Introduction to Signals and Systems: Properties of systems

In this article, we will go through Signal and systems, First, we will define what is a signal and what is a system, then we will go through the calculation of the Energy and Power of Signals with their transformation, At last we will conclude our article with Properties, Applications and some FAQs.

What are Signals ?

Signal is an electric or electromagnetic current carrying data, that can be transmitted or received. 

Mathematically represented as a function of an independent variable e.g. density, depth, etc. Therefore, a signal is a physical quantity that varies with time, space, or any other independent variable by which information can be conveyed. Here the independent variable is time.

 Example:  It includes time-depending voltages and currents in an electric circuit, the variation in a gross national product, music waveforms, and the variation of atmospheric temperature.

 Types of Time signals

• Continuous time signals x(t): If a signal is represented at all instants of time, it is said to be a continuous-time signal or simply a continuous signal.
• Discrete-time signals x[n] : A signal that is specified at discrete instants of time is said to be a discrete-time signal or simply a discrete signal. Discrete signals occur either due to the nature of the process, e.g. the variation in the number of cars crossing the border every day, or due to the sampling process. 

What is a System ?

A System is any physical set of components or a function of several devices that takes a signal in input, and produces a signal as output. These components interact with each other and with the environment to perform the function or process that will contribute to the system.

Calculating the Energy and Power of the Signals

The Energy of the Signals can be calculated as

Square of amplitude/magnitude(if complex) over entire time domain. 
for a continuous time signal
  E=\int_{-\infty}^{\infty} |x(t)|^2 dt
for a discrete time signal
  E=\sum_{-\infty}^{\infty} |x[n]|^2

The power of the Signal can be calculated as

Rate of change of energy. 
for a continuous time signal. 
P=\lim_{T\to\infty} 1/(2T) (\int_{-T}^{T} |x(t)|^2 dt)  
for a discrete time signal-
 P=\lim_{N\to\infty} 1/(2N+1) (\sum_{-N}^{N} |x[n]|^2)   

Classes of signals on the basis of their power and energy

• Energy signal: generally converging signals, aperiodic signals or signals that are bounded.
    E < \infty\ and\ P=0
• Power signal: generally periodic signals, as they encompass infinite area under their graph and extend from +\infty  to -\infty  . 
  E \rightarrow \infty\ and\ P=constant
• Neither energy nor power signal
    E \rightarrow \infty\ and\ P \rightarrow \infty

Transformation of the Independent Variable

• Shifting: the signal can be delayed ( x(t-T) ) or advanced ( x(t+T) ) by incrementing or decrementing the independent variable (time here). The shape of the graph remains same only shifted on the time axis.
• Scaling: the signal can be compressed ( x(at), a>1 ) or expanded ( x(t/a), a>1 or x(at), 1>a>0 ). 
  Here the shape/behaviour of the graph of the signal changes as the fundamental time period changes. In compression the time period decreases and in expansion the time period increases.
• Reversal: also called folding as the graph is folded about the Y-axis or T if given x(T-t).

Properties of Systems

Given Below are the Properties of Systems :

• Periodicity
• Even and Odd
• Linearity
• Time Invariant
• LTI Systems
• BIBO Stability
• Causality
• Reflection

Periodicity

The signal's behavior/graph repeats after every T. Therefore, 
x(t)=x(t+nT)\ or\ x(t)=x(t-nT)

Here, T is the fundamental period 
So we can say signal remains unchanged when shifted by multiples of T.

Even and Odd

An even signal is symmetric about the Y-axis. 
x(t)=x(-t) even 
x(t)=-x(-t) odd 
A signal can be broken into it's even and odd parts to make certain conversions easy.
 Even(x(t)) = (x(t)+x(-t))/2 Odd(x(t)) = (x(t)-x(-t))/2 

Linearity

It constitutes of two properties- 

(i) Additivity/Superposition
if x1(t) -> y1(t) 
and x2(t) -> y2(t)
 x1(t) + x2(t)\ \rightarrow y1(t) + y2(t)

(ii) Property of scaling 
if x1(t) -> y1(t) 
then 
a*x1(t)\ \rightarrow a*y1(t)
If both are satisfied, the system is linear.

Time Invariant

Any delay provided in the input must be reflected in the output for a time invariant system.
  take \ x2(t)=x(t-T) then\ y(x2(t))\ must\ be\ =x2(y(t)) 
Here, x2(t) is a delayed input. 
We check if putting a delayed input through the system is the same as a delay in the output signal.

LTI Systems

A linear time invariant system. A system that is linear and time-invariant.

BIBO Stability

The bounded input bounded output stability. We say a system is BIBO stable if-
 \int_{-\infty}^{\infty} |x(t)| dt\ < \infty 

Causality

Causal signals are signals that are zero for all negative time. 
If any value of the output signal depends on a future value of the input signal then the signal is non-causal.

Reflection

If a  signal is given by x(t) then the reflected signal is described by x(-t). Thus, the reflected signal assumes at time –t the value of the original signal that occurs at time t.

Applications of Signals and Systems

The Signals and Systems have various applications such as

• Communications: The signals and systems has applications in communications such as Transmission and reception of voice, data, and video signals.
• Data Transmission: The Techniques such as error detection and correction, encoding, and decoding uses basic signals and system principle.
• Circuit Design: The Design of circuit in electrical system such as filters, amplifiers, and oscillators requires basic
• Control Systems: The Signals and systems are used in Control system for Automation and Aerospace.
• Medical Imaging: Techniques such as MRI, CT scans, and ultrasound imaging uses signal processing for image reconstruction and enhancement.

Conclusion

In this Article we have gone through What is a signal and a system, we have gone through calculation of energy and power of the signal, then we have seen through the Properties and Applications of the Signal and System.