November 20, 1997

Specialized test judges
DMT-amplifier performance

Richard Roberts, Harris Semiconductor

Traditional analog test techniques and specifications, which are based on sine-wave testing, don't reflect the true performance of devices that process the discrete-multitone waveform. A unique specification--the multitone power ratio--is a better performance indicator for this application.

Designing the analog channel for systems that process the discrete-multitone (DMT) waveform requires selecting the appropriate components, including amplifiers, whose specifications accurately indicate performance in the system. You need to know how various amplifier impairments, such as slew-rate limiting, cross-over dead zone, and gain compression, impact the DMT amplifier's performance.

Unfortunately, traditional measures of amplifier linearity, such as SFDR (spurious-free dynamic range) and the related THD and SINAD (SNR-plus-distortion) specifications, are based on sine-wave testing. These tests suit many applications but do not necessarily indicate an amplifier's performance when processing the DMT waveform. Statistically, the DMT waveform radically differs from a sinusoid, and, therefore, an analog impairment affects the waveform differently.

A unique performance metric, the multitone power ratio (MTPR), is a better test of performance for circuits that process the DMT waveform. Correlation data on simulated SFDR and MTPR performance supports this conclusion. Although you can sometimes see a correlation between SFDR, which is based on a sinusoidal stimulus, and MTPR performance, which is based on a DMT test vector, depending on the location of the analog impairment, adequate evidence exists that this correlation is generally untrue.

The nature of the DMT waveform

Before you analyze the correlation data, however, clear descriptions of the DMT waveform and MTPR are necessary. You can describe every communications channel in terms of bandwidth, and the frequency response over that bandwidth somewhat determines how well digitally modulated data passes over that channel. For example, designers commonly think that a telephone line has only 3 kHz or so of bandwidth. But, in fact, if you remove the loading coils, the telephone line can actually support bandwidth in the megahertz range if the modulated waveform can survive tens of decibels of gain slope. For example, the overall signal bandwidth can be as wide as 1.1 MHz for the emerging asymmetric-digital-subscriber-line (ADSL) waveform over existing phone lines.

This fact forms the basis of DMT modulation. DMT modulation partitions the available bandwidth into frequency sub-bands, or bins, and assigns a low-baud-rate QAM (quadrature AM) carrier to each bin's center frequency (Figure 1). The assumption is that over the sub-band bandwidth, the channel looks somewhat benign with minimal gain slope, even though the overall channel has significant gain slope. This frequency-bin approach is a natural one for modulation by the inverse FFT and demodulation by the FFT. Because the frequency-domain waveform comprises modulated multiple tones, it is natural to call it discrete-multitone modulation.

A drawback to DMT modulation is a bothersome peak-to-average ratio (PAR) associated with the channel waveform because of possible subcarrier instantaneous summation. For ADSL, with a possible 255 carriers, this peak can be large but fortunately occurs infrequently.

For systems that process the DMT waveform, you have to determine if traditional specifications, such as SFDR, indicate true performance or if new tests, such as MTPR, are necessary. SFDR describes the ratio between an assumed sinusoidal-stimulus signal and the first locatable spurious response. The SFDR number is somewhat loosely related to the THD and SINAD.

Alternatively, the MTPR test is a modification of the classic noise-power-ratio (NPR) technique for testing system linearity in analog frequency-division-multiplexed (FDM) systems. The ANSI standard committee T1E1.4 includes the term "MTPR" in the issue 2 release of standard T1.413 to describe the performance testing for ADSL noise and distortion.

As the standard stipulates, the MTPR test waveform comprises frequency-domain impulses uniformly spaced over a bandwidth of interest. This series of impulses periodically includes a "missing" impulse, giving the appearance of a spectral notch. Thus, the spectrum of this comblike test vector includes periodic suppressed tones (Figure 2). The time-series representation of this waveform, for the case in which every 16th tone is absent, is

where L=256 for ADSL, ohm_i=2 pi i/L, and theta,lc_i represents the starting phase of the ith tone. The DMT-carrier spacing determines the separation between the frequency impulses, and each carrier of the comb has a controlled starting phase, theta,lc_i, to constrain the PAR. Specifically, to generate the waveform, you adjust each tone's starting phase to establish a desired PAR, and you adjust the test vector's average signal level for a certain "back-off level" below full-scale. For example, the worst-case PAR occurs when summing cosine waveforms that all have a zero starting phase. By scrambling the starting phase of each term in the summation, you can establish a desired PAR between deterministic limits. (Note that the inverse FFT is a more efficient technique for generating this time series.)

