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47-568: Digital signal processing and analog signal processing are subfields of signal processing. DSP applications include audio and speech processing , sonar , radar and other sensor array processing, spectral density estimation , statistical signal processing , digital image processing , data compression , video coding , audio coding , image compression , signal processing for telecommunications , control systems , biomedical engineering , and seismology , among others. DSP can involve linear or nonlinear operations. Nonlinear signal processing

94-412: A Fourier series . A sinusoidal function can be represented in terms of an exponential by the application of Euler's Formula . An impulse ( Dirac delta function ) is defined as a signal that has an infinite magnitude and an infinitesimally narrow width with an area under it of one, centered at zero. An impulse can be represented as an infinite sum of sinusoids that includes all possible frequencies. It

141-479: A real-time computing requirement and the signal data (either input or output) exists in data files, processing may be done economically with a general-purpose computer. This is essentially no different from any other data processing , except DSP mathematical techniques (such as the DCT and FFT ) are used, and the sampled data is usually assumed to be uniformly sampled in time or space. An example of such an application

188-489: A voltage , electric current , or electric charge around components in the electronic devices. An error or noise affecting such physical quantities will result in a corresponding error in the signals represented by such physical quantities. Examples of analog signal processing include crossover filters in loudspeakers, "bass", "treble" and "volume" controls on stereos, and "tint" controls on TVs. Common analog processing elements include capacitors, resistors and inductors (as

235-512: A linear combination of the outputs. An example of a linear system is a first order low-pass or high-pass filter. Linear systems are made out of analog devices that demonstrate linear properties. These devices don't have to be entirely linear, but must have a region of operation that is linear. An operational amplifier is a non-linear device, but has a region of operation that is linear, so it can be modeled as linear within that region of operation. Time-invariance means it doesn't matter when you start

282-406: A range of frequencies. Though most precisely referring to time in physics , the term time domain may occasionally informally refer to position in space when dealing with spatial frequencies , as a substitute for the more precise term spatial domain . The use of the contrasting terms time domain and frequency domain developed in U.S. communication engineering in the late 1940s, with

329-573: A system, the same output will result. For example, if you have a system and put an input into it today, you would get the same output if you started the system tomorrow instead. There aren't any real systems that are LTI, but many systems can be modeled as LTI for simplicity in determining what their output will be. All systems have some dependence on things like temperature, signal level or other factors that cause them to be non-linear or non-time-invariant, but most are stable enough to model as LTI. Linearity and time-invariance are important because they are

376-500: A table of transform pairs is used to find the Fourier transform of a signal or system. The inverse Fourier transform is used to go from frequency domain to time domain: Each signal or system that can be transformed has a unique Fourier transform. There is only one time signal for any frequency signal, and vice versa. The Laplace transform is a generalized Fourier transform . It allows a transform of any system or signal because it

423-561: A tool for analyzing stability issues of digital IIR filters. It is analogous to the Laplace transform , which is used to design and analyze analog IIR filters. A signal is represented as linear combination of its previous samples. Coefficients of the combination are called autoregression coefficients. This method has higher frequency resolution and can process shorter signals compared to the Fourier transform. Prony's method can be used to estimate phases, amplitudes, initial phases and decays of

470-469: Is a transform into the complex plane instead of just the jω line like the Fourier transform. The major difference is that the Laplace transform has a region of convergence for which the transform is valid. This implies that a signal in frequency may have more than one signal in time; the correct time signal for the transform is determined by the region of convergence . If the region of convergence includes

517-450: Is a type of signal processing conducted on continuous analog signals by some analog means (as opposed to the discrete digital signal processing where the signal processing is carried out by a digital process). "Analog" indicates something that is mathematically represented as a set of continuous values. This differs from "digital" which uses a series of discrete quantities to represent signal. Analog values are typically represented as

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564-464: Is a versatile field supported by a wide array of software tools. From high-level environments like MATLAB and Python to low-level programming with C/C++, these tools cater to various needs, whether for research, education, or industry applications. As DSP continues to evolve, these software tools play a critical role in advancing the capabilities and efficiencies of signal processing technologies. Analog signal processing Analog signal processing

