Discrete-time signal
A discrete signal or discrete-time signal is a time series consisting of a sequence of qualities. In other words, it is a type series that is a function over a domain of discrete integral.
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, and then each value in the sequence is called a sample. When a discrete-time signal obtained by sampling a sequence corresponding to uniformly spaced times, it has an associated sampling rate; the sampling rate is not apparent in the data sequence, and so needs to be associated as a separate data item.
System clock
One of the fundamental concepts behind discrete time is an implied (actual or hypothetical) system clock.[1] If one wishes, one might imagine some atomic clock to be the de facto system clock.
Time signals
Uniformly sampled discrete-time signals can be expressed as the time-domain multiplication between a pulse train and a continuous time signal. This time-domain multiplication is equivalent to a convolution in the frequency domain. Practically, this means that a signal must be bandlimited to less than half the sampling frequency, i.e. Fs/2 - ε, in order to prevent aliasing. Likewise, all non-linear operations performed on discrete-time signals must be bandlimited to Fs/2 - ε. Wagner's book Analytical Transients proves why equality is not permissible.[2]
Acquisition
Discrete signals may have several origins, but can usually be classified into one of two groups:[3]
- By acquiring values of an analog signal at constant or variable rate. This process is called sampling.[4]
- By recording the number of events of a given kind over finite time periods. For example, this could be the number of people taking a certain elevator every day.
Digital signals
A digital signal is a discrete-time signal for which not only the time but also the amplitude has been made discrete; in other words, its samples take on only values from a discrete set (a countable set that can be mapped one-to-one to a subset of integers). If that discrete set is finite, the discrete values can be represented with digital words of a finite width. Most commonly, these discrete values are represented as fixed-point words (either proportional to the waveform values or companded) or floating-point words.
The process of converting a continuous-valued discrete-time signal to a digital (discrete-valued discrete-time) signal is known as analog-to-digital conversion. It usually proceeds by replacing each original sample value by an approximation selected from a given discrete set (for example by truncating or rounding, but much more sophisticated methods exist), a process known as quantization. This process loses information, and so discrete-valued signals are only an approximation of the converted continuous-valued discrete-time signal, itself only an approximation of the original continuous-valued continuous-time signal.
Common practical digital signals are represented as 8-bit (256 levels), 16-bit (65,536 levels), 32-bit (4.3 billion levels), and so on, though any number of quantization levels is possible, not just powers of two.
See also
- Aliasing
- Anti-aliasing filter
- Bernoulli process
- Digital
- Digital-to-analog converter
- Digital control
- Digital frequency
- Discrete system
- Nyquist frequency
- Nyquist–Shannon sampling theorem
- Whittaker–Shannon interpolation formula
- Sampling (signal processing)
Notes
References
- Gershenfeld, Neil A. (1999). The Nature of mathematical Modeling. Cambridge University Press. ISBN 0-521-57095-6.
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- Wagner, Thomas Charles Gordon (1959). Analytical transients. Wiley.
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