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# Improved DDS sign era utilizing interpolation

## Improved DDS sign era utilizing interpolation

Posted Date: 2023-06-11

The time period “interpolating DDS approach” used to explain the workings behind a few of at the moment’s arbitrary waveform mills could also be unfamiliar or complicated to some engineers and even skilled customers of perform mills. To clarify what the time period means and talk about its benefits, let’s evaluate conventional direct digital synthesis (DDS) with interpolated DDS.

The time period “interpolating DDS approach” used to explain the workings behind a few of at the moment’s arbitrary waveform mills could also be unfamiliar or complicated to some engineers and even skilled customers of perform mills. To clarify what the time period means and talk about its benefits, let’s evaluate conventional direct digital synthesis (DDS) with interpolated DDS.

Custom and Interpolation

Within the primary design of conventional DDS know-how (Determine 1), the output interval of digital waveform information is set by the frequency of the reference clock of the DDS circuit. Every of those discrete information samples is then transformed constantly by the D/A converter utilizing the identical clock. That's, within the conventional DDS construction, the DDS and D/A converter clocks run on the similar frequency.

Determine 1: In a conventional DDS construction, the output of a DDS circuit managed by a reference clock is fed to a D/A converter managed by the identical clock. The output of the D/A is smoothed by a reconstruction filter.

The output of the D/A converter is then fed to a reconstruction filter, which smoothes the discrete step values ​​of the D/A output to supply a extra steady “analog” sign.

In an interpolating DDS system (Determine 2), the primary distinction is that an interpolator is inserted between the DDS circuit and the D/A converter. On this configuration, the digital waveform information is output in accordance with the reference clock of the DDS circuit and interpolated by some numerical issue I, thereby rising the variety of discrete digital waveform information samples by an element I.

Determine 2: The construction of an interpolating DDS system contains two further parts: a phase-locked loop (PLL) and an interpolator, proven right here in inexperienced.

The ensuing digital information is then D/A transformed and is now clocked by a sampling clock whose frequency is “I” occasions the frequency of the DDS reference clock. As with standard DDS, reconstruction filters are utilized to the D/A converter output.

As a real-world instance of the way it works, think about the interpolating DDS approach utilized within the Siglent SDG2000X. The SDG2000X makes use of an interpolation issue (I) of 4, and because the reference clock of the DDS circuit of the SDG2000X is 300 MHz, the sampling clock frequency of the D/A is elevated to 1.2 GHz. Subsequently, the D/A outputs at a sampling price of 1.2 Gsamples/s.

Break the bandwidth restrict

Whereas the previous paragraphs clarify the mechanics behind the interpolating sampling approach, the query stays: “Why use this distinctive sampling approach?” To reply this query, we first time the D/A’s output utilizing two sampling charges Area comparability.

Trying on the D/A’s output waveform, the interpolated 1.2-Gsamples/s pattern price ends in smaller (increased decision) steps than utilizing a 300-Msamples/s pattern clock (Determine 3).

Determine 3: The interpolator will increase the variety of samples despatched to the D/A by an element of I, working on the sampling clock frequency created by the PPL, which is I occasions the frequency of the bottom DDS reference clock; for the case proven above, I = 4. You is likely to be pondering, “If each waveforms undergo the reconstruction filter to easy the steps, then the pattern price shouldn’t matter.” However the step measurement does have an effect on the ultimate smoothed waveform.

To grasp why extra clearly, let’s take a look at two waveforms within the frequency area. Contemplate an instance through which a traditional DDS generates a sine wave at a frequency of 80 MHz (f out ) utilizing a D/A clocked at a sampling frequency (fs ) of 300 MHz. The spectrum of the D/A output (Determine 4) contains the basic frequency (f out ) and its picture frequency (N xf s ) ± f out, the place N = 1, 2, … . The magnitudes of all spectral components conform to the sin(x)/x envelope.

Determine 4: The above graph reveals the spectrum of an 80-MHz waveform created by a traditional DDS utilizing a 300-Msamples/s A/D converter. To easy the waveform, the reconstruction filter should ideally filter out all picture frequencies exterior the band from 0 Hz to half the sampling frequency—that's, exterior the Nyquist bandwidth—and maintain all frequencies within the Nyquist Indicators throughout the bandwidth. In different phrases, the frequency response of the filter needs to be in keeping with the Nyquist bandwidth (shaded space in Determine 4). On this case, the utmost frequency of the retained sign can attain the Nyquist restrict (i.e. half the sampling clock frequency).

