# Basic signal analysis (system invertible)

In summary, the conversation discusses the invertibility of a system with the signal y(t) = x(t-4) and how it can be shown mathematically. It is suggested to make a substitution and use function composition to demonstrate invertibility. However, the inverted function is non-causal and requires knowledge of future values.
Ok I have this really simple question that is bugging me.

Lets say you have the signal:

y(t) = x(t-4)

where y(t) corresponds to the output, and x(t) the input.

I know this system is invertible, but I don't really know how to show that this is the case. I see that the output is x(t) with an independet variable transformation such that the input shifted by 4 units to the right. So if we shift the output four units to the left then we get the input without the independent variable transformation. I just don't know how to express what is going on here mathematically.

Maybe I don't understand invertibility well enough to apply it.
From what I gather it can be shown by,

x(t) --> [system] --> y(t) = T{x(t)}
y(t) --> [invert] --> T{y(t)} = x(t)

I'm getting confused since the problem has x(t-4) in this case. I'm guessing I can show it with some type of function composition, but I need some help.

thanks

You can make the substitution
t' = t - 4
so that
x(t') = y(t'+4)
The inverted function is non-causal, since you need to know future values of y to find the present value of x.

## 1. What is signal analysis?

Signal analysis is the process of studying and understanding the characteristics and properties of a signal. This includes identifying patterns, trends, and relationships within the signal.

## 2. Why is signal analysis important?

Signal analysis plays a crucial role in many scientific and engineering fields, including telecommunications, digital signal processing, and control systems. By analyzing signals, we can gain valuable insights and make informed decisions based on the data.

## 3. What is a system invertible in signal analysis?

A system is considered invertible if it is possible to reconstruct the input signal from the output signal. In other words, an invertible system preserves all the information of the original signal.

## 4. How do you determine if a system is invertible?

To determine if a system is invertible, we can perform a basic signal analysis by comparing the input and output signals. If the output signal contains all the same information as the input signal, then the system is considered invertible.

## 5. What are the benefits of using an invertible system in signal analysis?

An invertible system allows for accurate and reliable signal reconstruction, which can be useful in applications such as data compression, noise reduction, and signal filtering. It also allows for easier and more efficient signal processing and analysis.

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