Example Code

Overview

A variety of communication protocols implement quadrature amplitude modulation, or QAM. Current protocols such as 802.11b wireless Ethernet (Wi-Fi) and Digital Video Broadcast (DVB), for example, both utilize 64-QAM modulation. In addition, emerging wireless technologies such as WiMAX, 802.11n, and HSDPA/HSUPA (a new cellular data standard) will implement QAM as well. Thus, understanding the QAM modulation scheme is important because of its widespread use in current and emerging technologies. QAM modulation involves sending digital information by periodically adjusting the phase and amplitude of a sinusoidal electromagnetic wave. Each combination of phase and amplitude is called a symbol and represents a digital bitstream. First, we will discuss the hardware implementation required to constantly adjust the phase and amplitude of a carrier wave. Second, we will discuss the binary value associated with each symbol.

Hardware Implementation:

On a hardware level, quadrature amplitude modulation (QAM), requires changing the phase and amplitude of a carrier sine wave.   One of the easiest ways to do this is to generate and mix two sine waves which are 90° out of phase with one another.  By adjusting only the amplitude of either signal, we are able to affect the phase and amplitude of the resulting mixed signal. 

These two carrier waves represent the I and Q components of our signal.  Individually each of these signals can be represented as:

Above, the signal I is the “in-phase” component and Q is the “quadrature” component.  Note that these are represented as sin and cos because the two signals are 90° out of phase with one another.  As a result of the two identities above, we can subtract the two signals to get:

As the equation above illustrates, the resulting identity is a periodic signal whose phase can be adjusted by changing the amplitude of I and Q.   Thus, it is possible to perform digital modulation on a carrier signal by adjusting the amplitude of the two mixed signals.

Below, we show a block diagram of the hardware required to generate the IF (intermediate frequency) signal.   In the “Quadrature Modulator” block, we can see how the I and Q signals are mixed with LO (local oscillator) before being added together.  Again, the two local oscillators are exactly 90° out of phase with one another. 

Next, we will discuss exactly how the I and Q components are used to represent actual digital data.   To do this, we will discuss the relationship between the QAM symbol map and the actual real-world signal.

The QAM Symbol Map:

Again, QAM modulation involves sending digital information by periodically adjusting the phase and amplitude of a sinusoidal electromagnetic wave.   With 4-QAM modulation uses exactly four combinations of phase and amplitude.  Moreover, each combination has an assigned 2-bit digital pattern.  For example, suppose we wish to generate the bitstream: (1,0,0,1,1,1).  Because each symbol has a unique 2-bit digital pattern, these bits are grouped in two’s so that they can be mapped to the corresponding symbols.  Here, the original bitstream, (1,0,0,1,1,1), is grouped into the three symbols (10,01,11).  Below, we show a modulated waveform (without pulse-shape filtering), such that each symbol is represented by one period of the modulated carrier.  Here we can see that digital information is transmitted by changing the phase and amplitude of the carrier signal. 

One common way to visualize the transitions in phase and amplitude of our carrier wave is with a constellation plot.   The constellation plot, shown below, shows each possible phase and amplitude of a carrier signal in polar coordinate form.

Below, we show the symbol map of 4-QAM.   As the image also illustrates, each symbol can be represented by a unique phase(Θ) and amplitude(A).  Here, 4-QAM consists of four unique combinations of phase and amplitude.  The combinations are called “symbols” and are shown as the white dots on the constellation plot.  The red lines represent the phase and amplitude transitions from one symbol to another.  Note that we have also labeled the digital bit pattern that be represented by each symbol.    Thus, a digital bit pattern can be set over a carrier signal by through generating unique combinations of phase and amplitude.

