It is said that noise spectral density is more useful than signal-to-noise ratio?
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The following article is from Analog Devices, author ADI
Have you noticed that when comparing systems operating at different speeds, or looking at how a software-defined system handles signals of different bandwidths, the noise spectral density (NSD) is arguably more useful than the signal-to-noise ratio (SNR). While it cannot replace other specifications, it is a useful parameter to have in your analysis toolbox.
explore--
How much noise is in my frequency band of interest?
The SNR on a data converter's data sheet represents the ratio of the full-scale signal power to the total noise power at all other frequencies.
Now consider a simple case to compare SNR and NSD, as shown in Figure 1. Assume that the ADC clock frequency is 75 MHz. Running a fast Fourier transform (FFT) on the output data shows a spectrum from dc to 37.5 MHz. In this case, the signal of interest is the only large signal and happens to be located around 2 MHz. For white noise (which in most cases consists of quantization noise and thermal noise), the noise is evenly distributed within the Nyquist band of the converter, which in this case is dc to 37.5 MHz.
Figure 1. Graphical representation of 9 dB modulation gain: all signal retained, 78 noise discarded.
Since the signal of interest is between DC and 4 MHz, it is relatively simple to apply digital post-processing to filter out or discard everything above 4 MHz (keeping only what is in the red box). This would require discarding 78 of the noise, keeping all of the signal energy, resulting in an effective SNR improvement of 9 dB. In other words, if you know the signal is in half of the band, you can actually discard the other half while removing only the noise.
A useful rule of thumb is that in the presence of white noise, the modulation gain improves the SNR of the oversampled signal by an additional 3 dB/octave. In the example of Figure 1, this technique can be applied over three octaves (by a factor of 8), resulting in a 9 dB improvement in SNR.
Of course, if the signal is somewhere between DC and 4 MHz, then you don’t need a fast 75 MSPS ADC to capture the signal. Only 9 MSPS or 10 MSPS is needed to meet the bandwidth requirement of the Nyquist sampling theorem. In fact, the 75 MSPS sampled data can be decimate by 1/8, yielding an effective data rate of 9.375 MSPS while preserving the noise floor in the band of interest.
It is important to do the decimation correctly. If you simply discard 7 out of every 8 samples, the noise will fold or alias back into the band of interest and you will not get any SNR improvement. You must filter first and then decimate to achieve modulation gain.
That said, while an ideal filter would eliminate all noise and achieve a perfect 3 dB/octave modulation gain, real filters do not have such characteristics. In practice, the amount of filter stopband rejection required is a function of how much modulation gain you are trying to achieve. Also note that the “3 dB/octave” rule of thumb is based on the assumption of white noise. This is a reasonable assumption, but it does not apply in all cases.
An important exception is when the dynamic range is affected by nonlinear errors or other spurious intermodulation products in the passband. In these cases, the “filter and discard” approach may not remove the spurious components and a more detailed frequency algorithm may be required.
method--
SNR and sampling rate
Convert to noise spectral density
The situation becomes more complicated when there are multiple signals in the spectrum, such as the many stations in the FM band. To recover any one signal, it is not the total noise of the data converter that is more important, but the amount of converter noise that falls within the frequency band of interest. This requires digital filtering and post-processing to remove any out-of-band noise.
There are several ways to reduce the amount of noise that falls within the red box. One way is to choose an ADC with better SNR (lower noise). Alternatively, you can use an ADC with the same SNR and provide a faster clock (such as 150 MHz), which will spread the noise over a wider bandwidth and result in less noise within the red box.
question--
Quickly compare converters' performance in filtering noise,
Is there a better specification than SNR?
This is where noise spectral density (NSD) comes in. Characterizing noise in terms of spectral density (usually expressed in decibels per full scale per hertz of bandwidth, or dBFS/Hz) allows ADCs with different sampling rates to be compared to determine which device is likely to have the lowest noise in a particular application.
