Sampling (signal processing)

Sub-sampling ( sampling in English ) is understood in the signal processing, the recording of measurement values ​​to discrete, generally equidistant instants. As a discrete-time signal is obtained from a time-continuous signal.

In multi-channel signals, each sample gives a " sample " of several samples. The number of samples per second is called the sample rate. In digital telephony ( ISDN), the sampling rate is 8 kHz, for example.

• 4.1 Conversion of discrete data into an analog signal
• 4.2 converting an analog signal to discrete data
• 4.3 The process signal data signal
• 4.4 The process data signal data

Demarcation

Digitizing an analog signal in the time domain comprises as an additional term to the sampling a further conversion, quantization, the two transformations can be carried out in any sequence to obtain a digital signal:

• Sampling a continuous-time signal to be converted into a discrete-time signal.
• The quantization of a continuous-value signal to convert it into a value-discrete signal.

Sampling in the time domain

Ideal sampling

For a simpler mathematical description of the ideal sampling is defined. Here the signal is not accumulated over time to the sampling time, but is evaluated precisely at the sampling time nT.

Mathematically, this is represented by the signal s (t ) with the Dirac comb, a sequence of Dirac shocks multiplied:

For the frequency spectrum Sa representative of the Fourier series of signal Sa, is obtained by means of the inversion of the convolution theorem:

The spectrum is thus the spectrum of the input signal s ( t), which is periodically repeated with period 1 / T - which expresses the convolution property of the Dirac pulse from. As a result, the spectrum of s up to 1 / (2T) can be wide, so as not to overlap the shifted spectra.

Is the spectrum of S is narrower than 1 / ( 2T), the original signal s (t ) according to the ideal low-pass filtering of the time discrete spectrum is completely reconstructed. This fact is the basis of the Nyquist -Shannon sampling theorem. However, the spectrum of the input signal s (t ) is wider than 1 / ( 2T), aliasing occurs and the original signal s (t ) can not be retrieved from SA (t).

Real sampling

In reality, the two conditions are unachievable ideal sampling:

Therefore, the real sampling conducted under the following modifications:

To 1: The Dirac comb function is replaced by a rectangular (rect ) of square wave pulses of length T0. The sampling is carried out by a sample - and-hold circuit which holds the value of a scan of the length of the rectangular pulse constant. Mathematically, this corresponds to a convolution with the rectangular function:

The spectrum is obtained therefrom

This is the spectrum of the ideal scan, weighted by a factor which includes the sinc function ( sinc function). This is distortion of the signal which can be corrected by an additional distortion in the reconstruction filter in the recovery of the original signal. This distortion does not occur in the natural sample.

To 2: In order to recover also with a non-ideal reconstruction filter the continuous signal from the spectrum with the smallest possible error, the sampling frequency can be increased. By the over-sampling, the individual spectra graphically move farther apart, causing the low-pass filter for the reconstruction in the image spectrums having higher attenuation values ​​.

Sampling in the spectral range

Due to the symmetry properties of the Fourier transform can be inverted and a function of frequency S (f) in the spectral range, with the ideal scan form through a sequence Sp ( f) with frequency- discrete values ​​:

The spectral sequence Sp ( f ) consists of weighted Dirac pulses, which describe individual, discrete frequencies. Such a discrete spectrum is also called the line spectrum.

Thus, by the inverse Fourier transform of it are formed the associated periodic form of the time function sp (t ):

And in the scanning in the spectral domain the sampling theorem applies to " reverse " the form: if the duration of a signal s ( t) is less than 1 / F, then the periodic components of SP (t) do not overlap each other. The object of the reconstruction filter in the time domain takes over a gate circuit, in the simplest case a switch which turns on for the duration of 1 / F and locks the rest of the time. However, if the signal s (t ) for more than 1 / F, there is temporal overlap and the original waveform can no longer be reconstructed.