611-452: Is an open-source software development toolkit that provides signal processing blocks to implement software-defined radios (SDRs) and signal processing systems. GNU Radio is used in academic research, prototyping of communication systems, and hobbyist projects involving radio and wireless communications. GNU Octave is an open-source alternative to MATLAB, providing a similar environment for numerical computations and signal processing. Octave

658-612: 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. Frequency domain analysis is also called spectrum- or spectral analysis . Filtering, particularly in non-realtime work can also be achieved in the frequency domain, applying the filter and then converting back to the time domain. This can be an efficient implementation and can give essentially any filter response including excellent approximations to brickwall filters . There are some commonly used frequency domain transformations. For example,

705-516: 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. The accuracy of the joint time-frequency resolution is limited by the uncertainty principle of time-frequency. Empirical mode decomposition is based on decomposition signal into intrinsic mode functions (IMFs). IMFs are quasiharmonical oscillations that are extracted from

752-466: Is applicable to both streaming data and static (stored) data. To digitally analyze and manipulate an analog signal, it must be digitized with an analog-to-digital converter (ADC). Sampling is usually carried out in two stages, discretization and quantization . Discretization means that the signal is divided into equal intervals of time, and each interval is represented by a single measurement of amplitude. Quantization means each amplitude measurement

799-432: Is approximated by a value from a finite set. Rounding real numbers to integers is an example. 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 component in the signal. In practice, the sampling frequency is often significantly higher than this. It is common to use an anti-aliasing filter to limit

846-501: Is closely related to nonlinear system identification and can be implemented in the time , frequency , and spatio-temporal domains . The application of digital computation to 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 . Digital signal processing is also fundamental to digital technology , such as digital telecommunication and wireless communications . DSP

893-914: Is done by additional third-party DSP chips located on extension cards or external hardware boxes or racks. Many digital audio workstations such as Logic Pro , Cubase , Digital Performer and Pro Tools LE use native processing. Others, such as Pro Tools HD, Universal Audio 's UAD-1 and TC Electronic 's Powercore use DSP processing. General application areas for DSP include Specific examples include speech coding and transmission in digital mobile phones , room correction of sound in hi-fi and sound reinforcement applications, analysis and control of industrial processes , medical imaging such as CAT scans and MRI , audio crossovers and equalization , digital synthesizers , and audio effects units . DSP has been used in hearing aid technology since 1996, which allows for automatic directional microphones, complex digital noise reduction , and improved adjustment of

940-431: Is in either degrees or radians. The frequency axes are in a [logarithmic scale]. These are useful because for sinusoidal inputs, the output is the input multiplied by the value of the magnitude plot at the frequency and shifted by the value of the phase plot at the frequency. This is the domain that most people are familiar with. A plot in the time domain shows the amplitude of the signal with respect to time. A plot in

987-442: Is not, in reality, possible to generate such a signal, but it can be sufficiently approximated with a large amplitude, narrow pulse, to produce the theoretical impulse response in a network to a high degree of accuracy. The symbol for an impulse is δ(t). If an impulse is used as an input to a system, the output is known as the impulse response. The impulse response defines the system because all possible frequencies are represented in

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1034-585: Is one of the most widely used software tools for DSP. It offers a high-level programming environment with built-in functions for signal processing, making it accessible for both beginners and experts. MATLAB is used for research, algorithm development, and prototyping in various fields such as telecommunications, audio processing, and biomedical engineering. Python is an open-source programming language that has gained popularity in scientific computing. Libraries such as NumPy and SciPy extend Python’s capabilities for numerical computations and signal processing. Python

1081-541: Is particularly useful for educational purposes, allowing students to learn DSP concepts without the cost of MATLAB. For high-performance DSP applications, C and C++ are often used, especially when low-level control over hardware is required. Libraries such as Intel’s IPP (Integrated Performance Primitives) and ARM’s CMSIS-DSP provide optimized functions for signal processing. C/C++ is used in applications requiring real-time processing, such as telecommunications, embedded systems, and video processing. Digital signal processing

