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Upsampling
In digital signal processing, upsampling, expansion, and interpolation are terms associated with the process of sample rate conversion, resampling in a multi-rate digital signal processing system. ''Upsampling'' can be synonymous with ''expansion'', or it can describe an entire process of ''expansion'' and filtering (''interpolation''). When upsampling is performed on a sequence of samples of a ''signal'' or other continuous function, it produces an approximation of the sequence that would have been obtained by sampling the signal at a higher rate (or Dots per inch, density, as in the case of a photograph). For example, if compact disc audio at 44,100 samples/second is upsampled by a factor of 5/4, the resulting sample-rate is 55,125. Upsampling by an integer factor Rate increase by an integer factor L can be explained as a 2-step process, with an equivalent implementation that is more efficient: #Expansion: Create a sequence, x_L[n], comprising the original samples, x[n], separat ...
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Downsampling (signal Processing)
In digital signal processing, downsampling, compression, and decimation are terms associated with the process of sample rate conversion, ''resampling'' in a multi-rate digital signal processing system. Both ''downsampling'' and ''decimation'' can be synonymous with ''compression'', or they can describe an entire process of bandwidth reduction (Lowpass filter, filtering) and sample-rate reduction. When the process is performed on a sequence of samples of a ''signal'' or a continuous function, it produces an approximation of the sequence that would have been obtained by sampling the signal at a lower rate (or Dots per inch, density, as in the case of a photograph). ''Decimation'' is a term that historically means the ''Decimation (punishment), removal of every tenth one''. But in signal processing, ''decimation by a factor of 10'' actually means ''keeping'' only every tenth sample. This factor multiplies the sampling interval or, equivalently, divides the sampling rate. For example, i ...
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Interpolation
In the mathematics, mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has a number of data points, obtained by sampling (statistics), sampling or experimentation, which represent the values of a function for a limited number of values of the Dependent and independent variables, independent variable. It is often required to interpolate; that is, estimate the value of that function for an intermediate value of the independent variable. A closely related problem is the function approximation, approximation of a complicated function by a simple function. Suppose the formula for some given function is known, but too complicated to evaluate efficiently. A few data points from the original function can be interpolated to produce a simpler function which is still fairly close to the original. The resulting gai ...
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Data Conversion
Data conversion is the conversion of computer data from one format to another. Throughout a computer environment, data is encoded in a variety of ways. For example, computer hardware is built on the basis of certain standards, which requires that data contains, for example, parity bit checks. Similarly, the operating system is predicated on certain standards for data and file handling. Furthermore, each computer program handles data in a different manner. Whenever any one of these variables is changed, data must be converted in some way before it can be used by a different computer, operating system or program. Even different versions of these elements usually involve different data structures. For example, the changing of bits from one format to another, usually for the purpose of application interoperability or of the capability of using new features, is merely a data conversion. Data conversions may be as simple as the conversion of a text file from one character encoding ...
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Sampling (information Theory)
In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a sound wave to a sequence of "samples". A sample is a value of the signal at a point in time and/or space; this definition differs from the term's usage in statistics, which refers to a set of such values. A sampler is a subsystem or operation that extracts samples from a continuous signal. A theoretical ideal sampler produces samples equivalent to the instantaneous value of the continuous signal at the desired points. The original signal can be reconstructed from a sequence of samples, up to the Nyquist limit, by passing the sequence of samples through a reconstruction filter. Theory Functions of space, time, or any other dimension can be sampled, and similarly in two or more dimensions. For functions that vary with time, let s(t) be a continuous function (or "signal") to be sampled, and let sampling be performed by measuring the ...
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Oversampling
In signal processing, oversampling is the process of sampling (signal processing), sampling a signal at a sampling frequency significantly higher than the Nyquist rate. Theoretically, a bandwidth-limited signal can be perfectly reconstructed if sampled at the Nyquist rate or above it. The Nyquist rate is defined as twice the Bandwidth (signal processing), bandwidth of the signal. Oversampling is capable of improving Resolution (audio), resolution and signal-to-noise ratio, and can be helpful in avoiding aliasing and phase distortion by relaxing anti-aliasing filter performance requirements. A signal is said to be oversampled by a factor of ''N'' if it is sampled at ''N'' times the Nyquist rate. Motivation There are three main reasons for performing oversampling: to improve anti-aliasing performance, to increase resolution and to reduce noise. Anti-aliasing Oversampling can make it easier to realize analog anti-aliasing filters. Without oversampling, it is very difficult to impl ...
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Digital Signal Processing
Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are a sequence of numbers that represent Sampling (signal processing), samples of a continuous variable in a domain such as time, space, or frequency. In digital electronics, a digital signal is represented as a pulse train, which is typically generated by the switching of a transistor. Digital signal processing and analog signal processing are subfields of signal processing. DSP applications include Audio signal processing, 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 ...
