Introduction to Signals and Systems: Properties of systems

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.

Types of time signals:

  1. Continuous time signals x(t)- defined at every point in time
  2. Discrete time signals x[n] – defined only at a discrete set of values of time (integer).

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)

         $$a*x1(t)\ \rightarrow a*y1(t)$$

    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.

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