1128-400: Is processing digital photographs with software such as Photoshop . When the application requirement is real-time, DSP is often implemented using specialized or dedicated processors or microprocessors, sometimes using multiple processors or multiple processing cores. These may process data using fixed-point arithmetic or floating point. For more demanding applications FPGAs may be used. For

1175-436: Is the basic concept in signal processing that states an input signal can be combined with the system's function to find the output signal. It is the integral of the product of two waveforms after one has reversed and shifted; the symbol for convolution is *. That is the convolution integral and is used to find the convolution of a signal and a system; typically a = -∞ and b = +∞. Consider two waveforms f and g. By calculating

1222-514: Is widely used in research, machine learning, and data analysis, making it suitable for DSP applications in various domains. LabVIEW (Laboratory Virtual Instrument Engineering Workbench) is a system-design platform and development environment from National Instruments. It is particularly popular in industry for automated testing and measurement. LabVIEW is commonly used in embedded systems, instrumentation, and control systems, particularly in industries like telecommunications and automotive. GNU Radio

1269-455: The cepstrum converts a signal to the frequency domain through Fourier transform, takes the logarithm, then applies another Fourier transform. This emphasizes the harmonic structure of the original spectrum. Digital filters come in both infinite impulse response (IIR) and finite impulse response (FIR) types. Whereas FIR filters are always stable, IIR filters have feedback loops that may become unstable and oscillate. The Z-transform provides

1316-502: The frequency domain shows either the phase shift or magnitude of a signal at each frequency that it exists at. These can be found by taking the Fourier transform of a time signal and are plotted similarly to a bode plot. While any signal can be used in analog signal processing, there are many types of signals that are used very frequently. Sinusoids are the building block of analog signal processing. All real world signals can be represented as an infinite sum of sinusoidal functions via

1363-529: The frequency response . Digital Signal Processing (DSP) involves the manipulation of signals after they have been converted into a digital format. This field is supported by a variety of software tools that enable engineers, researchers, and hobbyists to design, analyze, and implement DSP algorithms. This article explores some of the most popular software tools used in DSP, highlighting their features, advantages, and common applications. MATLAB (Matrix Laboratory)

1410-531: The abstract process of sampling . Numerical methods require a quantized signal, such as those produced by an ADC. The processed result might be a frequency spectrum or a set of statistics. But often it is another quantized signal that is converted back to analog form by a digital-to-analog converter (DAC). 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

1457-431: The analysis of signals with respect to time. Similarly, space domain refers to the analysis of signals with respect to position, e.g., pixel location for the case of image processing. 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

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1504-584: The components of signal. Components are assumed to be complex decaying exponents. A time-frequency representation of signal can capture both temporal evolution and frequency structure of analyzed signal. Temporal and frequency resolution are limited by the principle of uncertainty and the tradeoff is adjusted by the width of analysis window. Linear techniques such as Short-time Fourier transform , wavelet transform , filter bank , non-linear (e.g., Wigner–Ville transform ) and autoregressive methods (e.g. segmented Prony method) are used for representation of signal on

1551-499: The convolution, we determine how much a reversed function g must be shifted along the x-axis to become identical to function f. The convolution function essentially reverses and slides function g along the axis, and calculates the integral of their (f and the reversed and shifted g) product for each possible amount of sliding. When the functions match, the value of (f*g) is maximized. This occurs because when positive areas (peaks) or negative areas (troughs) are multiplied, they contribute to

1598-465: The current sample of the input or output signal. The surrounding samples may be identified with respect to time or space. The output of a linear digital filter to any given input may be calculated by convolving the input signal with an impulse response . Signals are converted from time or space domain to the frequency domain usually through use of the Fourier transform . The Fourier transform converts

1645-441: The domain in which to process a signal by making an informed assumption (or by trying different possibilities) as to which domain best represents the essential characteristics of the signal and the processing to be applied to it. A sequence of samples from a measuring device produces a temporal or spatial domain representation, whereas a discrete Fourier transform produces the frequency domain representation. Time domain refers to

