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In the context of digital signal processing (DSP), a **digital signal** is a discrete-time signal for which not only the time but also the amplitude has discrete values; in other words, its samples take on only values from a discrete set (a countable set that can be mapped one-to-one to a subset of integers). If that discrete set is finite, the discrete values can be represented with digital words of a finite width. Most commonly, these discrete values are represented as fixed-point words (either proportional to the waveform values or companded) or floating-point words.^{ [1] }^{ [2] }^{ [3] }^{ [4] }^{ [5] }

**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 signals processed in this manner are a sequence of numbers that represent samples of a continuous variable in a domain such as time, space, or frequency.

In mathematics, an **injective function** or **injection** or **one-to-one function** is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the function's codomain is the image of *at most* one element of its domain. The term *one-to-one function* must not be confused with *one-to-one correspondence*, which uniquely maps all elements in both domain and codomain to each other.

An **integer** is a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5 1/2, and √2 are not.

The process of analog-to-digital conversion produces a digital signal.^{ [6] } The conversion process can be thought of as occurring in two steps:

- sampling, which produces a continuous-valued discrete-time signal, and
- quantization, which replaces each sample value by an approximation selected from a given discrete set (for example by truncating or rounding).

It can be shown that for signal frequencies strictly below the Nyquist limit that the original continuous-valued continuous-time signal can be almost perfectly reconstructed, down to the (often very low) limit set by the quantisation.

Common practical digital signals are represented as 8-bit (256 levels), 16-bit (65,536 levels), 24-bit (16.8 million levels) and 32-bit (4.3 billion levels). But the number of quantization levels is not necessarily limited to powers of two. A floating point representation is used in many DSP applications.

In computer architecture, **8-bit** integers, memory addresses, or other data units are those that are 8 bits wide. Also, 8-bit CPU and ALU architectures are those that are based on registers, address buses, or data buses of that size. **8-bit** is also a generation of microcomputers in which 8-bit microprocessors were the norm.

In computer architecture, **16-bit** integers, memory addresses, or other data units are those that are 16 bits wide. Also, 16-bit CPU and ALU architectures are those that are based on registers, address buses, or data buses of that size. 16-bit microcomputers are computers in which 16-bit microprocessors were the norm.

In computer architecture, **24-bit** integers, memory addresses, or other data units are those that are 24 bits wide. Also, 24-bit CPU and ALU architectures are those that are based on registers, address buses, or data buses of that size.

In electronics and telecommunications, **modulation** is the process of varying one or more properties of a periodic waveform, called the *carrier signal*, with a modulating signal that typically contains information to be transmitted. Most radio systems in the 20th century used frequency modulation (FM) or amplitude modulation (AM) for radio broadcast.

**Signal processing** is a subfield of mathematics, information and electrical engineering that concerns the analysis, synthesis, and modification of signals, which are broadly defined as functions conveying "information about the behavior or attributes of some phenomenon", such as sound, images, and biological measurements. For example, signal processing techniques are used to improve signal transmission fidelity, storage efficiency, and subjective quality, and to emphasize or detect components of interest in a measured signal.

In electronics, an **analog-to-digital converter** is a system that converts an analog signal, such as a sound picked up by a microphone or light entering a digital camera, into a digital signal. An ADC may also provide an isolated measurement such as an electronic device that converts an input analog voltage or current to a digital number representing the magnitude of the voltage or current. Typically the digital output is a two's complement binary number that is proportional to the input, but there are other possibilities.

In electronics, a **digital-to-analog converter** is a system that converts a digital signal into an analog signal. An analog-to-digital converter (ADC) performs the reverse function.

Sound can be recorded and stored and played using either digital or analog techniques. Both techniques introduce errors and distortions in the sound, and these methods can be systematically compared. Musicians and listeners have argued over the superiority of digital versus analog sound recordings. Arguments for analog systems include the absence of fundamental error mechanisms which are present in digital audio systems, including aliasing and quantization noise. Advocates of digital point to the high levels of performance possible with digital audio, including excellent linearity in the audible band and low levels of noise and distortion.

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.

In communication systems, signal processing, and electrical engineering, a **signal** is a function that "conveys information about the behavior or attributes of some phenomenon". In its most common usage, in electronics and telecommunication, this is a time varying voltage, current or electromagnetic wave used to carry information. A signal may also be defined as an "observable change in a quantifiable entity". In the physical world, any quantity exhibiting variation in time or variation in space is potentially a signal that might provide information on the status of a physical system, or convey a message between observers, among other possibilities. The *IEEE Transactions on Signal Processing* states that the term "signal" includes audio, video, speech, image, communication, geophysical, sonar, radar, medical and musical signals. In a later effort of redefining a signal, anything that is only a function of space, such as an image, is excluded from the category of signals. Also, it is stated that a signal may or may not contain any information.

