# Digital signal processing

The digital signal processing is a branch of telecommunications and is engaged in the production and processing of digital signals using digital systems. In a narrower sense it focuses on the storage, transmission and transformation of information in the sense of information theory in the form of digital, discrete-time signals.

In practical use today are based almost all the recording, transmission and storage procedures for photos and film (photography, television, video) and audio ( music, communications technology) in the digital processing of the corresponding signals. It provides a variety of conversion and processing of such data types, such as the compression of audio and video data, NLE or image editing for photos. In addition, digital signal processing is used - used in the measurement, control and regulation technology and medical technology, such as magnetic resonance imaging - among many other industrial applications.

The digital signal processing based on electronic components, such as digital signal processors ( DSP), or efficient microprocessors, memory elements, and interfaces to the corresponding signal input and output. The signal processing algorithms may be supplemented with additional software for a programmable hardware that controls the flow of signals. Digital signal processing provides opportunities and processing facilities, which are not feasible or only with great effort in the past has analog circuitry.

The methods of digital signal processing are of mathematics, such as the branches of number theory or coding theory, much closer than the classical electrical engineering. The starting point was the notoriety of the fast Fourier transform ( FFT) from the year 1965, by publication of JW Cooley and John Tukey. In addition, improved in the same period the practical possibilities of digital circuit technology, so that the newly developed method could be applied.

## Overview

The processing of the signal is always the same: Analog → Digital → Analog. The changes in the signal are made exclusively in the digital domain. Using the example of an audio CD, the procedure will be explained:

## Construction of a digital signal processing system,

The graph shows the typical structure of a signal processing system, which also always has analogous components to the interface to the "outside world". For digital signal processing system in the strict sense are only the red-colored components in the lower part.

If we follow the path of the signals in the graphic means of a sensor are converted to physical quantities, often faint electrical signal. This signal is increased for further processing, for example, with the aid of an operational amplifier at the level required for the subsequent steps. From the amplified analog signal, the sample-and -hold stage samples values at certain time intervals, and keep it at a constant interval. From an analog continuous-time curve as a discrete-time analog signal. A constant signal for a certain time is needed by the analog-to- digital converter to determine the discrete digital values . These can then be processed by the digital signal processor. The signal then takes the opposite approach and can be an actor again if necessary in the technical process incorporated.

## Object: What is a signal?

A digital signal, in contrast to the continuous function of the analog signal processing in discrete -time and range of values, that is, a sequence of elementary signals ( for example, square wave pulses ). This sequence usually occurs in a time or location periodic measurement process. Thus, for example, is converted by the deflection or bending of a membrane of a piezoelectric crystal to an electrical voltage and converts this voltage by means of an AD converter into digital data in a periodically repeating time sound. Such a realistic measurement process is finite, the resulting sequence has a beginning and an end index index α ω.

We can use the signal So as a data structure ( δ, α, ω, s) define, with the distance δ between the two data points, the indices α < ω and the finite sequence (array) S = ( S? ..., Sω ) of the data.

The data are instances of a data structure. The simplest structure is the data bit, the most common (1, 2, 4-byte ) integer and floating point data. But it is also possible that the individual date itself is a vector or a sequence, such as in the coding of the color information as RGB or RGBA triple quad, or that the signal S contains the columns sk of a raster image. The single column is again a signal including, for example gray or color values as data.

## Abstraction of a signal

To the theory signals do not have to be considered separately by the beginning and end, the finite sequences in the abstract signal space, a Hilbert space embedded. Condition: The basis functions are orthogonal to each other, their cross-correlation results therefore zero. An abstract signal is therefore by a pair ( δ, s), δ > 0, given.

Here, the Euclidean vector space V models the data type of the signal, for example, for simple data for RGB color triad. An element in a doubly infinite sequence. The defining characteristic of the sequence space is that the so-called energy of the signal at last (see also signal energy ), that is

## Methods: transformation of signals

The processing of digital signals is performed by signal processors.

The theoretical model of the electronic circuit is the algorithm. In the digital signal processing algorithms, such as mixers, filters, discrete Fourier transform, discrete wavelet transform, PID control may be used. The algorithm is composed of elementary operations; Such are for example the term by the addition of signal values , the term by term multiplication of signal values by a constant, for the delay, that is time difference of a signal, and more mathematical operations which periodically a section of a ( or more ) signal (s) a new generate value and from these values a new signal.

