) = as(0) + bx(0)s(2) = as(1) + bx(1) = a2s(0) + abx(0) + bx(1)…s(n) = ans(0) + an-1bx(0) + an-2bx(1) + … + bx(n-1)
Output response
y = (y(0), y(1), …) [Ints0 R]
is obtained from state response and y(n) = cs(n) + dx(n)
zero-inputresponse
zero-stateresponse
(total)response
= +
Zero-state response is
Define
Then the zero-state response is the convolution sum
h: Integers0 R is the (zero-state) impulse response
Define a state machine by
Substitute from differential equation above to get
Note: A is 22, b is 21, cT is 12, d is 11
Recall Zero-state response is
Define
Then the zero-state response is the convolution sum
h: Integers0 R is the (zero-state) impulse response
The Kronecker delta or impulse at time k is the inputsignal
nk
graph of impulse at k
The impulse signal
1
If k = 0, write instead of 0, and call it the impulse
Recall the general formula for the zero-state responseto 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.
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 n0
k k+nk
Time-invariance
50 1
graph ofm h(m)
m
0-1-5m
10-4m
graph ofm h(4- m)
40m
-1
graph ofm h(-m) flip
graph ofm h(1- m)
flip & drag
Convolution mechanics by flip and drag
0 1 2 3
x(m)
m 0 1 2
h(m)
m
1
2
1/32/3
1
0-1-2m
h(0-m)
y(0) = 1/3 x 1 = 1/3
10-1m
h(1-m)y(1) = 2/3 x 1 + 1/3 x 2 = 4/3
210m
h(2-m)y(2) = 1 x 1 + 2/3 x 2 + 1/3 x 1= 8/3