Signals & Systems : Basics of Signals & System

By Yash Bansal|Updated : October 1st, 2021

In this article, you will find the study notes on Representation of Continuous and Discrete-Time Signals which will cover the topics such as Different types of Basic Continuous/Discrete Signals, Energy & Power Signal.                                                                                                                                                                             

In this article, you will find the study notes on Representation of Continuous and Discrete-Time Signals which will cover the topics such as Different types of Basic Continuous/Discrete Signals, Energy & Power Signal

Properties of Signals

A signal can be classified as periodic or aperiodic; discrete or continuous time; discrete of continuous-valued; or as a power or energy signal. The following defines each of these terms. In addition, the signal-to-noise ratio of a signal corrupted by noise is defined.

Periodic / Aperiodic:

A periodic signal repeats itself at regular intervals. In general, any signal x(t) for which

x(t) = x(t+T)

for all t is said to be periodic.

The fundamental period of the signal is the minimum positive, non-zero value of T for which above equation is satisfied. If a signal is not periodic, then it is aperiodic.

Symmetric / Asymmetric:

There are two types of signal symmetry: odd and even. A signal x(t) has odd symmetry if and only if x(-t) = -x(t) for all t. It has even symmetry if and only if x(-t) = x(t).

 Continuous and Discrete Signals and Systems

A continuous signal is a mathematical function of an independent variable, which represents a set of real numbers. It is required that signals are uniquely defined in except for a finite number of points.

  • A continuous time signal is one which is defined for all values of time. A continuous time signal does not need to be continuous (in the mathematical sense) at all points in time. A continuous-time signal contains values for all real numbers along the X-axis. It is denoted by x(t).
  • Basically, the Signals are detectable quantities which are used to convey some information about time-varying physical phenomena. some examples of signals are human speech, temperature, pressure, and stock prices.
  • Electrical signals, normally expressed in the form of voltage or current waveforms, they are some of the easiest signals to generate and process.

Example: A rectangular wave is discontinuous at several points but it is continuous time signal.

01-Representation (9)

Discrete / Continuous-Time Signals:

A continuous time signal is defined for all values of t. A discrete time signal is only defined for discrete values of t = ..., t-1, t0, t1, ..., tn, tn+1, tn+2, ... It is uncommon for the spacing between tn and tn+1 to change with n. The spacing is most often some constant value referred to as the sampling rate,

Ts = tn+1 - tn.

It is convenient to express discrete time signals as x(nTs)=x[n].

That is, if x(t) is a continuous-time signal, then x[n] can be considered as the nth sample of x(t).

Sampling of a continuous-time signal x(t) to yield the discrete-time signal x[n] is an important step in the process of digitizing a signal.

 

Energy and Power Signal:

When the strength of a signal is measured, it is usually the signal power or signal energy that is of interest.

The signal power of x(t) is defined as                         

01-Representation (1)

and the signal energy as

  • A signal for which Px is finite and non-zero is known as a power signal.
  • A signal for which Ex is finite and non-zero is known as an energy signal.
  • Px is also known as the mean-square value of the signal.
  • Signal power is often expressed in the units of decibels (dB).
  • The decibel is defined as  01-Representation (3)
  • where P0 is a reference power level, usually equal to one squared SI unit of the signal.
  • For example if the signal is a voltage then the P0 is equal to one square Volt.
  • A Signal can be Energy Signal or a Power Signal but it can not be both. Also a signal can be neither a Energy nor a Power Signal.
  • As an example, the sinusoidal test signal of amplitude A,

       x(t)=Asin(ωt)

       has energy Ex that tends to infinity and power , 01-Representation (4)

       or in decibels (dB):    20log(A)-3

       The signal is thus a power signal.

Signal to Noise Ratio:

Any measurement of a signal necessarily contains some random noise in addition to the signal. In the case of additive noise, the measurement is

x(t)=s(t)+n(t)

where s(t) is the signal component and n(t) is the noise component.

The signal to noise ratio is defined as

       01-Representation (5)

 or in decibels, 01-Representation (6)

The signal to noise ratio is an indication of how much noise is contained in a measurement.

 

Standard Continuous Time Signals

  • Impulse Signal

   

where ∞ is the height of impulse signal having unit area.                                        

and     01-Representation (11) When A = 1 (unit impulse Area) 

  • Step Signal

01-Representation (13)

Unit Step Signal  if A =1, 01-Representation (14)

  • Ramp Signal

     01-Representation (16)

   Unit Ramp Signal (A=1)

   01-Representation (18)

  • Parabolic Signal

    01-Representation (20)

   Unit Parabolic Signal when A = 1,

    01-Representation (21)

01-Representation (22)

  • Unit Pulse Signal

     01-Representation (23)

 

Sinusoidal Signal

  • Co-sinusoidal Signal:

    01-Representation (25)

    Where, ω0 is the angular frequency in rad/sec

    f0 = frequency in cycle/sec or Hz

    T = time period in second

    When  01-Representation (27)

   When ϕ = positive, 01-Representation (28)

   When ϕ = negative, 01-Representation (29)

 

  • Sinusoidal Signal:

    01-Representation (30)

    Where,01-Representation (26)  Angular frequency in red/sec

    f0 = frequency in cycle/sec or Hz

    T = time period in second

    When 01-Representation (31)

    When ϕ = positive, 01-Representation (32)

   When ϕ = negative, 01-Representation (33)

01-Representation (34)

    Exponential Signal: 

  • Real Exponential Signal

    01-Representation (35) where, A and b are real.

