EE222

Nota:

• \( f\left( t\right) =f\left( -t\right),\forall t. \)

Nota:

• [\[f(t)=-f(-t)\]

1. Static(memoryless)

   Nota:

   • A system is static (memoryless) if its output at any arbitrary time depends on the input at exactly the same time.
2. Dynamic(memory)

   Nota:

   • A system which has memory (i.e., it is not memoryless) is a dynamic system.

1. Causal

   Nota:

   • A system is causal if the output at any time t1 (n1) depends on values of the input at t <= t1 (n< n1).
   • In other words, a system is causal if its output is generated during or after the application of input and not before! Causal systems are also called non-anticipatory. A system which is not causal is called noncausal.
2. Noncausal

1. Linear
   1. Additivity

      Nota:

      • The input x = x1 + x2 yields the response y = y1 + y2.
      1. Homogeneity

         Nota:

         • The input ax1 yields the response ay1 for any constant a.
2. Nonlinear
   1. Additivity Fails
   2. Homogeneity Fails

1. Time-Invarient

   Nota:

   • Combining the two conditions stated in the definition of the linearity property, we obtain the superposition principle
2. Time Varying

   Nota:

   • A system is time-varying if it is not time-invariant

1. Invertible

   Nota:

   • A system is invertible if distinct inputs yield distinct outputs. Inother words, in an invertible system whenever two inputs x1 and x2 yield the output y, then x1 = x2.

Nota:

• A relaxed system is bounded-input, bounded-output (BIBO) stable if every bounded input yields a bounded output.

1. Relaxed
   1. BIBO

1. A CT LTI system is memoryless if its impulse-response is given by h(t) = K (t) ,for some constant K .

Nota:

• Apart from easy cases, it is often hard to verify whether a system property holds or not. This statement is not true if we restrict our attention to LTI systems. In fact, it is possible to verify whether an LTI system is memoryless, causal, or BIBO stable simply by inspecting its so-called impulse-response.

1. Simplified

   Nota:

   • \[\overline {h}^{\left( n-1\right) }\left( 0+\right) =1\]

1. A DT LTI system is memoryless if its impulse-response is given by h[n] = K [n] ,for some constant K .

1. A CT LTI system is causal if its impulse-response satisfies the condition h(t) = 0 , 8 t < 0 .

1. A DT LTI system is causal iffits impulse-response satisfies the condition h[n] = 0 , 8 n < 0 . 37

1. A CT LTI system is BIBO stable if its impulse-response satisfies the condition Integral between -inf & + inf of |h(t)| dt < inf

1. A DT LTI system is BIBO stable if its impulse-response satisfies the condition sum between n=−inf & +inf |h[n]| < inf.