## Abstract transformations: Filter

A mapping F between two signal spaces is generally called system. A first limitation is the requirement of time invariance (TI for engl. Time invariance ) of Figure F. This occurs roughly considered the fact that a discrete-time signal processing system composed of a shift register that stores a limited past, and a function f, the saved from the values creates a new, there. Considering also location-dependent signals, such as in image processing, so stand next to the previous values and subsequent disposal. To protect the public, as a two-sided neighborhood of the current data point is considered.

The area had a radius d, at the time? N are the values of a discrete-time input signal ( δ, a) in the environment space. From these, by means of the circuit embodying function f of the value bn at time nδ of the output signal ( δ, b) = F ( δ, a) determines

The f function can also be independent of some of the arguments. For time- dependent signals, it would make little sense if f values of the signal at time points (n 1) δ depended, ..., (n d) δ in the future. Examples of such functions are

- F (ak -d, ..., ak, ..., ak d ) = max { ak -d, ..., ak } produces a system that smoothes the signal,
- F (ak -1, ak, ak 1 ) = ak -1 produces a shift of the signal in the direction of increasing indices, ie a delay.

You can combine time-invariant systems arbitrarily and can be daisy chained and receives time-invariant systems again.

TI systems F, F which are generated by a linear transformation, such as

Called convolution filter. They are a special case of linear time-invariant filters (LTI ) and can be written as F ( a) = f * a. In this case, * represents the convolution operator.

LTI systems may be defined and analyzed in the spatial or in the frequency domain or the time domain. Non-linear time-invariant filter or not such rules can be regarded as real-time systems only in the time domain.

An LTI system F can be in the time domain by means of its impulse response function f = { f k }: = F ( δ0 ) or in the frequency domain by means of its transfer function ( Response Amplitude Operator Sheet, RAO ), analyzed and realized. The impulse response of the convolution filter Q (a) = f * A is just F ( δ0 ) = f One can construct LTI systems that suppress certain frequencies while leaving others invariant. If you want to emphasize the frequency- selective effect of such a system, it is called filter.

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

Filters in general:

- Bandpass
- High-pass
- Low-pass

Special filters:

- Boxcarfilter

- CIC ( Cascaded Integrated Comb) filter
- Goertzelfilter

- Hilbert filter

The realization of the filter types, there are several possibilities.

- FIR filters (finite impulse response)

- IIR ( Infinite Impulse Response)

- Fast Convolution

- QMF ( Quadrature Mirror Filter)

## Applications

Exemplary applications of digital signal processing are:

- Automotive sector: ABS, EPS, collision avoidance, active noise reduction, engine running control, parking assistance, navigation aid, voice control, airbag, GPS
- Industry: Motor control, robotics, computer vision, servo control systems, barcode reader, metrology
- Medical technology: magnetic resonance imaging, positron emission tomography, computed tomography, optical coherence tomography, sonography
- Military and research: sonar and radar systems, seismic analysis, missile guidance systems, aircraft control and control system, nuclear magnetic resonance spectroscopy
- Telecoms: mobile phone, DSL, ISDN, Voice over IP, Modem, Wireless LAN, Bluetooth, satellite communications,
- Consumer electronics, DVD players, MP3 players, digital TV, digital radio, video technology, sound engineering

## Advantages of digital signal processing over conventional techniques

In contrast to conventional filter systems in communications technology that have to be implemented individually in hardware (eg, decoding ) with the help of software on or off with the digital signal processing any filter simply in "real time" if necessary.

This may, depending on the performance of the system as many filters and elaborate filtering curves and even phase shifts as a function of other parameters in "real time " is generated and then the origin signal can be processed.

Therefore, with the digital signal processing by DSP a much more effective signal processing than with conventional filter systems (eg for noise reduction analog signals) are possible, see noise filter.

### Demonstrate the benefits of an audio CD

The example of the CD, some advantages of digital over analog signal processing can be seen: The digitally stored on a CD measurement values do not change even after many years, there is no " crosstalk " from one track to another, it will go no high frequencies lost. Even for arbitrary frequent play the CD the data is not changed as in a record: There " grinds " the needle of the pick- up automatically every time a little material away and smooth the edges with the result that the high -frequency components are changed.