digital signal processing

Smartphone functions such as video recording, photography, video telephony and voice telephony itself are all based on digital processing of the respective image and sound signals from the built-in sensors (camera and microphone). The touchscreen also works by digitally processing the signals generated with the finger gestures .
CD player from 1983. With the compact disc , digital signal processing began to enter the private sector.
Digital compression of video data made compact, high-resolution camcorders possible.

The digital signal processing is a branch of the communications technology and deals with the production and processing of digital signals by means of digital systems . In a narrower sense, her focus is on the storage, transmission and transformation of information in the sense of information theory in the form of digital, time-discrete signals . It has diverse and far-reaching applications in today's world and has a strong influence on almost all areas of life, as it is one of the technical foundations of the digitization of all modern communication technology and entertainment electronics . This is also known as the digital revolution .

In practical use nowadays, almost all transmission, recording and storage methods for image and film ( photo , television , video ) and sound (music, telephony, etc.) are based on digital processing of the corresponding signals. Digital signal processing enables a variety of types of conversion and processing for digital data, e.g. B. the compression of audio and video data, non-linear video editing or image processing for photos. In addition, digital signal processing is also used - in addition to many other industrial areas of application - in measurement, control and regulation technology and in medical technology, for example in magnetic resonance imaging . These developments are a result of the rapid advances in digital and computer technology ( information technology ) over the past few decades. With the introduction of the music CD in the early 1980s, the above-mentioned “digitization” began to influence people's everyday lives, which is most evident today in the universal spread of versatile, multimedia- capable smartphones .

Digital signal processing is based on electronic components , such as digital signal processors ( DSP ) or powerful microprocessors , corresponding memory elements and interfaces for signal input and output. In the case of programmable hardware, the algorithms for signal processing can be supplemented by additional software that controls the signal flow. The digital signal processing offers possibilities and processing possibilities, which in the earlier common analog circuit technology could not be realized or only with great effort.

The methods of digital signal processing are much closer to mathematics , such as the sub-areas of number theory or coding theory , than to classical electrical engineering . The starting point was the general awareness of the fast Fourier transform (FFT) from 1965 through a publication by JW Cooley and John Tukey . In addition, the practical possibilities of digital circuit technology improved in the same period , so that the newly developed processes could be used.

Basic technical principle

Processing scheme of digital signal processing
Integrated electronic circuit (chip, in the center of the picture) with analog-digital converters for converting analog audio signals into a digital data stream, here on a PC sound card .

The digital processing of a signal always follows the scheme Analog → Digital → Processing → Analog. The changes to the signal are only made in the digital area. The procedure will be explained using the example of an audio CD :

1. When recording a sound , the sound pressure is converted into an analog alternating voltage via a microphone . This alternating voltage is converted into a sequence of digital values with the aid of an analog-digital converter .
The following values ​​are used for audio CDs:
• a sampling rate of 44.1 kHz, i.e. H. the signal is sampled 44,100 times per second
• a word length of 16 bits, i.e. H. the sampled, continuous value is mapped to one of 65,536 discrete values
2. In an intermediate step, the digital audio signal can now be processed, e.g. B. be filtered or provided with effects before it is saved.
3. To store the audio signal, the individual values ​​are written to the audio CD one after the other.
4. In order to play back the sound recording later, the data is read from the CD and converted back into a continuous alternating voltage by a digital-to-analog converter. This is then transmitted to the loudspeakers or an amplifier.

Construction of a digital signal processing system

System for digital signal processing

The diagram shows the typical structure of a signal processing system that always has analog components at the interface to the "outside world". Only the red-colored components in the lower part of the picture belong to the digital signal processing system in the narrower sense .

Let us follow the path of the signals in the graphic: Using a sensor , physical quantities are converted into an, often weak, electrical signal. This signal is used for further processing e.g. B. raised to the level required for the following steps with the help of an operational amplifier . The sample-and-hold stage samples values ​​from the amplified analog signal at specific time intervals and keeps them constant during an interval. An analog time-continuous curve becomes a time-discrete analog signal. A signal that is constant for a certain time is required by the analog-digital converter in order to determine the discrete digital values. These can then be processed by the digital signal processor. The signal then takes the opposite route and can optionally flow back into the technical process via an actuator .

Object: What is a signal?

