# 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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