MDO4000 Mixed Domain Oscilloscope Structure Decoding (Part 2)

Figure 25 below illustrates the process of generating a displayed spectrum curve:

Figure 25 Generated spectrum curve

In this process, the data is first multiplied by a window function. Since the FFT assumes that the signal is constant over the entire period, discontinuities at the end of the sampling interval will appear as spectral leakage in the resulting spectrum. The window function is designed to reduce these discontinuities. For more information on the various window functions and their use, see Appendix 1.

One implication of the assumption that the signal is constant over time is that if the signal changes amplitude during the time interval covered by the RF time domain data, it will show up in the resulting spectrum at a reduced power level. The only way to avoid this effect is to adjust the RBW resolution bandwidth setting to ensure that the signal is stable over the entire time interval.

Since FFT processing is more efficient in data lengths that are powers of 2, the input data is zero-padded until the nearest power of 2. Zero padding increases the spectral resolution without changing the frequency content.

It should be noted that the FFT length used is completely determined by the span/RBW ratio. This can be easily seen in the above formula:

FFT length = (window factor * filter factor * (1/2 * span)) / RBW (Formula 10)

For the MDO4000 Mixed Domain Oscilloscope, the default Kaiser window has a window factor of 2.23. As mentioned above, the filter factor is approximately equal to 3. The default span/RBW ratio is 1000:1. With these default settings, the resulting FFT length is approximately 3345 points. This will zero pad until a 4096 point FFT.

The more samples there are in each transform frame, the better the frequency resolution after the transform is completed. Unfortunately, this also means that a greater number of data calculations are required to transform the frame. The FFT transform process is also known for its intensive computation requirements.

Figure 26. Increasing the number of time samples improves frequency domain resolution.

We then use FFT to convert the RF time domain data into frequency domain data in the form of a spectrum. We then further modify this spectrum:

The entire spectrum is multiplied by a set of coefficients that adjust for flatness. These coefficients are determined during factory calibration. There is no phase calibration in the MDO4000 Mixed Domain Oscilloscope.

As mentioned previously, the FFT process can involve 1,000 - 2,000,000 points. The spectrum record can be compressed to fit on a 1000 point display. This data compression (decimation) process is called detection and is used to aggregate multiple FFT bins into a single displayed bin. The user has control over the detection method selected and compression is as follows:

• + Peak: retain the largest data point in the compression interval
• -Peak: retain the smallest data point in the compression interval
• Average: averages the data in the entire compression interval
• Sample: Keep the last data point in the compression interval

The final spectrum can then be logarithmized to obtain the final picture.

Generating RF time domain data

Another use of IQ data is to generate RF time domain data. Recall that in the digital down-conversion above, the IQ data is just a Cartesian representation of the signal plotted as a vector in the imaginary IQ data plane. Therefore, the IQ data can be transformed as follows:

Figure 27. Generating RF time domain data

RF time domain data can be plotted in the time domain graticule along with other time domain plots. All time domain data (including analog data, digital data, and RF channels) are time aligned in the graticule, allowing the user to evaluate the timing relationship between the channels.

Note that both the phase and frequency calculations are independent of the amplitude calculations. If the amplitude is low, then the IQ data will become increasingly dominated by noise. This effect is shown in the screenshot below:

Figure 28. Phase vs. time without blanking.

To avoid this problem, the MDO4000 mixed domain oscilloscope has a squelch control function that allows the user to blank the phase and frequency curves when the amplitude drops below a user-defined threshold. The screenshot below shows the result. [page]

Figure 29. Blanking phase variation over time.

Generate a waterfall spectrum

Another use of the spectrum is to plot a waterfall plot.

The process is relatively simple, color-coding the spectral amplitudes and plotting the result in the spectrogram display as a line of pixels. Each new "slice" pushes existing data up in the display until the data at the top of the display is discarded. A "slice" represents one FFT frame that has been processed according to the span and RBW settings in the spectrum display.

Figure 30. The spectrogram display shows the spectral history of a signal record.

Figure 31. A “segment” shows a previously recorded signal.

Time resolution

The final topic to be discussed is the temporal resolution of the data.

The time resolution of the spectrum is relatively poor and the reason for this can be seen in the figure below.

Figure 32. Time resolution

First, as described above in "Generating a Spectrum", the FFT is generated from data covering the time interval defined by the RBW setting. Therefore, the changes in the signal's spectral components within this time interval cannot be distinguished, but rather converge into one spectrum.

Second, as can be seen from the graph, there is a delay between acquisition events. Changes that occur between acquisition events will not be seen.

To reduce the time to calculate the spectrum, the RBW should be increased. Since the default setting links the RBW to the span, increasing the span will have the desired effect. In addition, this will also reduce the time between acquisitions because the time required to perform the digital downconversion is reduced.

To further reduce the time between acquisitions, the span/RBW ratio should be reduced, which can speed up the FFT processing time.

The time resolution of RF time domain data is relatively fine compared to the spectrum. As mentioned earlier in “Digital Down Conversion”, the sampling rate of IQ data depends on the span setting, so the time resolution is much finer than that of the spectrum. This is one of the main advantages of RF time domain traces.

