signals

Any quantity that can be represented in terms of time axis or function of independent variable time is called signal. Signal may contain data or information. Weathers are also form of signal they continuously changes with time.

systems

They are used to process signals. They can modify them and can provide us a lot of information. Example a radar system if any missile launched from enemy radars tells us about there velocity there source and also by calculating speed and direction there destination. Systems consists of physical components.

signal energy

To understand logic about signal energy we suppose area under a signal is g(t) it contains amplitude and its duration. If signal size is very large then there can be defective measure but if its positive and negative areas cancel each other it means signal is of small size. This difficulty can be corrected by defining the signal size by area under g2  which is always positive.

signal power

First we need to understand difference between energy and power. Power is defined as rate of producing or consuming energy but in signals energy is a essential thing for power. If signal amplitude approaches to 0 and t → ∞ then signal energy is finite. If signal amp does not → 0 and t does not → ∞ then signal energy is infinite. We can define it as

The following diagrams showing that signal energy is finite or infinite,

From this thing we can observe that the power that we obtained is average (mean) power. If we take its square then it is called mean square value of g(t) .The square root of g(t) is called root mean square (rms) value of g(t)

Example 1: Determine the energy of the signal

Solution: 

Example 2: Determine the Power of the signal

Solution

classification of signals

There are following classes of signals

  1. Continuous-time and Discrete-time Signals
  2. Analog and Digital Signals
  3. Periodic and APeriodic Signals
  4. Energy and Power Signals
  5. Deterministic and Probabilistic Signals
1) continuous -time and discrete-time signal

A signal which can be defined for any value of time interval is called continuous-time signal e.g sine wave. A signal which can be define for some specific values called discrete-time signal.

2) analog and digital signal

We often got confused with analog and continuous signals so a signal whose amplitude can take on any value in a continuous range is an analog signal the figure given above in example-1 is good example for analog signal. A signal whose amplitude can take on finite number of values is called digital signal.

3) periodic and aperiodic signal

A signal which repeat itself after a certain time is called periodic signal it can be continuous or discrete. The opposite of periodic signal is aperiodic signal.

Fig. Aperiodic signal

Fig. Periodic signal

4) energy and power signal

We already discussed that topic but for short review a signal with finite energy is energy signal and signal with finite power is power signal.

5) deterministic and random signals

This topic is very new for us so a signal whose physical description is known completely in either mathematical form or graphical form is called a deterministic signal. If a signal known in terms of random terms such as mean value,mean squared value rather than its mathematical or graphical form is called random signal.

useful signal operations

1) time shifting

Time scaling is very important operation on signals in simple words time scaling is moving a signal forward or backward e.g if g(t) is a signal then g(t-T) is moving a signal forward and g(t+T) is moving a signal backward.

2) time scaling

In simple words time scaling means the compression and expansion of a signal. For a signal g(t) the term g(t)/2 means to expand the signal by factor 2 and 2g(t) means to compress the signal by factor 2.

3) time inverse

As the name stated this operation is used to invert any signal if signal is g(t) then its inverse is g(-t)

signals and vectors

We study about vectors mostly in Physics then why are we using this concept in Signals? because both signals and vectors have direction and magnitude. We mostly represent a vector by bold letters or arrow-head. Consider two vectors and let the component along x be cx and when we draw a straight line from tip of g to x we got perpendicular line so  the vector g can be represent in terms

g = cx + e

we got minimum value of at perpendicular line for different cases g is represented in terms of x plus another vector called error vector e.

Note: g and x are orthogonal if there inner product of the two vectors is zero.

g . x = 0