After all, in engineering, there is no such thing as a best filter with a “brick wall” response (full sign attenuation above a sure frequency and no attenuation earlier than that frequency). In the actual world, filters even have some extent of roll-off. (Observe that in Determine 4, the latest picture seems at 220 MHz, so for the roll-off slope proven in blue, the roll-off ought to begin at 140 MHz to make sure most attenuation.)

Nonetheless, if the frequency of the output waveform was 149.5 MHz as a substitute of 80 MHz, the closest mirror could be at 150.5 MHz (1f s – f out ), and the remaining roll-off headroom could be 1 MHz, making it nearly impractical to filter and never accomplish. Sometimes, the bandwidth of the reconstruction filter is restricted to 40% of the sampling clock. Subsequently, for a 300-Msamples/s D/A, the utmost usable output frequency is 120 MHz.

In Determine 5, the 2 roll-offs computed for the ninth order elliptic reconstruction filter are plotted. The distinction is that the calculation on the left plot makes use of best parts, whereas the calculation on the proper makes use of actual parts with parasitic parameters. As proven, the precise filter has an extra ~3 dB attenuation on the nook frequency (120 MHz), and its attenuation efficiency is degraded (~18 dB) on the stopband cutoff frequency (~180 MHz).

Determine 5: Computational efficiency plot of a ninth-order elliptic reconstruction filter with best parts (left) could be very completely different from a filter utilizing actual parts with parasitic parameters.

As well as, the sin(x)/x envelope of the D/A response itself will increase the attenuation of the sign. At 40% of the pattern clock, the attenuation as a result of D/A is about 2.4 dB. Often an inverse sin(x)/x filter is required to compensate for this attenuation.

Now think about the case of utilizing a D/A sampling price of 1.2 GHz (1.2 Gsamples/s) to generate an interpolated DDS for an 80 MHz sign. The newest picture is 1.12 GHz (1.2 GHz – 180 MHz), so the utmost roll-off of the reconstruction filter will be 1.04 GHz (Fig. 6), and the design of the reconstruction filter is significantly simplified (Fig. 7).

Determine 6: For the above spectrum for the 80-MHz output, the waveform was generated utilizing an interpolating DDS generator with a 1.2-Gsamples/s D/A converter; notice how the roll-off of the reconstruction filter will be relaxed.

Determine 7: The true reconstruction filter design with roll-off from 120 MHz to 1.08 GHz is way simpler than the design with roll-off from 120 to 180 MHz.

Alternatively, as the primary lobe width of the sin(x)/x envelope will increase, the attenuation of the D/A contribution decreases. At 120 MHz, the attenuation attributable to a 1.2-Gsamples/s D/A is about 0.14 dB, which is negligible generally. Subsequently, inverse sin(x)/x filtering isn't required.

To sum up, a 300-Msamples/s D/A can solely output a most frequency of 120-MHz as a result of limitations of the reconstruction filter. However the 1.2-Gsamples/s D/A can obtain the next higher frequency restrict. After all, in an interpolating DDS construction, the digital filter within the interpolator will restrict the frequency to the Nyquist restrict of the DDS clock (say 150 MHz), however digital filters are simpler to design than analog reconstruction filters. Utilizing interpolating DDS, it's straightforward to extend the higher restrict of the output frequency from 120 MHz to 130 MHz or increased.

keep away from spurs

Spurious indicators attributable to intermodulation distortion are unavoidable in D/A converters. Within the conventional DDS construction, it's tough to take away some intermodulation distortion parts (known as spurs) between the clock and the output sign, such because the spurs at fs – 2f out and fs – 3f out. At an output frequency of 120 MHz and a sampling price of 300 Msamples/s, the fs – 2f out distortion of standard DDS thus happens at 60 MHz (Determine 8) and thus falls into the passband of the reconstruction filter. It can't be deleted.

Determine 8: If the ensuing intermodulation distortion between the output sign frequency and the reference clock frequency is throughout the Nyquist bandwidth, it can't be filtered out.

