The arbitrary waveform generator (AWG) is a useful tool for a wide range of testing applications. Integrated into an ATE system, it allows powerful and flexible testing of complex, mixed-signal devices. Some devices, however, require AWG performance that is either unavailable or excessively expensive.

By bending the normal rules for determining the available performance of an AWG, we can extend its useful performance. For example, the AWG3™ in the Credence Duo™ has a 1-GHz maximum sample rate, which normally is used for frequencies below the Nyquist rate of 500 MHz. But by playing some interesting games, it is easy to use this instrument to generate useful signals in* *excess of 600 MHz.

Review of Sampling Theory

A continuous time-domain sinusoid has Fourier spectral components (complex sinusoids) at plus and minus its frequency, *f,* as shown in **Figure 1**. For a real signal, the phases of the real parts of the components add, and their imaginary parts cancel.

After sampling to the discrete time domain, the result is Fourier spectral components or images at all multiples (including zero) of the sampling frequency, Fs, plus and minus *f*, as shown in **Figure 2**. It is impossible to determine if the original frequency was *f,* or Fs – *f*, or (N × Fs) ±* f*.

**Figure 3** shows that the samples at 1-ns intervals occur at the same points for sine waves at 100 MHz and 900 MHz. This effect is called aliasing, the reason for the Nyquist criterion. To represent and uniquely reconstruct a continuous signal, the signal must be sampled at a rate at least twice the highest frequency in that signal. Sampling instruments usually have low-pass (anti-aliasing) filters to eliminate frequencies above Fs ÷ 2 before sampling.

Breaking the Rules

In reality, the Nyquist rules state that the sampling rate must be at least twice the bandwidth of the signal. If we work with signals that have narrow bandwidths at high frequencies, then we can violate the typical interpretation of the sampling theorem.

Since the sampled waveform contains all of the possible frequencies (N × Fs) ± *f*, we can reverse this process to generate any desired frequency using any lower sampling frequency.

Normally, AWGs have reconstruction (low-pass) filters to eliminate frequencies above Fs ÷ 2 from the output. Substituting bandpass filters isolates the desired frequencies from the myriad of frequencies present.

Danger Zones

While this sounds simple, there are practical limitations. First, the math assumes that the sampling process involves impulse samples. Most digital-to-analog converters (DACs) generate stepped, not impulse, outputs. Stepwise reconstruction produces a low-pass filtering effect, known as sinx ÷ x, that limits higher frequency spectral images.

**Figure 4** shows the stepwise reconstruction of the 100- or 900-MHz sampled waveform and the frequency response from 0 to 2 GHz of the sinx ÷ x effect at the 1-GHz sampling rate. The 100-MHz component is present, but not the 900-MHz component.

Second, the output bandwidth of a low-sample-rate DAC may not be very high, adding another low-pass filtering effect. Third, as the N in (N × Fs) ± *f* increases, the band-pass filter becomes more problematic as its total pass and transition bandwidth of Fs ÷ 2 become increasingly narrow with respect to N × Fs.

Filter sharpness also makes it difficult to operate near multiples of Fs and Fs ÷ 2 due to the proximity of the other images. As a result, the most practical frequency band for applying this technique is approximately 0.6 Fs to 0.9 Fs.

Example Application

This technique was used with a 1-GHz sampling rate to generate sinusoidal clocks at 622.08 MHz (0.62208 × Fs), with and without a sinusoidal jitter (phase modulation) component. This allows testing of OC-12 ATM/SONET devices for clock jitter tolerance per Bellcore TR-NWT-000253.

Phase modulation adds side lobes to the spectrum of the sine wave, spaced at the modulation frequency, with amplitudes determined by the modulation amplitude. For example, with 10 Unit Intervals (UIs) peak-to-peak of jitter at 960 kHz, the significant side lobes extend from 585 MHz to 659 MHz.

Hardware and Software Used

The primary instrument was the Credence AWG3 1-GS/s (1-ns sample interval) AWG installed in a Duo tester. It was placed in the ultra-high-bandwidth mode, with no reconstruction filtering and a raw output bandwidth of 750 MHz, excluding sinx ÷ x.

Cascading two RF filters produced a nominal passband from 545 MHz to 770 MHz. Results were sampled with the tester’s Analog Capture Port (ACP) and analyzed in the time and frequency domains with the Analog Wavetool™ (AWT). Measurements also were made with the WaveCrest DTS-2070 Digital Time Scope.

Results

**Figure 5** shows the captured sine wave before and after filtering. Since the filtered version appears very pure when viewed in the time domain, the frequency-domain analysis provides more useful results. Specifically, the relative strength of the 622.08-MHz component vs the 377.92-MHz image component is 33.2 dB.

For the jittered signals, graphical time-domain viewing, superimposed over the pure sine, allows easy verification of the amplitude and frequency of the phase-modulation component. The frequency-domain results were as expected, as shown in **Figure 6**.

The measurements obtained with the DTS-2070 verified good performance and correlated well with the results obtained with the ACP and AWT. A much more detailed analysis of the results is available.^{2}

Summary

The performance of an AWG may be extended to produce signals above the Nyquist rate. These frequencies, normally filtered out, can be isolated with a bandpass filter. Factors such as bandwidth, filter performance, and the sinx ÷ x effect limit the practical range of this technique to 0.6 to 0.9 of the sampling frequency.

This application showed how a 1-GS/s AWG produced 622.08-MHz sinusoidal clocks with and without jitter for ATM/SONET jitter tolerance testing. Two methods verified performance of the methodology.

References

1. Jacob, G., “Easily Generated Complex Waveforms Accommodate Many Applications,” *EE-Evaluation Engineering*, August 1997, pp. 35-38.

NOTE: This article can be accessed on **EE**’s TestSite at www.nelsonpub.com/ee/. Select EE Archives and use the key word search.

2. Kulp, B., “‘Nyquist Voodoo’—Creating 622.08 MHz Clock Signals with Jitter for ATM/SONET Device Test Using a 1 GSample/s AWG,” Semicon Taiwan ‘97 Test Seminar, September 1997, pp. 63-85.

*About the Author*

* Barry Kulp is a senior applications engineer in the Communications Market Sector at Credence Systems. He has written and presented papers at ITC 1996 and SEMICON Taiwan 1997. Mr. Kulp holds a B.S.E.E. from MIT and a patent in the field of medical electronics. Credence Systems, 9000 S.W. Nimbus Ave., Beaverton, OR 97008, (503) 350-7537, e-mail: [email protected].*

**Copyright 1998 Nelson Publishing Inc.**

September 1998