In the real sample in the spectral range is in lieu of a sequence of Dirac pulses, a spectral sequence of rectangular pulses, which each cover a band-limited detail of the spectrum itself. This function can take bandpass filter in the technical reference.

General mathematical representation of the sampling

When storing a piece of music on a CD, the sampled signal for transmission and storage of the analog output signal is used. The method used for sampling depends in this case on the method used for analog reconstruction. This view is also advantageous for the mathematical treatment.

The composition of sample and playback in the reverse direction in the message transmission occurs, for example, when a binary-coded message is converted into an analog radio signal. Through a sampling process, the original binary string is then reconstructed.

Conversion of discrete data into an analog signal

In the simplest case, the transformation of a sequence of real numbers, that is a time-discrete signal, performed by a single core operation. That is, into a sequence by means of a function h, and a time increment T which, in the broadest sense, interpolating function

Formed. Its Fourier transform is

Where H ( f) is the Fourier transform of h ( t).

Converting an analog signal to discrete data

A more realistic model of the measurement of a time varying process is the formation of a weighted mean value over a certain period. This can be accomplished mathematically by convolving with a weight function w. Let x ( t) is the signal to be measured, and V (t) is the measured value at time t (which is assigned to the center of gravity of the weight function, for example ), then

Taking the Fourier transform of the convolution goes to the multiplication. Are W, V and X, the Fourier transform of w, v, and x, then, V (f) = W ( f) X (f).

The process signal data signal

You are now determined a sequence of measured values ​​with time step T, in order to use them in the interpolation rule, we obtain an analog signal rekonstruktiertes

To estimate the error of the whole process of discretization and playback, you can apply this process in a simple frequency- limited test signals. This may be abbreviated by the determination of the Fourier transform of the model. For this purpose, however, is the Fourier series

To determine more precisely. According to the Poisson summation formula this periodic function is identical to the periodization of V ( f). Be the sampling frequency, then applies

In summary, therefore

A frequency component at the frequency f so suffers at a distortion factor and the aliasing of the starch to the frequency.

To approximate baseband signals as well as possible, it is necessary to apply that in a neighborhood of f = 0 and for the same f and for all. As part of a mathematically exact theory these requirements are satisfied and all operations are well defined if

• And with a B > 0, f is valid for all and
• ( the Kronecker delta) for f = 0 is continuous and there is a zero.

One then obtains the square value of the norm function space L ² is a base band function x (t) with the highest frequency according to the Parseval Identity, an estimate of the relative error, i.e., the signal -to-noise ratio than

Examples: If w and h rectangular functions of width T centered at 0, then

And it is true.

Conversely, if w and h is the cardinal sine functions, their Fourier transform of the corresponding rectangular functions, and the resulting reconstruction formula is the cardinal number of the Whittaker - Kotelnikov - Nyquist -Shannon sampling theorem.

In any case, functions with frequency components above lead to errors in the alias frequency range, the frequency limit of the sampling theorem is therefore necessary, but not sufficient for error-free reconstruction.

The process data signal data

In this direction, the " interpolating " function a (t ) is sampled. So it turns out

For the Fourier series one obtains

According to the Poisson summation formula applies in this case

If the Fourier series of the sequence c, and thus the result are retained so that sum the value 1 must have everywhere. The maximum deviation therefrom is also in this case a sensor for the relative error in the data transmission.

From a mathematical point of view the functions W and H must again comply with the above mentioned barrier of periodization of the magnitude squared.

Other types of sampling

Traditionally, the equidistant (periodic ) sampling is the most used because it has already been studied very extensively and implemented in many applications. In recent decades, other types of sampling were examined, which dispense with equal time intervals between samples, which promises several advantages such as effective utilization of the communication channel. These include, inter alia, end -on -delta sample.

Literature sources

• Hans Dieter Lüke: signal transmission. 11th edition. Springer Verlag, 2010, ISBN 978-3-642-10199-1.
• Hans Dieter Lüke: signal transmission (online version ). 11th edition. Springer Verlag, 2010, ISBN 978-3-642-10200-4.