7.5 Fungsi Singular (3 dari 3)

Summary

This tutorial explains how to analyze a pulse function represented using unit step functions in mathematical terms. The speaker walks through the process of decomposing the function into its component step functions, integrating them, and then sketching the resulting graph. The function is broken down into three parts: a constant value, a downward shift, and an upward shift, which are represented by unit step functions. The video highlights the importance of understanding step functions and integration to solve such problems, concluding with a sketch of the integrated ramp functions.

Takeaways

• 😀 The task involves determining the integration of a pulse signal represented by unit step functions.
• 😀 The pulse signal is made up of three components, each corresponding to a unit step function.
• 😀 The first unit step function starts from 0 and jumps to 5, represented as 5u(t).
• 😀 The second unit step function is used to shift the signal downward, from 5 to -5, represented as -10u(t-2).
• 😀 The third unit step function shifts the signal back to 0, adding +5 at t=4, represented as 5u(t-4).
• 😀 These components are combined to form the complete pulse signal function.
• 😀 The integration of each unit step function results in ramp functions, as demonstrated by the process.
• 😀 After integrating the unit step functions, the final result is represented as a sum of ramp functions.
• 😀 The graphical representation of the integrated signal requires sketching the ramp functions based on their slope values.
• 😀 The final sketch includes the calculation of the signal at various time points, showing the interaction between the ramp functions.

Q & A

What is the main concept discussed in the video?

The main concept discussed is how to analyze and integrate a piecewise function consisting of unit step functions, and how to visualize its result through a graphical sketch.

What is a unit step function (U(t))?

A unit step function, denoted as U(t), is a mathematical function that is zero for t < 0 and one for t ≥ 0. It is commonly used to represent signals that switch on or off at specific points in time.

What are the three components that make up the signal in the example?

The signal in the example is made up of three components: U1(t) with a magnitude of 5, U2(t) with a magnitude of -10 (shifted to t = 2), and U3(t) with a magnitude of 5 (shifted to t = 4).

How are the unit step functions represented in the script?

The unit step functions are represented as expressions: U1(t) = 5U(t), U2(t) = -10U(t-2), and U3(t) = 5U(t-4). These represent different steps in the signal at different times.

Why is the magnitude of U2(t) -10 instead of a positive value?

The magnitude of U2(t) is -10 to shift the signal downward from +5 to -5 starting at t = 2. The negative value ensures that the signal decreases as required.

What is the significance of integrating the step function?

Integrating the step function allows us to calculate the cumulative area under the curve, which represents the total accumulated value of the signal. The result of the integration is a piecewise linear function.

How is the integration of a unit step function performed?

When a unit step function is integrated, it results in a ramp function. For example, the integration of U(t) gives tU(t), and the integration of U(t-a) gives (t-a)U(t-a). This reflects a shift in the ramp function by the specified time a.

What are the mathematical steps for integrating the given function?

The integration of the function is done by integrating each component. The integral of 5U(t) is 5tU(t), the integral of -10U(t-2) is -10(t-2)U(t-2), and the integral of 5U(t-4) is 5(t-4)U(t-4).

What does the graphical sketch of the integrated function look like?

The graphical sketch of the integrated function is a piecewise linear graph that changes slope at t = 2 and t = 4. The slope is initially 5, then changes to -10, and finally back to 5 after t = 4.

How do we interpret the graph of the integrated function in terms of slope?

The graph of the integrated function shows different slopes at different intervals. Initially, it has a slope of 5, then changes to -10 at t = 2, and back to 5 at t = 4. This reflects the changes in the unit step functions.