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1. Classification of Discrete-
Time Systems

2. Static VS Dynamic Systems
Static System
 Also called memoryless system
 If the response is due to the present alone
 The output at any instant ‘n’ depends only the
input at the same instant ‘n’ but not the past or
future values.
 Examples:
y(n)=ax(n)
𝑦 𝑛 = 𝑛𝑥𝑛 + 𝑏𝑥2
(𝑛)
 A purely resistive electrical circuit is a static system
Dynamic System
 Also called memory system
 Depends upon past or future inputs or past
outputs.
 A summer or accumulator, a delay element is
discrete-time system with memory
 Examples:
y(n)=x(2n)
y(n)+4y(n-1)+4y(n-2)=x(n)
𝑦 𝑛 = 𝑘=0
𝑛
𝑥(𝑛 − 𝑘)
 Electrical circuits with inductors and/or capacitors
is a dynamic system
 Any discrete-time system described by a
differential equation is a dynamic system

3. Time Invariant VS Time Varying System
 A system is called time invariant if its input-output characteristics do not change with time
 A relaxed system 𝒯 is time invariant if and only if
𝑥 𝑛
𝒯
𝑦 𝑛
 Implies that
𝑥(𝑛 − 𝑘)
𝒯
𝑦 𝑛 − 𝑘
 For every input signal x(n) and every time shift k
 In general, we can write the output as
𝑦 𝑛, 𝑘 = 𝒯 𝑥 𝑛 − 𝑘
 If y(n,k)=y(n-k), it is time invariant
 If 𝑦 𝑛, 𝑘 ≠ 𝑦 𝑛 − 𝑘 , it is time varying

4. Example:
 𝑦 𝑛 = 𝒯 𝑥 𝑛 = 𝑥 𝑛 − 𝑥(𝑛 − 1)
 𝑦 𝑛, 𝑘 = 𝒯 𝑥 𝑛 − 𝑘 = 𝑥 𝑛 − 𝑘 − 𝑥(𝑛 − 1 − 𝑘)
 𝑦 𝑛 − 𝑘 = 𝑥 𝑛 − 𝑘 − 𝑥 𝑛 − 𝑘 − 1 = 𝑥 𝑛 − 𝑘 − 𝑥(𝑛 − 1 − 𝑘)
 Time Invariant

5. Example:
 𝑦 𝑛 = 𝒯 𝑥 𝑛 = 𝑛𝑥(𝑛)
 𝑦 𝑛, 𝑘 = 𝒯 𝑥 𝑛 − 𝑘 = 𝑛𝑥 𝑛 − 𝑘
 𝑦 𝑛 − 𝑘 = 𝑛 − 𝑘 𝑥(𝑛 − 𝑘)
 Time varying

6. Example:
 𝑦 𝑛 = 𝒯 𝑥 𝑛 = 𝑥(−𝑛)
 𝑦 𝑛, 𝑘 = 𝒯 𝑥 𝑛 − 𝑘 = 𝑥(−𝑛 − 𝑘)
 𝑦 𝑛 − 𝑘 = 𝑛 − 𝑘 𝑥(𝑛 − 𝑘)
 Time variant

7. Additivity
 An additive system is one for which the response to a sum of inputs is equal to the
sum of the inputs individually
 The system is said to be additive if:
𝑇[𝑥1 𝑛 + 𝑥2 𝑛 ] = 𝑇[𝑥1 𝑛 ] + 𝑇[𝑥2 𝑛 ]

8. Homogeneity
 A system is said to be homogeneous if
𝑇 𝑐𝑥 𝑛 = 𝑐𝑇[𝑥 𝑛 ]
 c is for any complex constant

9. Example: 𝑦 𝑛 =
𝑥2(𝑛)
𝑥(𝑛−1)
 𝑇[𝑥1 𝑛 + 𝑥2 𝑛 ] =
[𝑥1 𝑛 +𝑥2 𝑛 ]2
𝑥1 𝑛−1 +𝑥2 𝑛−1
 𝑇 𝑥1 𝑛 + 𝑇 𝑥2 𝑛 =
𝑥1
2 (𝑛)
𝑥1(𝑛−1)
+
𝑥2
2 (𝑛)
𝑥2(𝑛−1)
=
𝑥1
2 𝑛 𝑥2 𝑛−1 +𝑥2
2 (𝑛)𝑥1(𝑛−1)
𝑥1(𝑛−1)𝑥2(𝑛−1)
 Not additive
 𝑇 𝑐𝑥 𝑛 =
[𝑐𝑥 𝑛 ]2
𝑐𝑥(𝑛−1)
=
𝑐2𝑥2(𝑛)
𝑐𝑥(𝑛−1)
= 𝑐
𝑥2(𝑛)
𝑥(𝑛−1)
 𝑐𝑇 𝑥 𝑛 = 𝑐
𝑥2(𝑛)
𝑥(𝑛−1)
 Homogenous