As we have already mentioned, it is possible to send up to two bits per symbol when using 4-QAM modulation.   However, it is also possible to send data at even higher rates by increasing the number of symbols in our symbol map.  However, it is also possible to send data at even higher rates by increasing the number of symbols in our symbol map.  By convention, the number of symbols in a symbol map is called its “M” and is considered the “M-ary” of the modulation scheme.  In other words, 4-QAM has an M-ary of 4 and 256-QAM has an M-ary of 256.  Moreover, the number of bits that can be represented by a symbol has a logarithmic relationship to the M-ary.  For example, we know that 2 bits can be represented by each symbol in 4-QAM.  While this makes sense intuitively, it is actually defined by the equation shown below:

Bits per Symbol = log₂ (M)

Thus, using the equation above, each symbol in 256-QAM can be used to represent an 8-bit digital pattern (log₂ (256) = 8).  Because the M-ary of a QAM modulation schem affects the number of bits per symbol, it has a significant affect on the actual data transmission rate.

Demonstration:

The following demonstration will take the user through a three-step process of modulating a digital bitstream onto a carrier sinusoidal waveform.

1. First, open the VI entitled “QAM symbol map.vi.”  Click on the run button in LabVIEW to begin execution.  As the diagram below illustrates, the user can select a custom bitstream by modifying the Boolean array control at the top of the screen.  As the second control indicates, every four bits of the initial bitstream is grouped four bits at a time.  These 4-bit digital patterns can then be used to represent unique symbols.  To observe this behavior, click on the initial bit pattern array to change the initial bitstream.  Then, observe, how this affects each of the 4-bit digital patterns.

   

2. Next, we will observe the relationship between each of the four bit digital patterns relate to the corresponding QAM symbols.  To see this, click on the initial bit pattern array again to change the bitstream.  This time, observe the changes in the “IQ in Polar Form” indicators.  As you can see, each 4-bit digital pattern can be represented by a different symbol with a unique phase and amplitude.  The graph indicator illustrates all 16 possible symbol locations for 16-QAM.  In addition, the red dots represent each symbol that is generated with the selected digital pattern.  Finally, numeric indicator below the polar graph gives the polar representation of the symbol being generated.

   

3. Again, note that each of these symbols represents a sinusoidal carrier signal at a specific phase and amplitude.  Thus, it is natural to represent these in polar form.

4. We should also observe the relationship between the digital bit pattern and the actual carrier waveform.   Again, click on the initial bit pattern array again to change the symbols that are being generated.  In this specific case, we have chosen a carrier frequency that is equal to the symbol rate.  Thus, the phase and amplitude of our carrier represents a new symbol at the rate of exactly one symbol per period.  In the graph indicator, we have labeled the first three periods of our carrier signal to correspond with the first three symbols of the IQ waveform.  Now, as we modify our bitstream, we see the IQ waveform also change as well.  We can see from the resulting waveform below that each symbol produces a different phase and amplitude in the time domain of the carrier signal.

   

5. Finally, we will evaluate how specific changes in the symbol map correspond to a different signal in the time domain.  To start with, choose the bit pattern: “0,0,1,1,0,1,1,0,0,1,1,1.”  As we can see on the front panel, this pattern is grouped into “0011,0110,0111.”  In addition, we can see on our polar graph that each of the symbols being generated represent the maximum amplitude (1) of our carrier waveform.  This is evident in the image below:

   

6. Next, we will choose a bit pattern that produces a signal that is smaller in amplitude.  To do this, enter the bit pattern of: “1,0,0,1,1,1,0,0, 1,1,0,1.”  Again, this bitstream is broken into 4-bit groupings to produce: “1001,1100,1101.”  Observe the new symbols on the polar graph, which are also shown below.

   

   As we can see from the polar representation above, the phase of the three new symbols has not changed.  However, the amplitude of the three new symbols is smaller than the original set.  Thus, we can see this effect in the resulting waveform, which has the same phase as the original.

   

   To observe this affect in full, enter several different bitstreams to observe the changes in the phase and amplitude of the upconverted waveform.  One key characteristic that you will observe is that each set of four bits affect only the first symbol of the waveform.

Conclusion:

As this demonstration illustrates, digital modulation occurs by changing characteristics such as the phase and amplitude of a carrier sinusoid.   In addition, we have seen how with QAM modulation, the original bitstream can be broken into log₂(M) bit groupings.  Finally, we have seen how each set of polar coordinates can translate into an actual real world signal.