# Notes on modulation

Sine waves propagate better than square waves.  A square wave with perfectly vertical sides is actually impossible.  The energy required to change the voltage on a wire is proportional to the time needed to make the change.  To make an immediate voltage change in zero time requires and infinite amount of energy.  Other factors also determine what can theoretically be sent.  A square wave can be considered the infinite sum of a sin * cos product.  The square wave is built from many harmonics or sine waves at higher frequencies.  Each product in the square wave sum represents another harmonic at an integer multiple of the original frequency.

You may be familiar with harmonics from the world of music.  If a piano and a flute play the note B flat, it sounds different even though it is the same note.  This is because each instrument has different intensities for the different harmonics.  If you have unlimited bandwidth, you can transmit all of the harmonics for a square wave and the wave will appear square.  In a bandwidth limited channel (such as a phone line), the higher frequencies or harmonics cannot be transmitted.  Therefore the wave is distorted.  Music does not sound very good over a telephone line because it filters the higher harmonics.

In the diagram below, the red wave represents the value transmitted for the corresponding square wave when only 4 harmonics are sent.  Many more harmonics or frequencies must be sent to make the transmitted wave actually look square. You can enter the number of harmonics to display in the simulation below to see what the square wave will look like when restricted to the specified number of harmonics.

It becomes difficult to send square waves through a bandwidth limited channel, such as a phone line, because the higher frequencies are filtered out. Without the higher frequencies, the wave no longer looks like a square wave. This makes it difficult for the receiver to determine the actual bit value. To avoid this problem, the data is send only as sine waves.  A sine wave can be represented by the “sum” of only one sin*cos product.  Therefore no higher frequency harmonics need to be sent. The data is transmitted by modulating the sine wave with a modulator-demodulator (modem).  A modem is needed to send the digital information over a bandwidth limited phone line that doesn’t support the higher frequencies needed to construct a square wave.

Different modulation techniques include:

Amplitude— the amplitude of the sine wave is changed.  In the three diagrams below, each represents the transmission of five values where a value is transmitted during one wavelength.  Each diagram shows the transmission of 01100.  The red waveform represents the carrier frequency while the blue waveform represents the wave that would actually be sent.  For amplitude modulation, a 0 is represented by a wave that has half the energy or height in the graph.  A 1 is represented by a full sized wave. Frequency— the frequency of the sine wave is changed.  In the diagram below, 01100 is transmitted in blue where 0 is sent using a low frequency and 1 is sent using a higher frequency. Phase shift — the phase of the sine wave is changed.  The value 01100 is sent below by shifting the wave 180° to represent a 1 bit while sending the wave unshifted to represent a 0. Combinations of AM and PSM are used in Quadrature Amplitude Modulation.

The following simulation will show the waveform generated by different modulation techniques.  Select the modulation technique you would like demonstrated, the length of a signal (the number of wavelengths used to represent a transmitted signal) and the number of different possible states that can be sent at a time (the “V” in the Nyquist formula).  Enter a short series of 0 and 1 bits that are to be transmitted using this technique and then press the Display button.  You can easily view the different waveforms generated by the different techniques.

The blue wave represents the base or carrier wave and the red wave represents the signal that would be sent to transmit the specified bit pattern.

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An interesting animation of modulated waves can be found at http://www.cs.williams.edu/~tom/courses/105/outlines/CS105_122.html

Modems

The purpose of a modulator-demodulator is to convert the digital signal to a modulated sine wave.  Dialup modems can also perform phone functions such as dial, answer and hang-up.

Modems over the public phones use two carrier frequencies (full duplex) in the audio range.

There are many different types of modems in use operating at different speeds and conforming to different standards.  To function, the modems on both sides of the connection must agree to use the same standard. When modems connect, they must negotiate:

o        quality of the line

o        speed

o        options such as encoding and compression.

o        disable echo suppression

This negotiation takes about 10 seconds for a V.34 modem and is usually heard as a series of squeals from your modem.  Some systems may use a modem to connect to distant computers over a dedicated or leased line.  This line is always connected between the two points.  No dialing or hang-up is required.  Frequently these modems can operate without the initial parameter negotiation since they can be configured during installation to be compatible.