Understanding Signals: Definition, Classification, and Representation - Prof. Khan, Assignments of Signal Processing and Analysis

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Introduction to Signals

Dr. Waqas Ahmed Imtiaz

January 24, 2018

4 CONTENTS

Chapter 1

Signals

1.1 Signal

Signal can be defined in number of ways based on the the type of application it is used to perform. For example

“Something that carries information about the behavior or attributes of some phenomenon”

“signal is a function of independent variables that can carry some informa- tion”

Signal is sometimes referred to a means for carrying information (meaning full data or a stimuli that has some meaning for a particular receiver) between the source (where the information is generated) and destination (where it is meant to receive). Now the parameters that are used to generate and transmit information between the source and receiver translates signal as

“Description of how one parameter is related to another”

“Pattern of variation of a physical quantity that can be generated, manipu- lated, stored, or transmitted by different physical process”

Signal itself is formed the combination of dependent variables (voltage, cur- rent, etc.) along Y axis, and independent (time, frequency, etc.) variables along X axis shown in Fig. 1.1(a). Consequently, the variation of voltage with respect to time can result in a particular signal, shown in Fig. 1.1(b) that will carry some meaningful data between two parties. The physical parameters along X and Y-axis can be linked mathematically, which translates the signal as

“Function of one or more variables”

Mathematically, signal can be represented as a function e.g. (x(t), v(t), etc.)

1.2. CLASSIFICATION OF SIGNALS 7

1.2.1 Classification w.r.t Time t

This type of classification consider the t along X-axis to determine the type of signal. There are two basic types in this family

1. Continuous Time (CT) Signals
2. Discrete Time (DT) Signals

Continuous Time Signals

A CT signal is a varying function (signal) of dependent and in-dependent vari- ables whose domain, which is often time, is a continuum (e.g., a continuous set of values). That is, the function’s domain t is an uncountable set. The function x(t) itself need not be continuous. For further elaboration a CT signal is one in which

“The independent variable changes continuously with t”

“The dependent variable t is defined for each and every value of the inde- pendent variable x(t) and vice versa”

“Has infinite number of points (and each point has a distinct value) between any two intervals”

That is, the function’s independent domain t is an uncountable set of values. For example if two instants are considered along the t axis, there lies infinite number of values between these two intervals. Figure 1.3 shows continuous time signals where the t consists of an uncountable set of values. For example if an instant between t = 1 and t = 2 contains infinite number of values i.e. 1 .0, 1 .001, 1.0001, 1.00001,... and so on. Furthermore, each and every instant on t axis will have a definite function x(t) along with y-axis.

0

t 1 2 3 4

(a)

0 1 2 3 4

t

(b)

Figure 1.3: Continuous time signals

Discrete Time Signals

Unlike CT, DT signals are represented for a given number of defined value (pure integer, no fraction) over a period of time n. Furthermore, values of the signal

8 CHAPTER 1. SIGNALS

exists at equally spaced portion/intervals of time. It is evident from the fact that discrete intervals are created due to sampling, which yields equidistant points along the X-axis. For further elaboration a DT signal is one in which

“The intensity levels of the signal are defined at discrete (discontinuous) intervals of time n”

n

(a)

n

(b)

Figure 1.4: Discrete time signals

Figure 1.4 shows DT signals where the time n consists of a discontinuous set of values i.e. the function/signal x[n] is defined at specific (usually integer) values of time. For example Fig. 1.4(a) shows DT signal with x[n] defined at [1234] only and no other points/instants are used to express the signal. Same phenomenon can be observed in Fig. 1.4(b), which represents the output of a natural sampling process, where the signal is defined for discrete (discontinuous) set intervals only.

1.2.2 Classification w.r.t Function

This type of classification considers value of the function along Y-axis to deter- mine if the signal is

1. Analog Signal
2. Digital Signal

Analog Signal

This type of signal uses a continuous set of function x(t) values to represent the information for a given signal. “Analog signal corresponds to the continuous set of possible function values between given intervals”

“An Analog signal has infinite number of possible values with in a range or any given intervals”

10 CHAPTER 1. SIGNALS

Equation 1.1 shows that for a signal to be even the value of the function along with positive t must be equal to the value of the function along the negative t axis. For example if x(t) = 1 at t = 2, then x(−t) = 1 at t = −2 for a signal to be even. In order words the signal is a mirror image of itself between the negative and positive t axis.

-t t

(a)

O π/2 π

1

x -3π/2 -π - π/2 3 π/

-x

(b)

Figure 1.7: Even signals

1.2.4 Odd Signal

The signal x(t) is said to be odd if

x(t) = −x(−t) (1.2) Equation 1.2 shows that an odd signal contains equal and opposite values along positive and negative t axis. For example if x(t) = 1 at t = 2, then x(−t) = −1 at t = −2 for a signal to be even as shown in Fig. 1.8.