Table 1 uses a 70 dB SNR data converter as an example to show how NSD improves as the sampling rate increases from 100 MHz to 2 GHz.
Table 1. Changing the sampling rate for a 70 dB SNR ADC
Table 2 shows a variety of SNR and sampling rate combinations for some very different converters, but all have the same NSD, so each will have the same total noise in a 1 MHz channel. Note that the actual resolution of the converter can be much higher than the effective number of bits, because many converters want extra resolution to ensure that the effect of quantization noise on NSD is negligible.
Table 2. Several very different converters all provide 95 dBSNR in a 1 MHz bandwidth; SNR calculation assumes a white noise floor (no spurious effects)
In a traditional single-carrier system, it may seem funny to use a 10 GSPS converter to capture a 1 MHz signal, but in a multi-carrier software-defined system, that may be exactly what the designer will do. An example is a cable set-top box, which may use a 2.7 GSPS to 3 GSPS full-band tuner to capture the cable signal containing hundreds of TV channels, each with a bandwidth of several MHz. For data converters, the units of noise spectral density are usually dBFS/Hz, that is, dB relative to full scale per Hz. This is a relative measurement that provides a kind of "output referred" measurement of the noise level. There are also units of dBm/Hz or even dB mV/Hz that provide a more absolute measurement, that is, an "input referred" measurement of the data converter noise.
SNR, full-scale voltage, input impedance, and Nyquist bandwidth can also be used to calculate the effective noise figure of the ADC, but this involves fairly complex calculations, as described in the ADI tutorial, ADC Noise Figure—A Often Misunderstood Parameter.
http://www.analog.com/media/cn/training-seminars/tutorials/MT-006_cn.pdf
think--
Oversampling Alternatives
Using an ADC at a higher sampling rate generally means higher power consumption—both in the ADC itself and in the subsequent digital processing. Table 1 shows that oversampling has a benefit on NSD, but the question remains: “Is oversampling really worth it?”
Using a lower noise converter can also achieve better NSD, as shown in Table 2. Systems that capture multiple carriers need to operate at a higher sampling rate, thus oversampling each carrier. However, there are still many advantages to oversampling.
Simplified anti-aliasing filtering - Oversampling will alias higher frequency signals (and noise) into the Nyquist band of the converter. So to prevent aliasing effects, these signals need to be filtered out before AD conversion. This means that the filter transition band must be between the highest target capture frequency (F IN ) and the alias of that frequency (F SAMPLE , F IN ).
As F IN gets closer to F SAMPLE /2, the transition band of this antialiasing filter becomes very narrow, requiring a very high order filter. 2 to 4 times oversampling greatly reduces this limitation in the analog domain and places the burden in the relatively easier to handle digital domain.
Even with a perfect anti-aliasing filter, minimizing the effects of converter distortion product folding can be a drawback, resulting in spurious and other distortion products in the ADC, including some very high-order harmonics. These harmonics will also fold within the sampling frequency, potentially back in-band, limiting the SNR in the band of interest. At higher sampling rates, the band of interest becomes a small fraction of the Nyquist bandwidth, thus reducing the probability of folding. It is worth noting that oversampling also helps with frequency planning for other system spurs that may fold in-band, such as device clock sources.
Modulation gain has an effect on any white noise, including thermal and quantization noise, as well as noise from certain types of clock jitter.
As higher speed converters and digital processing products mature, system designers more frequently use some amount of oversampling to take advantage of the benefits of noise floor and FFT.
It may be tempting to compare converters by examining the spectral plots and looking at the depth of the noise floor, as shown in Figure 2. When making such comparisons, it is important to remember that the spectral plots are dependent on the size of the Fast Fourier Transform. A larger FFT divides the bandwidth into more frequency bins, with less noise accumulated in each bin. In this case, the spectral plot will show a lower noise floor, but this is just a plotting artifact. In reality, the noise spectral density has not changed (this is the signal processing equivalent of changing the resolution bandwidth of a spectrum analyzer).
