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# Digital signal processing

Digital signal processing (DSP) is the study of signals in a digital representation and the processing methods of these signals. DSP and analog signal processing are subfields of signal processing. DSP has three major subfields: audio signal processing, digital image processing and speech processing.

Since the goal of DSP is usually to measure or filter continuous real-world analog signals, the first step is usually to convert the signal from an analog to a digital form, by using an analog to digital converter. Often, the required output signal is another analog output signal, which requires a digital to analog converter.

The algorithms required for DSP are sometimes performed using specialized computers, digital signal processors (also abbreviated DSP). These process signals in real time. They are optimized for DSP computations.

 Contents

## DSP Domains

In DSP, engineers usually study digital signals in one of the following domains: time domain (one-dimensional signals), spatial domain (multidimensional signals), frequency domain, autocorrelation domain, and wavelet domains. They choose the domain in which to process a signal by making an educated guess (or by trying different possibilities) as to which domain best represents the essential characteristics of the signal. A sequence of samples from a measuring device produces a time or spatial domain representation, whereas a discrete Fourier transform produces the frequency domain information, that is the frequency spectrum. Autocorrelation is defined as the cross-correlation of the signal with itself over varying intervals of time or space.

## Signal sampling

A digital signal is often a discrete representation of a continuous analog signal (for example, a real world signal that might represent a pressure or a velocity). The continuous signal is usually sampled at regular intervals by an analog to digital converter (ADC) and the value of the continuous signal in that interval is represented by a discrete value. The representation often introduces some error into the data. The error depends mostly on the sampling frequency, and the number of bits used for the representation. The sampling frequency or sampling rate is then the rate at which new samples are taken from the continuous signal. It represents the temporal or spatial accuracy of the discrete signal. The number of bits used for one value of the discrete signal indicates how accurately the signal magnitude is represented.

In ideal sampling, each sample is a measurement of the signal at an instantaneous point in time. Of course, real ADCs cannot measure a signal at a point of exactly zero duration; the measurement they return will be some kind of mean over a nonzero duration. This averaging effectively applies a low-pass filter to the signal, which needs to be compensated for in high-quality applications.

The Nyquist-Shannon sampling theorem, a fundamental theorem of signal processing, states that a sampled signal cannot unambiguously represent signal components with frequencies above half the sampling frequency. This frequency (half the sampling frequency) is called the Nyquist frequency. Frequencies above the Nyquist frequency N can be observed in the digital signal, but their frequency is ambiguous. That is, a frequency component with frequency f cannot be distinguished from another component with frequency 2N-f, 2N+f, 4N-f, etc. This is called aliasing. To handle this problem as gracefully as possible, most analog signals are filtered with an anti-aliasing filter (usually a low-pass filter) at the Nyquist frequency before conversion to the digital representation.

Within the limitations of the sampling theorem, the original signal can be completely reconstructed (to within the resolution of the sample values) from the set of ideal samples by expanding each sample into a signal component constructed from the sinc function, using the Nyquist-Shannon interpolation formula.

## Time and space domains

The most common processing approach in the time or space domain is enhancement of the input signal through a method called filtering. Filtering generally consists of some transformation of a number of surrounding samples around the current sample of the input or output signal. There are various ways to characterize filters; for example:

• A "linear" filter is a linear transformation of input samples; other filters are "non-linear." Linear filters satisfy the superposition condition, i.e. if an input signal is a weighted linear combination of different input signals, the output is an equally weighted linear combination of the corresponding output signals.
• A "causal" filter uses only previous samples of the input or output signals; while a "non-causal" filter uses future input samples. A non-causal filter can be changed into a causal filter by adding a delay to it.
• A "time-invariant" filter has constant properties over time; other filters such as adaptive filters change in time.
• Some filters are "stable", others are "unstable".
• A "finite impulse response" (FIR) filter uses only the input signal, while an "infinite impulse response" filter (IIR) uses both the input signal and previous samples of the output signal. FIR filters are always stable, while IIR filters may be unstable.

Most filters can be described in Z-domain (a superset of the frequency domain) by their Transfer functions. A filter may also be described as a difference equation, a collection of zeroes and poles or, if it is an FIR filter, an impulse response or step response. The output of an FIR filter to any given input may be calculated by convolving the input signal with the impulse response.

## Frequency domain

Signals are converted from time or space domain to the frequency domain usually through the Fourier transform. The Fourier transform converts the signal information to a magnitude and phase component of each frequency. Often the Fourier transform is converted to the power spectrum, which is the magnitude of each frequency component squared.