-2 -1^0 1

t

-t

(a)

0

1

π/2 (^) π

V

• π (^) - π/

-x x

(b)

Figure 1.8: Odd signals

1.2.5 Periodic Signal

A signal is said to be periodic if

1.3. SIGNAL REPRESENTATION 11

“It repeats itself over regular interval of time”

Mathematically, if x(t) represent a signal and T 0 is the, then a periodic signal can be written as

x(t) = x(t + T 0 ) (1.3)

1.3 Signal Representation

Three techniques can be used to represent a signal namely

1. Graphical
2. Mathematical
3. Vector space

Graphical and mathematical representations can be linked together as both are used simultaneously to represent a particular signal.

1.3.1 Mathematical Representation

Mathematically any signal x(t) is given by

x(t) = Acos(2πf t + φ) (1.4)

where A = Amplitude f = F requency φ = Phase or Phase shift

Amplitude

Amplitude of any signal can be defined in a number of ways for example:

“The largest possible value at a given instant that a signal can take”

“The peak value of a signal at any instant”

“The height of a signal from the center line (x-axis) to the peak ”

“The maximum height, force or power of the wave/ signal ”

“Maximum displacement of a particular wave/ signal from its equilibrium position”

It is evident from the above-mentioned definitions that Amplitude of any given signal simply represents the maximum value (height) that the signal can attain at any given instant. For example Fig. 1.9 shows a simple signal with numerous intensity level, but amplitude A = 2, which represents the highest or peak value that the signal can achieve.

1.3. SIGNAL REPRESENTATION 13

1

-1 (^) T

1

(a)

1

t

T (b)

Figure 1.11: (a)AC and (b)DC signals

Phase or Phase Shift

Phase shift for any given signal x(t) can be defined as:

“Starting point of a signal w.r.t to zero”

“How far the signal is horizontally to the right of original position at zero”

In order to demonstrate the concept of phase, refer to Fig. Figure 1.12 shows a simple sine wave with different phases given by a dotted point representing the starting point of a sine wave. For further elaboration of the phase/ phase-shift concept consider the Figure which shows a simple co- sine wave with different phase shifts. Fig. 1.13(a) represents a simple cosine wave with no phase shift or φ = 0, thus x(t) = cos(2πf t) with A = 1. Figure 1.13(b) represents a cosine wave with different phase (represented by dot, start- ing point). It is evident from the fact that a change in phase has introduce a shift in the signal along x-axis, and due to this π/2 shift x(t) = cos(2πf t − π 2 ). Same phenomenon is observed in Fig. 1.13(c) where cosine wave is shift to right of the references resulting in x(t) = cos(2πf t + π 2 ). It is observed from Fig. 1.12, and 1.13 that a change in phase slides the signal along x-axis. This sliding or time shift Ts operation can be related to phase shift as

−φ = w × Ts (1.5)

14 CHAPTER 1. SIGNALS

X(t)

t 1

(a)

t 1

X(t)

(b)

t 1

X(t)

(c)

Figure 1.12: Sine wave with different phases

16 CHAPTER 1. SIGNALS

Example 1.3.2 Plot the signal x(t) = cos(2πt). Before plotting the signal, it is necessary to determine the required parame- ters including A, f, φ. So compare the given signal with the basic equation x(t) = Acos(2πf t + φ) comparison elaborates that A = 1 2 πf = 2π , which yields f = 1 after simplification. Now the time period T = (^) f^1 becomes equal to T = 1 φ = 0

Example 1.3.3 Plot the signal x(t) = cos(4πt).

Solution Before plotting the signal, it is necessary to determine the required parame- ters including A, f, φ. So compare the given signal with the basic equation x(t) = Acos(2πf t + φ) comparison elaborates that A = 1 2 πf = 4π , which yields f = 2 after simplification. Now the time period T = (^) f^1 becomes equal to T = (^12) φ = 0

Example 1.3.4 Plot the signal x(t) = cos(0. 2 πt).

Solution Before plotting the signal, it is necessary to determine the required parame- ters including A, f, φ. So compare the given signal with the basic equation x(t) = Acos(2πf t + φ) comparison elaborates that A = 1 2 πf = 0. 2 π , which yields f = 101 after simplification. Now the time period T = (^) f^1 becomes equal to T = 10 φ = 0

1.3. SIGNAL REPRESENTATION 17

−1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.

−0.

−0.

−0.

0

1

Time

x(t)