Y Y N N Y Y Y Y Time Invariant vs Time Variant Systems CO2035 Discrete Time

Y y n n y y y y time invariant vs time variant

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Y Y N N Y Y Y Y Time-Invariant vs. Time-Variant Systems
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CO2035 – Discrete-Time Signals and Systems 35 § Linear System Obey superposition principle Definition A system is linear if and only if: T[a 1 x 1 (n) + a 2 x 2 (n)] = a 1 T[x 1 (n)] + a 2 T[x 2 (n)] " a i , " x i (n) Homogeneity Let a 2 = 0 ® T[a 1 x 1 (n)] = a 1 T[x 1 (n)] Additivity Let a 1 = a 2 = 1 ® T[x 1 (n) + x 2 (n)] = T[x 1 (n)] + T[x 2 (n)] § Non-Linear System The system does not obey superposition principle Linear vs. Non-Linear Systems
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CO2035 – Discrete-Time Signals and Systems 36 § Note: Linearity = Homogeneity + Additivity If a system is not homogeneous , it is not linear . If a system is not additive , it is not linear . Linear vs. Non-Linear Systems
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CO2035 – Discrete-Time Signals and Systems 37 § Example 1 The system is described by the input-output equation y(n) = T[x(n)] = nx(n) For two input sequences x 1 (n) and x 2 (n), the corresponding output are: y 1 (n) = nx 1 (n) (I) y 2 (n) = nx 2 (n) (II) A linear combination of the two input sequences in the output y 3 (n) =T[a 1 x 1 (n)+a 2 x 2 (n)] = n[a 1 x 1 (n)+a 2 x 2 (n)]=na 1 x 1 (n) + na 2 x 2 (n) On the other hand, a linear combination of the two outputs (I)&(II) results in the output. a 1 y 1 (n)+a 2 y 2 (n)=a 1 nx 1 (n) + a 2 nx 2 (n) Obviously, the system obeys superposition principle. Therefore, the system is Linear . Linear vs. Non-Linear Systems
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CO2035 – Discrete-Time Signals and Systems 38 § Example 2 The system is described by the input-output equation y(n) = T[x(n)] = x 2 (n) For two input sequences x 1 (n) and x 2 (n), the corresponding output are: y 1 (n) = x 1 2 (n) (I) y 2 (n) = x 2 2 (n) (II) A linear combination of the two input sequences in the output y 3 (n) =T[a 1 x 1 (n)+a 2 x 2 (n)] = [a 1 x 1 (n)+a 2 x 2 (n)] 2 (III) On the other hand, a linear combination of the two outputs (I)&(II) results in the output. a 1 y 1 (n)+a 2 y 2 (n)=a 1 x 1 2 (n) + a 2 x 2 2 (n) (IV) From (III) & (IV) , the system does not obey superposition principle. Therefore, the system is Non-Linear . Linear vs. Non-Linear Systems
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CO2035 – Discrete-Time Signals and Systems 39 § Quiz: Linear or not? Y N Y Y Y N N Y Linear vs. Non-Linear Systems
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CO2035 – Discrete-Time Signals and Systems 40 § Causal System Definition A system T is said to be causal if the output of the system at any time n [i.e. y(n)] depends only on present and past inputs [i.e. x(n), x(n – 1), x(n – 2) …]. In mathematical term, the output of a causal system satisfies an equation of the form y(n) = F[x(n), x(n–1), x(n–2), …] § Noncausal System The system is said to be Noncausal if the output of the system does not abbey the above definition. Causal vs. Noncausal Systems
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CO2035 – Discrete-Time Signals and Systems 41 § Quiz: Causal or not? Y Y Y N N N Y Y Causal vs. Noncausal Systems
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CO2035 – Discrete-Time Signals and Systems 42 § Stable System BIBO: Bounded Input-Bounded Output Definition A relaxed system is said to be BIBO Stable if and only if very bounded input produces a bounded output.
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