The Interleaving Process in Digital Bandwidth Interleaving (DBI) Scopes December, 009 Summary The signal translation aspects of Digital Bandwidth Interleaving have been explained in the White Paper Digital Bandwidth Interleaving By Peter J. Pupalaikis. One question which was not addressed there and is often asked is about the upsampling process which increases the sampling rate from 40 to 80 GS/s. Upsampling is a well documented technique for increasing the sampling rate of sampled data. To understand the similarities and differences between upsampling in the time and frequency domains, it is instructive to begin by considering the most straightforward form of upsampling--the familiar approach in which the outputs of multiple digitizers are interleaved in the time domain.. Signal Input Sampling Clock Delay Digitizer B Figure : Time interleaving to increase sampling rate using two digitizers Time interleaving uses two digitizers. Note that the input to digitizer B is delayed by one half the sample period. Consequently, digitized data from digitizer B is delayed in time by one half of the sampling period. The output data from both digitizers is read alternately with a time delay of half the sampling period. This can be done by inserting alternate zeros in each data stream and then summing them. This process is outlined in Figure. LeCroy Corporation The Interleaving Process in DBI Scopes page of 5
Figure : Interleaving in the time domain involves upsampling the acquired waveforms by inserting intermediate zeros and then adding them together with a half sample period delay for digitizer B s samples In preparation for our discussion of upsampling in DBI scopes let s look at time interleaving in the frequency domain as shown in Figure 3. We sample each input at what we will call the old sampling frequency. components which lie below fall exactly as they should. Components above are folded about the frequency into the range below. If we upsample by writing interleaved zeros, doubling the effective sampling rate we still have a problem because the original data was sampled at the old sampling rate and components above fold back. In addition, we get a lower sideband image about the new sampling frequency. So the output of digitizer A shows the original spectral component in the baseband (below old ) along with the frequency reversed image of (, the complex conjugate of ). In the band between old and the old sampling rate(also the new if we are doubling) we see the complex conjugate of and the original spectral component marked. The same thing happens with the spectrum of digitizer B with the exception that it is still delayed is indicated by the superscript D. At this point we apply a delay adjustment to make the samples from digitizer B line up with the inserted zeros in A and vice versa. The spectral diagram marked C shows this operation, indicated by the U operator. If you look at the rules you see that a spectral component subjected to both delay and delay adjust is identical to the original spectral component. Similarly, the delay/delay adjust operation applied to the complex conjugate elements results in an inversion of the conjugate. LeCroy Corporation The Interleaving Process in DBI Scopes page of 5
Amplitude A B D Digitizer B * Upsample Both Digitizers by Zero Insertion D * D D * Sample Rate Now let s return to the upsampling operation in the DBI scopes, referring to Figure 4. The upsampling process takes place in the digital signal processing circuits following the digitizers. At this point the data is numeric so all the operations are implemented in software. The key elements are shown in the block diagram below: C DU Digitizer B Adjust Delay D * U D * U DU D A Figure 3: A view of time interleaving in the frequency domain. Apply Rules - * Add A and D & Normalize Rules D - Delay U - Adjust Delay * - Conjugate X DU = X X D * U = -X* If we add the spectrum output from with the delay adjusted spectrum of Digitizer B the resulting spectrum consists of the original components and. The conjugate components have been cancelled. At this point we have effectively increased the sample rate to times the original sampling rate. There are no aliased spectral components due to under sampling the data. They have been eliminated in the summing operation. -* * New Figure 4: The digital signal processing subsystem of a DBI scope. Courtesy the presentation Recent Advances in Waveform Digitizer Technology by Peter J. Pupalaikis. The upsampling process is a function of the upsampler blocks along with the filters and mixer which follow them. Digital data entering this system have been sampled at 40 GS/s. The upsampler inserts alternate zeros into the data stream increasing the effective sampling rate to 80 GS/s. Inserting the alternate zeros mixes the original data with a 40 GHz signal. As in any mixing process this results in the original baseband data being imaged around the mixing frequency. This process is best understood looking at the frequency domain as shown in Figure 5. LeCroy Corporation The Interleaving Process in DBI Scopes page 3 of 5
Amplitude Low Band Sample Rate The input signal is spectrally separated using a diplexer into upper and lower bands each containing one half of the spectral content. The upper band with signals from 6 to 30 GHz is down converted to the baseband. Note that it is spectrally reversed. High Band Down Converted * Upsample Both Digitizers by Zero Insertion Low Pass Filter both Bands We insert alternate zeros into the waveforms in both bands. This process up-converts the sampling rate to 80 GS/s. This is also equivalent to mixing the signals with a 40 GHz local oscillator and as a result we image the baseband spectra about 40 GHz, and harmonics thereof. At this point we low pass filter the signals in both bands and we are left with the baseband spectra and a new frequency of 40 GHz due to the 80 GS/s sampling rate. The high band spectrum is at baseband and is still spectrally reversed. We mix the high band with a local oscillator in order to remove the spectral reversal and also to restore it to its original frequency range. The output of this mixer is bandpass filtered eliminating the upper sideband image, as shown in Figure 6. Up convert and Band Pass Filter Upper Band Sum Upper and Lower Bands Figure 6: A detailed look at the up conversion of the high band spectrum New The result of this process is to translate the high band up to its original position and open the baseband region. Figure 5: The frequency domain view showing the steps is processing the high and low band signals in DBI. l LeCroy Corporation The Interleaving Process in DBI Scopes page 4 of 5
Keep in mind that, due to the filtering operations, the low band only has components that extend to 6 GHz the translated highband components now extend from 6 GHz to 30 GHz this overlaps the old frequency at 0 GHz. At this point we can sum the low and high band spectra, with suitable crossover phase correction. The resulting spectrum is continuous from DC to 30 GHz sampled at 80 GS/s. So you can see that upsampling by time sample interleaving is based on phase cancellation while Digital Bandwidth Interleaving relies on filtering; but, in either case it is possible to double the sampling rate without incurring aliasing. at this sample rate the frequency is 0 GHz. If the combined inputs were not truly upsampled at 80 GS/s then we would expect the GHz signal to be aliased down to 9 GHz. The spectrum of a GHz sine input is shown in Figure 7. The scaling on the FFT in Figure 7 is 3 GHz/division horizontally and 0 db/division vertically. The GHz spectral line is 47 db above the line at 9 GHz and greater than 40 db above the other spurious responses. This is the most convincing argument that the upsampling process works well in LeCroy s DBI oscilloscopes. Figure 7: The spectrum of a GHz sine showing little evidence of a 9 GHz alias The best way to assess the performance of the DBI upsampling technique is to look at a display of a WaveMaster 830 Zi with a sine input of GHz. The input of each digitizer was sampled at 40 GS/s, References: Digital Bandwidth Interleaving By Peter J. Pupalaikis, http://www.lecroy.com/tm/library/whitepapers/pdf/dbi_e xplained.pdf Recent Advances in Waveform Digitizer Technology By Peter J. Pupalaikis, NORTH JERSEY SECTION MTT-Society & AP-Society Joint Chapter 4th ANNUAL SYMPOSIUM AND MINI SHOW OCTOBER, 009 LeCroy Corporation The Interleaving Process in DBI Scopes page 5 of 5