EECS 216 LECTURE NOTES
BASIC SYSTEM PROPERTIES
What: Input x(t)→ |SYSTEM| → y(t) output.
Why: Design the system to filter input x(t).
DEF: A system is LINEAR if the following two properties hold:
1. Scaling: If x(t)→ |SYS| → y(t), then ax(t)→ |SYS| → ay(t)
for: any constant a. NOT true if avaries with time (i.e., a(t)).
2. Superposition: If x1(t)→ |SYS| → y1(t) and x2(t)→ |SYS| → y2(t),
Then: (ax1(t) + bx2(t)) → |SYS| → (ay1(t) + by2(t)) (linear combinations)
for: any constants a, b. NOT true if aor bvary with time (i.e., a(t), b(t)).
EX: y(t) = 3x(t−2); 2dy
dt + 3y(t) = 5dx
dt + 6x(t); y(t) = sin(t)x(t).
NOT: y(t) = x2(t); y(t) = sin(x(t)); y(t) = |x(t)|;y(t) = x(t)/x(t−1).
NOT: y(t) = x(t) + 1 (try it). This is an affine (linear+constant) system.
HOW: If any nonlinear function of x(t), not linear. Nonlinear of just tOK.
DEF: A system is TIME-INVARIANT if this property holds:
If x(t)→ |SYS| → y(t), then x(t−T)→ |SYS| → y(t−T)
for: any constant time delay T. NOT true if Tvaries with time (e.g., T(t)).
EX: y(t) = 3x(t−2); y(t) = sin(x(t)); y(t) = x(t)/x(t−1).
NOT: y(t) = tx(t); y(t) = x(t2); y(t) = x(2t); y(t) = x(−t).
HOW: If tappears anywhere other than in x(t), not time-invariant. Else OK.
DEF: A system is CAUSAL if it has this form for some function F(·):
y(t) = F({x(τ), τ ≤t}) (depends only on present and past inputs)
Note: Physical systems must be causal. But DSP filters need not be causal!
DEF: A system is MEMORYLESS if y(t) = F(x(t)) (present input only).
DEF: LTI system is (BIBO) STABLE iff: Let x(t)→ |SYSTEM| → y(t)
If |x(t)| ≤ Mfor some constant M, then |y(t)| ≤ Nfor some N.
i.e.: “Every bounded input (BI) produces a bounded output (BO).”
HOW: BIBO stable ⇔R∞
−∞ |h(τ)|dτ ≤Lfor some L(absolutely integrable)
where: Impulse δ(t)→ |SYS| → h(t)=impulse response of the system.
EX: A discrete time-invariant system has these two responses:
{0,0,3} → |SYS| → {0,1,0,2}and {0,0,0,1} → |SYS| → {1,2,1}.
Prove: The system is nonlinear.
Proof: By contradiction. Suppose the system is linear. But then:
{0,0,0,1} → |SYS| → {1,2,1}implies {0,0,3} → |SYS| → {3,6,3}
since we know it is time-invariant. Then {0,0,3}produces two outputs!
