Signal processing is a discipline dedicated to the engineering end of stochastic process inference and prediction, especially linear time series
There are various translation difficulties for statisticians; “Testing”=“Detection”, “Linear Filter”=“ARIMA model”, estimation of parameters is system identification, estimation of hidden states is filtering and so on.
This is a very general note to mention that the field exists. Most useful information is under subfields, e.g. machine listening, for some signal processing tricks for audio, or feedback systems, for some models particularly appropriate to systems that accept their own output as input etc.
See also orthogonal decompositions. There are close connections to optimal control.
Anyway, I don’t need to explain that here; there are so many software engineers involved with it. the internet is full of interactive diagrammy textbooks to fill that niche.
But here are some notes on some nuggets of interest that I wasn’t sure where else to file.
Signal processing on graphs
Nothing to say here yet but I feel I should raid the literature of the EPFL Signal processing lab 2 who make a specialty of it.
Stochastic decomposition
Model for decomposing harmonic sound into pure tones plus other stuff (aside: why not other periodic functions?) This is just some kind of parametric state or system inference, right?
Sampling
Signal sampling is the art of turning continuous signals into discrete ones and back again.
Resources
See also the slightly more specialised and overlapping list of filter design resources

Tom O’Haver has a free online textbook with extensive OCTAVE/MATLAB code, A Pragmatic Introduction to Signal Processing. Very skewed towards pure Fourier domain techniques.

Textbook: Paolo Prandoni and Martin Vetterli, Signal Processing for Communications is available online. Vetterli is very smart at unexpected and enlightening perspectives; I’m a fan.

Textbook: Antoniou has been generally recommended if you want to get handson ASAP. (Anto05)

Textbook: Orfandis’ opus is free online. (Orfa96)

Course notes/textbook: Oppenheim and Verghese, Signals, Systems, and Inference is free online.

Numerical tours of signal processing gives python, julia and matlab tours of signal processing. Better consumed through their github repo.
Refs
 PaRa15: Sameer Pawar, Kannan Ramchandran (2015) A robust sublinear time RFFAST algorithm for computing a sparse DFT. ArXiv:1501.00320 [Cs, Math].
 GrDa10: Robert M. Gray, Lee D. Davisson (2010) An introduction to statistical signal processing. Cambridge: Cambridge University Press
 Nyqu28: H. Nyquist (1928) Certain Topics in Telegraph Transmission Theory. Transactions of the American Institute of Electrical Engineers, 47(2), 617–644. DOI
 NaIV02: M. J. Narasimha, A. Ignjatovic, P. P. Vaidyanathan (2002) Chromatic derivative filter banks. IEEE Signal Processing Letters, 9(7), 215–216. DOI
 Shan49: C. E. Shannon (1949) Communication in the Presence of Noise. Proceedings of the IRE, 37(1), 10–21. DOI
 Anto05: Andreas Antoniou (2005) Digital signal processing: signals, systems and filters. New York: McGrawHill
 Marp87: S. Lawrence Marple Jr. (1987) Digital spectral analysis with applications
 Ther92: Charles W. Therrien (1992) Discrete random signals and statistical signal processing. Englewood Cliffs, NJ: Prentice Hall
 OpSB99: Alan V. Oppenheim, Ronald W. Schafer, John R. Buck (1999) Discretetime signal processing. Upper Saddle River, N.J: Prentice Hall
 Kay93: Steven M. Kay (1993) Fundamentals of statistical signal processing. Englewood Cliffs, N.J: PrenticeHall PTR
 Smit07: Julius O. Smith (2007) Introduction to Digital Filters with Audio Applications. http://www.w3k.org/books/: W3K Publishing
 Orfa96: Sophocles J. Orfanidis (1996) Introduction to signal processing. Englewood Cliffs, N.J: Prentice Hall
 KaSH00: Thomas Kailath, Ali H. Sayed, Babak Hassibi (2000) Linear estimation. Upper Saddle River, N.J: Prentice Hall
 MoSt00: Todd K. Moon, Wynn C. Stirling (2000) Mathematical methods and algorithms for signal processing. Upper Saddle River, NJ: Prentice Hall
 Cham85: Hal Chamberlin (1985) Musical applications of microprocessors. Hasbrouck Heights, N.J: Hayden Book Co.
 Laro07: Jean Laroche (2007) On the Stability of TimeVarying Recursive Filters. Journal of the Audio Engineering Society, 55(6), 460–471.
 Bart46: M. S. Bartlett (1946) On the Theoretical Specification and Sampling Properties of Autocorrelated TimeSeries. Supplement to the Journal of the Royal Statistical Society, 8(1), 27–41. DOI
 PrVe08: Paolo Prandoni, Martin Vetterli (2008) Signal processing for communications. Lausanne: EPFL Press
 QiCh94: Shie Qian, Dapang Chen (1994) Signal representation using adaptive normalized Gaussian functions. Signal Processing, 36(1), 1–11. DOI
 StMo05: Petre Stoica, Randolph L. Moses (2005) Spectral Analysis of Signals. Upper Saddle River, N.J: Prentice Hall
 Scar81: Jeffrey D Scargle (1981) Studies in astronomical time series analysis IModeling random processes in the time domain. The Astrophysical Journal Supplement Series, 45, 1–71.
 RaZa52: J. R. Ragazzini, L. A. Zadeh (1952) The analysis of sampleddata systems. Transactions of the American Institute of Electrical Engineers, Part II: Applications and Industry, 71(5), 225–234. DOI
 HoLD10: Scott H. Holan, Robert Lund, Ginger Davis (2010) The ARMA alphabet soup: A tour of ARMA model variants. Statistics Surveys, 4, 232–274. DOI
 PrLu13: Tommaso Proietti, Alessandra Luati (2013) The Exponential Model for the Spectrum of a Time Series: Extensions and Applications (SSRN Scholarly Paper No. ID 2254038). Rochester, NY: Social Science Research Network
 Moor74: J.A Moorer (1974) The optimum comb method of pitch period analysis of continuous digitized speech. IEEE Transactions on Acoustics, Speech and Signal Processing, 22(5), 330–338. DOI
 BJRL16: George E. P. Box, Gwilym M. Jenkins, Gregory C. Reinsel, Greta M. Ljung (2016) Time series analysis: forecasting and control. Hoboken, New Jersey: John Wiley & Sons, Inc
 Wish14: Aaron Wishnick (2014) TimeVarying Filters for Musical Applications. In DAFx (pp. 69–76).