# <Signals and Systems> Chapter 1 Study Notes

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<Signals and Systems> Chapter 1

Make up your mind to complete this signal and system!

This person's name I typed n times on the keyboard... Alan v. alppenhiem

Two major signal types:

Discrete and continuous Signals

The left and right sides are typical continuous and discrete periods.

Note the parity of the Signal Functions

Even: X (-T) = x (t)

Odd: X (-T) =-x (t)

The discrete form of the unit step function above. below is the discrete form of the Unit Impulse Function (Unit Impulse Function)

This is the most important basic signal type. Other types change around these basic signal types!

A very important fact is that any signal can be decomposed into a singular function signal and an even function signal linear superposition and.

I may wonder why?

Think about a signal x (t). What does it mean if it is expressed as X (-T?

X (-T) and X (t) are "image symmetric" on the timeline T, for example, X and Y. That is to say, any signal x (t) can obtain its "image signal" through X (-T"

This will lead to a property,

(X (t) + x (-t)/2. This produces an even function.

(X (t)-X (-t)/2. This produces a strange function.

After adding these two functions, we can obtain the linear superposition and

`x = [0 0 0 1 1 1 1];y = [1 1 1 1 0 0 0];figure(1);subplot(1,2,1);scatter(-3:3,x,"filled" );title('x = [0 0 0 1 1 1 1];');subplot(1,2,2);scatter(-3:3,y,'r',"filled");title('y = [1 1 1 1 0 0 0];');figure(2);subplot(1,2,1);scatter(-3:3,(x+y)/2,"filled");axis([-3,3,-1,1]);title('Even signal function');subplot(1,2,2);scatter(-3:3,(x-y)/2,'g',"filled");axis([-3,3,-1,1]);title('Odd signal function');`

Basic System Properties

System with and without memory.

Be sure to seize this definition:

A system is said to be memoryless if its output for each value of the independent variable at a given time is dependentonly on the input at that same time.

Y [x] = x [N] * X [N] is the memoryless system.

Once the output y [N] involves X [N] prior to X [n-1], X [N-2]... any signal on memory system!

The typical discrete-time system with memory is an accumulator)

Y [N] =

Sum = 0;

For-Nan to n

Sum = sum + X [k];

End

(You can only use pseudocode to accumulate ...)

In addition, the typical memory system operation is delay.

Y [N] = x [n-1];

Causality (causal)

Since the output of Y [N] may depend on the input signal of X [n-1] and so on, the formation of memory system... what if y [N] depends on X [n + 1] and so on ??

Here we will talk about another system nature-Causality

A system is causal if the output at any time dependsonly on values of the input at the present time and in the past.

Y [N] = Y [n-1] + X [N]; meets the condition of causality.

Y [N] = x [N]-X [n + 1] does not meet the causality condition!

All memoryless systems are causality, because the current output only depends on the current input.

Invertibility and Inverse Systems

A system is said to be invertible if distinct inputs lead to distinct outputs.

For example, the inverse system corresponding to Y (t) = 2 * x (t) is W (t) = (1/2) * Y (t );

Stability

A stable system is one in which small inputs lead to responses that do not diverge.

Simply put, a stable system is to be able to converge at the end, and a non-converged system is unstable.

Time Invariance

A system is time invariant if the behavior and characteristics of the sytem are fixed over time.

To put it bluntly, the results of today's tests on the system are consistent with those of yesterday's tests.

The following is a typical proof process of a time-unchanged system:

Linearity

The system is linear if

1. The response to x1 (t) + X2 (t) is Y1 (t) + y2 (t)

2. The response to ax1 (t) is Ay1 (t), where A is any complex constant.

The first is addition, and the second is homogeneous.

For example:

Y (t) = x (t) * x (t); this system is not a linear system.

The input 5x (t) is not 5y (t), but 25 x Y (T). It does not satisfy the uniformity.

Octave drawing prompt:

Stem function scatter plot

X =; y = 2 * X; stem (X, Y, 'filened ')

Scatter can also be used.

Room with balcony

<Signals and Systems> Chapter 1 Study Notes

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