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A discrete signal or discrete-time signal is a time series consisting of a sequence of quantities. Unlike a continuous-time signal, a discrete-time signal is not a function of a continuous argument; however, it may have been obtained by sampling from a continuous-time signal.
The term discrete-time refers to the fact that the transform operates on discrete data, often samples whose interval has units of time. From uniformly spaced samples it produces a function of frequency that is a periodic summation of the continuous Fourier transform of the original continuous function.
Signals can be classified as continuous or discrete time. In the mathematical abstraction, the domain of a continuous-time signal is the set of real numbers (or some interval thereof), whereas the domain of a discrete-time (DT) signal is the set of integers (or other subsets of real numbers). What these integers represent depends on the nature ...
A digital signal is an abstraction that is discrete in time and amplitude. The signal's value only exists at regular time intervals, since only the values of the corresponding physical signal at those sampled moments are significant for further digital processing. The digital signal is a sequence of codes drawn from a finite set of values. [10]
The discrete-time Fourier transform, on the other hand, maps functions with discrete time (discrete-time signals) to functions that have a continuous frequency domain. [2] [3] A periodic signal has energy only at a base frequency and its harmonics; thus it can be analyzed using a discrete frequency domain. A discrete-time signal gives rise to a ...
Instead of using the Laplace transform (which is better for continuous-time signals), discrete-time signals are dealt with using the z-transform (notated with a corresponding capital letter, like () and ()), so a discrete-time system's transfer function can be written as:
Systems with this property are known as IIR systems or IIR filters. In practice, the impulse response, even of IIR systems, usually approaches zero and can be neglected past a certain point. However the physical systems which give rise to IIR or FIR responses are dissimilar, and therein lies the importance of the distinction.
In particular, the DTFT of the product of two discrete sequences is the periodic convolution of the DTFTs of the individual sequences. And each DTFT is a periodic summation of a continuous Fourier transform function (see Discrete-time Fourier transform § Relation to Fourier Transform). Although DTFTs are usually continuous functions of ...