Signal Processing

Signal processing is the study and application of methods for analyzing, modifying, generating, and interpreting signals: representations of information that vary over time, space, or another independent variable. Examples include sound recordings, electrical measurements, images, and sensor observations. Rooted in electrical engineering, it draws on mathematics, statistics, and computational methods. Its objectives include recovering useful information, reducing unwanted components, changing signal representations, and enabling efficient transmission or storage. Signal processing encompasses both physical devices and mathematical algorithms. (signalprocessingsociety.org)

Signals and their representations

A signal can be modeled as a function, such as $x(t)$ for a continuous-time waveform or $x[n]$ for a discrete-time sequence. The independent variable need not be time: an image varies across spatial coordinates, while a video combines spatial and temporal variation. Signals may have scalar values or several components, such as multiple microphone channels. Their representation determines which operations and analytical tools are appropriate. (ocw.mit.edu)

Continuous-time and discrete-time describe the signal’s domain, whereas analog and digital also concern how its values are represented. Sampling creates a sequence of measurements; quantization assigns measured amplitudes to a finite set of levels. A discrete-time mathematical signal can therefore have continuously valued amplitudes without being a finite-precision digital representation. In practical digital systems, both sampling and quantization influence the information retained. (dspguide.com)

Deterministic models specify signal values directly. Statistical models describe uncertain signals using stochastic processes, with properties such as expected values, variances, and correlations. These models are particularly important when observations contain noise or when the underlying source cannot be described exactly. (ocw.mit.edu)

Systems, convolution, and filtering

A system maps an input signal to an output. An important class consists of linear time-invariant systems. Linearity means that weighted combinations of inputs produce the corresponding weighted combinations of outputs; time invariance means that shifting the input shifts the output by the same amount. Such systems can be characterized by their impulse response. (ocw.mit.edu)

For a discrete-time linear time-invariant system, the output is expressed through convolution:

where $x[n]$ is the input and $h[n]$ is the impulse response. The continuous-time counterpart uses an integral. This relationship connects a system’s local response to its behavior for general inputs. (ocw.mit.edu)

Filtering changes selected properties of a signal. A low-pass filter preserves lower-frequency components while attenuating higher-frequency components; other filters select different frequency ranges. A moving average, for example, smooths a sequence by combining neighboring samples. Filtering can suppress unwanted variation, but it can also remove useful detail when the desired and unwanted components overlap. (ocw.mit.edu)

Frequency-domain analysis

The Fourier transform represents a signal in terms of frequency components. Its complex-valued coefficients describe both magnitude and phase. For linear time-invariant systems, convolution in the time domain corresponds to multiplication in the frequency domain, making frequency-domain analysis especially useful for understanding filtering and system response. (ocw.mit.edu)

For finite sequences, the discrete Fourier transform provides a finite set of frequency coefficients. The fast Fourier transform is a family of efficient algorithms for computing that transform, rather than a different mathematical transform. These methods support spectral analysis and computationally efficient implementations of convolution. Their outputs describe the analyzed record, whose length and treatment affect the resulting spectral representation. (ocw.mit.edu)

Sampling and digital processing

The sampling theorem establishes conditions under which a continuous-time signal can be reconstructed from regularly spaced samples. In its standard low-pass form, a signal with no frequency components above $B$ can be reconstructed ideally when the sampling frequency exceeds $2B$. The result assumes an appropriately band-limited signal and ideal reconstruction; it does not imply that arbitrary waveforms can be recovered from any sampling rate. (ocw.mit.edu)

Insufficient sampling can cause aliasing, in which different continuous-time frequencies become indistinguishable in the sampled sequence. Practical acquisition systems use an antialiasing filter before conversion to attenuate frequencies that would otherwise fold into the retained band. Quantization introduces a separate error by limiting amplitude resolution. A typical processing chain therefore includes analog filtering, analog-to-digital conversion, numerical processing, and, when an analog output is required, digital-to-analog conversion followed by reconstruction filtering. (dspguide.com)

Statistical inference and learned methods

Statistical signal processing distinguishes estimation—inferring an unknown quantity—from detection—deciding among hypotheses about an observation. Examples include estimating a source waveform from noisy measurements or deciding whether a transmitted signal is present. Performance depends on the assumed signal and noise models and on the criterion used to compare possible solutions. (ocw.mit.edu)

A Wiener filter constructs a linear estimate using second-order statistical information, with the objective of minimizing mean squared error. Its optimality is relative to the specified model and estimator class. Machine learning provides another set of methods for learning relationships from data, including mappings between observed signals and desired outputs. These approaches overlap with established signal-processing methods rather than replacing the field’s mathematical foundations. (ocw.mit.edu)

Applications

In telecommunications, signal processing supports encoding, modulation, reception, and recovery of transmitted information. Audio systems use it for noise reduction, echo control, and compression, while speech recognition uses signal-derived representations to identify spoken content. (signalprocessingsociety.org)

Image processing extends these ideas to spatial data, including enhancement, restoration, and compression. Biomedical applications include processing physiological recordings and reconstructing magnetic resonance images. Across these applications, acquisition, representation, filtering, and inference form interconnected parts of the processing task. (signalprocessingsociety.org)