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Hertz
The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), often described as being equivalent to one event (or Cycle per second, cycle) per second. The hertz is an SI derived unit whose formal expression in terms of SI base units is 1/s or s−1, meaning that one hertz is one per second or the Inverse second, reciprocal of one second. It is used only in the case of periodic events. It is named after Heinrich Hertz, Heinrich Rudolf Hertz (1857–1894), the first person to provide conclusive proof of the existence of electromagnetic waves. For high frequencies, the unit is commonly expressed in metric prefix, multiples: kilohertz (kHz), megahertz (MHz), gigahertz (GHz), terahertz (THz). Some of the unit's most common uses are in the description of periodic waveforms and musical tones, particularly those used in radio- and audio-related applications. It is also used to describe the clock speeds at which computers and other electronics are driven. T ...
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Poisson Summation Formula
In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function (mathematics), function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples of the original function's Fourier transform. And conversely, the periodic summation of a function's Fourier transform is completely defined by discrete samples of the original function. The Poisson summation formula was discovered by Siméon Denis Poisson and is sometimes called Poisson resummation. For a smooth, complex valued function s(x) on \mathbb R which decays at infinity with all derivatives (Schwartz function), the simplest version of the Poisson summation formula states that where S is the Fourier transform of s, i.e., S(f) \triangleq \int_^ s(x)\ e^\, dx. The summation formula can be restated in many equivalent ways, but a simple one is the following. Sup ...
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Signal (information Theory)
A signal is both the process and the result of Signal transmission, transmission of data over some transmission media, media accomplished by embedding some variation. Signals are important in multiple subject fields including signal processing, information theory and biology. In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The ''IEEE Transactions on Signal Processing'' includes audio signal, audio, video, speech, image, sonar, and radar as examples of signals. A signal may also be defined as observable change in a quantity over space or time (a time series), even if it does not carry information. In nature, signals can be actions done by an organism to alert other organisms, ranging from the release of plant chemicals to warn nearby plants of a predator, to sounds or motions made by animals to alert other animals of food. Signa ...
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Half-band Filter
In digital signal processing, half-band filters are widely used for their efficiency in multi-rate applications. A half-band filter is a low-pass filter that reduces the maximum bandwidth of sampled data by a factor of 2 (one octave). When multiple octaves of reduction are needed, a cascade of half-band filters is common. And when the goal is downsampling, each half-band filter needs to compute only half as many output samples as input samples. It follows from the filter's definition that its transition region, or ''skirt'', can be centered at frequency  f_s/4,  where  f_s  is the input sample-rate. That makes it possible to design a FIR Firs are evergreen coniferous trees belonging to the genus ''Abies'' () in the family Pinaceae. There are approximately 48–65 extant species, found on mountains throughout much of North and Central America, Eurasia, and North Africa. The genu ... filter whose every other coefficient is zero, and whose non-zero coeffic ...
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Multi-rate Digital Signal Processing
Sample-rate conversion, sampling-frequency conversion or resampling is the process of changing the sampling rate or sampling frequency of a discrete signal to obtain a new discrete representation of the underlying continuous signal. Application areas include image scaling and audio/visual systems, where different sampling rates may be used for engineering, economic, or historical reasons. For example, Compact Disc Digital Audio and Digital Audio Tape systems use different sampling rates, and American television, European television, and movies all use different frame rates. Sample-rate conversion prevents changes in speed and pitch that would otherwise occur when transferring recorded material between such systems. More specific types of resampling include: ''upsampling'' or ''upscaling''; ''downsampling'', ''downscaling'', or ''decimation''; and ''interpolation''. The term multi-rate digital signal processing is sometimes used to refer to systems that incorporate sample-rate co ...
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Normalized Frequency (unit)
In digital signal processing (DSP), a normalized frequency is a ratio of a variable frequency (f) and a constant frequency associated with a system (such as a ''sampling rate'', f_s). Some software applications require normalized inputs and produce normalized outputs, which can be re-scaled to physical units when necessary. Mathematical derivations are usually done in normalized units, relevant to a wide range of applications. Examples of normalization A typical choice of characteristic frequency is the ''sampling rate'' (f_s) that is used to create the digital signal from a continuous one. The normalized quantity, f' = \tfrac, has the unit ''cycle per sample'' regardless of whether the original signal is a function of time or distance. For example, when f is expressed in Hz (''cycles per second''), f_s is expressed in ''samples per second''. Some programs (such as MATLAB toolboxes) that design filters with real-valued coefficients prefer the Nyquist frequency (f_s/2) as t ...
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