1692-459: The input A unit step function, also called the Heaviside step function , is a signal that has a magnitude of zero before zero and a magnitude of one after zero. The symbol for a unit step is u(t). If a step is used as the input to a system, the output is called the step response. The step response shows how a system responds to a sudden input, similar to turning on a switch. The period before

1739-473: The integral. The Fourier transform is a function that transforms a signal or system in the time domain into the frequency domain, but it only works for certain functions. The constraint on which systems or signals can be transformed by the Fourier Transform is that: This is the Fourier transform integral: Usually the Fourier transform integral isn't used to determine the transform; instead,

1786-403: The jω axis, jω can be substituted into the Laplace transform for s and it's the same as the Fourier transform. The Laplace transform is: and the inverse Laplace transform, if all the singularities of X(s) are in the left half of the complex plane, is: Bode plots are plots of magnitude vs. frequency and phase vs. frequency for a system. The magnitude axis is in [Decibel] (dB). The phase axis

1833-408: The most demanding applications or high-volume products, ASICs might be designed specifically for the application. Parallel implementations of DSP algorithms, utilising multi-core CPU and many-core GPU architectures, are developed to improve the performances in terms of latency of these algorithms. Native processing is done by the computer's CPU rather than by DSP or outboard processing, which

1880-548: The only types of systems that can be easily solved using conventional analog signal processing methods. Once a system becomes non-linear or non-time-invariant, it becomes a non-linear differential equations problem, and there are very few of those that can actually be solved. (Haykin & Van Veen 2003) Time domain Time domain refers to the analysis of mathematical functions , physical signals or time series of economic or environmental data, with respect to time . In

1927-451: The output stabilizes is called the transient part of a signal. The step response can be multiplied with other signals to show how the system responds when an input is suddenly turned on. The unit step function is related to the Dirac delta function by; Linearity means that if you have two inputs and two corresponding outputs, if you take a linear combination of those two inputs you will get

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1974-520: The passive elements) and transistors or op-amps (as the active elements). A system's behavior can be mathematically modeled and is represented in the time domain as h(t) and in the frequency domain as H(s), where s is a complex number in the form of s=a+ib, or s=a+jb in electrical engineering terms (electrical engineers use "j" instead of "i" because current is represented by the variable i). Input signals are usually called x(t) or X(s) and output signals are usually called y(t) or Y(s). Convolution

2021-444: The signal bandwidth to comply with the sampling theorem, however careful selection of this filter is required because the reconstructed signal will be the filtered signal plus residual aliasing from imperfect stop band rejection instead of the original (unfiltered) signal. Theoretical DSP analyses and derivations are typically performed on discrete-time signal models with no amplitude inaccuracies ( quantization error ), created by

2068-551: The signal. DSP algorithms may be run on general-purpose computers and digital signal processors . DSP algorithms are also implemented on purpose-built hardware such as application-specific integrated circuit (ASICs). Additional technologies for digital signal processing include more powerful general purpose microprocessors , graphics processing units , field-programmable gate arrays (FPGAs), digital signal controllers (mostly for industrial applications such as motor control), and stream processors . For systems that do not have

2115-437: The time domain, the signal or function's value is known for all real numbers , for the case of continuous time , or at various separate instants in the case of discrete time . An oscilloscope is a tool commonly used to visualize real-world signals in the time domain. A time-domain graph shows how a signal changes with time, whereas a frequency-domain graph shows how much of the signal lies within each given frequency band over

2162-405: The time or space information to a magnitude and phase component of each frequency. With some applications, how the phase varies with frequency can be a significant consideration. Where phase is unimportant, 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

2209-421: The time-frequency plane. Non-linear and segmented Prony methods can provide higher resolution, but may produce undesirable artifacts. Time-frequency analysis is usually used for analysis of non-stationary signals. For example, methods of fundamental frequency estimation, such as RAPT and PEFAC are based on windowed spectral analysis. In numerical analysis and functional analysis , a discrete wavelet transform

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