Quadrature Amplitude Modulation

QAM is a communication scheme that uses two carriers at the same frequency, but 90 degrees out of phase, i.e., in “quadrature,” to modulate two separate message signals:

s(t)=m1(t)cos(ωct)+m2(t)sin(ωct)s(t)=m1(t)cos(ωct)+m2(t)sin(ωct) size 12{s \( t \) =m rSub { size 8{1} } \( t \) `"cos" \( ω rSub { size 8{c} } t \) +m rSub { size 8{2} } \( t \) `"sin" \( ω rSub { size 8{c} } t \) } {},

where m1(t)m1(t) size 12{m rSub { size 8{1} } \( t \) } {} is the “in-phase” message and m2(t)m2(t) size 12{m rSub { size 8{2} } \( t \) } {} is the “quadrature” message. In the literature, these two signal components are often denoted “I(t)” and “Q(t),” respectively.

In practice, m1(t)m1(t) size 12{m rSub { size 8{1} } \( t \) } {} and m2(t)m2(t) size 12{m rSub { size 8{2} } \( t \) } {} can be simple analog signals such as speech or music, or they can represent digital information coded using pulse-amplitude modulation. Forward error correction, pulse shaping, and differential coding techniques are frequently applied to m1(t)m1(t) size 12{m rSub { size 8{1} } \( t \) } {} and m2(t)m2(t) size 12{m rSub { size 8{2} } \( t \) } {} to improve efficiency. The term M-QAM indicates that M voltage levels are used to encode n bits in each symbol, where M=2nM=2n size 12{M=2 rSup { size 8{n} } } {}. Often, n is chosen to be an even integer so that I and Q are each coded with n/2 bits or 2n/22n/2 size 12{2 rSup { size 8{n/2} } } {} voltage levels. For example 256-QAM represents 8 bits/symbol, with 4 bits represented by 16 in-phase voltage levels and the remaining 4 bits represented by 16 quadrature voltage levels.

An alternative representation for the QAM signal is in terms of magnitude and phase:

s(t)=E(t)cos(ωct−θ(t))s(t)=E(t)cos(ωct−θ(t)) size 12{s \( t \) =E \( t \) `"cos" \( ω rSub { size 8{c} } t - θ \( t \) \) } {},

where E(t)=I2(t)+Q2(t)E(t)=I2(t)+Q2(t) size 12{E \( t \) = sqrt {I rSup { size 8{2} } \( t \) +Q rSup { size 8{2} } \( t \) } } {} and θ(t)=tan−1Q(t)I(t)θ(t)=tan−1Q(t)I(t) size 12{θ \( t \) ="tan" rSup { size 8{ - 1} } { {Q \( t \) } over {I \( t \) } } } {}. This leads to a useful base-band or “phasor” representation of the signal: S(t)=E(t)ejθ(t)=I(t)+jQ(t)S(t)=E(t)ejθ(t)=I(t)+jQ(t) size 12{S \( t \) =E \( t \) `e rSup { size 8{jθ \( t \) } } =I \( t \) + ital "jQ" \( t \) } {}, where E(t) is the “envelope” of the signal.

Noise and Bit Errors

In digital communication systems, the objective at the receiver is to correctly select the transmitted message symbols out of a finite set. The presence of channel noise complicates the task and causes bit errors. One important performance criterion in a digital communication system is the bit error ratio (BER), which is the ratio of the number of incorrectly received bits to the total number of transmitted bits. Usually, BER (abscissa) is plotted against signal-to-noise ratio (SNR) (ordinate). SNR is typically measured in terms of Eb/N0Eb/N0 size 12{E rSub { size 8{b} } /N rSub { size 8{0} } } {}, where EbEb size 12{E rSub { size 8{b} } } {} is the energy per bit and N0N0 size 12{N rSub { size 8{0} } } {} is the noise power spectral density.

Example BER curves for different values of M are shown in Figure 1. These are easily generated in MATLAB using the “Bit Error Rate Analysis Tool” (“bertool”).

Figure 1

Figure 1. M-QAM BER Curve.

The figure shows that for low noise levels (i.e., SNR large), the BER is extremely small. However, as noise increases beyond a certain threshold level, the BER rapidly becomes unacceptable. For example, 4-QAM is reasonably robust for SNR values of 10 dB or more, but for smaller values, the BER quickly approaches 50% bit errors, at which point the signal is entirely lost. It’s also interesting to note that larger values of M, i.e., more bits per symbol, require significantly higher SNR to provide comparable bit-error performance.

Pulse Shaping

Real-world communication channels have limited bandwidth, either because of physical reasons such as a band-pass frequency response, or because of regulatory constraints imposed by the FCC. Restricting the bandwidth causes pulses to spread out in time, and causes them to overlap with neighboring pulses. The result is called “inter-symbol interference,” or ISI. The conventional way to deal with ISI is to accept overlap, but carefully shape the pulses so that overlapping pulse waveforms have zero amplitude at each decision-making instant.

