shiftsfrom

Shiftsfrom is a term used in digital signal processing to denote an operator that generates a set of time-shifted versions of a base signal starting from a chosen origin. The concept is used to illustrate how a single signal can be expanded into a bank of shifted copies for comparison, alignment, or augmentation.

Formally, for a discrete-time signal x[n] and a finite set of integers S, shiftsfrom(x, S) yields the collection { x[n - s] : s in S }. If the shifts are combined linearly as sum_{s in S} a_s x[n - s], the operator can emulate a finite impulse response filter with predetermined taps. The operation is time-invariant with respect to x, and the shifts are separable.

Applications include pattern matching, template tracking, echo detection, and training data augmentation in machine learning where shifted variants of a template are used to improve robustness to timing variations. Shiftsfrom helps in constructing feature representations that are invariant to small timing offsets and in generating synthetic examples for model training.

Implementation notes: In software, shiftsfrom can be implemented by indexing or through convolution with a shifted delta basis. For continuous signals, the same idea extends with appropriate interpolation. Memory considerations apply when S is large; streaming implementations avoid materializing all shifts and compute results on demand.

See also: time shift operator, delay, convolution, cross-correlation, data augmentation.