Question

We are given a time-invariant and linear system of which we know that the input \(u(t)\) yields an output \(y(t)\), where $$ u(t)=\left\\{\begin{array}{ll} 1 & 0 \leq t<2 \\ 0 & \text { otherwise } \end{array}, y(t)= \begin{cases}t & 0 \leq t<2 \\ 4-t & 2 \leq t<4 \\ 0 & \text { otherwise }\end{cases}\right. $$ Determine the output function \(\tilde{y}(t)\) which corresponds to the input \(\tilde{u}(t)\), where $$ \tilde{u}(t)= \begin{cases}1 & 0 \leq t<1 \\ 0 & \text { otherwise }\end{cases} $$ 147

Short answer

\( \tilde{y}(t) = t \) for \( 0 \leq t < 1 \); \( \tilde{y}(t) = 2 - t \) for \( 1 \leq t < 2 \).

Step-by-step solution

Analyze the Input-Output Relation

The given input-output relation indicates a specific response of the system to a pulse input from 0 to 2. During this period, the system produces the output \( y(t) = t \) for \( 0 \leq t < 2 \) and \( y(t) = 4-t \) for \( 2 \leq t < 4 \). This confirms the system's response to an input pulse of unit amplitude between 0 and 2.

Use Linearity of the System

Since the system is linear, we can use scaling properties. If the input pulse from 0 to 2 produces the given output, then an input pulse from 0 to 1 should yield half the magnitude during each interval of output, i.e., from 0 to 2 and 2 to 4. This scaling is possible because each section of the output is linearly dependent on the input duration.

Determine Output for Modified Input \\(\tilde{u}(t)\\)

Given the input \( \tilde{u}(t) = 1 \) for \( 0 \leq t < 1 \), we know \( u(t) \) linearly stretched \( y(t) \) over its defined range. Therefore, we can calculate: first interpolation for \( 0 \leq t < 1 \): \( \tilde{y}(t) = t \); and for \( 1 \leq t < 2 \), \( \tilde{y}(t) = 2 - t \). This adjustment of \( y(t) \) is due to its contraction in time to fit the input interval size.

Write Down the Complete Output Function \\(\tilde{y}(t)\\)

Combining the linear time contraction effects in each segment with zero values outside interval results in:- \( \tilde{y}(t) = t \) for \( 0 \leq t < 1 \)- \( \tilde{y}(t) = 2 - t \) for \( 1 \leq t < 2 \)- \( \tilde{y}(t) = 0 \) otherwise, i.e., for \( t \geq 2 \). Thus, our adjusted output sequence reflects the half-duration of the input sequence efficiently.

Key concepts

Time-Invariant Systems

Time-invariant systems are special types of systems where the behavior and characteristics do not change over time. This means that if you apply the same input at different times, the output will always be the same, apart from any time shift. For example, in our problem, the given system is time-invariant. Thus, an input of 1 between 0 and 2 produces an output of given form; regardless of when this input happens, the shape of the output will remain consistent.
Understanding time-invariance is crucial because it makes analyzing systems much easier. You can predict future behavior based solely on past performance, without worrying about any timing changes affecting the results.

System Response

The system response refers to how a system reacts to a given input, especially in terms of the output it generates. In linear systems, the response can often be predicted using known inputs, thanks to the system's linearity. For instance, in this exercise, when the input is a pulse between 0 and 2, the system's response is defined as:

• For 0 ≤ t < 2: The output, y(t), is equal to t, indicating a linearly increasing response.
• For 2 ≤ t < 4: The output decreases as 4-t.

This response gives us insight into how the system transforms a given input over time into an output signal, reflecting its dynamic behavior.

Input-Output Relation

The input-output relation in a system defines how each specific input results in a particular output. This relationship is crucial for understanding and predicting system behavior. For our linear system, the input-output relation is clearly defined, allowing us to comprehend how input pulses are transformed into output sequences.
In our exercise, we started with an input pulse of 1 from 0 to 2 and received a very specific output from the system. The relationship allowed us to predict the output for a different input, specifically from 0 to 1; therefore, the input-output relation is vital for recognizing patterns and making calibrated adjustments.

Scaling Properties

Scaling properties are essential features of linear systems that allow us to manipulate the input or output without affecting the overall system behavior. In our problem, these properties were used to adjust the output when the input pulse duration changed from 0 to 2 originally, and then to 0 to 1.

• By understanding the scaling, the system's response to the modified input pulse \(\tilde{u}(t)\) is determined.
• The principle that the output scales proportionally to the input means: when input duration is halved, output scales accordingly, maintaining proportional changes in amplitude and intervals.

The linearity and scaling properties together make it manageable to predict system outputs even when inputs vary, enhancing understanding and controlling dynamic systems.