Classification of Discrete-Time Systems. System Properties. Terminology: Implication. Terminology: Equivalence

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1 Classification of Discrete-Time Systems Professor Deepa Kundur University of Toronto Why is this so important? mathematical techniques developed to analyze systems are often contingent upon the general characteristics of the systems being considered for a system to possess a given property, the property must hold for every possible input to the system to disprove a property, need a single counterexample to prove a property, need to prove for the general case / 24 2 / 24 Terminology: mplication f A then B Shorthand: A = B Example : it is snowing = it is at or below freezing temperature Example 2: α 5.2 = α is positive Note: For both examples above, B A Terminology: Equivalence f A then B Shorthand: A = B and f B then A Shorthand: B = A can be rewritten as A if and only if B Shorthand: A B We can also say: A is EQUVALENT to B A = B = 3 / 24 4 / 24

2 Terminology: Systems Stability A cts-time system processes a cts-time input signal to produce a cts-time output signal. Bounded nput-bounded output (BBO) stable system: every bounded input produces a bounded output a cts-time system is BBO stable iff y (t) = H{x(t)} A dst-time system processes a dst-time input signal to produce a dst-time output signal. x(t) Mx < = y (t) My < y [n] = H{x[n]} for all t. a dst-time system is BBO stable iff x[n] Mx < = y [n] My < for all n. Note: iff = if and only if 5 / 24 6 / 24 Bounded Signals Stability Examples: Are each of the following systems BBO stable?. y (t) = A x(t), note: A < 2. y (t) = A x(t) + B,, note: A, B <, B 6= 0 3. y [n] = n x[n] 4. y (t) = x(t) cos(ωc t) 5. y [n] = 3 (x[n] + x[n ] + x[n 2]) n 6. y [n] = r x[n], note: r > 7. y [n] = x[n+2] 8. y (t) = e 3x(t) BOUNDED SGNAL UNBOUNDED SGNAL 7 / 24 Ans: Y, Y, N, Y, Y, N, N, Y 8 / 24

3 Memory Memory Memoryless system: output signal depends only on the present value of the input signal cts-time: y(t) only depends on x(t) for all t dst-time: y[n] only depends on x[n] for all n Note: a system that is not memoryless has memory System with Memory: output signal depends on past or future values of the input signal Examples: Do each of the following systems have memory?. y(t) = A x(t) 2. y(t) = t C x(τ)dτ 4. y(t) = x(t) cos(ω c (t )) 6. y(t) = a 0 + a x(t) + a 2 x 2 (t) + a 3 x 3 (t) 7. y[n] = 3 (x[n] + x[n ] + x[n 2]) Ans: N, Y, N, N, Y, N, Y, N 9 / 24 0 / 24 Causality Causality Causal system: present value of the output signal depends only on the present or past values of the input signal a cts-time system is causal iff for all t a dst-time system is causal iff for all n y(t) = F [x(τ) τ t] y[n] = F [x[n], x[n ], x[n 2],...] Examples: Are each of the following systems causal?. y(t) = A x(t) 2. y(t) = A x(t) + B, B 0 3. y[n] = (n + ) x[n] 4. y(t) = x(t) cos(ω c (t + )) 6. y[n] = 3 (x[n + ] + x[n] + x[n ]) 7. y[n] = x[n+2] Ans: Y, Y, Y, Y, N, N, N, Y / 24 2 / 24

4 nvertibility nvertible system: input of the system can always be recovered from the output a system is invertible iff there exists an inverse system as follows DENTTY SYSTEM nvertibility A system that is invertible has a one-to-one mapping between input and output. That is, a given output can be mapped to a single possible input that generated it. DENTTY SYSTEM Consider A system that is not invertible can be shown to have two or more input signals that produce the same output signal. x(t) = H inv {y(t)} = H inv {H{x(t)}} x(t) = H inv {H{x(t)}} DENTTY SYSTEM 3 / 24 4 / 24 nvertibility nvertibility Examples: Are each of the following systems invertible?. y(t) = A x(t), note: A 0 2. y(t) = A x(t) + B, note: A, B 0 4. y(t) = t L x(τ)dτ 6. y(t) = x 2 (t ) 7. y[n] = n k= x[k] Ans: Y, Y, N, Y, Y, N, Y, Y Examples: The associated inverse systems are:. y(t) = x(t) A, note: A 0 2. y(t) = x(t) B A, note: A, B 0 3. N/A; x [n] = δ[n] and x 2 [n] = 2δ[n] give the same output y[n] = 0 4. y(t) = dx(t) dt 6. N/A; x (t) = and x 2 (t) = give the same output y(t) = 7. y[n] = x[n] x[n ] 8. y(t) = ln(x(t)) 3 5 / 24 6 / 24

5 Time-invariance Time-invariance The characteristics of H do not change with time. Time-invariant system: a time delay or time advance of the input signal leads to an identical time shift in the output signal; a cts-time system H is time-invariant iff y(t) = H{x(t)} = y(t t 0 ) = H{x(t t 0 )} for every input x(t) and every time shift t 0. a dst-time system H is time-invariant iff y[n] = H{x[n]} = y[n n 0 ] = H{x[n n 0 ]} for every input x[n] and every time shift n 0. 7 / 24 8 / 24 Time-invariance Examples: Are each of the following systems time-invariant?. y(t) = A x(t) 2. y(t) = A x(t) + B 4. y(t) = x(t) cos(ω c t) 6. y(t) = t L x(τ)dτ Linear system: obeys superposition principle = Homogeniety + Additivity Homogenous system: Additive system: 7. y[n] = x[n+2] Ans: Y, Y, N, N, N, Y, Y, Y 9 / / 24

6 a cts-time system H is linear iff a dst-time system H is linear iff y (t) = H{x (t)} y 2 (t) = H{x 2 (t)} = a y (t) + a 2 y 2 (t) = H{a x (t) + a 2 x 2 (t)} y [n] = H{x [n]} y 2 [n] = H{x 2 [n]} = a y [n] + a 2 y 2 [n] = H{a x [n] + a 2 x 2 [n]} 2 / / 24 Final Words Examples: Are each of the following systems linear?. y(t) = A x(t) 2. y(t) = A x(t) + B, B 0 4. y(t) = x(t) cos(ω c t) 6. y(t) = x 2 (t ) 7. y[n] = x[n+2] Ans: Y, N, Y, Y, Y, N, N, N To prove a property, you must show that it holds in general. For instance, for all possible inputs and/or time instants. To disprove a property, provide a simple counterexample to the definition. 23 / / 24

ECE 301 Division 1 Exam 1 Solutions, 10/6/2011, 8-9:45pm in ME 1061.

ECE 301 Division 1 Exam 1 Solutions, 10/6/011, 8-9:45pm in ME 1061. Your ID will be checked during the exam. Please bring a No. pencil to fill out the answer sheet. This is a closed-book exam. No calculators