In contrast to the continuous functions of analog signal processing, a digital signal is discrete in terms of time and value, i.e. a sequence of elementary signals (e.g. square-wave pulses ). This sequence usually arises in a time or space-periodic measurement process. For example, sound is converted into an electrical voltage by deflecting a membrane or bending a piezocrystal and this voltage is repeatedly converted into digital data by means of an AD converter. Such a realistic measurement process is finite, the resulting sequence has an initial index and an final index . ${\ displaystyle \ alpha}$${\ displaystyle \ omega}$

So we can define the signal as a data structure , with the distance between two data points, the indices and the finite sequence (array) of the data. ${\ displaystyle (\ delta, \ alpha, \ omega, s)}$${\ displaystyle \ delta}$${\ displaystyle \ alpha <\ omega}$${\ displaystyle s = (s _ {\ alpha}, \ dots, s _ {\ omega})}$

The data are instances of a data structure. The simplest data structure is the bit, the most common are (1, 2, 4 byte) integer and floating point number data. However, it is also possible that the individual datum itself is a vector or a sequence, such as when coding color information as RGB triples or RGBA quadruples, or that the signal contains the columns of a raster image . The individual column is again a signal that contains, for example, gray or color values ​​as data. ${\ displaystyle s}$${\ displaystyle s_ {k}}$

Abstraction of a signal

In order not to have to consider signals separately according to their beginning and end in theory, the finite sequences are embedded in the abstract signal space , a Hilbert space . Condition: The basis functions are orthogonal to one another, their cross-correlation therefore results in zero. An abstract signal is given by a pair . ${\ displaystyle \ ell _ {2} (V)}$${\ displaystyle (\ delta, s), \ \ delta> 0, \ s \ in \ ell _ {2} (V)}$

The Euclidean vector space models the data type of the signal, for example for simple data, for RGB color triples. An element in is a doubly infinite sequence . The defining property for the sequence space is that the so-called energy of the signal is finite ( see also energy signal ), that is ${\ displaystyle V}$${\ displaystyle V = \ mathbb {R}}$${\ displaystyle V = \ mathbb {R} ^ {3}}$${\ displaystyle \ ell _ {2} (V)}$${\ displaystyle s \ colon \ mathbb {Z} \ to V, k \ mapsto s_ {k}}$

${\ displaystyle E: = \ | s \ | ^ {2}: = \ sum _ {n = - \ infty} ^ {\ infty} \ | s_ {n} \ | ^ {2} <\ infty}$

Methods: Transformation of signals

The processing of digital signals is done by signal processors .

The theoretical model of the electronic circuit is the algorithm . Algorithms such as mixers , filters , discrete Fourier transforms , discrete wavelet transforms and PID control are used in digital signal processing . The algorithm is composed of elementary operations; Such are, for example, the addition of signal values ​​in terms of terms, the multiplication of signal values ​​in terms of terms with a constant, the delay, i.e. time shift, of a signal, as well as other mathematical operations that periodically generate a new value from a section of a signal (or several signals) and from these values ​​a new signal.

Abstract Transformations: Filters

A mapping between two signal spaces is generally called a system . A first limitation is the demand of time invariance (TI for English. Time invariance ) in the illustration . Roughly speaking, this arises from the fact that a time-discrete signal processing system consists of a shift register that stores a limited history and a function that generates a new one from the stored values. If one also considers location-dependent signals, such as B. in image processing, the following values ​​are available in addition to the previous values. In order to protect the general public, a two-sided environment of the current data point is to be considered. ${\ displaystyle F}$${\ displaystyle F}$${\ displaystyle f}$

The environment has a radius , at the point in time the values ​​of a time-discrete input signal are in the environment memory. From these, the value at the time of the output signal is determined by means of the function embodying the circuit , ${\ displaystyle d}$${\ displaystyle \ delta n}$${\ displaystyle (a_ {nd}, \ dots, a_ {n}, \ dots, a_ {n + d})}$${\ displaystyle (\ delta, a)}$ ${\ displaystyle f}$${\ displaystyle b_ {n}}$${\ displaystyle n \ delta}$${\ displaystyle (\ delta, b) = F (\ delta, a)}$

${\ displaystyle b_ {n} = f (a_ {nd}, \ dots, a_ {n}, \ dots, a_ {n + d})}$.

The function can also be independent of some of the arguments. In the case of time-dependent signals, it would make little sense if the values ​​of the signal were dependent on points in time in the future. Examples of such functions are ${\ displaystyle f}$${\ displaystyle f}$${\ displaystyle (n + 1) \ delta, \ dots, (n + d) \ delta}$

• ${\ displaystyle f (a_ {kd}, \ dots, a_ {k}, \ dots, a_ {k + d}) = \ max \ {a_ {kd}, \ dots, a_ {k} \}}$ creates a system that smooths the signal,
• ${\ displaystyle f (a_ {k-1}, a_ {k}, a_ {k + 1}) = a_ {k-1}}$produces a shift of the signal in the direction of increasing indices, i.e. H. a delay.

