Why Do You Need To Know About Signals?You are at: Basic Concepts - Signals - IntroductionWhat Are Signals?Signal Representation - Sinusoidal Signals
Amplitude
Frequency
PhaseWhat 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.
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.
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:
Microphone voltage = V_{mike}(t)
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:
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.
V(t) = V_{max}sin(wt + f)
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:
f = 1/T
f is in Hertz (Hz)
T is in seconds (sec)
For the signal above, the frequency is 0.25 Hz. We get that from:
f = 1/T = 1/2 = 0.25 Hz.
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.
V(t) = A cos(2pft)
In this expression:
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.
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.
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.