Signals A Preliminary Discussion EE442 Analog & Digital Communication Systems Lecture 2

Signals A Preliminary Discussion EE442 Analog & Digital Communication Systems Lecture 2 The Fourier transform of single pulse is the sinc function. EE 442 Signal Preliminaries 1

Communication Systems and Signals Information converted into an electrical waveform suitable for transmission is called a signal. Signals are time-varying. A communication system is a collection of devices used to send messages or information from a source (i.e., a transmitter) to a destination (i.e., a receiver) over a communication channel (i.e., a propagation medium). In communication systems a source generates the message or information to be communicated. The message or information typically falls into one of three general categories: Voice/audio Data Video ES 442 Signal Preliminaries 2

Definition and Classification of Signals I A signal is a function of time that represents a physical quantity. For EE442 a signal is a waveform containing or encoding information. A signal may be a voltage, current, electromagnetic field, or another physical parameter such as air pressure in an acoustic signal. A signal may be either deterministic or random (i.e., stochastic). Deterministic signals are by far the most common. There are two domains in which to describe signals: (1) Time-domain waveform (2) Frequency-domain spectrum This requires us to use Fourier analysis. Read: Chapter 2 of Agbo & Sadiku; Sections 2.1 & 2.2 ES 442 Signal Preliminaries 3

General Classifications of Signals Deterministic Random (Stochastic) Periodic Aperiodic Stationary Quasiperiodic Nonstationary Sinusoidal, Triangular, Rectangular Transient, Unit pulse response ECG waveform, Temperature record Noise in Electronic Circuits Language, Music, etc. Mathematical representation possible Often can calculate the waveform Roughly approximate mathematically White Gaussian Noise Voice waveform Not mathematically calculable Which of these categories don t contain information? EE 442 Signal Preliminaries 4

Physically Realizable Signals (Waveforms) Meaning the signal (waveform) is measurable in a laboratory. 1. Waveform has significant non-zero values over a finite composite time interval. 2. Spectrum of the waveform has significant non-zero values over a finite frequency interval. 3. The waveform is a continuous function of time. 4. The waveform is limited to finite peak values. 5. The waveform takes on only real values (as compared to complex values of the format a + jb). ES 442 Signal Preliminaries 5

Definition and Classification of Signals II A signal may be either analog or digital. Sometimes the terminology of continuous or discrete is used to distinguish between analog and digital signals. A signal may be periodic if it has a uniform period of repetition, T, or non-periodic (i.e., aperiodic) when no uniform period of repetition exists or it does not repeat its form or shape. Signals may be differentiated as either baseband or carrier-based. Signals can also be distinguished by being energy signals or a power signals. But we must be careful with this terminology. Refer to Agbo & Sadiku on page 19 of Chapter 2 for their discussion. ES 442 Signal Preliminaries 6

Energy Signal versus Power Signal Given signal g(t) (can either be current or voltage) Energy Signal 2 Eg = g() t dt Power Signal T 2 1 2 Pg = lim g( t) dt T T T 2 0 < E g < Deterministic & non-periodic signals 0 < P g < Periodic & random signals We shall call this a Finite Energy Signal We shall call this a Finite Power Signal Refer to page 19 of Agbo & Sadiku ES 442 Signal Preliminaries 7

Why So Much Emphasis on Sinusoidal Signals? All practical waveforms can be analyzed and constructed from many harmonically-related sinusoidal waveforms. Example: Rectangular waveform synthesized from the sum of sinusoidal signals fundamental f third harmonic 3f fifth harmonic 5f with many more harmonics added of decreasing amplitude. EE 442 Signal Preliminaries 8

Fourier Series Expressing a Periodic Square Waveform Trigonometric format: ( ) f ( t) = a + a cos ( n ) + b sin ( n ) 0 n 0 n 0 n= 1 = 0 DC 2 ; T T is the period AC We compute the coefficients using T T T 1 2 2 a = f ( t) dt, a = f ( t)cos( n t) dt, b = f ( t)sin( n t) dt 0 n 0 n 0 T T T 0 0 0 ES 442 Signal Preliminaries 9

https://slideplayer.com/slide/1663100/ ES 442 Signal Preliminaries 10

Signal Spectra of Periodic Square Waveform A T fundamental f 3 rd harmonic Refer to Section 2.5 of Agbo & Sadiku; Pages 26 to 33 5 th harmonic.... f (1/T) 3f 5f 7f 9f Frequency f Trigonometric Fourier series ES 442 Signal Preliminaries 11

Signal Spectra of Periodic Square Waveform https://gifer.com/en/cuas ES 442 Signal Preliminaries 12

Sinusoidal Signals Constructing a Periodic Waveform Two Viewpoints: Time Domain and Frequency Domain Oscilloscope Display Spectrum Analyzer Display ES 442 Signal Preliminaries 13

Sinusoidal Signals Generating a Periodic Square Wave = t Recommended: https://www.youtube.com /watch?v=k8fxf1kjzy0 https://en.wikipedia.org/wiki/fourier_series ES 442 Signal Preliminaries 14

Review of Phasors Phasors are used only to represent sinusoidal waveshapes. Definition: A complex number c is a phasor if it represents a sinusoidal Waveform; for example where the phasor is j 0 ( ) t g( t) = c cos 0t + c = Re c e j c c = c e Static phasor c c j 0t j c ce + is a rotating phasor and is distinguished from phasor c. Note: Magnitude c is usually a peak value, but sometimes an RMS value, where RMS stands for root-mean square. ES 442 Signal Preliminaries 15

time Rotating Phasor Generating a Cosine Waveform Time evolving projection onto horizontal axis yields cosine waveform Rotating Phasor: -A Im A A Re Complex Plane Note: CCW rotation is a positive angle or positive frequency Rotating Phasor EE 442 Signal Preliminaries 16