The object of the test is to pass this test waveform through an amplifier and observe at the output the depth of the notches with respect to the level of the adjacent carriers; the greater the notch, the greater the MTPR. Intermodulation characteristics of the analog amplifier and the residual noise floor contribute to an undesirable filling-in of the output notches. This test technique better represents the actual scenario in the DMT spectrum, for which it is important to maintain a high signal-to-distortion ratio in each of the frequency bins. Typical aggregate MTPR requirements for ADSL are on the order of 65 dB; that is, the notches need to be at least 65 dB deep. (You may note the similarity between the MTPR test and a technique used in cable television called the composite-triple-beat test.)

Consider three analog impairments

You can determine the validity of MTPR and SFDR measurements by how well these tests gauge the effect of three well-known analog impairments--slew-rate limiting, dead zone, and gain compression--on a DMT waveform. Slew-rate limiting occurs when the dV/dt of the processed signal exceeds the amplifier's capability to respond (Figure 3a). The effects of slew-rate limiting are frequency-dependent, because dV/dt is frequency-dependent. In hardware, the slew rate can be frequency-dependent when time delays exist, so that a plot of slew rate vs frequency is not necessarily a flat line. Hence, characterizing this impairment and its effects on signal distortion can be challenging.

Dead-zone impairments come in many forms. You can model the traditional flat dead zone as a flat spot in the transfer function with a related width that you can reference to the input signal (Figure 3b). Whenever the input signal falls within this dead zone, the output becomes nonresponsive. Other forms of dead zones may not be flat and may exhibit a voltage step or even a reverse gain-slope characteristic. These characteristics can also be frequency-dependent.

Gain compression results in a transfer function for which the gain of the device decreases with increasing signal input (Figure 3c). Gain compression can be a soft nonlinearity or a hard nonlinearity. The soft nonlinearity shows up as the traditionally slight reduction in gain, whereas the hard nonlinearity represents a definite signal clip.

Analyze statistical distributions

To analyze the impact of any amplifier impairment, you must first understand the statistical nature of the stimulus signal; the distortion is directly proportional to the time a signal waveform spends in the region of the anomaly. In other words, an impaired region that the signal seldom passes through is of less significance than an impaired region in which the signal spends a lot of time.

The statistical distributions of a DMT signal and a sine wave show that the signal distributions of these waveforms are entirely different (Figure 4). The most likely signal level for the DMT waveform is 0 (Figure 4a), and for the sine wave, it is a magnitude of 1 (Figure 4b). This difference strongly implies that basing design decisions on an SFDR number or a related THD or SINAD number that uses a sinusoidal stimulus to actually process a DMT waveform is questionable. Taking a look at how the three analog impairments affect these two waveforms justifies this suspicion.

Because the DMT waveform's bandwidth can be wide (as previously mentioned, ADSL has a bandwidth of basically 1.1 MHz), comparing the DMT slew-rate requirements with those of a sine wave at the highest DMT frequency (which for ADSL would be at 1.1 MHz) is undoubtedly invalid. For example, a full-scale sine wave at 1.1 MHz is a much more severe slew-rate load than the DMT waveform of the same bandwidth.

A slew-rate histogram shows the slew-rate requirements of the ADSL DMT waveform on a per-sample basis (Figure 5). The most likely slew rate for the DMT waveform is actually around 0V/µsec, and the three-sigma point is around 2.1V/µsec. You can compare this slew-rate number for DMT with that required for the same full-scale, 1.1-MHz sine wave, which is around 4.5V/µsec. You could argue that a design that meets the sine-wave slew-rate requirement is satisfactory for DMT, which is true. But the design would also probably result in excessive die size, cost, and power dissipation.

