07/22

## Algebra Structure of Signals and Systems in Time Domain (信号与系统的时域理论结构)

This post contains $\LaTeX$ equations.

Signals and Systems is a compulsory course for EE major. It gives the formal definition of  signals and systems, and introduces the conceptions of convolutions. This course also includes several applications of convolutions together with transformations, which are commonly implemented in signal processing, signal detection, and telecommunications. This post shows the mathematical principle, or algebra structure, of the signals world proposed by Signals and Systems, though it does not use this way to present.

### Vectors: Signals

I am going to use linear algebra to explain the principles of the signal world. Vector is one of the elements of a vector space defined in linear algebra. It is the set being operated. Number is called vector in a number space (maybe a number field or just a number group).

Signals and Systems models signals by functions. Such function set $\left \{ f(t) \right \}$ can be used to defined as the vector set, and then we get an abstract space consisting of functions. It is obvious that we can define addition, subtraction, and multiplication of the signals.

$(f+g)(t):=f(t)+g(t)$
$(f-g)(t):=f(t)-g(t)$
$(f\times g)(t):=f(t)\times g(t)$

### Mapping: Systems

To make it easy, we would like to discuss single-input-single-output (SISO) systems. There is no difference in theory to generalize the result of SISO systems to multiple-input-multiple-output (MIMO) systems.

Signals and Systems abstracts SISO systems into "something" yields an output signal when it gets an input signal. Based on the modeling of signals above, the system $\mathcal{S}$ then outputs a function with a function in, which is called mapping in math.

$x(t)\overset{\mathcal{S}}{\rightarrow} y(t)$

#### Operator: Convolution

Signals and Systems models linear time-invariant (LTI) systems into an expression containing convolution, "$*$".

$x(t)\overset{\mathcal{S}}{\rightarrow}y(t)=x(t)*h(t)$

where, $h(t)$ is the impulse response of the LTI system $\mathcal{S}$.

Because the systems represented by convolution are linear, the convolution is linear operator of signal set. There are also proofs about the communicable and associative principles of convolution. Convolution and signal set make a group $\left < *, \left \{ f(t)\right \} \right >$.

##### Convolution and Multiplication

Convolution is not communicable with multiplication. There is a counter-example of pulse amplitude sampling, which is widely used in telecommunications. There are two methods, natural PAM and flat PAM, which have a different result with exchange of the two operators in expression. I would like to explain this phenomenon formally in a future post.

This is the algebra structure of signal world in Time Domain. I would like to explain the Transformed Domain in the future.