CASPER Memo 007

Effects of Mismatches Between Interleaved Analog-to-Digital Converters

D. H. E. MacMahon

June 19, 2006

Introduction

High speed analog to digital converters often achieve maximum sampling rates by interleaving slower ADCs. While this allows for very high effective sample rates, it also provides opportunities for the introduction of less desirable features into the system if the interleaved ADCs are not ideally matched in terms of amplitude response, frequency response, DC bias, etc. Through analysis and simulation, this memo explores the effects introduced by two types of potential mismatches in a dual interleaved ADC configuration: amplitude response and DC bias.

Amplitude Response Mismatch

An amplitude response mismatch between the ADCs can be thought of as multiplying the ideal sample stream by two interleaved impulse trains: one with unity gain, the other with non-unity gain. These interleaved impulse trains are equivalent to a sinusoidal waveform at the Nyquist frequency (i.e. half the sampling frequency) with a DC offset. Multiplying the ideal sample stream by this Nyquist frequency "carrier" tone introduces a frequency-reversed image of the input signal into the spectrum.

The magnitude of the image depends on the degree of gain mismatch. In the case of every other sample being zero (e.g. if one of the interleaved ADCs is "dead"), the magnitude of the image will be the same as the magnitude of the desired signal. The image cannot be detected by looking at an integrated power spectrum of noise that is band limited with a center frequency at half the Nyquist frequency because in that case the image, although reversed in frequency, exactly overlays the actual spectrum. The effect can be seen with a pure tone (not at half Nyquist) or band limited noise that is not centered at half the Nyquist frequency.

DC Bias Mismatch

A DC bias mismatch between the ADCs can be thought of as adding an impulse train at the Nyquist frequency to the ideal sample stream. This has the effect of introducing a small overall DC bias to the input signal as well as a small sinusoidal component at the Nyquist frequency. This pollutes a quite limited part of the band (only two frequencies) and is therefore a much less nefarious problem than the mismatched amplitude response.

Simulation

The above conclusions have been simulated using Octave -- an open source, largely Matlab-compatible mathematics program. The simulation script (shown in Appendix A and can be obtained from http://seti.berkeley.edu/casper/memos/adcmismatch/adcmismatch.m) performs these steps:

  1. Generates 1024*1024 (i.e. 2^20) input signal of normally distributed random numbers.
  2. Generates a 50% to 75% bandpass digital filter and creates a band limited version of the input signal.
  3. Simulates both a 20 % ADC amplitude response mismatch by copying the broadband and band limited signals and then multiplying every other sample by 1.2.
  4. Simulates a 1% of dynamic range DC offset by adding 0.05 of the RMS value to every other sample of copies of the two inputs. This presumes that the level of the input signals is 20% of the overall dynamic range.
  5. Simulates both effects simultaneously by doing steps 3 and 4 to additional copies of the input signals.
  6. Plots the integrated power spectrum (1024 points per spectrum) of all the various signals. These plots are shown in Figure 1 (http://seti.berkeley.edu/casper/memos/adcmismatch/figure1.png).
  7. Computes an "intra-spectrum correlation" by multiplying the positive frequencies of the spectra by the reversed order positive frequencies for both the ideal broadband input and the mismatched gain broadband input.
  8. Plots the magnitude and phase of the two intra-spectrum correlations. These plots are shown in Figure 2 (http://seti.berkeley.edu/casper/memos/adcmismatch/figure2.png). The intra-spectrum correlation of the ideal input shows no coherence, while that of the mismatched gain signal does show coherence (as well as a larger magnitude relative to the ideal input's intra-spectrum correlation). This demonstrates that there is an image in the mismatched gain broadband input even though the integrated power spectrum showed no obvious image.

Recommendations

When testing systems that include interleaved ADCs, a measure of the amplitude response mismatch between the ADCs should be performed. Various means of making this measurement can be contrived, but capturing raw ADC samples and analyzing them offline is perhaps the most versatile.

If the interleaved ADCs provide a means of controlling the relative amplitude response (i.e. gain) of the ADCs, then creation of an on-line calibration feedback loop is possible. In this scheme, samples of a known input are captured, image detection (e.g. via the intra-spectrum correlation) is performed, and gain compensation is calculated/adjusted until the image is minimized (e.g. coherence in the intra-spectrum correlation is minimized).

Frequency response mismatch, not discussed in this memo, is essentially a frequency dependent amplitude response mismatch, so the potential exists for multiple images depending on the the frequency response mismatch and the frequency content of the input.