Propriedades de Sistemas | Sinais e Sistemas

Introduction to Systems and Signals

Overview of the Video

• Ester introduces the topic of the video, focusing on properties of systems in signals and systems.

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Review of Previous Lesson

• A system is defined as a process where an input signal x[n] undergoes transformations to produce an output signal y[n] .

• The first mathematical representation shows that y[n] = 2x[n] , indicating that the output is double the input.

Mathematical Representations of Systems

Examples of System Transformations

• The second example illustrates y[n] = x[n - 1]^2 , where the input is delayed by one sample and then squared.

• The third example presents y[n] = x[n] + y[n - 1] , demonstrating how current output depends on both current input and previous output.

Causal vs Non-Causal Systems

Definition of Causal Systems

• A causal system's output depends only on present or past inputs, never future ones.

• Example: For y[n] = 2x[n - 1] , it relies solely on past inputs, confirming it's a causal system.

Non-Causal System Characteristics

• In contrast, for y[n] = x[n + 1], it depends on future inputs, making it non-causal.

Further Analysis of Causality

Identifying Causality in Examples

• Another example shows that if outputs depend on future values (e.g., using x), it indicates a non-causal system.

Exploring More Causal Systems

Additional Examples

• Analyzing another case with y[n] = x[-n], this remains causal since outputs rely only on past values.

• When examining negative indices, such as when n equals -1 or -2, outputs still reference prior values from earlier time steps.

Understanding System Stability and Linearity

System Behavior with Negative Inputs

• The discussion begins with the behavior of systems when dealing with negative inputs, emphasizing that while positive inputs only consider past values, negative inputs also incorporate future values. This distinction is crucial for understanding system causality.

Conditions for Stability in Systems

• A system is deemed stable if finite inputs lead to finite outputs. This means the output will not diverge to infinity, ensuring controlled and predictable behavior. An example provided illustrates how a specific input results in a bounded output.

Example of Controlled Output

• When analyzing a system where the output y[n] equals twice the input minus one, it is shown that even with infinite input values, the output remains finite and controlled (e.g., y[-3] = 0 , y= 0 ). This demonstrates stability as outputs do not explode to infinity.

Accumulation Leading to Instability

• In contrast, another example shows that when an accumulation occurs (e.g., adding previous outputs), the system can become unstable as it leads to increasing sums without bounds (e.g., y= 7 ). This indicates divergence towards infinity and thus instability.

Characteristics of Linear Systems

Understanding Linear Systems and Time Invariance

Introduction to Inputs and Outputs

• The discussion begins with the introduction of two inputs, x_1[n] and x_2[n] , leading to an output y[n] = x_1[n + 1] .

• The relationship between inputs and outputs is further explored, where the output for x_2 is defined as y_2[n] = x_2[n + 1] .

Linear Combinations in System Behavior

• A new input, x_3[n] , is introduced as a linear combination of previous inputs, emphasizing that for a system to be linear, both inputs and outputs must maintain this property.

• The output for the new input follows the same pattern: it combines previous outputs linearly, confirming the system's linearity.

Time Invariance Concept

• The concept of time invariance is introduced; a system is invariant if its behavior does not change over time. This means that delaying an input signal results in a correspondingly delayed output.

• An example illustrates this principle: if an input signal generates an output at one time instance, delaying that input should yield the same output delayed by the same amount.

Properties of Linear Systems

• Key properties such as linearity and time invariance are highlighted. These concepts will be revisited with practical exercises in future lessons.