Systems

Systems has many applications. The system is as general tools for analyze, design the system for different applications. Such as circuits or car system.

There are different systems for different use. For example adding up function is one operation for a system. Increase the size of function could be an operation of a system as well. Lastly integration and differentiation is also a function.

Feed back system is like looping, after you finish one operation and you go back to start of the operation using the value that you obtain from the previous result.

Linearity
Youtube: Resource 1 and 2 There are 3 requirements for linearity. y(t)=x(t)+5
 * 1) Additivity: An input of  results in an output of .
 * 2) Homogeneity: An input of  results in an output of
 * 3) If x(t) = 0, y(t) = 0.

This function is not linear, because when x(t) = 0, y(t) = 5 (fails requirement 3). This may surprise people, because this equation is the equation for a straight line!

Linear need to satisfy the principle of superposition". Superposition is a fancy term for the system is additive and homogeneous. The terms linearity and superposition can be used interchangably, but in this book we will prefer to use the term linearity exclusively.

We can combine the three requirements into a single equation: In a linear system, an input of  results in an output of .

Additivity

A system is said to be additive if a sum of inputs results in a sum of outputs. To test for additivity, we need to create two arbitrary inputs, x1(t) and x2(t). We then use these inputs to produce two respective outputs:

y1(t)=f(x1(t))

y2(t)=f(x2(t))

AND ! We prove it by y1(t)+y2(t)= f(x1(t)+f(x2(t)

Homogeneity

a system is homogeneous if a scaled input (multiplied by a constant) results in a scaled output. If we have two inputs to a system:

y1(t)=f(x1(t))

y2(t)=f(x2(t))

x1(t)=cx2(t) so y1(t)= cy2(t)

Time invariance
If the input signal x(t) produces an output y(t) then any time shifted input, x(t + δ), results in a time-shifted output y(t + δ).

This property can be satisfied if the transfer function of the system is not a function of time except expressed by the input and output.

It is also when the time shift in the input only produces the same time shift in the output.

X(t) with the system H will become y(t). A system H is time invariant.

System describe by differential equation
many system are describe as linear constant. The coefficient ordinary differential equation. (LCCODE)

The LCCODE systems are linear and time invariant where initial con conditions are all 0.

This is a linear and time invariant system, such as integrator or differentiation.

System memory
Have memory if the output from the system is dependent on past inputs (or future inputs) to the system. A system is called memoryless if the output is only dependent on the current input. Memoryless systems are easier to work with, but systems with memory are more common in digital signal processing applications. A memory system is also called a dynamic system whereas a memoryless system is called a static system.

Non-causal system
Causality is a property that is very similar to memory. A system is called causal if it is only dependent on past or current inputs. A system is called non-causal if the output of the system is dependent on future inputs. Most of the practical systems are causal.

Linear time invariant system (LTI)
The system is linear time-invariant (LTI) if it satisfies both the property of linearity and time-invariance. This book will study LTI systems almost exclusively, because they are the easiest systems to work with, and they are ideal to analyze and design.

Practice Problems
Problem Solution

Extra Resources
MIT course