Figure 2. ADC with 524,288-sample FFT and 8192-sample FFT
Ultimately, if the sampling rate is equal to the FFT size (or in proper proportion), then comparing the noise floor is acceptable, otherwise it can be misleading. Here, the NSD specification can be used for direct comparison.
special case--
When the noise floor is not flat…
So far, the discussion of modulation gain and oversampling has assumed that the noise is flat within the Nyquist band of the converter. This is a reasonable approximation in many cases, but there are some cases where this assumption does not apply.
For example, it has been mentioned before that modulation gain does not apply to spurs, although oversampled systems may have some advantages in frequency planning and spurious handling. In addition, 1/f noise and some types of oscillator phase noise have spectral shaping properties, and modulation gain calculations are not applicable in such cases.
An important case where the noise is not flat is when using sigma-delta converters.
The ∑-Δ modulator modulates the feedback loop (quantizer output) to shape the quantization noise, thereby reducing the noise in the target frequency band, but at the cost of increasing the out-of-band noise, as shown in Figure 3.
Figure 3. Target frequency band and noise shaping.
Even without doing a full analysis, it can be seen that for sigma-delta modulators, using NSD as a specification for determining the in-band usable dynamic range is particularly effective. Figure 4 shows a zoomed-in noise floor plot of a high-speed bandpass sigma-delta ADC. In the 75 MHz band of interest (centered at 225 MHz), the noise is around -160 dBFS/Hz, and the SNR is over 74 dBFS.
Figure 4. AD6676 —noise floor
Example:
A concluding example
To summarize and reinforce what we have discussed, let’s now look at the curves shown in Figure 5. This example considers six ADCs:
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12-bit, 2.5 GSPS ADC (purple curve);
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14-bit, 1.25 GSPS ADC, clocked at 500 MSPS (red curve);
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14-bit, 1.25 GSPS ADC, clocked at 1 GSPS (green curve);
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14-bit, 3 GSPS ADC, clocked at 3 GSPS (grey curve);
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14-bit, 500MSPS ADC, clocked at 500 MSPS (blue curve);
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Bandpass ∑-Δ ADC.
The first five cases are characterized by a nearly white (flat) noise floor, while the sigma-delta ADC has a bathtub-shaped noise spectral density with low noise in the band of interest, as shown in Figure 4.
In each case, the sampling rate is kept fixed and the signal bandwidth is swept by varying the cutoff frequency of the digital filter (which removes out-of-band noise after digitization). Several conclusions can be drawn from this:
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Reducing the signal bandwidth increases the dynamic range. However, the slope of the purple, red, and green lines is always 3 dB/octave because their NSD curves are flat. The slope of the blue curve (sigma-delta ADC) is quite steep. This is especially true when sweeping the decimation filter cutoff frequency on the steep slope of the channel, because each increment/decrement of this frequency results in a rapid change in the amount of noise power filtered out.
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Each curve has a different vertical offset, depending on the NSD of the converter . For example, the red and green curves correspond to the same ADC. But the green curve (1 GSPS) is higher than the red curve (500 MSPS) because its NSD is 3 dB/Hz lower than the other case and its clock is twice that of the red curve.
Figure 5 shows the SNR vs. signal bandwidth tradeoff for several different high speed ADCs: the five slopes follow a 3 dB/octave modulation gain with a flat noise floor, while the AD6676 exhibits a steeper modulation gain due to noise floor shaping.
Figure 5. Relationship between SNR and signal bandwidth for different ADCs
Final Thoughts
The growing availability of high-speed and very high-speed ADCs and digital processing products is making oversampling a practical architectural approach for wideband and RF systems. Advances in semiconductor technology have contributed to increasing speeds and reducing costs (e.g., price, power consumption, and board area), allowing us to explore a variety of methods for converting and processing signals—whether using wideband converters with flat noise spectral density or band-limited sigma-delta converters with high dynamic range in the frequency band of interest. These technologies have changed the way we think about signal processing and how we define product specifications.
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