Why Do You Need To Know About Signals?

What Are Signals?

Signal Representation - Sinusoidal Signals
     Amplitude
     Frequency
     Phase

What If Things Are A Little Different?

Problems

        Electricity has been with us for a while now, and we use it in many ways that were never anticipated when we began to use electricity.

• After electricity was discovered and understood, almost all of the first applications were ones that involved power.  The application may have been an electric light or it may have been a motor.  In either case, the application of electricity involved the use of large amounts of energy and power.
• However, other early applications of electricity were used to transmit information.  The telegraph, and later the telephone, had a very significant effect on the history of the United States.
• As things progressed it became necessary to design an build systems to distribute large amounts of electrical power.
• The city of Sunbury, PA, was the first to have an electrical distribution system for lighting on July 4, 1883.  That system was a DC system.
• Later, AC systems were developed.  Edison, who installed that first DC system in Sunbury, and who had begun installing DC systems in many other cities bitterly fought the introduction of AC systems.  Edison put out advertisements that showed an electric chair and asked the public if they wanted AC in their home since it was used in the electric chair for killing people.

        Today, we have vast industries that use electricity to distritute information.  Those industries are the TV networks (over the air, cable and satellite), phone systems, the internet and other radio communciation systems, for example.

        If you want to understand the basics of how the information distribution industry works you will need to know about the various forms of electrical signals.

        Goals for this lesson are simple.

   Given a signal,

   Be able to describe the signal when possible.
   Be able to use time-varying currents and voltages in KCL and KVL.

        There are many different kinds of electrical signals.  If we look at how signals are generated, we find that there are many different kinds of electrical signal sources.  Here are some.

• Microphones
• Fax machines
• Thermocouples
• Remote controls for television sets

• If you are a sound engineer, you will have to deal with the small voltages produced by microphones and you will have to worry about processing the signal so that you can reproduce the sound signals accurately.
• If you design fax machines you will need to ensure that your signals are transmitted accurately so that information is not corrupted when it is sent.
• If you need to control temperature in an industrial process, you will need to worry about the small signal from a thermocouple - how to amplify it - how to process it.
• TV remotes are designed to set the channel accurately.  They set the channel, adjust volume, etc. by sending signals to the TV.

• Amplify a signal - make it larger
• Remove noise from a signal
• Change a signal to emphasize certain characteristics - for example adding bass boost to a sound signal.

        If you will be dealing with signals, then you will need to have some sort of model for the signals you work with.  Usually that will be some sort of mathematical representation.

        The simplest representation for a signal is to represent the signal as a function of time.  For example, the voltage that appears across the terminals of a microphone will vary in time when the microphone "picks up" a sound.  Then, we might say something like:

 Representing a signal as a time function is so common that there are many instruments and data gathering devices that give you a picture of a voltage time function.  The most common instrument that gives a picture of a voltage time function is the oscilloscope.

        At this point, we can consider some specific kinds of signals.  We'll start with periodic signals, and in particular we'll start with sinusoidal signals.

        Periodic signals are signals that repeat in time with a certain period.  The most fundamental periodic signal is the sinusoidal signal.  Any other periodic signal can be thought of as a combination of sinusoidal signals added together.  That approach is based on the Fourier Series and you can go to that topic when you know enough about sinusoidal signals.

        In this section you will begin to learn about sinusoidal signals.  Sinusoidal signals, based on sine and cosine functions, are the most important signals you will deal with.  They are important because virtually every other signal can be thought of as being composed of many different sine and cosine signals.  They form the basis for many other things you will do in signal processing and information transmission.  Eventually you will deal with signals as different as voice signals, radar signals, measurement signals and entertainment signals like those found in television and radio.  Sinusoidal signals are the starting point for almost all work in signal processing and information transmission.

        In this part of this lesson you will learn:

• What Sinusoidal Signals are and How they are Represented
• The Parameters of a Sinusoidal Signal
• How to Measure the Parameters of a Sinusoidal Signal.

        If you put a voltage signal into an oscilloscope you can get a picture of how the signal varies in time.  Sinusoidal signals are often voltages which vary sinusoidally in time.  (Sinusoidal signals could be, however, other physical variables like current, pressure, or virtually any other physical variable.)  Here's a simulator that will let you put various kinds of signals on a simulated oscilloscope.  Click the Start button to see a typical sinusoidal signal.

You can write a mathematical expression for the voltage signal as a function of time.  Call that mathematical expression V(t).  V(t) will have this general form.

This signal has three parameters, the maximum voltage, or amplitude, denoted by V_max, the angular frequency denoted by w and the phase angle denoted by f.  We'll examine these separately.

