Summary of Statistics for Digital Signal Processing
Statistics for Digital Signal Processing: A Student Guide
Introduction
Digital Signal Processing (DSP) statistics provide the tools to describe, quantify and interpret signals measured in engineering and science. This material covers basic statistical descriptors used in DSP: mean, standard deviation, variance, root-mean-square, histograms and how to treat signals produced by underlying processes. These concepts help you summarize signal behavior, estimate noise effects, and prepare signals for further processing.
Definition: A signal is a description of how one parameter is related to another; for example, amplitude as a function of time.
1. Types of signals
Continuous vs. discrete
- Continuous signal: Both independent and dependent variables can assume continuous values.
- Discrete (digitized) signal: One or both parameters are quantized into discrete values (samples).
Definition: The domain of a signal is the type of parameter on the horizontal axis (e.g., time, frequency, space).
2. Mean and measures of spread
Mean (average)
- The mean $mu$ of a finite signal with $N$ samples $x_i$ is $$mu = \frac{1}{N} \sum_{i=1}^{N} x_i$$
- The mean describes the DC (constant) component or central tendency of the signal.
Standard deviation and variance
- Variance measures the average squared deviation from the mean: $$\sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i - \u0003mu)^2$$
- Standard deviation is the square root of variance: $$\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \u0003mu)^2}$$
- These quantify how much the signal fluctuates around its mean (the AC portion).
Definition: Variance $\sigma^2$ is the power of the fluctuation about the mean.
Root-mean-square (RMS)
- RMS measures the combined effect of AC and DC components: $$\text{rms} = \sqrt{\frac{1}{N} \sum_{i=1}^{N} x_i^2}$$
- For signals with zero DC (mean zero), $ \text{rms} = \sigma$.
Quick comparison table
| Quantity | Formula | Measures | Notes |
|---|---|---|---|
| Mean $\u0003mu$ | $\frac{1}{N}\sum_{i=1}^{N} x_i$ | DC / central tendency | Sensitive to offsets |
| Variance $\sigma^2$ | $\frac{1}{N}\sum_{i=1}^{N} (x_i-\u0003mu)^2$ | Power of fluctuations | Always nonnegative |
| Std. dev. $\sigma$ | $\sqrt{\sigma^2}$ | Spread of values | Same units as $x$ |
| RMS | $\sqrt{\frac{1}{N}\sum x_i^2}$ | AC+DC magnitude | Equals $\sigma$ if mean $=0$ |
3. Relationships and practical considerations
Relationship to peak-to-peak
- Different waveform shapes (sine, square, triangular) have different relationships between $\sigma$ and peak-to-peak amplitude. Know the waveform shape to relate statistical to amplitude measures.
Numerical limitations when computing mean and variance
-
Two main issues when computing statistics on finite-precision machines:
- When $\u0003mu \gg \sigma$, subtracting two nearly equal numbers in variance computation can cause large round-off error.
- Naive running statistics that recompute sums over all samples on each update are inefficient.
-
Stable incremental formulas (running statistics) avoid catastrophic cancellation and unnecessary recomputation. Use numerically stable algorithms (for example, Welford's method) in practice.
Definition: Running statistics are algorithms that update mean and variance incrementally as new samples arrive, without recomputing over the entire dataset.
Signal-to-noise measures
- When the mean represents the signal of interest and fluctuations are considered noise:
- Signal-to-noise ratio (SNR): $\text{SNR} = \frac{\u0003mu}{\sigma}$ (often expressed in linear units)
- Coefficient of variation (CV): $\text{CV} = \frac{\sigma}{\u0003mu} \times 100%$ — useful to express variability relative to mean.
4. Signals vs underlying processes
- Statistics of acquired signals change each repetition due to statistical noise from the measurement.
- Probability characterizes the underlying pro
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DSP Statistics Basics
Klíčové pojmy: Mean $\u03BC$ is $\frac{1}{N}\sum_{i=1}^{N} x_i$, Variance $\sigma^2$ is $\frac{1}{N}\sum_{i=1}^{N}(x_i-\u03BC)^2$, Standard deviation $\sigma$ equals $\sqrt{\sigma^2}$, RMS $=\sqrt{\frac{1}{N}\sum x_i^2}$ measures AC+DC, Use running/stable algorithms to avoid numerical cancellation, SNR $=\u03BC/\sigma$, CV $=\sigma/\u03BC\times100\%$, Segment nonstationary signals into windows before averaging statistics, Normalized histogram $p_i=H_i/N$ approximates probability