1.2: Discrete-Time Systems
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- 126928
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Learning Objectives: Discrete-Time System
- Translate verbal descriptions of systems into equations.
- Demonstrate whether systems satisfy the properties of linearity, time-invariance, causality, and stability.
- Argue whether a system can be linear and time-invariant (LTI) by examining a collection of inputs and outputs.
YouTube Videos
Systems Fundamentals
Connection of DT Systems
Systems Properties: Linearity
DT System Properties: Time Invariance
System Properties: Stability (Revisit)
Revisiting Linearity Proofs
Revisiting Time-Invariance Proofs
Proving Linearity
Proving Time Invariance
System Properties: Causality
Summary
Discrete-Time Systems
Definition: A discrete-time (DT) system transforms an input signal \(x[n]\) into an output signal \(y[n]\).
Figure 1.2.1: Basic discrete system flow diagram.
The system S is an equation relating the input signal \(x[n]\) to the output signal \(y[n]\).
\[\displaystyle y[n] = \frac{1}{7}\big(x[n] + x[n-1] + x[n-2] + x[n-3] + x[n-4] + x[n-5] + x[n-6]\big)\]
The output signal is given by the average of the most recent input signal values (a running average). For example, if \(x[n]\) represents the number of new cases of a disease on a given day, then \(y[n]\) is the running average of new cases. This system is non-recursive because \(y[n]\) does not depend on past or future values of \(y[n]\).
\[\displaystyle y[n] = 1.03\,y[n-1] + x[n]\]
This system can be interpreted as a bank balance with positive 3% interest and daily transactions. This system is recursive because the current value of \(y[n]\) depends on a past value of the output \(y[n-1]\).
Connections of Discrete-Time Systems
There are three ways to connect systems:
Series (Cascade) - The output from one system is fed into another system.
\[y[n] = S_2\{S_1\{x[n]\}\}\]
Figure 1.2.2: Basic discrete series (cascade) system flow diagram.
Parallel - The same input signal is applied to both systems and then combined.
\[y[n] = y_1[n] + y_2[n] = S_1\{x[n]\} + S_2\{x[n]\}\]
Figure 1.2.3: Basic discrete parallel system flow diagram.
Feedback - One system output is fed back and combined with the input to repeat the process.
Figure 1.2.4: Basic discrete feedback system flow diagram.
System Properties
Linearity
A linear system must satisfy both the scaling and superposition properties.
Scaling:
\[T\{a\,x[n]\} = a\,T\{x[n]\}\]
Figure 1.2.5: Discrete flow diagram showing a visual proof for the scaling property.
Superposition:
\[T\{x_1[n] + x_2[n]\} = T\{x_1[n]\} + T\{x_2[n]\}\]
Figure 1.2.6: Discrete flow diagram showing a visual proof for the superposition property.
Time-Invariance
If \(y[n] = S\{x[n]\}\), then the system is time-invariant if: \(S\{x[n-n_0]\} = y[n-n_0]\)
Figure 1.2.5: Discrete flow diagram showing a delay in series with a system.
Casuality
A system is causal if all output values depend only on past and present values. I,e, the index for the output must always be greater than the index of the system.
Stability
For a system to be stable, given any input that is bounded, the output must always be bounded as well.
Figure 1.2.6: Graphical stability representation.
Inverteability
A system is invertible if distinct inputs lead to distinct outputs. A system is invertible if an inverse system exists that recovers the input from the output.
Figure 1.2.7: Flow diagram showing a system and it's inverse system in parallel producing the initial input.