Digital signal processing

Digital signal processing (DSP) is concerned with the representation of discrete time signals by a sequence of numbers or symbols and the processing of these signals. Digital signal processing and analog signal processing are subfields of signal processing. DSP includes subfields like: audio and speech signal processing, sonar and radar signal processing, sensor array processing, spectral estimation, statistical signal processing, digital image processing, signal processing for communications, control of systems, biomedical signal processing, seismic data processing, etc.

The goal of DSP is usually to measure, filter and/or compress continuous real-world analog signals. The first step is usually to convert the signal from an analog to a digital form, by sampling and then digitizing it using an analog-to-digital converter (ADC), which turns the analog signal into a stream of numbers. However, often, the required output signal is another analog output signal, which requires a digital-to-analog converter (DAC). Even if this process is more complex than analog processing and has a discrete value range, the application of computational power to digital signal processing allows for many advantages over analog processing in many applications, such as error detection and correction in transmission as well as data compression.^[1]

DSP algorithms have long been run on standard computers, on specialized processors called digital signal processor on purpose-built hardware such as application-specific integrated circuit (ASICs). Today there are additional technologies used for digital signal processing including more powerful general purpose microprocessors, field-programmable gate arrays (FPGAs), digital signal controllers (mostly for industrial apps such as motor control), and stream processors, among others.^[2]

1 Signal sampling
2 DSP domains
3 Applications
4 Implementation
5 Techniques
6 Related fields
7 References
8 Further reading

Signal sampling

Main article: Sampling (signal processing)

With the increasing use of computers the usage of and need for digital signal processing has increased. In order to use an analog signal on a computer it must be digitized with an analog-to-digital converter. Sampling is usually carried out in two stages, discretization and quantization. In the discretization stage, the space of signals is partitioned into equivalence classes and quantization is carried out by replacing the signal with representative signal of the corresponding equivalence class. In the quantization stage the representative signal values are approximated by values from a finite set.

The Nyquist–Shannon sampling theorem states that a signal can be exactly reconstructed from its samples if the sampling frequency is greater than twice the highest frequency of the signal; but requires an infinite number of samples. In practice, the sampling frequency is often significantly more than twice that required by the signal's limited bandwidth.

A digital-to-analog converter is used to convert the digital signal back to analog. The use of a digital computer is a key ingredient in digital control systems.

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, and wavelet domains. They choose the domain in which to process a signal by making an informed 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.

Time and space domains

Main article: Time domain

The most common processing approach in the time or space domain is enhancement of the input signal through a method called filtering. Digital filtering generally consists of some linear 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 is a weighted linear combination of different 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 usually 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 stable filter produces an output that converges to a constant value with time, or remains bounded within a finite interval. An unstable filter can produce an output that grows without bounds, with bounded or even zero input.

A "finite impulse response" (FIR) filter uses only the input signals, 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.

Filters can be represented by block diagrams which can then be used to derive a sample processing algorithm to implement the filter using hardware instructions. 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 a digital filter to any given input may be calculated by convolving the input signal with the impulse response.

Frequency domain

Main article: 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 determine which frequencies are present in the input signal and which are missing.

In addition to frequency information, phase information is often needed. This can be obtained from the Fourier transform. With some applications, how the phase varies with frequency can be a significant consideration.

Filtering, particularly in non-realtime work can also be achieved by converting to the frequency domain, applying the filter and then converting back to the time domain. This is a fast, O(n log n) operation, and can give essentially any filter shape including excellent approximations to brickwall filters.

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 emphasizes the frequency components with smaller magnitude while retaining the order of magnitudes of frequency components.

Frequency domain analysis is also called spectrum- or spectral analysis.

Z-plane analysis

Main article: Z-transform

Whereas analog filters are usually analysed in terms of transfer functions in the s plane using Laplace transforms, digital filters are analysed in the z plane in terms of Z-transforms. A digital filter may be described in the z plane by its characteristic collection of zeroes and poles. The z plane provides a means for mapping digital frequency (samples/second) to real and imaginary z components, were $z=e^{j\Omega}$ for continuous periodic signals and $\Omega = 2 \pi F$ ( $F$ is the digital frequency). This is useful for providing a visualization of the frequency response of a digital system or signal.

Wavelet

Main article: Discrete wavelet transform

In numerical analysis and functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal resolution: it captures both frequency and location information (location in time).

Applications

The main applications of DSP are audio signal processing, audio compression, digital image processing, video compression, speech processing, speech recognition, digital communications, RADAR, SONAR, seismology and biomedicine. Specific examples are speech compression and transmission in digital mobile phones, room correction of sound in hi-fi and sound reinforcement applications, weather forecasting, economic forecasting, seismic data processing, analysis and control of industrial processes, medical imaging such as CAT scans and MRI, MP3 compression, computer graphics, image manipulation, hi-fi loudspeaker crossovers and equalization, and audio effects for use with electric guitar amplifiers.

Implementation

Depending on the requirements of the application, digital signal processing tasks can be implemented on general purpose computers (e.g. super computers, mainframe computers, or personal computers) or with embedded processors that may or may not include specialized microprocessors call digital signal processors.

Often when the processing requirement is not real-time, processing is economically done with an existing general-purpose computer and the signal data (either input or output) exists in data files. This is essentially no different than any other data processing except DSP mathematical techniques (such as the FFT) are made use of and the sampled data is usually assumed to be uniformly sampled in time or space. For example: processing digital photographs with software such as Photoshop.

However, when the application requirement is real-time, DSP is often implemented using specialised microprocessors such as the DSP56000, the TMS320, or the SHARC. These often process data using fixed-point arithmetic, although some versions are available which use floating point arithmetic and are more powerful. For faster applications FPGAs^[3] might be used. Beginning in 2007, multicore implementations of DSPs have started to emerge from companies including Freescale and Stream Processors, Inc. For faster applications with vast usage, ASICs might be designed specifically. For slow applications, a traditional slower processor such as a microcontroller may be adequate. Also a growing number of DSP applications are now being implemented on Embedded Systems using powerful PCs with a Multi-core processor.

Techniques

Related fields

References

^ James D. Broesch, Dag Stranneby and William Walker. Digital Signal Processing: Instant access. Butterworth-Heinemann. p. 3.
^ Dag Stranneby and William Walker (2004). Digital Signal Processing and Applications (2nd ed. ed.). Elsevier. ISBN 0750663448. http://books.google.com/books?id=NKK1DdqcDVUC&pg=PA241&dq=called+digital+signal+processor+hardware+application-specific+integrated+circuit+general-purpose+microprocessors+field-programmable+gate+arrays+dsp+asic+fpga+stream.
^ JpFix. "FPGA-Based Image Processing Accelerator". http://www.jpfix.com/About_Us/Articles/FPGA-Based_Image_Processing_Ac/fpga-based_image_processing_ac.html. Retrieved 2008-05-10.

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Digital systems

Components	Logic gate Combinational logic Sequential logic Digital circuit Integrated circuit (IC)

Theory	Boolean logic Logic design Digital signal processing Computer architecture

Applications	Digital audio Digital photography Digital video Electronic literature

Digital signal processing

Theory	discrete-time signal · sampling theorem · estimation theory · detection theory

Sub-fields	audio signal processing · digital image processing · speech processing · statistical signal processing

Techniques	Discrete Fourier transform (DFT) · Discrete-time Fourier transform (DTFT) · Impulse invariance · bilinear transform · pole–zero mapping · Z-transform · advanced Z-transform

Sampling	oversampling · undersampling · downsampling · upsampling · aliasing · anti-aliasing filter · sampling rate · Nyquist rate/frequency