Agilent Technologies E1441A user service Signal Imperfections, Appendix C

The maximum output frequency, with the condition that every waveshape point in

RAM is output every waveform cycle, is defined by:

Fout = Fclk / Points

The minimum number of points required to accurately reproduce a waveshape will determine the maximum useful output frequency using the same equation.

The rule governing waveforms is referred to as the Nyquist Sampling Theorem, which states that you must include at least two points from the highest frequency component of the signal you are attempting to reproduce.

Signal Imperfections

Most signal imperfections are easiest to observe in the frequency domain using a spectrum analyzer. Sampling theory predicts the location and size of spurious signals resulting from the sampling processes used by DDS generators. In fact, since DDS generators use a fixed sampling rate (40 MHz for the Agilent E1441A), spurious signals can be removed with a fixed frequency “anti-alias” filter. A 17 MHz, ninth-order elliptical filter providing a sharp cut-off (in excess of 60 dB attenuation for signals greater than 19 MHz) is used for sine wave outputs. A 10 MHz, seventh-order Bessel filter is used for non-sine wave outputs. The Bessel filter provides slower amplitude roll-off for anti-alias filtering, but maintains linear phase response to minimize shape distortion for complex waveshapes. The Agilent E1441A automatically selects the appropriate filter when the output function is selected.

All digital-to-analog converters, including those used in DDS generators, produce spurious signals resulting from non-ideal performance. These spurious signals are harmonically related to the desired output signal. At lower frequencies, the Agilent E1441A's 12-bit waveform DAC produces spurious signals near the -74 dBc level (decibels below the carrier or output signal) as described by the equation on the following page. The Agilent E1441A uses the complete vertical resolution (N=1) of the DAC for all internal waveshapes, thus minimizing amplitude quantization error.

At higher output frequencies, additional DAC errors produce non-harmonic spurious outputs. These are signals “folded back” or aliased to a frequency within the signal bandwidth. A “perfect” DAC will also produce a wideband noise floor due to amplitude quantization. The noise floor for a 12-bit DAC will be near the -74 dBc level; this corresponds to a noise density of -148 dBc/Hz for sine wave outputs from the Agilent E1441A.

Amplitude Quantization ≤ – ( 20 x log10( N x 4096 ) + 1.8 ) dBc

where “N” is the fraction of available DAC codes used to describe the signal waveshape (0 ≤N ≤1).

Another type of waveform error visible in the frequency domain is phase truncation error. This error results from time quantization of the output waveform. Whenever a waveshape is described by a finite number of horizontal points (length), it has been sampled in time (or quantized) causing a phase truncation error. Spurious signals caused by phase truncation introduce jitter into the output waveform. This may be regarded as time (and phase) displacement of output zero crossings.

Phase truncation causes phase modulation of the output signal which results in spurious harmonics (see the equation below). For lower output frequencies, the phase accumulator periodically does not advance RAM addresses, causing the DAC to deliver the same voltage as recorded on the previous clock cycle. Therefore, the phase “slips” back by 360° / points before continuing to move forward again. When RAM address increments are the same on each cycle of the output, phase truncation error (and jitter) are essentially zero. All standard waveshapes in the Agilent E1441A are generated with at least 16,000 waveform points which results in spurious signals below the wide-band noise floor of the DAC.

Appendix C Agilent E1441A Function Generator Tutorial 155