In many common situations in engineering a function f(t) is sampled. When a function is evaluated by numerical procedures, it is always necessary to sample the function in some manner, because digital computers cannot deal with anologue, continuous functions (except by sampling them!). Often the function is not even explicitly defined, but only known as a series of values recorded on tape, data logger or computer.
If the signal to be analysed is anologue in nature then it must be converted into digital form, as it is sampled, by an analogue to digital (A/D) converter.
If delta is the time interval between consecutive samples, then the sampled time data can be represented as
Examples of where sampled time domain data is used in engineering include simple and complex vibration analysis of machinery, as well as measurements of other variables such as boiler pressures, temperatures, flow rates and turbine speeds and many other machine parameters. It is also used in areas where computers and microcontrollers are used to automate processes and react to input data from the processes.
Consider an analogue signal x(t) that can be viewed as a continuous function of time, as shown below. We can represent this signal as a discrete time signal by using values of x(t) at intervals of nTs to form x[n] as shown. We are "grabbing" points from the function x(t) at regular intervals of time, Ts , called the sampling period.
It is usual to specify a sampling rate or frequency fs rather than the sampling period. The frequency is given by fs = 1/Ts, where fs is in Hertz. If the sampling rate were high enough, then the signal x(t) could be constructed from x[n] by simply joining the points by small linear portions. This approximates to the analogue signal.
One would expect that if the signal has significant variation then Ts must be small enough to provide an accurate approximation of the signal x(t). Significant signal variation usually implies that high frequency components are present in the signal. It could therefore be inferred that the higher the frequency of the components present in the signal, the higher the sampling rate should be. If the sampling rate is not high enough to sample the signal correctly then a phenomenon called aliasing occurs.
This figure shows the effect of different sampling rates when sampling the function cos(60 * t). Refer the M-files List for an m-file which illustrates the phenomenon.
The term aliasing refers to the distortion that occurs when a continuous time signal has frequencies larger than half of the sampling rate. The process of aliasing describes the phenomenon in which components of the signal at high frequencies are mistaken for components at lower frequencies.
The Nyquist Sampling Theorem states that to avoid aliasing occuring in the sampling of a signal the samping rate should be greater than or equal to twice the highest frequency present in the signal. This is refered to as the Nyquist sampling rate.
When a continuous time signal is sampled, its spectrum will show the aliasing effect if aliasing occurs because regions of the frequency domain will be shifted by an amount equal to the sampling frequency (see Tutorial 5 on the frequency spectrum and filtering).