The most common purpose for analysis of signals in the frequency domain is analysis of signal properties. The engineer can study the spectrum to get information of which frequencies are present in the input signal and which are missing.

There are some commonly used frequency domain transformations. For example, the cepstrum converts a signal to the frequency domain through Fourier transform, takes the logarithm, then applies another Fourier transform. This emphasises the frequency components with smaller magnitude while retaining the order of magnitudes of frequency components.

## Applications

The main applications of DSP are audio signal processing, audio compression, digital image processing, video compression, speech processing, speech recognition and digital communications. Specific examples are speech compression and transmission in digital mobile phones, equalisation of sound in Hifi equipment, weather forecasting, economic forecasting, seismic data processing, analysis and control of industrial processes, computer-generated animations in movies, medical imaging such as CAT scans and MRI, and image manipulation. A further application is very low frequency (VLF) reception with a PC soundcard [1].

## References

• Alan V. Oppenheim , Ronald W. Schafer , John R. Buck  : Discrete-Time Signal Processing, Prentice Hall, ISBN 0-13-754920-2
• Richard G. Lyons : Understanding Digital Signal Processing, Prentice Hall, ISBN 0131089897
• Sen M. Kuo , Woon-Seng Gan : Digital Signal Processors: Architectures, Implementations, and Applications, Prentice Hall, ISBN 0130352144
• Bernard Mulgrew , Peter Grant, John Thompson: Digital Signal Processing - Concepts and Applications, Palgrave Macmillan, ISBN 0-333-96356-3
• Steven W. Smith : Digital Signal Processing - A Practical Guide for Engineers and Scientists, Newnes, ISBN 0-7506-7444-X
• Paul A. Lynn , Wolfgang Fuerst : Introductory Digital Signal Processing with Computer Applications, John Wiley & Sons, ISBN 0-471-97984-8
• James D. Broesch : Digital Signal Processing Demystified, Newnes, ISBN 1878707167
• John Proakis , Dimitris Manolakis : Digital Signal Processing - Principles, Algorithms and Applications, Pearson, ISBN 0133942899
• Hari Krishna Garg : Digital Signal Processing Algorithms, CRC Press, ISBN 0849371783
• P. Gaydecki : Foundations Of Digital Signal Processing: Theory, Algorithms And Hardware Design, Institution of Electrical Engineers, ISBN 0852964315
• Paul M. Embree , Damon Danieli : C++ Algorithms for Digital Signal Processing, Prentice Hall, ISBN 0131791443
• Anthony Zaknich : Neural Networks for Intelligent Signal Processing, World Scientific Pub Co Inc, ISBN 9812383050
• Vijay Madisetti , Douglas B. Williams : The Digital Signal Processing Handbook, CRC Press, ISBN 0849385725
• Stergios Stergiopoulos : Advanced Signal Processing Handbook: Theory and Implementation for Radar, Sonar, and Medical Imaging Real-Time Systems, CRC Press, ISBN 0849336910
• Joyce Van De Vegte : Fundamentals of Digital Signal Processing, Prentice Hall, ISBN 0130160776
• Ashfaq Khan : Digital Signal Processing Fundamentals, Charles River Media, ISBN 1584502819
• Jonathan M. Blackledge , Martin Turner : Digital Signal Processing: Mathematical and Computational Methods, Software Development and Applications, Horwood Publishing, ISBN 1898563489
• Bimal Krishna , K. Y. Lin , Hari C. Krishna : Computational Number Theory & Digital Signal Processing, CRC Press, ISBN 0849371775
• Doug Smith : Digital Signal Processing Technology: Essentials of the Communications Revolution, American Radio Relay League, ISBN 0872598195
• Henrique S. Malvar: Signal Processing with Lapped Transforms, Artech House Publishers, ISBN 0890064679
• Charles A. Schuler : Digital Signal Processing: A Hands-On Approach, McGraw-Hill, ISBN 0078297443
• James H. McClellan , Ronald Schafer: Signal Processing First, Prentice Hall, ISBN 0130909998
• Artur Krukowski , Izzet Kale : DSP System Design: Complexity Reduced Iir Filter Implementation for Practical Applications, Kluwer Academic Publishers, ISBN 1402075588
• John G. Proakis : A Self-Study Guide for Digial Signal Processing, Prentice Hall, ISBN 0131432397