However for the I = 4 interpolating DDS construction, for a frequency output of 120 MHz and 1.2 Gsamples/s, the distortion parts fs – 2f out are 960 MHz and fs – 3f out are 840 MHz, effectively past the passband of the reconstruction filter. Subsequently, within the case of interpolated DDS, the spurs don't have an effect on the ultimate output waveform.

The time period “interpolating DDS approach” used to explain the workings behind a few of at the moment’s arbitrary waveform mills could also be unfamiliar or complicated to some engineers and even skilled customers of perform mills. To clarify what the time period means and talk about its benefits, let’s evaluate conventional direct digital synthesis (DDS) with interpolated DDS.

Custom and Interpolation

Within the primary design of conventional DDS know-how (Determine 1), the output interval of digital waveform information is set by the frequency of the reference clock of the DDS circuit. Every of those discrete information samples is then transformed constantly by the D/A converter utilizing the identical clock. That's, within the conventional DDS construction, the DDS and D/A converter clocks run on the similar frequency.

Determine 1: In a conventional DDS construction, the output of a DDS circuit managed by a reference clock is fed to a D/A converter managed by the identical clock. The output of the D/A is smoothed by a reconstruction filter.

The output of the D/A converter is then fed to a reconstruction filter, which smoothes the discrete step values ​​of the D/A output to supply a extra steady “analog” sign.

In an interpolating DDS system (Determine 2), the primary distinction is that an interpolator is inserted between the DDS circuit and the D/A converter. On this configuration, the digital waveform information is output in accordance with the reference clock of the DDS circuit and interpolated by some numerical issue I, thereby rising the variety of discrete digital waveform information samples by an element I.

Determine 2: The construction of an interpolating DDS system contains two further parts: a phase-locked loop (PLL) and an interpolator, proven right here in inexperienced.

The ensuing digital information is then D/A transformed and is now clocked by a sampling clock whose frequency is “I” occasions the frequency of the DDS reference clock. As with standard DDS, reconstruction filters are utilized to the D/A converter output.

As a real-world instance of the way it works, think about the interpolating DDS approach utilized within the Siglent SDG2000X. The SDG2000X makes use of an interpolation issue (I) of 4, and because the reference clock of the DDS circuit of the SDG2000X is 300 MHz, the sampling clock frequency of the D/A is elevated to 1.2 GHz. Subsequently, the D/A outputs at a sampling price of 1.2 Gsamples/s.

Break the bandwidth restrict

Whereas the previous paragraphs clarify the mechanics behind the interpolating sampling approach, the query stays: “Why use this distinctive sampling approach?” To reply this query, we first time the D/A’s output utilizing two sampling charges Area comparability.

Trying on the D/A’s output waveform, the interpolated 1.2-Gsamples/s pattern price ends in smaller (increased decision) steps than utilizing a 300-Msamples/s pattern clock (Determine 3).

Determine 3: The interpolator will increase the variety of samples despatched to the D/A by an element of I, working on the sampling clock frequency created by the PPL, which is I occasions the frequency of the bottom DDS reference clock; for the case proven above, I = 4. You is likely to be pondering, “If each waveforms undergo the reconstruction filter to easy the steps, then the pattern price shouldn’t matter.” However the step measurement does have an effect on the ultimate smoothed waveform.

To grasp why extra clearly, let’s take a look at two waveforms within the frequency area. Contemplate an instance through which a traditional DDS generates a sine wave at a frequency of 80 MHz (f out ) utilizing a D/A clocked at a sampling frequency (fs ) of 300 MHz. The spectrum of the D/A output (Determine 4) contains the basic frequency (f out ) and its picture frequency (N xf s ) ± f out, the place N = 1, 2, … . The magnitudes of all spectral parts conform to the sin(x)/x envelope.

Determine 4: The above graph reveals the spectrum of an 80-MHz waveform created by a traditional DDS utilizing a 300-Msamples/s A/D converter. To easy the waveform, the reconstruction filter should ideally filter out all picture frequencies exterior the band from 0 Hz to half the sampling frequency—that's, exterior the Nyquist bandwidth—and maintain all frequencies within the Nyquist Indicators throughout the bandwidth. In different phrases, the frequency response of the filter needs to be in keeping with the Nyquist bandwidth (shaded space in Determine 4). On this case, the utmost frequency of the retained sign can attain the Nyquist restrict (i.e. half the sampling clock frequency).

