Even though third-generation (3G) wireless deployments are still on the horizon, the wireless design sector has not slowed down its efforts to bring data capabilities to existing wireless handset and base station designs. This move-forward mentality sparked the development of so-called 2.5G technologies that would allow wireless operators to deliver voice and higher-speed data services (up to 384 kbps)without having to rip out the entire infrastructure.
One of the key 2.5G technologies capturing interest in the design community is the enhanced data rate for GSM evolution (EDGE) specification. Delivering data rates up to 384 kbps, EDGE is an evolutionary technology for current GSM systems, allowing these systems to offer both voice, data, Internet, and other connectivity solutions.
EDGE has stepped into the spotlight for several reasons.
The first is its ability to deliver higher data rates without the need for new infrastructure. Another key benefit is sheer data-rate performance. By offering up to 384-kbps data rates, EDGE provides a significant speed increase over current wireless architectures, which at best deliver 14.4-kbps data rates.
Packing more data traffic into a given bandwidth is requiring the use of more sophisticated signaling formats and receivers. Both the design of new systems and the testing of their prototypes have become correspondingly more complex.
To combat the design challenges provided by EDGE, system developers can turn to digital signal processing (DSP) techniques. Through digital simulation, developers can test new receiver designs before committing them to hardware. For receiver testing, classical statistics like error vector magnitude (EVM) may be accurately predicted for complex signals using DSP software techniques. To better illustrate this point, let's take a more in-depth look at the simulation of digital communication links.
Link simulation basics
The most often-used measure of performance for digital communication systems is average bit-error rate (BER). BER is simply the probability that a transmitted bit will be received in error at the destination. Channel disturbances can transform a logical 0 into a logical 1 and vice versa. This single measure of performance is often adequate for characterization of the end-to-end link quality by the end user.
A plot of BER versus signal-to-noise ratio (SNR), expressed as Eb/N0 (dB), is shown in Figure 1. This plot describes the decline of the probability of a bit error with increasing SNR at the receiver output. Four different types of curves are displayed in Figure 1.
Other statistics are often easier to gather, however, and can be used for early assessments of candidate system architectures. Communi-cation systems are increasingly employing complex algorithms such as error-control coding, equalization, multi-user detection, and many others. Gathering end-to-end BER for such systems requires a model for the entire link design, which may not be known in the early stages of the system development. Also, evaluating the entire design may be cumbersome and expensive. Measures of performance that can provide early quick-and-dirty assessments of a candidate system or subsystem designs are highly desirable.
As an example, in coded systems it is often possible to predict BER at the output of an error-correcting decoder by measuring the raw symbol error rate at the output of the receiver. For example, Figure 2 illustrates a simple quadrature phase-shift keying (QPSK) system employing error-correcting coding. In a system such as this, the demodulator could make hard decisions on each received symbol and declare its best estimate xi, of an original transmitted symbol, xi. This is sometimes called the symbol error rate or raw BER as opposed to coded BER.
In Figure 2, the raw BER is displayed in open squares overlaid on a theoretical curve for raw BER in the presence of additive white Gaussian noise (AWGN). Since there are no sources of distortion in the system used here, we would expect the curves to agree within statistical limits.
Predicting coded BER
In some cases, it is possible to predict coded BER based on knowledge of symbol error rate. On AWGN channels, bounds on coded BER can be developed using the symbol error rate and certain parameters of the error correcting code.
For example, in Figure 1 a bound on the coded BER performance of a rate r = 1/2, constraint length k = 7 convolutional code decoded with the Viterbi algorithm is shown. The bound is an upper bound (represented by a solid purple line). The actual Monte-Carlo simulation results of the end-to-end coded system are shown as purple boxes. Note that for interesting values of BER, the curves agree closely.
The type of bound shown in Figure 1 can be calculated given knowledge of only the code and the received SNR, Eb/N0. If Eb/N0 is known or can be measured, coded BER may be closely approximated. This can sometimes obviate the need for extensive Monte-Carlo simulations of the end-to-end coded link at the target BER.
