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Introduction to Signals and Systems: Properties of systems

Last Updated : 18 May, 2023
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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 independent variable is time.

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

 Types of time signals:

  1. 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.
  2. Discrete time signals x[n] – . A signal which 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. 

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.

Calculating Energy and Power of signals: 
Energy– 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-
 

    $$ E=\sum_{-\infty}^{\infty} |x[n]|^2 $$


Power- 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:

  1. Energy signal– generally converging signals, aperiodic signals or signals that are bounded.
     

        $$ E < \infty\ and\ P=0 $$

  2. Power signal– generally periodic signals, as they encompass infinite area under their graph and extend from +\infty  to -\infty

        $$ E \rightarrow \infty\ and\ P=constant $$

  3. Neither energy nor power signal
     

        $$ E \rightarrow \infty\ and\ P \rightarrow \infty $$

Transformation of the independent variable:

  1. 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.
  2. 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.
  3. Reversal- also called folding as the graph is folded about the Y-axis or T if given x(T-t).

Properties of systems:

  1. 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.
  2. 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.
     

        $$Even(x(t)) = (x(t)+x(-t))/2$$ $$Odd(x(t)) = (x(t)-x(-t))/2$$

  3. Linearity- 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.
  4. 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.
  5. LTI systems- A linear time invariant system. A system that is linear and time-invariant.
  6. BIBO stability- The bounded input bounded output stability. We say a system is BIBO stable if-
     

        $$\int_{-\infty}^{\infty} |x(t)| dt\ < \infty$$

  7. 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.
  8. 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.


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