Certainly You Remember Euler s Identity jx e = cos( x) jsin( x) Let x = 2 ft = t, then jt e = cos( t) + jsin( t) jt e = cos( t) + jsin( t) = cos( t) jsin( t) Because cosine is an even function and sine is an odd function. 1 cos( t) = 2 e + e 1 sin( t) = 2 j e e ( jt jt ) ( jt jt ) EE 442 Signal Preliminaries 17

Sine and Cosine Waves are in Quadrature https://te.m.wikipedia.org/wiki/దస త ర :Simple_harmonic_motion_animation_2.gif ES 442 Signal Preliminaries 18

Conjugate Phasor Representation of Sines & Cosines cos(2 ft) = e j2 ft j2 ft + 2 e sin(2 ft) = e j2 ft j2 ft e 2 j Complex Plane Im CCW e j2 ft e j2 ft CW Im CCW j 2 ft e CW e Re j2 ft Rotating Phasors Positive frequency (CCW) Negative frequency (CW) Re Counter rotating vectors (or phasors) EE 442 Signal Preliminaries 19

Forming a Cosine Signal With Conjugate Phasors cos(2 ) = 1 + 1 2 2 j2 ft j2 ft ft e e Euler s formula Im j 2 ft e Counter rotating vectors (phasors) Re Projection onto real-axis: 0 e j2 ft Amplitude Time t evolution EE 442 Signal Preliminaries 20

Forming a Sine Signal With Conjugate Phasors Counter rotating vectors (phasors) Im Amplitude e j2 ft sin(2 ) = 1 1 2j 2j j 2 ft e Re j2 ft j2 ft ft e e 0 Time t evolution Projection onto imaginary-axis: EE 442 Signal Preliminaries 21

How Do We Explain Negative Frequencies? The existence of the spectrum at negative frequencies is somewhat disturbing to some people because, by definition, the frequency (number of repetitions per second) is a positive quantity. How do we interpret a negative frequency f 0? https://www.researchgate.net/post/can_anyone_explain_the_concept_of_ne gative_frequency EE 442 Signal Preliminaries 22

How Do We Explain Negative Frequencies? The existence of the spectrum at negative frequencies is somewhat disturbing to some people because, by definition, the frequency (number of repetitions per second) is a positive quantity. Answer: How do we interpret a negative frequency f 0? Negative frequencies are a mathematical construct to analyze real signals using a complex number framework. It requires the use of double-sided spectra. A complex number can be made real by adding its conjugate to it (e.g., (a + jb) + (a - jb) = 2a. A real sinusoid can be represented using complex exponentials by using the sum of e (jωt) and its complex conjugate e (-jωt). This is where the negative frequency idea comes from. https://www.researchgate.net/post/can_anyone_explain_the_concept_of_ne gative_frequency ES 442 Signal Preliminaries 23

Analog Signals and Digital Signals Analog Signals (1) A parameter of the signal represents a physical parameter (2) Physical parameter is time-varying (3) Parameter takes on any value within a defined range (said to be continuous valued) Digital Signals (1) Represents a sequence of numbers or states (2) Numbers change in discrete time (said to be time-varying) (3) Numbers are restricted to a finite set of discrete values Waveforms: Analog signal Digital signal Waveforms as commonly drawn in textbooks All signals are analog signals the differentiator is what they represent! EE 442 Signal Preliminaries 24

Analog & Digital Signals: Continuous versus Discrete Valued Analog & continuous Digital & continuous t Analog & discrete t 0 1 0 0 1 1 0 1 1 1 0 0 1 1 0 1 1 0 Digital & discrete t n t n 0 1 0 0 1 1 0 1 1 1 0 0 1 1 0 1 1 0 String of values EE 442 Signal Preliminaries 25

Amplitude Quiz: How would you classify waveform A and waveform B? (1) Continuous, (2) Discrete, (3) Analog, (4) Digital Waveform A (gray) Waveform B (red) time EE 442 Signal Preliminaries 26

Time variation alone is not sufficient to communicate information Example A: Bit Sequence of 10101010101... Square Waveform shown Amplitude +½A -½A 1 0 1 0 1 0 1 0 1 0 NO communication. Why? Time Example B: Bit Sequence of 10001010111... Amplitude +½A High state Represents a 1 Low state Represents a 0 1 0 0 0 1 0 1 0 1 1 1 This is a periodic waveform. A pure sinusoidal waveform or a square waveform doesn t transmit information. Information is Being transmitted. Why? -½A Time Not a periodic waveform. EE 442 Signal Preliminaries 27

EE 442 Signal Preliminaries 28

Bandwidth Definitions The bandwidth of a signal provides a measure of the extent of spectral content of the signal for positive frequencies. What does significant mean? 1. 3-dB Bandwidth The separation (along positive frequency axis) between The points where the amplitude drops to of its peak value (½ power points). 2. Null-to-null Bandwidth For example, for the sinc function the bandwidth would be the frequency width from -1/T to 1/T (null-to-null points). 1 2 3. Root-mean-square (RMS) Bandwidth Defined as BW RMS f = 2 2 G( f ) G( f) 2 And there are numerous other bandwidth definitions... EE 442 Signal Preliminaries 29