Now, consider how varying an amplifier's slew rate impacts MTPR and SFDR values. Simulated curves of performance degradation vs slew-rate limiting for both a 1.1-MHz sine wave and the DMT waveform (the sine wave's frequency is the frequency of the DMT waveform's highest tone) show that lower slew rates predictably produce much lower values of SFDR than of MTPR (Figure 6). In other words, lowering the slew rate disturbs the sine wave far more than it does the DMT waveform.

If you test a sine wave and DMT waveform of the same peak value, the sine-wave frequency that corresponds to the same level of slew-rate-induced degradation of the DMT waveform is only 250 kHz. In other words, in terms of the effect of slew rate, a 1.1-MHz DMT waveform's performance is similar to that of a much slower sine wave.

Thus, if you base your decision about a DMT amplifier solely on SFDR levels, you could mistakenly assume that the amplifier has poor slew-rate-impact performance. However, MTPR tests may say that the same amplifier is adequate for the DMT application.

By reviewing the probability distribution of the waveform amplitudes in
Figure 4, you can also predictably determine that a dead zone near zero has more impact on the DMT waveform than it does on the sine wave. Likewise, you would suspect that a dead zone closer to the amplifier's upper peak-voltage range would affect the DMT waveform less than it would the sine wave. Figure 7 confirms these suspicions; note how much higher the MTPR levels are between 0.5 and 0.9V. Interestingly, notice that the SFDR is somewhat insensitive to the dead-zone location between values of 0 and 0.7. Because a flat-dead-zone model produced these results, this type of dead zone has a smaller impact on the sine wave than it does on the DMT waveform as the dead zone moves toward the peak value; the sine wave is somewhat "flat" at the peak value anyway.

Compression performance

To understand the impact of gain compression on the DMT waveform, you need first to consider the PAR, or crest factor, associated with each waveform, because gain compression most strongly affects the waveform's peak value at the onset of the compression. Specifically, you need to consider how frequently this peak value occurs. For a sine wave, the peak value occurs every cycle and is related to the average value by the square root of 2. A DMT waveform's "peak" value occurs rather infrequently, but the peak value can be extreme. For example, for ADSL, a PAR of 5.3 occurs about once every 10⁷ samples, based on typical ADSL sample rates.

Plotting the theoretical probability density function (PDF) of the PAR for the ADSL waveform reveals that the ADSL DMT PAR can be large compared with that of the sine wave (Figure 8). (Note that the PAR for a sine wave is 1.414 and occurs every cycle.) The PDF shows that the extreme peak values for DMT occur somewhat infrequently, with the most likely DMT PAR value being about 3.2. So again, a difference exists between the statistical nature of the two waveforms. Specifically, although designing an amplifier to handle the peak value of the DMT waveform is more difficult than designing for the same rms-value sine-wave peak, the impact on the waveform distortion of failing to cleanly handle the peak value is radically different.

Thus, if an amplifier lacks adequate head room to cleanly handle a given peak value, this amplifier will have a greater impact on the sine wave than it will on the DMT waveform. That is, for the same peak value, the sine wave is more distorted than the DMT waveform. This statement is true for cases of hard compression, or "clipping" (Figure 9a), and soft compression (Figure 9b). The soft-compression simulation is based on a soft limiting arctangent nonlinearity. These results are no surprise when you consider how often the peak occurs for the sine wave.

To learn more about MTPR testing, visit Harris' Web site, www.semi.harris.com/appnotes/ and click on application note AN9718.

References

1. Asymmetrical Digital Subscriber Line (ADSL) Metallic Interface, ANSI T1.413, 1995.

2. Gimlin, DR, and CR Patisaul, "On minimizing the peak-to-average power ratio for the sum of N sinusoids," IEEE TCOM, April 1993, pg 631.

3. Bingham, JAC, "Multicarrier modulation for data Transmission: an idea whose time has come," IEEE Communications Magazine, May 1990, pg 5.

Author's biography

Richard Roberts is a senior principal engineer with Harris Semiconductor (Palm Bay, FL), where he has worked for 17 years. His primary role is creating "systems on silicon" by translating customer needs into specifications for Harris' IC designers. He has helped develop the Prism WLAN, ADSL, and VDSL chip sets. He holds a BSEE from the University of Wisconsin--Madison, and an MSEE and PhD from the Florida Institute of Technology (Melbourne, FL). His hobbies include playing with ham radio, coaching little-league baseball, and studying French.

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