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:
- Continuous time signals x(t)- defined at every point in time
- 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-
for a discrete time signal-
Power- Rate of change of energy.
for a continuous time signal.
for a discrete time signal-
Classes of signals on the basis of their power and energy:
- Energy signal– generally converging signals, aperiodic signals or signals that are bounded.
- Power signal– generally periodic signals, as they encompass infinite area under their graph and extend from
to
.
- Neither energy nor power signal
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:
- Periodicity- the signal’s behavior/graph repeats after every T. Therefore,
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. - Linearity- constitutes of two properties-
(i) Additivity/Superposition-
if x1(t) -> y1(t)
and x2(t) -> y2(t)(ii) Property of scaling-
if x1(t) -> y1(t)
thenIf 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.
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- - 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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