Signal Processing Fundamentals

This section collects the definitions and elementary facts from signal processing that recur throughout the documentation. Each page is self-contained and refers to no particular implementation; the elements that realize these operations cite the definitions given here, so that the mathematics is stated in one place.

We work in continuous time. A signal is a function of a real time variable, and a filter acts by an integral. The discrete-time realization obtained by sampling, together with the numerical examples, is set apart in admonitions labeled In discrete time.

Notation

Unless stated otherwise:

• \(x(t)\) is an input signal and \(y(t)\) an output signal, with \(t \in \mathbb{R}\) measured in seconds.
• \(h\) is a filter, given by its impulse response \(h(\tau)\). The filter has finite impulse response of duration \(T\) when \(h\) is supported on \([0, T]\).
• \(\omega\) is an angular frequency, and \(i\) is the imaginary unit.
• \(f_s\) is the sampling frequency, used only in discrete-time admonitions, where time and sample index are related by \(t = n / f_s\).

Contents

• Impulse Response and Convolution: the impulse response, convolution, and its relation to cross-correlation.
• The Transfer Function and Frequency Response: the Laplace transform of the impulse response and its restriction to the imaginary axis.
• Phase Delay and Group Delay: the two measures of the time shift a filter imposes.
• Linear-Phase and Minimum-Phase Filters: two distinguished classes of filter and their delay properties.
• Latency: the kinds of latency in a real-time stream (physical and algorithmic), and how the output timestamp is chosen.

References

• A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, 3rd ed., Pearson, 2009.
• B. P. Lathi and R. A. Green, Linear Systems and Signals, 3rd ed., Oxford, 2018.
• Signal processing (Wikipedia).