The first assignment for AD Converters (ADC) was all about noise, aliasing and distortion. Corrections to the exercises will be processed and marked in red.
1. Sampling and Jitter
How much SNR can be obtained if a signal of
To calculate the SNR for a 10 MHz sampled signal, we use the SNR formula:
We take our signal for
Substituting Eq. (2) into Eq. (1) for
If we change the sample rate, the SNR would be unaffected, since our equation is not influenced by the sample rate. It depends on the input frequency. Our full bandwidth jitter noise would just be spread across a larger Nyquist band, so the layer of jitter noise would become thinner. This means that the spectral noise density, however, does go down.
Chapter 7.6 of Marcel Pelgrom’s A/D Converters states that the jitter has a flat spectrum, similar to thermal noise. Thus, the jitter error power is inversely related to the sample rate;
where
Now that we have solved for the noise density it can be proven that for a BW equal to the Nyquist bandwidth, the jitter error power rate will remain the same resulting in equal SNR since the term
Now we can calculate the
Simplified and in dB:
For a sample rate of 80 MS/s and 310 MS/s we get that the SNR is roughly equal to 83 dB and 89 dB respectively.
2. Aliasing
A signal source delivers a signal that consists of three components: at
To find aliasing we can use:
where
Let us analyze the spacing of the absolute frequencies to determine the sampling frequency:
The relationship between input and output is given by:
Assuming the most direct folding case with
| 3 | 3 | 0.27 | |
| 3 | 4 | 0.64 | |
| 4 | 4 | 0.36 | |
| 4 | 5 | 0.55 | |
| 5 | 5 | 0.45 | |
| 5 | 6 | 0.46 |
3. RF Oscillator
An RF oscillator at
To view signals within a 10 MHz BW we use subsampling and intentional aliasing to map higher frequencies down to the lower end of the spectrum.
For a fundamental frequency
Setting the target alias
Fundamental (2.45 GHz):
2nd Harmonic (4.90 GHz):
3rd Harmonic (7.35 GHz):
All three components fall within the 10 MHz bandwidth and are spaced 1 MHz apart, allowing for clear identification.
4. Subsampling
A very fast ADC samples a
If the ADC samples at
Our Nyquist frequency is
As we can see, this is close to the input sine with frequency
Fundamental (750.366 MHz):