In the simulation shown in Figure 2, the AWGN level is exactly controlled by adjusting the standard deviation of Gaussian noise samples that are added to the sampled-data representation of the signal waveform. Thus, the SNR conditions at the receiver output are known and BER can be tabulated as these conditions are varied. This is not the general case in either a software simulation or a laboratory test of a communication system.
In the case of a software simulation, there may be various additional sources of noise and distortion that can affect the characteristics of the receiver decision statistic, ri. Even if we assume that the noise statistics are Gaussian at the receiver output, the actual SNR must often be measured. In the laboratory, we have less accurate control over the effect and calibration of the various subsystems, so an accurate measurement of receiver output characteristics is essential.
The receiver output statistic, {ri}, is plotted in the display shown in the lower right of Figure 2. This is the classical scatter diagram and is constructed by plotting the quadrature versus the in-phase demodulator outputs. In this particular plot, 256 symbol outputs are overlaid.
In Figure 2, the AWGN has been adjusted to provide a received Es/N0 of 30 dB. At this value of Es/N0, even raw BER coherent detection is extremely small. Under these conditions, an estimate of received Es/N0 can be made by simply subtracting constellation points corresponding to the receiver's hard decisions from the receiver's normalized decision statistic.
"Normalized" refers to the fact that the receiver outputs have been scaled so that Es = 0 dB. Since the system model used here is linear, the result is just a sequence of complex Gaussian variables with variance N0/2. The value of N0 can be estimated by taking the average of the square of the magnitude of these complex numbers. The average should be taken over a sufficient number of symbols so that the statistics of any random phenomena inherent in the system or the channel (in this case AWGN) are averaged out. This may require hundreds, thousands, or even millions of symbols.
The magnitude of the deviation of the receiver's complex output statistic from the ideal signal constellation points refers to an EVM. EVM has proven useful for characterizing the distortion effects of non-linear circuit components such as amplifiers.
EDGE signal basics
The GSM mobile communication standard is now in place throughout most areas of the world.1 Developed in the mid-80s, it is a mature communication technology and provides reliable service. GSM systems operate using a Gaussian minimum-shift keying (GMSK) modulation scheme that has a constant envelope.2
In order to provide a more bandwidth-efficient signal for high data rate applications, the EDGE signaling format has been developed as an upgrade for current GSM architectures.3 The EDGE format is an 8-ary MPSK signal that is implemented as a linear I&Q modulation with special pulse shaping filters. The EDGE signaling format was designed so that spectral and other characteristics would be compatible with and suitable for overlaying on existing GSM and TDMA systems employing GMSK.
Whereas the phase modulation in EDGE is a linear function of the input coefficients, the continuous phase modulation (CPM) of GMSK has a non-linear dependency on these same coefficients. Depending on how GMSK is to be demodulated/detected, different types of precoding and interleaving of transmitted bits may be employed.4 Our interest here is in describing the transmitted output waveform characteristics only.
Constructing the signal
The EDGE signal is constructed by mapping incoming bits into 8-ary symbols, which correspond to points in an ordinary 8-level phase-shift keying (8PSK) signal constellation. These signal points are taken as I&Q modulating coefficients, which are input to the corresponding rails in an I&Q modulator. An additional step in the EDGE modulation is a rotation of 3/8 
for succeeding symbols.
Simulated transmitter output waveforms for both GMSK and EDGE signals are shown in Figure 3 . Complex-envelope representation is employed in this simulation. I&Q envelopes of the RF signal are shown overlaid in the bottom left of the figure. A sampled-data approximation to the transmitted analog waveform is employed, with 8 samples per channel signaling interval. Note that both GMSK and EDGE waveforms are smooth.
Transitions between symbols occur gradually in EDGE and GMSK signals, producing a compact spectrum. The spectral characteristics of the modulation schemes used in both systems are very similar, as shown by the overlaid traces in the bottom spectral plot. In this case, spectral characteristics were averaged over 10,000 randomly generated symbols.
