Signal and System

Signals and systems are frequently discussed without knowing the exact parameters being represented. This is the same as using x and y in algebra, without assigning a physical meaning to the variables. This brings in a fourth rule for naming signals. If a more descriptive name is not available, the input signal to a discrete system is usually called: x[n], and the output signal: y[n]. For continuous systems, the signals: x(t) and y(t) are used.

There are many reasons for wanting to understand a system. For example, you may want to design a system to remove noise in an electrocardiogram, sharpen an out-of-focus image, or remove echoes in an audio recording. In other cases, the system might have a distortion or interfering effect that you need to characterize or measure. For instance, when you speak into a telephone, you expect the other person to hear something that resembles your voice. Unfortunately, the input signal to a transmission line is seldom identical to the output signal. If you understand how the transmission line (the system) is changing the signal, maybe you can compensate for its effect. In still other cases, the system may represent some physical process that you want to study or analyze. Radar and sonar are good examples of this. These methods operate by comparing the transmitted and reflected signals to find the characteristics of a remote object. In terms of system theory, the problem is to find the system that changes the transmitted signal into the received signal.
At first glance, it may seem an overwhelming task to understand all of the possible systems in the world. Fortunately, most useful systems fall into a category called linear systems. This fact is extremely important. Without the linear system concept, we would be forced to examine the individual characteristics of many unrelated systems. With this approach, we can focus on the traits of the linear system category as a whole. Our first task is to identify what properties make a system linear, and how they fit into the everyday notion of electronics, software, and other signal processing systems.

Signal

A signal is a function representing a physical quantity or variable, and typically it
contains information about the behavior or nature of the phenomenon. For instance, in a
RC circuit the signal may represent the voltage across the capacitor or the current flowing
in the resistor. Mathematically, a signal is represented as a function of an independent
variable t. Usually t represents time. Thus, a signal is denoted by x(t).

System

A system is a mathematical model of a physical process that relates the input (or
excitation) signal to the output (or response) signal.
Let x and y be the input and output signals, respectively, of a system. Then the system
is viewed as a transformation (or mapping) of x into y. This transformation is represented
by the mathematical notation

y = Tx

where T is the operator representing some well-defined rule by which x is transformed
into y.

Convolution

Convolution is a powerful way of characterizing the input-output relationship of time invarient linear systems.
There are two convolution theorems

1) Time Convolution
Time convolution theorem is for time domain

2) Frequency Convolution
this theorem is for frequency domain

Energy and Power Signals

x(t) (or x[n]) is said to be an energy signal (or sequence) if and only if 0 < E < infinity, and
so P = 0.

x(t) (or x[n]) is said to be a power signal (or sequence) if and only if 0 < P < infinity, thus
implying that E = infinity.

Signals that satisfy neither property are referred to as neither energy signals nor power
signals.

Note that a periodic signal is a power signal if its energy content per period is finite, and
then the average power of this signal need only be calculated over a period

Real and Complex Signals

Real and Complex Signals:
A signal x(t) is a real signal if its value is a real number, and a signal x(t) is a complex
signal if its value is a complex number. A general complex signal x( t) is a function of the form

x ( t ) = x , ( t ) + i x 2 ( t )


where x,( t ) and x2( t ) are real signals and j = m.
Note that in Eq. t represents either a continuous or a discrete variable.

Analog and Digital Signals

If a continuous-time signal x(t) can take on any value in the continuous interval (a, b),
where a may be - infinity and b may be + infinity, then the continuous-time signal x(t) is called an
analog signal. If a discrete-time signal x[n] can take on only a finite number of distinct
values, then we call this signal a digital signal