Understanding the Graph of Sint.Ut-1.U9-T in Continuous Time Domain

Understanding the concept of the unit step function and its applications in the continuous time domain is crucial in fields such as signal processing, control systems, and communication engineering. One specific function that often requires analysis is Sint.Ut-1.U9-T, where ut and u9-t are unit step functions. This article aims to explain the graph of Sint.Ut-1.U9-T and provide a detailed analysis in the continuous time domain.

Definition and Properties of Unit Step Functions

The unit step function, ut, is a fundamental tool in the analysis of systems in the continuous time domain. It is defined as:

ut begin{cases} 1 text{for } t geq 0 0 text{for } t 0 end{cases}

When shifted by 1 unit to the right, the function becomes ut-1, defined as:

ut-1 begin{cases} 1 text{for } t geq 1 0 text{for } t 1 end{cases}

Similarly, a shifted unit step function with a delay of 9 units to the left becomes u9-t, which is defined as:

u9-t begin{cases} 1 text{for } 9 leq t 0 text{for } t 9 end{cases}

Graph of Sint.Ut-1.U9-T

The function Sint.Ut-1.U9-T is the product of the unit step functions and a sine function, and it is defined as:

Sint.Ut-1.U9-T Sint times Ut-1 times U9-t

To understand the graph of this function, we need to consider the intervals in which the step functions are non-zero:

Ut-1 1 for t geq 1 U9-t 1 for t leq 9

The product of these functions, Sint.Ut-1.U9-T, will be non-zero in the intersection of these intervals, i.e., 1 leq t leq 9.

Sint.Ut-1.U9-T Sint text{ for } 1 leq t leq 9

For t 1 and t 9, the function Sint.Ut-1.U9-T is zero.

Graphical Analysis

To visualize the graph of Sint.Ut-1.U9-T, consider the sine function between 1 and 9. The graph will be:

A sine curve from t 1 to t 9 (the interval where both step functions are active). The sine curve will not exist for t 1 and t 9 because the step functions are zero in these intervals.

Thus, the graph of Sint.Ut-1.U9-T can be summarized as:

Zero for t 1 A sine curve for 1 leq t leq 9 Zero for t 9

Conclusion

Understanding the graph of Sint.Ut-1.U9-T is critical for various applications in the field of signal processing and control systems. By breaking down the function into its constituent parts and analyzing the intervals where the step functions are defined, we can construct a clear graph of the function. This analysis provides a deeper insight into the behavior of the sine function in the presence of time-domain manipulations.

Keywords

Unit step function, continuous time domain, graph analysis