The final plot in Figure 3 is the I&Q trajectory plot for the two modulations. The GMSK trajectory has a constant envelope. Since the EDGE signal is not constrained to possess a constant envelope, the I&Q plot shows variation in both magnitude and phase. This signal format places more stringent requirements on amplifiers in terms of linearity.
ISI effects
Like the GMSK signal employed in GSM, the EDGE signal employs a modulating pulse with leading and trailing skirts, which extend into neighboring symbol intervals. Unless special care is taken, this type of pulse can cause successive received symbol statistics to interfere with each other at the receiver output. Such interference causes one symbol to interfere with the voltage level of some of its predecessors and successors.
It is possible, with some pulses called Nyquist pulses, to achieve independence between receiver outputs {ri} even though the pulse extends for more than one symbol.5 These pulses allow trailing and leading skirts of successive pulses to cancel each other. This theory has been developed within the context of linear modulation types, such as EDGE.
When considering the characteristics of the pulse shape of a linear modulation, it is the cascade of the modulator and demodulator filters that must be considered. In order to optimize the available Es/N0 at the receiver output, the demodulator filter should also be matched to the modulator filter. To satisfy both goals, it is the square root of the frequency response of the Nyquist filter that must be employed in both modulator and demodulator. The use of a raised cosine response for the cascade and the resulting root-raised cosine terminal filter responses is a favorite choice.
Unfortunately, the selection of the EDGE modulator pulse filter is based on achieving compatibility with existing GMSK signals, not Nyquist filtering considerations. Clearly, the EDGE pulse shape does not correspond to the square root of a Nyquist filter. Therefore, it is not possible to select a demodulator filter that simultaneously matches the modulator filter and does not produce intersymbol interference (ISI).
Many options are possible for the choice of a sub-optimum filter. In the case when the modulator filter does not have nulls in its spectrum, a demodulator filter may be constructed, causing the cascade to have a Nyquist response. In this case, available Eb/N0 may be compromised, but ISI will be eliminated at the demodulator output.
A matched filter might also be employed. Using a matched filter, however, can create substantial ISI at the demodulator output, making the raw receiver outputs unusable.
DSP receiver techniques
In addition to the traditional RF approaches, DSP techniques have also emerged as a key method for performing filtering within modern receivers. The signal processing employed in typical EDGE receiver designs often includes algorithms that deliver sophisticated equalization and coding schemes. The algorithms employed within these functions can process raw received symbols in an optimum fashion to actually improve end-to-end BER performance. These algorithms can also be employed to perform traditional measurement techniques, such as EVM.
EVM measurements become troublesome, however, for signals with significant ISI. In this case, even on a linear channel with negligible noise or distortion, the scatter diagram of the demodulator may be highly dispersed.
The ISI that results from matched filtering of an EDGE signal is severe. Because the EDGE pulse is 5 symbols in duration, complete equalization requires consideration of the effect of four preceding and four succeeding symbols on the current receiver decision statistic. The total number of symbols (nine in this case) is sometimes referred to as the ISI span.
A maximum likelihood sequence estimator employing the Viterbi algorithm requires a computational complexity that is proportional to the modulation alphabet size raised to this power.5 It is obviously impractical to implement an equalizer that provides full equalization in this case. Since we know the original transmitted symbols (having generated them), it is a simple matter to construct an ideal equalizer that undoes the dependency on the neighboring symbols that was imposed by the cascade of the modulation/demodulation filters.
The ideal equalizer
An ideal equalizer is simply a tapped delay line, or transversal filter, with all but the center tap connected to symbol positions across the ISI span. The tapped delay line is fed with the known symbol stream, with the ISI coefficients {In}: n = 0, +-1, +-2..., serving as the tap gains. In this case, the center tap, corresponding to n = 0, is not connected.