10. Linear VS Nonlinear Systems
 A linear system is one that satisfies both additivity and homogeneity or called the
superposition principle.
𝒯 𝑎1𝑥1 𝑛 + 𝑎2𝑥2 𝑛 = 𝑎1𝒯 𝑥1 𝑛 ] + 𝑎2𝒯[𝑥2 𝑛
• In general, a relaxed, linear system with zero input
produces a zero output. If a system produces a
nonzero output with a zero input, the system may
either be nonrelaxed or nonlinear. If a relaxed
system does not satisfy the superposition
principle, it is called nonlinear.

11. Example: 𝑦 𝑛 = 𝑛𝑥(𝑛)
 𝒯 𝑎1𝑥1 𝑛 + 𝑎2𝑥2 𝑛 = 𝑛[𝑎1𝑥1 𝑛 + 𝑎2𝑥2 𝑛 ]
 𝑎1𝒯 𝑥1 𝑛 ] + 𝑎2𝒯[𝑥2 𝑛 = 𝑎1𝑛 𝑥1 𝑛 ] + 𝑎2𝑛[𝑥2 𝑛
 linear

12. Example: 𝑦 𝑛 = 𝑥2
(𝑛)
 𝒯 𝑎1𝑥1 𝑛 + 𝑎2𝑥2 𝑛 = 𝑎1𝑥1 𝑛 + 𝑎2𝑥2 𝑛 2
 𝑎1𝒯 𝑥1 𝑛 ] + 𝑎2𝒯[𝑥2 𝑛 = 𝑎1 𝑥1 𝑛 2 + 𝑎2 𝑥2 𝑛 2
 Non-linear

13. Example: 𝑦 𝑛 = 𝑒𝑥(𝑛)
 The system is non-relaxed. If x(n)=0, the output is 1. This is an indication that the
system is nonlinear.

14. Causal VS Noncausal Systems
 A system is said to be causal if the output of the system at any time ‘n’ depends
only on the present (n) and past (n-k) inputs, but does not depend on future
inputs (n+k).
 If the system does not satisfy the statement above, it is noncausal.

15. Example: y(n)=x(-n)
 It is noncausal.
 By testing n=-1, the output y(n) would have to rely on a future value at instance
x(1), which is 2 units from the future

16. Stable VS Unstable Systems
 An arbitrary relaxed system is said to be bounded input bounded output (BIBO) if and only if every
bounded input produces a bounded output
 The condition that the input sequence x(n) and the output y(n) are bounded is translated mathematically to
mean that there exist some finite numbers, say 𝑀𝑥 and 𝑀𝑦, such that
𝑥 𝑛 ≤ 𝑀𝑥 < ∞, 𝑦 𝑛 ≤ 𝑀𝑦 < ∞
for all n. If for some bounded input sequence x(n), the output is unbounded (infinite), the system is
classified as unstable.
 For a linear shift-invariant (LSI) system, stability is guaranteed if the unit sample response is absolutely
summable:
𝑛=−∞
∞
|ℎ 𝑛 | < ∞

17. Example: 𝑦 𝑛 = 𝑦2
𝑛 − 1 + 𝑥(𝑛) with an input
sequence of 𝑥 𝑛 = 𝐶𝛿(𝑛) and y(-1)=0
 Output is 𝑦 0 = 𝐶, 𝑦 1 = 𝐶2
, 𝑦 2 = 𝐶4
, … , 𝑦 𝑛 = 𝐶2𝑛
 Unstable when 1 < 𝐶 < ∞
 Stable when 0 ≤ 𝐶 < 1

18. Invertible VS Non-invertible Systems
 Invertibility is a property that’s important in applications such as channel
equalization and deconvolution
 A system is said to be invertible if the input to the system may be uniquely
determined from the output.
 In order for the system to be invertible, it is necessary for distinct inputs to
produce distinct outputs.

19. Discrete-Time Classifications
 Static or dynamic
 Linear or nonlinear
 Time invariant or time varying
 Causal or noncausal
 Stable or unstable