To improve the time resolution of amplitude, phase, or frequency versus time plots, the span should be increased.

Summarize

The MDO4000 Mixed Domain Oscilloscope is the biggest technological breakthrough and innovation in the oscilloscope market in the past 20 years. During the invention and design process of Tektronix, 26 patents were applied for, proving that it contains many technological innovations. It is not just an oscilloscope and a spectrum analyzer integrated together, it also provides the industry's first "mixed domain" analysis and many world "firsts".

To learn more about the performance and functions of the MDO4000 mixed domain oscilloscope and its applications, and how it can help design engineers solve various problems of wireless embedded systems more effectively and quickly, please visit: http://www.tek.com/en/scoperevolution/

Appendix A Window Functions

window

The mathematical calculation of Discrete Fourier Transform (DFT) analysis itself has an assumption that the data to be processed is a period of a periodically repeating signal. [page]

Figure A1 depicts a series of time domain samples. For example, when DFT processing is applied to the second frame in Figure A1, the signal will be periodically expanded. Discontinuities generally occur between multiple consecutive frames, as shown in Figure A2.

Figure A1/A2. Three frames of a sampled time-domain signal (a) and discontinuities caused by regularly spreading samples within a frame (b).

These spurious discontinuities generate spectral artifacts that were not present in the original signal. This effect produces an inaccurate representation of the signal, known as spectral leakage. Spectral leakage not only produces signals in the input that were not present, but also reduces the ability to observe small signals in the presence of nearby large signals.

The MDO4000 Series spectrum analyzer function applies windowing to reduce the effects of spectral leakage. Before performing the DFT, the DFT frame is multiplied by a window function of the same length, sample by sample. The window function is usually bell-shaped, which reduces or eliminates the discontinuities at the end of the DFT frame.

The choice of window function depends on the frequency response characteristics, such as side lobe level, equivalent noise bandwidth and amplitude error. The window shape also determines the effective RBW resolution bandwidth filtering.

Like other spectrum analyzers, MDO mixed domain oscilloscopes allow the user to select the RBW resolution bandwidth filter. MDO mixed domain oscilloscopes also allow the user to select between several common window types. With the added flexibility of being able to directly specify the window shape, the user can optimize for specific measurements. For example, special attention should be paid to the spectrum analysis of pulsed or transient RF signals. Table A1 provides some suggestions on the use of different window functions.

Table A1 FFT window options available on the MDO4000

The frequency response amplitude of the window function determines the shape of the RBW resolution bandwidth. For example, the RBW resolution bandwidth on an MDO mixed domain oscilloscope is defined as a 3 dB bandwidth, which has the following relative relationship with the sampling frequency and number of samples in the DFT:

Where k is a coefficient related to the window, N is the number of time domain samples used in the DFT calculation, and Fs is the sampling frequency. For a Kaiser window, k is approximately 2.23. The RBW resolution bandwidth shape factor is defined as the frequency ratio of the spectrum amplitude at 60 dB and 3 dB, which is approximately 4:1. On an MDO mixed domain oscilloscope, spectrum analysis measurements use Equation 2 to calculate the number of samples required for the DFT based on the input span and RBW setting.

Figures A3 and A4 show the time domain and spectrum of the Kaiser window used in spectrum analysis on an MDO Mixed Domain Oscilloscope. This is the default window used by the MDO4000 Mixed Domain Oscilloscope in spectrum analysis.

Figure A3: Kaiser window in the time domain, the horizontal axis is the time domain sampling points, the vertical axis is the linear scale

Figure A4: Kaiser window in the frequency domain, with the horizontal axis being the frequency binary (Fs/N) and the vertical axis being dB

The example of a frequency hopping signal in Figure A5 illustrates how different windows affect the spectral representation of a time varying signal. The spectrum time associated with this acquisition is 1.12 ms when using the default Kaiser window. The Frequency vs. Time display shows that the spectrum time is centered around the middle frequency of the three hopping sequences for most of the hopping time. The time associated with the upper and lower frequency "on frequency" cycles is roughly equal, and the window function described in Figure A3 shows that the level of the time samples near the beginning and edge of the acquisition drops off because the window function uses a Gaussian distribution of samples in the center of the acquisition. Looking at the amplitude of the four peaks in the frequency domain display (center frequency, high frequency, low frequency, and maximum overshoot peak), the center peak exceeds the rest of the signal by nearly 30 dB.

Figure A5. Kaiser window at 2 kHz RBW.

In Figure A6, the window type selected is now rectangular. Since the window function of the rectangular window is different from the Kaiser window, the RBW becomes 750 Hz, so the spectrum time is roughly the same as the acquisition time in the previous example.

The spectrum is again time aligned to the same points in the three frequency hopping sequences, but the spectral representation is very different.

Figure A6. Rectangular window at 750 Hz RBW

Since the rectangular window function does not filter the time samples in the acquisition time, and the dwell time at each of the three frequencies is substantially equal, the spectrum amplitudes of the three peak signals displayed by the spectrum using the rectangular window are substantially equal.

Other windows (such as Blackman-Harris, Rectangular, Hanning) can also be selected by the user to meet special measurement requirements, and the instrument can also use these windows when performing some of the measurements provided by the instrument.