In general, the coefficients may be complex. We must know the ISI coefficients between symbols at the receiver output in order to construct such a device. These coefficients can be constructed by calculating a cross-correlation between transmit and receive pulse shapes. For the case of the EDGE pulse employed in both terminals, the ISI coefficients are {1.1E-11, 2.5E-07, 6.63E-04, 5.65E-02, 5.13E-01, 5.65E-02, 6.63E-04, 2.50E-07, 1.1E-11}.
A sub-optimum approach
Since many of the ISI coefficients are very small, it may be possible to implement a sub-optimum equalizer that performs well, yet has a much reduced complexity. In this case, the ideal equalizer will provide a useful performance benchmark with which to compare sub-optimum schemes.
The scatter diagram in Figure 4 shows the output of a matched filter demodulator for the EDGE signal. In this figure, 25,600 symbols are overlaid. There is no noise or any source of nonlinear distortion in this simulation. The red stars represent the raw, or unequalized, symbol outputs. Note that the pattern is highly dispersed even in this ideal test. In this case, the scatter diagram would be useless for even qualitative assessment of the distortion effects.
In Figure 4, the ideal constellation points gathered from the modulator are shown in open squares. The symbol values after passing through the ideal equalizer are shown in solid dots. Note that they fall well inside the ideal constellation points. In the example provided in Figure 4, the number of taps in the equalizer has been limited to five. The inclusion of additional taps does not make any significant difference, due to the small values of the end ISI coefficients. As can be seen from the scatter diagram highlighted in Figure 4, the capability to perform ideal equalization allows the scatter diagram to remain a useful predictor of system performance, even for communication systems that do not make final bit decisions based on raw symbol values alone.
By removing signal dispersion that is caused by ISI, the residual dispersion due to other distortions is more clearly revealed. This is critical in the evaluation of noise and distortion effects that may occur within a practical implementation of a system.
Figure 5 shows a simple model of an EDGE system operating with a power amplifier model. Power amplifiers can be modeled through a description of their AM/AM and AM/PM characteristics, which is the model employed here.
The I&Q plot on the left shows the input to and the output from the amplifier overlaid. This particular amplifier is not driven very deeply into saturation, operating mainly in its linear region. Some clipping is evident on the most extreme excursions of the signal.
The scatter plot to the right displays the dispersion in the output of the receiver. This amplifier is causing moderate dispersion about the nominal received constellation points.
The final plot at the bottom of the figure plots a histogram and amplitude probability distribution for the EVM. From this plot, accurate details about the root mean square (RMS) EVM, percentile levels, and other useful characteristics of this statistic can be gleaned.
Eliminating errors
As EDGE-based systems begin to rollout, designers will need to focus more attention on finding and eliminating errors in RF signals. Through the use of DSP technology applied to traditional RF measurement techniques, such as EVM, designers of wireless handsets and base stations can quickly track down errors and improve the performance of their 2.5G designs.
Dr. Kurt Matis is the director of systems research for Applied Wave Research Corp. Prior to joining AWR, he has held a number of technical positions in various companies, most recently as president of ICUCOM Corp. Matis graduated from Rensselaer Polytechnic Institute in 1984 with a Ph.D. He can be reached at kurt@mwoffice.com.
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References
- Recommendation GSM 01.04, version 3.0.1 : European Standard (Telecommunications series), Digital cellular telecommunications system (Phase 1), February, 1991.
- Implementing GSM Simulations Using ACOLADE, Applied Wave Research Application Note, 1998.
- Recommendation GSM 05.04, version 8.1.0; European Standard (Telecommunications series), Digital cellular telecommunications system (Phase 2+), Modulation, Release 1999.
- Pasupathy, S.,"Minimum shift keying; a spectrally efficient modulation," IEEE Communications Magazine, vol. 17, No. 4, pp. 14-22, July 1979.
- Proakis, J.G., Digital Communications, 3rd ed., McGraw-Hill, 1995.
