# Confronting Measurement Uncertainty in Signal Generation - Part 3: Waveform Sampling Images

In the digital world, a digital electronic system that is designed to process signals does not treat the signals continuously as they are in reality. Instead, the system uses “sample” to indicate the instantaneous value of a signal at a given instant in time, and a series of equally spaced samples to define a continuous signal. When converting the samples into an analog signal, hardware implementations lose some desired data and inject unwanted errors.

In the previous blog posts, we discussed the errors of waveform discontinuity and interpolation overshoot. In this post, we will address waveform sampling images and how to remove the images.

## Sampling Images and Filtering

Figure 1 shows how an ideal digital-to-analog converter (DAC) converts a sampled sine waveform (sample rate at Fs) to an analog signal. The DAC outputs a series of impulses, and the output spectrum of these time-domain impulses yields the frequency of the signal of interest (Fsig, red arrow), plus a family of image signals located at multiples of the sampling frequency (Fsig ± n*Fs, n=1, 2, 3…).

Figure 1. The output of an ideal DAC

In order to reconstruct the signal of interest, we can filter out these sampling images with a low pass filter (LPF). Because the LPF removes the higher-order images (which are an alias of the signal of interest) to reconstruct the desired waveform, the filter is commonly referred to as an “anti-aliasing filter” or a “reconstruction filter.”

Figure 2 illustrates a baseband generator that uses an LPF to remove the aliasing signals. The filter cutoff frequency must be low enough to reject the images and high enough to cover the full bandwidth of the signal of interest. The filter shown in this example cannot reject the images (dashed line) completely. This means you need to know what the bandwidths of reconstruction filters are before you decide the waveform sample rate (fs). You need to ensure that the filter can reject all the image signals.

Figure 2. Remove sampling images with a reconstruction filter

## Reject Image Signals with Oversampling

Another approach to reject image signals is to oversample a designed waveform that moves image signals far away from the LPF’s cutoff frequency. Figure 3 illustrates a 4x the oversampling ratio that moves the 2nd order image signals to fos (4x fs). Waveform oversampling adds more sampled points in the same waveform time period when creating the waveform and increases the size of the waveform file. However, the restriction of waveform memory usually is a bottleneck for waveform playback. You need to carefully balance between oversampling ratio and the size of the waveform file.

Figure 3. Move sampling images far away with 4x the oversampling ratio

## Benefits of DSP Waveform Resampling

Newer vector signal generators (Keysight N5182A MXG or N5172A EXG) support hardware resampling to remove sampling images with a single wideband reconstruction filter. Figure 4 illustrates the resampling process in a digital signal processor (DSP) before sending waveform data to the DACs. The resampling process (aka interpolator) includes oversampling, filtering, and decimation. With the oversampling and decimation in the DSP, the DACs can use a fixed clock rate and reconstruction filter without increasing playback memory. This helps you design and generate waveforms with ease and flexibility.

Figure 4. A waveform resampling process

## An Example of Hardware Oversampling

Let’s look at an example and compare the different baseband generator architectures. A digitally modulated signal has 614.4 kHz RF bandwidth and five times the oversampling ratio, so the sample rate of the waveform is 3.072 MSa/s. The cutoff frequency of the reconstruction filter is 8.5 MHz. You can see the image signals within two times of cutoff frequency and at the side lobe of the reconstruction filter as shown to the left of Figure 5. Some of the spur-like signals are a result of the intermodulation of the images.

Even with a 40-MHz cutoff frequency reconstruction filter, the spectrum is very clean without any image and spur by applying the DSP hardware oversampling ratio by 128 times as shown to the right of Figure 5. The hardware resampling process does not increase the usage of the playback memory.

Figure 5. Measuring sampling images with and without hardware oversampling

## Summary

Understanding the capabilities and performance of your signal generators is the first step to making accurate and consistent measurements. Using hardware waveform resampling helps you get an image-free signal and free up waveform playback memory. In my next post, I will discuss common sources of error in the I/Q modulator.

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