1.2. Fourier Fast transform.
Transform of Fourier for a discrete signal gives a continuous spectrum in a range [0, fS/2], hence, it is inconvenient from the point of view of calculations. In practice instead of transform of Fourier use discrete transform of Fourier (FFT). FFT it is equivalent to a number of Fourier for analog signals.
A number of Fourier for periodic analog signals:
Where Т – the period, and ω0 – the basic frequency.
Fourier discrete transform (FDT) for periodic discrete signals
It is necessary to notice that pair FDT and numbers of Fourier are defined for periodic signals. There Fourier at calculation FDT it is necessary to assume periodicity of a signal with the period in N readout. The discretizired variant is considered periodic if analog frequency satisfies to a following condition:
Where m - number of cycles for which it is calculated FDT.
Complexity of calculations (number of operations of addition and multiplication) FDT can be reduced with N2 to N LogN by means of algorithms Fourier fast transform (FFT). In these algorithms N usually is number 2 degree.
1.3. Quantization
ADC works with the limited quantity of bits (has the limited word length). As a result continuous values of amplitude are approximated by discrete levels. Process of transform of continuous values of amplitude in discrete is called as quantization. Such approach leads to discrepancies which are called as an error, o discretization and quantization r a quantization error.
The quantization interval depends on quantity of levels of quantization or permission levels. It is obvious that the size of an error of quantization ADC depends on a quantization interval. The more bits the interval of quantization is used for quantization, the already and, hence, quantization error there is less.
2.Study of processes of discretization and quantization
2.1. Discretization.
In this example the discrete signal is formed of readout of a sinusoidal signal. When нормированная frequency of a discrete signal f/fS becomes more than 0.5, or frequencies of Nyquist, there is obvious an effect of imposing.
For discretization process we will establish frequency of discretization fS = 1 kHz and number of readout N = 10. As a result we will receive a signal, discretizated for a time interval 10 мс. Frequency of a signal varies from 0 to 1000 Hz by means of management elements on the obverse panel. In a Figure 2 the sinusoidal signal with frequency of 300 Hz, discretizated is shown at frequency 1 кHz for 10 мс. 10 readout which are displayed on the schedule of oscillogram Waveform Graph Thus turn out.
Figure 2.
Besides, on the schedule the analog signal received as a result of discretization of a sinusoidal signal with frequency in 100 times more is presented. In other words, analog representation is received at frequency of discretization 100 kHz from 1000 readout.
The obverse panel of it VI contains horizontal slider regulator Horizontal Slide for management of frequency of a signal and two numerical indicators Numeric Indicator for rate setting and false frequencies. On the obverse panel the button stop, connected to a cycle on a condition on the block diagram also is located. This button stops work VI.
In a Figure 3 it is possible to see the block diagram of this system of discretization. For formation of analog and discrete sinusoidal signals three are used VI Sine Wave (Sinusoid) (Functions → Signal Processing → Signal Generation → Sine Wave (Functions → signal Processing →Generation signal →Sine Wave). These VI are located vertically in the middle of the block diagram. On their inputs the quantity of readout, amplitude, frequency and phase shift move. If Amplitude isn't connected to any terminal by default its value is considered equal 1. Frequency for a conclusion f should be expressed in cycles on readout, that is this size is return to quantity of readout for the period. The phase is set by the numerical constant 90 attached to input Phase in.
Figure 3. The example block diagram in which there is an effect of imposing – a case TRUE.
On an exit this VI gives a file of readout of a sinusoid. Function Build Waveform (to Create the oscillogram) (Functions - > Programming—> Waveform—> Build Waveform (Functions - > Programming → the Oscillogram—> to Create the oscillogram)) unites readout from input Y and time intervals between readout TS = 1/1000, submitted on an input dt. As the number of readout for analog representation of a signal in 100 times more, than for a discrete signal was told earlier. That the parity between frequency and readout remains same as for a discrete signal, the value coming on an input f, we will divide on 100. Besides, a time interval between readout of an analog signal we will set equal the one 100 - th interval of a discrete signal.
The top device from three VI from the block diagram in a Figure 3. Generates a discrete signal, average — an analog signal and bottom — a signal with false frequency which appears when frequency of a signal exceeds frequency of Nyquist.
To distinguish cases of discretization with the advent of false frequencies and without them, the structure of a choice of variant Case Structure is used. If normalized frequency more than 0.5 that there corresponds to a case TRUE the third VI Sine Wave generates a signal with effect of imposing. All inputs remain former, except frequency of a signal with imposing.
Pay attention that for frequency reception at which there is an effect of false frequency, the knot of expression Expression Node (Functions → Mathematics →Numeric → Expression Node (Functions → Mathematics Numerical → expression Knot)) is used. The expression knot is usually used for calculation of expressions from one variable. In it the set of the built in functions (for example, abs — the number module), the equations required for the decision is accessible.
For a case the False, that is discretization without effect of false frequencies, isn't required to generate a signal with false frequency. There Fourier the analog signal is connected to an exit of structure of a choice of a variant so that the same signal was displayed on the oscillogram schedule, and frequency of imposing of a signal is established equal 0. Don't forget that at use of structure of a choice of a variant it is necessary to connect conductors all exits for each case.
With imposing it is possible to see result of discretization in a Figure 4 on which readout of a sinusoid with frequency of 700 Hz are represented. From drawing it is visible that such readout also probably to receive from a sinusoid with frequency of 300 Hz, as shown a dashed line
Figure 4.
All three oscillograms are united by function Build Array (to Construct a file) and displayed on the same schedule. Properties Waveform Graph should be adjusted how is shown in drawing. Stretch the panel of editing of schedules downwards, and you will see three elements. Rename labels in appropriate way. Click the right button on the schedule of the oscillogram and choose from the menu point Properties (Properties). There will be dialogue window Waveform Graph Property (Property of the schedule of the oscillogram). Open window Plots (Schedules) and adjust style of the schedule. Choose Sampled Signal from dropping out menu Plot (Schedule) and set options Point Style (Style of a point), Plot Interpolation (schedule Interpolation), Fill to (Filling). For a signal with false frequency choose dotted type of a line.
Rename all elements of management and indicators and establish the maximum value horizontal sliding regulator Horizontal Slide equal 1000. After that given VI it is possible to consider ready.