        The amplitude of a sinusoidal signal is the largest value it takes (when the sine function has a value of +1 or -1).  Amplitude has whatever units the physical quantity has, so if it is a voltage signal, like the one below, it might have an amplitude of 10 volts.  On the other hand if the shock absorbers on your car are bad, your car can run down the road and vibrate up and down with an amplitude of three inches.  For the signal we saw above - repeated here - the amplitude is 100 volts.  However, you can change the amplitude of the sinusoidal signal here by typing in a different value.  Do that now.

P1.   Here is a sinusoidal signal.  Willy Nilly has measured this signal, and acquired it in a computer file and plotted it for you.  Determine the amplitude of this signal.

        Frequency is a parameter that determines how often the sinusoidal signal goes through a cycle.  It is usually represented with the symbol f, and it has the units hertz (sec^-1 ).  Here is the simulator you saw above.

This signal repeats every 4 seconds.  We say that this signal has a period of 4 seconds, and we usually represent the period of the signal as "T".  Here we have T = 4 sec.  The frequency of the signal is the reciprocal of the period.  That's why the frequency is indicated as 0.25 in the simulator.  When "f" is the frequency, we have:

For the signal above, the frequency is 0.25 Hz.  We get that from:

Now, you can also change the frequency in the simulator above.  Change the frequency and observe what happens.  Try a frequency of 0.5 Hz, 1 Hz, etc.

P2.   Here is Willy Nilly's signal again.  Determine the frequency of this signal.  Hint:  First determine the period of the signal.

In this expression:

• A is the amplitude,
• f is the frequency.
• w = 2pf is the angular frequency

        Notice how these sounds are what a musician would call an octave apart.

        Sinusoidal signals don't need to start at zero at t = 0.  There are other possibilities.  Here are two sinusoidal signals.

These two signals have the same amplitude and frequency, but they are not the same.  The difference in the two signals is in their phase.  Phase is another parameter of sinusoids.  Consider how we might write mathematical expressions for the signals in the plot above.

• For the "red" signal, we can write:
• v_red(t) = 150sin(2p60t) = 150cos(2p60t - p/2)
• This signal has a phase angle of -p/2.
• For the "blue" signal, we can write:
• v_blue(t) = 150sin(2p60t-p/2) = -150cos(2p60t)
• This signal has a phase angle of -p radians.
• In general, any sinusoidal signal can be written as:
• v_signal(t) = Asin(2pft + f), where:
• A = amplitude,
• f = frequency,
• f = phase.

• Any time you use a sinusoidal signal you have to make an arbitrary decision about where the time origin (t = 0) is located.
• If you have just one signal you can often choose the time origin to be the instant when the signal goes through zero.  Then
• If you have more than one signal, you can often choose one of the signals as a reference - with zero phase - and measure phase from that reference.
• In the example above, we chose the red signal as the reference, and the blue signal has a phase of -p/2 radians.
• We have used the sine function here, but we could also have done everything with cosines.

P3.   Here are two signals.  Both signals have the same amplitude.  Determine the amplitude of the two signals.

P4.   Next, determine the frequency of the two signals.

P5.   Now, determine the phase of the "blue" signal assuming that the "red" signal is the reference.  Give your answer in radians.

        Sinusoidal signals aren't always the most interesting kind of signal.  They keep doing the same thing over and over.  However, other signals which contain information can often be thought of as combinations of sinusoidal signals.  That includes periodic signals - which repeat in time but not sinusoidally - and even non-periodic signals.  Even totally random signals are often viewed as having frequency components and that concept is borrowed from concepts that first arise when you consider sinusoidal signals.

        So, even if you don't ever see a sinusoidal signal again, you may well be trying to deal with sinusoidal components.  As you get into the study of signals you'll deal with numerical algorithms - like the FFT - that decompose signals into sinusoidal components.  You'll find many uses for whatever you learn about sinusoids.

        There are many other kinds of signals besides sinusoidal signals.

• Some signals are periodic, but not sinusoidal.  Those signals will have a Fourier Series representation.  That's just a way of representing periodic signals as sums of sinusoids.  Fourier Series are at the root of numerical algorithms like the Fast Fourier Transform - the FFT - and are widely used in the analysis of signals.
• Other signals may not even be periodic.  Those signals are also of interest, and we can look at how those signals look.

Now, difficult as it is to imagine, this signal can be represented as a sum of sinusoidal signals.  While that is a subject for another lesson, you should be motivated to learn everything you can about sinusoidal signals.  The things you don't learn can prevent you from continuing when you get to concepts like representing this signal with a sum of sinusoidal signals.

        Information carrying signals will vary in time.  A constant signal really doesn't convey any information.  However, when you are dealing with time-varying signals things that you know about constant signals may still hold true.  In particular.

• KVL still holds for time-varying signals.  In fact, KVL holds at every instant of time.
• KCL also is true at every instant of time.

• Problem Signal1P01 - Millie needs a 300v source
• Problem Signal1P02 - Millie reverses the leads.
• Problem Signal1P03 - The Wall Plug Voltage