01-Representation (36)

  • Complex Exponential signal

    01-Representation (37)

    The complex exponential signal can be represented in a complex plane by a rotating vector, which rotates with a constant angular velocity of ω0 red/sec.

01-Representation (38)

  • Exponentially Rising/Decaying Sinusoidal Signal

    01-Representation (39)

01-Representation (40)

  • Triangular Pulse Signal

      01-Representation (41)

  • Signum Signal

    01-Representation (43)

  • SinC Signal

   01-Representation (45)

  • Gaussian Signal

    01-Representation (47)

    Important points:

  • The sinusoidal and complex exponential signals are always periodic.
  • The sum of two periodic signals is also periodic if the ratio of their fundamental periods is a rational number.
  • Ideally, an impulse signal is a signal with infinite magnitude and zero duration.
  • Practically, an impulse signal is a signal with large magnitude and short duration.

    Classification of Continuous Time Signal: The continuous time signal can be classified as

  1. Deterministic and Non-deterministic Signals:
    • The signal that can be completely specified by a mathematical equation is called a deterministic signal. The step, ramp, exponential and sinusoidal signals are examples of deterministic signals.
    • The signal whose characteristics are random in nature is called a non-deterministic signal. The noise signal from various sources like electronic amplifiers, oscillator etc., are examples of non-deterministic signals.
    • Periodic and Non-periodic Signals
    • A periodic signal will have a definite pattern that repeats again and again over a certain period of time.

                 x(t+T) = x(t)

   2. Symmetric (even) and Anti-symmetric (odd) Signals

     When a signal exhibits symmetry with respect to t = 0, then it is called an even signal.

      x(-t) = x(t)

    When a signal exhibits anti-symmetry with respect to t = 0, then it is called an odd signal.

     x(-t) = -x(t)

     Let 01-Representation (51)

     Where,01-Representation (52)  even part of 01-Representation (53)

     01-Representation (54)odd part of 01-Representation (53)

     01-Representation (55)

Discrete-Time Signals 

 The discrete signal is a function of a discrete independent variable. In a discrete time signal, the value of discrete time signal and the independent variable time are discrete. The digital signal is same as discrete signal except that the magnitude of the signal is quantized. Basically, discrete time signals can be obtained by sampling a continuous-time signal. It is denoted as x(n).

01-Representation (57)

   Standard Discrete Time Signals

  • Digital Impulse Signal or Unit Sample Sequence

      Impulse signal,01-Representation (58)

    

  • Unit Step Signal

     01-Representation (60)

   

  • Ramp Signal

        Ramp signal,   01-Representation (62)

       

  • Exponential Signal

     Exponential Signal,01-Representation (64)

01-Representation (65)

  • Discrete Time Sinusoidal Signal

    

01-Representation (70)

  • A discrete-time sinusoid is periodic only if its frequency is a rational number.
  • Discrete-time sinusoids whose frequencies are separated by an integer multiple of 2π are identical.

Operations in Continuous Time Signals: 

Periodic & Non-Periodic Signals:

  • A signal is a periodic signal if it completes a pattern within a measurable time frame, called a period and repeats that pattern over identical subsequent periods.
  • The period is the smallest value of T satisfying g(t + T) = g(t) for all t. The period is defined so because if g(t + T) = g(t) for all t, it can be verified that g(t + T') = g(t) for all t where T' = 2T, 3T, 4T, ... In essence, it's the smallest amount of time it takes for the function to repeat itself. If the period of a function is finite, the function is called "periodic".
  • Functions that never repeat themselves have an infinite period, and are known as "aperiodic functions".

Even & Odd Signals:

A function even function if it is symmetric about the y-axis. While, A signal is odd if it is inversely symmetrical about the y-axis.

Even Signal, f(x) = f(-x)

Odd Signal, f(x) = - f(-x)

Note: Some functions are neither even nor odd. These functions can be written as a sum of even and odd functions. A function f(x) can be expressed in terms of sum of an odd function and an even function.

Invertibility and Inverse Systems:

A system is invertible if distinct inputs results distinct outputs. As shown in the figure for the continuous-time case, if a system is invertible, then an inverse system exists that, when cascaded with the original system, results an output w(t) equal to the input x(t) to the first system.

An example of an invertible continuous-time system is y(t) = 2x(t),

for which the inverse system is w(t) = 1/2 y(t)

Causal System:

A system is causal if the output depends only on the input at the present time and in the past. Such systems are often referred as non anticipative, as the system output does not anticipate future values of the input. Similarly, if two inputs to a causal system are identical up to some point in time to or no the corresponding outputs must also be equal up to this same time.

y1(t) = 2x(t) + x(t-1) + [x(t)]2  ⇒ Causal Signal 

y1(t) = 2x(t) + x(t-1) + [x(t+2)] ⇒ Non-Causal Signal 

Homogeneity (Scaling):

A system is said to be homogeneous if, for any input signal X(t), i.e. When the input signal is scaled, the output signal is scaled by the same factor.

 Time-Shifting / Time Reversal / Time Scaling:

 Time-Shifting

Time Shifting can be understood as shifting the signal in time. When a constant is added to the time, we obtain the advanced signal, & when we decrease the time, we get the delayed signal.

Time Scaling: 

Due to the scaling in time the output Signal may shrink or stretch it depends on the numerical value of scaling factor.

Time Inversion:

Time Inversion referred as flipping the signal about the y-axis.

 

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Yash BansalYash BansalMember since Nov 2018
Content Manager (GATE)
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