After all, in engineering, there is no such thing as a best filter with a “brick wall” response (full sign attenuation above a sure frequency and no attenuation earlier than that frequency). In the actual world, filters even have some extent of roll-off. (Observe that in Determine 4, the latest picture seems at 220 MHz, so for the roll-off slope proven in blue, the roll-off ought to begin at 140 MHz to make sure most attenuation.)

Nonetheless, if the frequency of the output waveform was 149.5 MHz as a substitute of 80 MHz, the closest mirror could be at 150.5 MHz (1f s – f out ), and the remaining roll-off headroom could be 1 MHz, making it nearly impractical to filter and never accomplish. Sometimes, the bandwidth of the reconstruction filter is restricted to 40% of the sampling clock. Subsequently, for a 300-Msamples/s D/A, the utmost usable output frequency is 120 MHz.

In Determine 5, the 2 roll-offs computed for the ninth order elliptic reconstruction filter are plotted. The distinction is that the calculation on the left plot makes use of best parts, whereas the calculation on the proper makes use of actual parts with parasitic parameters. As proven, the precise filter has an extra ~3 dB attenuation on the nook frequency (120 MHz), and its attenuation efficiency is degraded (~18 dB) on the stopband cutoff frequency (~180 MHz).

Determine 5: Computational efficiency plot of a ninth-order elliptic reconstruction filter with best parts (left) could be very completely different from a filter utilizing actual parts with parasitic parameters.

As well as, the sin(x)/x envelope of the D/A response itself will increase the attenuation of the sign. At 40% of the pattern clock, the attenuation as a result of D/A is about 2.4 dB. Often an inverse sin(x)/x filter is required to compensate for this attenuation.

Now think about the case of utilizing a D/A sampling price of 1.2 GHz (1.2 Gsamples/s) to generate an interpolated DDS for an 80 MHz sign. The newest picture is 1.12 GHz (1.2 GHz – 180 MHz), so the utmost roll-off of the reconstruction filter will be 1.04 GHz (Fig. 6), and the design of the reconstruction filter is significantly simplified (Fig. 7).

Determine 6: For the above spectrum for the 80-MHz output, the waveform was generated utilizing an interpolating DDS generator with a 1.2-Gsamples/s D/A converter; notice how the roll-off of the reconstruction filter will be relaxed.

Determine 7: The true reconstruction filter design with roll-off from 120 MHz to 1.08 GHz is way simpler than the design with roll-off from 120 to 180 MHz.

Alternatively, as the primary lobe width of the sin(x)/x envelope will increase, the attenuation of the D/A contribution decreases. At 120 MHz, the attenuation attributable to a 1.2-Gsamples/s D/A is about 0.14 dB, which is negligible generally. Subsequently, inverse sin(x)/x filtering isn't required.

To sum up, a 300-Msamples/s D/A can solely output a most frequency of 120-MHz as a result of limitations of the reconstruction filter. However the 1.2-Gsamples/s D/A can obtain the next higher frequency restrict. After all, in an interpolating DDS construction, the digital filter within the interpolator will restrict the frequency to the Nyquist restrict of the DDS clock (say 150 MHz), however digital filters are simpler to design than analog reconstruction filters. Utilizing interpolating DDS, it's straightforward to extend the higher restrict of the output frequency from 120 MHz to 130 MHz or increased.

keep away from spurs

Spurious indicators attributable to intermodulation distortion are unavoidable in D/A converters. Within the conventional DDS construction, it's tough to take away some intermodulation distortion parts (known as spurs) between the clock and the output sign, such because the spurs at fs – 2f out and fs – 3f out. At an output frequency of 120 MHz and a sampling price of 300 Msamples/s, the fs – 2f out distortion of standard DDS thus happens at 60 MHz (Determine 8) and thus falls into the passband of the reconstruction filter. It can't be deleted.

Determine 8: If the ensuing intermodulation distortion between the output sign frequency and the reference clock frequency is throughout the Nyquist bandwidth, it can't be filtered out.

However for the I = 4 interpolating DDS construction, for a frequency output of 120 MHz and 1.2 Gsamples/s, the distortion parts fs – 2f out are 960 MHz and fs – 3f out are 840 MHz, effectively past the passband of the reconstruction filter. Subsequently, within the case of interpolated DDS, the spurs don't have an effect on the ultimate output waveform.

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