Time-invariant systems can be combined and connected one after the other and time-invariant systems are obtained again.

TI systems that are generated from a linear map , for example ${\ displaystyle F}$${\ displaystyle f}$

${\ displaystyle f (a_ {nd}, \ dots, a_ {n}, \ dots, a_ {n + d}) = f_ {d} (a_ {nd}) + \ dots + f_ {0} (a_ { n}) + \ dots + f _ {- d} (a_ {n + d}) = \ sum _ {k = -d} ^ {d} f_ {k} (a_ {nk}) \,}$

is called a convolution filter . They are a special case of the linear time-invariant filter (LTI) and can also be written as. Denotes the convolution operator . ${\ displaystyle F (a) = f * a}$${\ displaystyle *}$

LTI systems can be defined and analyzed in the space or time domain or in the frequency domain . Non-linear or non-time-invariant filters such as closed-loop controls can only be viewed as real-time systems in the time domain.

A LTI system can be in the time domain by means of its impulse response function or in the frequency domain by means of its transfer function (engl. Response Amplitude Operator , RAO) ${\ displaystyle F}$ ${\ displaystyle f = \ {f_ {k} \}: = F (\ delta ^ {0})}$

${\ displaystyle {\ hat {f}} (\ omega): = \ sum _ {k \ in \ mathbb {Z}} f_ {k} e ^ {- i \ omega k}}$,

analyzed and realized. The impulse response of a convolution filter is even . LTI systems can be constructed that suppress certain frequency ranges and leave others invariant. If you want to emphasize the frequency-selective effect of such a system, it is called a filter . ${\ displaystyle F (a) = f * a}$${\ displaystyle F (\ delta ^ {0}) = f}$

The FFT algorithm, which mediates between the representation of a signal in the time domain and in the frequency domain, plays a central role in the practical implementation of LTI systems . In particular, a convolution in the time domain can be implemented by a multiplication in the frequency domain.

General filters:

special filters:

• Boxcar filter
Each coefficient of the FIR filter is one. This makes the output the sum of all N input samples. This filter is very easy to implement, you only need adders
decimating bandpass filter
linear amplitude response
Signal phase can be changed, phase rotation of the signal by 90 °.

There are several options for implementing the filter types.

This corresponds to a convolution in the time domain with an impulse response
The impulse response reflects the coefficients of the filter
It has a linear phase with a symmetrical impulse response
Always stable
FIR filter with feedback (feedback)
Processing in blocks using the Overlap Add / Overlap Save method
Fourier transformation of the signal with subsequent multiplication of the transfer function in the frequency domain.

Applications

Exemplary areas of application for digital signal processing are:

Advantages of digital signal processing compared to conventional techniques

In contrast to conventional filter systems in communications technology , which have to be implemented individually in hardware, with digital signal processing any filters can be switched on or off with the aid of software in "real time" (e.g. for decoding).

Depending on the performance of the system, any number of filters and complex filter curves and even phase shifts depending on other parameters can be generated in "real time" and the original signal can be processed.

Therefore, with digital signal processing by DSPs, signal processing is much more effective than with conventional filter systems (e.g. for noise suppression of analog signals), see noise filter .

Advantages using the example of an audio CD

The example of the CD shows some advantages of digital over analog signal processing: The measured values ​​stored digitally on a CD do not change even after years, there is no " crosstalk " from one track to another, and no high frequencies are lost. Even if the CD is played any number of times, the data is not changed, as is the case with a vinyl record : There, the stylus of the pickup “grinds” away a little material with each playback and smooths the edges - with the result that the high frequency components are changed.

literature

• Alan V. Oppenheim, Ronald W. Schafer: Discrete-time signal processing . 3rd revised edition. R. Oldenbourg, Munich a. a. 1999, ISBN 3-486-24145-1 .
• Steven W. Smith: The Scientist and Engineer's Guide to digital Signal Processing . 2nd Edition. California Technical Publishing, San Diego CA 1999, ISBN 0-9660176-4-1 ( e-book ).
• Li Tan: Digital Signal Processing. Fundamentals and Applications . Elsevier Academic Press, Amsterdam u. a. 2008, ISBN 978-0-12-374090-8 .