Interpolation and Filtering to Improve Spectral Purity

In the following document, learn how you can use signal generators with analog and digital filters to improve the spectral purity of periodic signals. This tutorial is part of the National Instruments Signal Generator Fundamentals series. Each tutorial in this series examines basic concepts about the architecture, features, or applications of signal generators. For more information about applications that commonly use signal generators, see the Signal Generator Applications Main Page. Also read more about this topic in the multimedia tutorial Signal Generator Fundamentals: Use of Filtering and Interpolation to Improve Spectral Purity.

Challenges in Approximating Analog Signals 

Because analog-to-digital converters use a sample-and-hold output technique, they are only able to approximate analog signals. Moreover, because the stepped output of a DAC results in high-frequency images, modern signal generators implement both analog and digital filters to provide the best approximation of an ideal analog signal. As an example, the time domain of a simulated unfiltered signal is shown in Figure 1.

Figure 1. DAC Sample-and-Hold Output

As Figure 1 illustrates, the output of a signal generator has a stepped shape as a result of the sample-and-hold property of a DAC. Unfortunately, the steps visible in the time domain of a signal translated into high-frequency spectral images. These images occur at each multiple of the sampling rate, plus or minus the fundamental tone. Thus, when generating a 20 MHz sinusoid sampled at 100 MHz, you see images at 80, 120, 180, 220 ... MHz. Figure 2 shows the frequency domain of the simulated 20 MHz sine wave.

Figure 2. Spectral Images of a 20 MHz Sine Wave

As the graph illustrates, high-frequency spectral images can distort the frequency domain of the signal you are generating. For applications requiring good spectral performance, these images are unacceptable. For that reason, NI signal generators reduce the effects of high-frequency images through interpolation (digital filtering) and an analog filter.

Interpolation (Digital Filtering)

The digital filter on NI signal generators increases the effective sample rate of the instrument. This is particularly useful for smoothing signals such as sinusoids or baseband I/Q waveforms. For these signals, you can use an interpolation filter to accurately produce new samples of the waveform without reducing signal quality. Figure 3 compares a simulated signal with no interpolation to one that has been interpolated by 4X.  

Figure 3. Interpolated DAC Output

As the image illustrates, 4X interpolation has the effect of creating three new samples for every one sample that is taken from memory. As a result, you can represent a sine wave much more accurately. National Instruments signal generators use a digital filter to implement 2X, 4X, or 8X interpolation up to an effective sample rate of 400 MS/s.  

In the frequency domain, the effect of interpolation on a signal is evident when observing the high-frequency images of the signal. As discussed previously, high-frequency images occur at the sum and difference of the fundamental frequency and the sample rate. Because interpolation increases the effective sample rate, the images of an interpolated signal get shifted to higher frequencies. In Figure 4, you can observe the frequency domain of a simulated interpolated signal.

Figure 4. Spectral Images of a 20 MHz Sine Wave with 4X Interpolation

As Figure 4 illustrates, digital filtering (interpolation) shifts spectral images to be centered around the new, interpolated sample rate. This effect has two advantages. First, because the nonzero rise time of a signal generator acts like a natural lowpass filter, the high-frequency images experience slight attenuation. The second advantage of interpolation is that shifting spectral images to higher frequencies enables them to be more significantly attenuated by an analog lowpass filter on the instrument. For example, several NI signal generators use a lowpass filter of 153 MHz to attenuate high-frequency images without affecting the fundamental tone.

Analog Filtering

The analog filter of a signal generator is able to further smooth the output of the device. As a result, the instrument can more accurately represent a truly analog signal. In addition, analog filters provide the added effect of reducing spectral images of the signal. This is illustrated in Figure 5, which compares the time domain of two simulated 20 MHz sine waves: one that has only been digitally filtered and one after both digital and analog filtering.

Figure 5. Time Domain of a 20 MHz Sine Wave

Figure 5 shows that the individual steps, once evident in the time domain, are no longer visible when the signal is filtered. Instead, the output appears like a pure sinusoid. You should enable the analog filter when generating smooth signals such as a sinusoid and disable it when generating signals with sharp transitions such as a square wave.

The advantages of an analog filter are even more apparent in the frequency domain. As discussed previously, even interpolation (digital filtering) is unable to completely remove the high-frequency images of a signal. Thus, you also should apply an analog filter to remove any images greater than 154 MHz. Figure 6 shows the frequency domain of a signal once the analog filter has been applied.

Figure 6. Frequency Domain of an Interpolated and Filtered 10 MHz Sine Wave

As discussed in a previous section, by interpolating the signal to an effective sampling rate of 400 MS/s, you can shift all aliases to a higher-frequency range. In Figure 6, the alias of a 10 MHz sine wave appears at 390 MHz once the signal has been interpolated. As you can see from the frequency response of the filter (Figure 6), any spectral images above 154 MHz are severely attenuated. In the simulated signal above, the image actually drops below the noise floor of the spectrum.

Note that the design decision to use a lowpass filter with a 154 MHz cutoff frequency is not trivial. This filter was specifically designed to be used with the interpolation filter of NI 5406, NI 5421, and NI 5441 generators. Because the bandwidths of each of these generators do not exceed 43 MHz, the nearest spectral image is not lower than 357 MHz (400 to 43 MHz). In addition, the higher the cutoff frequency of the filter, the better flatness that the signal generator exhibits in lower-frequency ranges. As a result, using a filter with a high cutoff frequency helps you attenuate spectral images while preserving the passband flatness of the instrument. For more information on why passband flatness is important, read Understanding Dynamic Hardware Specifications.

Figure 7 compares the filter responses of two lowpass filters - one with a cutoff at 50 MHz and the other with a cutoff at 154 MHz.   

Figure 7. Lowpass Filter Comparison

In Figure 7, both filters shown on the graph are seventh-order elliptic lowpass filters. However, the red line shows a lowpass cutoff of 50 MHz, and the blue line shows a lowpass cutoff at 154 MHz. In this case, if you do not use interpolation to shift spectral images to higher frequencies, the nearest spectral image for a 43 MHz sinusoid occurs at the maximum sample rate minus the fundamental frequency. For an NI 5421 generator, the closest image without interpolation would be at 57 MHz (100 to 43 MHz). In this example, the fundamental tone and alias are too close in frequency to filter the alias without affecting the desired tone. Thus, you should use interpolation with analog filtering to move the alias to higher frequencies before applying the analog filter.  

Applications Requiring Spectral Purity

Using both digital and analog filters improves the quality of signals from arbitrary waveform and function generators. You should use these features for applications where it is important to preserve spectral purity. For example, consider using two arbitrary waveform generators to generate a baseband I/Q signal (see block diagram in Figure 8).

Figure 8. Direct Upconverter Characterization Test

When characterizing an RF upconverter, spectral images from the baseband input can translate to spectral images in resulting RF signals. To accurately characterize this type of RFIC (radio frequency integrated circuit), you need to ensure that the baseband signal is free of any unwanted spectral images. Using arbitrary waveform generators for baseband I/Q waveforms requires digital and analog filtering to produce the highest-quality signal.

For more information on the fundamental features you can use to generate high-quality signals, see Signal Generator Fundamentals. For more information on the applications that typically use signal generators, see Signal Generator Applications main page.

Related Links

• Webcast: Signal Generator Fundamentals - Use of Filtering and Interpolation to Improve Spectral Purity
• NI Signal Generators
• NI Modular Instruments

 

Reader Comments | Submit a comment »

Good document!
- Mark Franklin, L-3 Communications. mark.franklin@l-3com.com - Jul 12, 2007

Excellent article!
- Feb 22, 2007