**Quantization**, in mathematics and digital signal processing, is the process of mapping input values from a large set to output values in a (countable) smaller set, often with a finite number of elements. Rounding and truncation are typical examples of quantization processes. Quantization is involved to some degree in nearly all digital signal processing, as the process of representing a signal in digital form ordinarily involves rounding. Quantization also forms the core of essentially all lossy compression algorithms.

**Digitization**, less commonly **digitalization**, is the process of converting information into a digital format, in which the information is organized into bits. The result is the representation of an object, image, sound, document or signal by generating a series of numbers that describe a discrete set of its points or samples. The result is called *digital representation* or, more specifically, a *digital image*, for the object, and *digital form*, for the signal. In modern practice, the digitized data is in the form of binary numbers, which facilitate computer processing and other operations, but, strictly speaking, digitizing simply means the conversion of analog source material into a numerical format; the decimal or any other number system that can be used instead.

In numerical analysis, computational physics, and simulation, **discretization error** is the error resulting from the fact that a function of a continuous variable is represented in the computer by a finite number of evaluations, for example, on a lattice. Discretization error can usually be reduced by using a more finely spaced lattice, with an increased computational cost.

**Differential pulse-code modulation** (**DPCM**) is a signal encoder that uses the baseline of pulse-code modulation (PCM) but adds some functionalities based on the prediction of the samples of the signal. The input can be an analog signal or a digital signal.

**Noise shaping** is a technique typically used in digital audio, image, and video processing, usually in combination with dithering, as part of the process of quantization or bit-depth reduction of a digital signal. Its purpose is to increase the apparent signal-to-noise ratio of the resultant signal. It does this by altering the spectral shape of the error that is introduced by dithering and quantization; such that the noise power is at a lower level in frequency bands at which noise is considered to be less desirable and at a correspondingly higher level in bands where it is considered to be more desirable. A popular noise shaping algorithm used in image processing is known as ‘Floyd Steinberg dithering’; and many noise shaping algorithms used in audio processing are based on an ‘Absolute threshold of hearing’ model.

**Delta-sigma** modulation is a method for encoding analog signals into digital signals as found in an analog-to-digital converter (ADC). It is also used to convert high bit-count, low-frequency digital signals into lower bit-count, higher-frequency digital signals as part of the process to convert digital signals into analog as part of a digital-to-analog converter (DAC).

**Signal-to-Quantization-Noise Ratio** is widely used quality measure in analysing digitizing schemes such as PCM and multimedia codecs. The SQNR reflects the relationship between the maximum nominal signal strength and the quantization error introduced in the analog-to-digital conversion.

In digital audio using pulse-code modulation (PCM), **bit depth** is the number of bits of information in each sample, and it directly corresponds to the **resolution** of each sample. Examples of bit depth include Compact Disc Digital Audio, which uses 16 bits per sample, and DVD-Audio and Blu-ray Disc which can support up to 24 bits per sample.

**Mean square quantization error** (MSQE) is a figure of merit for the process of analog to digital conversion.

A **Bitcrusher** is a lo-fi digital audio effect, which produces a distortion by the reduction of the resolution or bandwidth of digital audio data. The resulting quantization noise may produce a “warmer” sound impression, or a harsh one, depending on the amount of reduction.

In mathematics and, in particular, mathematical dynamics, **discrete time** and **continuous time** are two alternative frameworks within which to model variables that evolve over time.

- ↑ Smith, Steven W. (2002-11-06). "3".
*Digital Signal Processing: A Practical Guide for Engineers and Scientists*. Demystifying Technology.**1**(1 ed.). Newnes. pp. 35–39. ISBN 075067444X. - ↑ Harris, Frederic J. (2004-05-24). "1.1".
*Multirate Signal Processing for Communication Systems*. Upper Saddle River, NJ: Prentice Hall PTR. p. 2. ISBN 0131465112. - ↑ Vaseghi, Saeed V. (2009-03-02). "1.4".
*Advanced Digital Signal Processing and Noise Reduction*(4 ed.). Chichester, West Suffix, United Kingdom: John Wiley & Sons. p. 23. ISBN 0470754060. - ↑ Diniz, Paulo S. R.; Eduardo A. B. da Silva; Sergio L. Netto (2010-09-13). "1.1".
*Digital Signal Processing: System Analysis and Design*(2 ed.). New York & UK: Cambridge University Press. p. 5. ISBN 0521887755. - ↑ Manolakis, Dimitris G.; Vinay K. Ingle (2011-11-21). "1.1.1".
*Applied Digital Signal Processing: Theory and Practice*. Cambridge, UK: Cambridge University Press. p. 5. ISBN 0521110025. - ↑ Ingle, Vinay K.; John G. Proakis (2011-01-01). "1.1".
*Digital Signal Processing Using MATLAB*(3 ed.). Stamford, CT: CL Engineering. p. 3. ISBN 1111427372.

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