Since pulse shaping allows efficient communication to take place within limited bandwidth, an equivalent viewpoint is that it provides a way to minimize the bandwidth of the digital signal. For example, the ideal zero-ISI pulse can be shown to be a sinc function. It has a rectangular low-pass frequency response with cut-off frequency of R/2, where R is the symbol rate. Although the sinc pulse is not realizable (it’s non-causal and has infinite duration), a related zero-ISI pulse shape, the “raised cosine” (RC) pulse, is often used in practice. The RC pulse spectrum is flat at low frequencies, and rolls off according to (1+cos(c f)), i.e., the roll off vs. frequency has a raised cosine shape. Its bandwidth ranges from R/2 Hz (at which point it is equivalent to the sinc pulse) to R Hz. The bandwidth is controlled by a roll-off parameter alpha, where alpha=1 gives the full bandwidth pulse (R Hz), and alpha=0 gives a sinc pulse (R/2 Hz).

The final pulse shape discussed here is the “root-raised cosine” or RRC pulse. Its Fourier transform is given by P(f)=Pr(f)P(f)=Pr(f) size 12{P \( f \) = sqrt {P rSub { size 8{r} } \( f \) } } {}, where Pr(f)Pr(f) size 12{P rSub { size 8{r} } \( f \) } {} is the transform of the raised cosine pulse described above. When used in a communication system, a pulse-shaping filter with response P(f) is used in both the transmitter and receiver. Thus, the overall response is the same as the raised cosine, Pr(f)=P(f)P(f)Pr(f)=P(f)P(f) size 12{P rSub { size 8{r} } \( f \) =P \( f \) `P \( f \) } {}. Receivers whose front-end filters are matched to the received pulse frequency response are called “matched filter” receivers, and these are known to provide near-optimal performance in channels with white Gaussian noise. This explains the popularity of root-raised cosine systems, since they provide both low ISI as well as good performance in noise.

Exercise: Understanding 4-QAM and waveform displays

Set the “M-QAM” parameter to be 4 (i.e., 2 bits per symbol), and change the “TX Filter” parameter to be “none.” Keep all other parameters at their default values and start the simulation. By clicking the tabs over the waveform display, the VI provides several different ways to view the signal. These are patterned after those used in practical communication test equipment such as vector signal analyzers, and provide a great deal of information about the signal’s characteristics. Please keep in mind that the displayed signal is measured following any processing such as integrators or filters in the receiver front end (such as the RRC filter discussed above).

Constellation Graph

The constellation graph at the transmitter end is shown below. The abscissa is the in-phase component and the ordinate is the quadrature component. Since M=4, there are four dots representing four possible combinations of amplitude and phase levels, which is equivalent to 2 bits/symbol. The vertical position of a signal sample (i.e., a dot) on the graph represents one bit of information, while its horizontal position represents the second bit.

Figure 3

Figure 3. Constellation graph of 4-QAM at the transmitter end.

Eye Diagram

An eye diagram shows either I(t) or Q(t) on a standard oscilloscope display. The display is synchronized to show two or more full pulse durations, and persistence is used to show the history of many pulses. As a result, the traces show the range of possible amplitude values and shapes that the receiver must be able to deal with. An eye diagram is thus an effective way to examine the effects of distortions such as channel noise and ISI. The best time to sample the waveform and decode it is where the “eye” is most open When there is no distortion, the “eye” has the maximum opening possible, showing high separation between the “0” and “1” waveforms. When noise is present, the eye closes, indicating that voltage ranges for 1’s and 0’s are beginning to overlap and that bit errors are more likely.

Figure 4

Figure 4. Eye diagram (in-phase) of 4-QAM at the receiver end with no noise.

Exercise: 16-QAM

Example: 4-QAM, 10 dB Eb/N0

After setting the SNR (i.e., Eb/N0Eb/N0 size 12{E rSub { size 8{b} } /N rSub { size 8{0} } } {}) to 10 dB, the following eye diagram is seen. Noise is causing a great deal of waveform variation, but the eye is still open, implying that most bits will be decoded without error.

Figure 5

Figure 5. Eye diagram (in-phase) of 4-QAM with 10 dB SNR.

Exercise: Noise in QAM

Example: 4-QAM with pulse shaping

Set up the simulation to give 4-QAM with “Raised Cosine” “TX Filter” and 40 dB Eb/N0Eb/N0 size 12{E rSub { size 8{b} } /N rSub { size 8{0} } } {}. Run the simulation. The eye diagram at the receiver end is as follows.

Figure 6

Figure 6. Eye diagram (in-phase) of 4-QAM at the receiver end with “Raised Cosine” pulse shaping filter.

Since the eye diagram is similar to an oscilloscope with persistence, we can get good idea of the pulse shape being used. This plot illustrates the important properties of zero-ISI pulses: First, note that the pulses are rounded and smooth – this results in minimal transmission bandwidth. Second, note that although the pulses are much wider than a single symbol duration, there is no interference from adjacent symbols as long as the receiver samples the waveform in the exact center of the eye.

Exercises: