Library / Signal processing and digital filters / DSP-04
Signal processing and digital filters
Aperture-jitter-limited sine SNR
Signal-to-noise ratio set by random sampling-time error alone, for a full-scale sine of frequency f and RMS timing jitter σt.
- Ref
- DSP-04
- Rev
- v1
- Field
- Signal processing and digital filters
- Result
- SNRj [dB]
Random error in the sampling instant turns the slope of the signal into an amplitude error. For a sine of frequency f, the RMS error is 2πfσt times its RMS amplitude, so the signal-to-noise ratio from jitter alone is −20 log₁₀(2πfσt). It grows worse by 6 dB for every doubling of the frequency or of the jitter.
Signal-to-noise ratio set by random sampling-time error alone, for a full-scale sine of frequency f and RMS timing jitter σt. It requires 2πfσt ≪ 1, and the jitter must be independent of the signal. Other noise sources are not included.
Calculator
- RMS phase error, 2πfσt6.283×10−4 rad
- Equivalent number of bits, (SNR − 1.76)/6.0210.34 bits
- Jitter allowed for a 12-bit (74 dB) SNR at this frequency3.176×10−11 s
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- With 100 ps of jitter on a 1 MHz sine, jitter alone limits the SNR to 64.04 dB, about 10.3 effective bits.
- For a 12-bit converter to reach its ideal 74 dB at this frequency, the clock jitter would have to be below 31.8 ps.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (dB) | Local step | Change in result |
|---|---|---|---|---|---|---|
| f Analogue sine frequency | 1 to 1e9 Hz | 4.036 | 184 | 180 | +10% | −0.828 dB |
| σt RMS timing jitter | 1e-15 to 1e-6 s | −15.96 | 164 | 180 | +10% | −0.828 dB |
Sweep of RMS timing jitter
| RMS timing jitter (s) | Jitter-limited SNR (dB) |
|---|---|
| 1e-15 | 164 |
| 1.334e-14 | 141.5 |
| 1.778e-13 | 119 |
| 2.371e-12 | 96.54 |
| 3.162e-11 | 74.04 |
| 4.217e-10 | 51.54 |
| 5.623e-9 | 29.04 |
| 7.499e-8 | 6.536 |
| 1e-6 | −15.96 |
Sweep one variable
Variables and how each one contributes
Jitter-limited SNR
What the equation gives you. Everything else feeds it.
Analogue sine frequency
The analogue sine frequency, in Hz. Doubling the frequency lowers the jitter-limited SNR by 6 dB. Raising it by 10 % (to 1,100,000) moves the result by −1.29 %, from 64.04 to 63.21 dB.
RMS timing jitter
The RMS timing jitter, in s. Doubling the jitter lowers the SNR by 6 dB. Raising it by 10 % (to 1.1×10−10) moves the result by −1.29 %, from 64.04 to 63.21 dB.
Worked example with the values above
- Phase error: 2πfσt = 2π × 1,000,000 × 1×10−10 = 6.2832×10−4 rad.
- Logarithm: log₁₀(6.2832×10−4) = −3.2018.
- SNR = −20 × −3.2018 = 64.036 dB.
Assumptions and limits
- Uniform sampling at a fixed, known rate with a clean, stable clock unless the page names a jitter term.
- Ideal, linear converters and a signal that stays inside full scale.
Common mistakes
- Using a frequency above fs/2, which aliases to a different baseband frequency.
- Mixing hertz with radians per second or rad/sample.
- Counting jitter or quantisation as the only noise source and forgetting thermal and clock noise.
Statistics and uncertainty
Real converters add offset, gain error, nonlinearity and thermal noise to the ideal quantisation and jitter limits, so a measured SNR is typically 3 dB to 10 dB below the ideal figure. Treat these formulas as upper bounds on performance.
Where it is used
- Generating test signals and predicting what a converter can resolve.
- Choosing a sampling rate, a bit depth and a clock source for an acquisition chain.
Maths you need
- Sampling theorem: a signal bandlimited below fs/2 is fully described by its samples
- Decibel: dB = 20 log₁₀ of an amplitude ratio
- Nyquist frequency: half the sampling rate