Pcm Quantization Error
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the original analog signal (green), the quantized signal (black dots), the signal reconstructed from the quantized signal (yellow) and the difference between the original signal and the reconstructed signal (red). The difference between
Quantization Error Formula
the original signal and the reconstructed signal is the quantization error and, in uniform quantization this simple quantization scheme, is a deterministic function of the input signal. Quantization, in mathematics and digital signal processing, is
Signal To Quantization Noise Ratio
the process of mapping a large set of input values to a (countable) smaller set. Rounding and truncation are typical examples of quantization processes. Quantization is involved to some degree in nearly all what is quantization 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. The difference between an input value and its quantized value (such as round-off error) is referred to as quantization error. A device or algorithmic function that performs quantization is called a quantizer. An analog-to-digital converter is an example of a how to reduce quantization error quantizer. Contents 1 Basic properties of quantization 2 Basic types of quantization 2.1 Analog-to-digital converter (ADC) 2.2 Rate–distortion optimization 3 Rounding example 4 Mid-riser and mid-tread uniform quantizers 5 Dead-zone quantizers 6 Granular distortion and overload distortion 7 The additive noise model for quantization error 8 Quantization error models 9 Quantization noise model 10 Rate–distortion quantizer design 11 Neglecting the entropy constraint: Lloyd–Max quantization 12 Uniform quantization and the 6 dB/bit approximation 13 Other fields 14 See also 15 Notes 16 References 17 External links Basic properties of quantization[edit] Because quantization is a many-to-few mapping, it is an inherently non-linear and irreversible process (i.e., because the same output value is shared by multiple input values, it is impossible in general to recover the exact input value when given only the output value). The set of possible input values may be infinitely large, and may possibly be continuous and therefore uncountable (such as the set of all real numbers, or all real numbers within some limited range). The set of possible output values may be finite or countably infinite. The input and output sets involved in quantization can be defined in a rather general way. For example, vector quantization i
Please help to improve this article by introducing more precise citations. (September 2011) (Learn how and when to remove this template message) Signal-to-Quantization-Noise Ratio (SQNR or SNqR) is widely used quality measure in analysing digitizing schemes such as PCM (pulse code modulation) and multimedia codecs. The SQNR reflects the relationship
Quantization Step Size Formula
between the maximum nominal signal strength and the quantization error (also known as quantization noise) introduced quantization example in the analog-to-digital conversion. The SQNR formula is derived from the general SNR (Signal-to-Noise Ratio) formula for the binary pulse-code modulated communication channel: S
Quantization Error Example
N R = 3 × 2 2 n 1 + 4 P e × ( 2 2 n − 1 ) m m ( t ) 2 m p ( t ) 2 {\displaystyle \mathrm {SNR} ={\frac {3\times 2^{2n}}{1+4P_{e}\times (2^{2n}-1)}}{\frac {m_{m}(t)^{2}}{m_{p}(t)^{2}}}} https://en.wikipedia.org/wiki/Quantization_(signal_processing) where P e {\displaystyle P_{e}} is the probability of received bit error m p ( t ) {\displaystyle m_{p}(t)} is the peak message signal level m m ( t ) {\displaystyle m_{m}(t)} is the mean message signal level As SQNR applies to quantized signals, the formulae for SQNR refer to discrete-time digital signals. Instead of m ( t ) {\displaystyle m(t)} , we will use the digitized signal x ( n ) {\displaystyle x(n)} . For N {\displaystyle N} quantization steps, each https://en.wikipedia.org/wiki/Signal-to-quantization-noise_ratio sample, x {\displaystyle x} requires ν = log 2 N {\displaystyle \nu =\log _{2}N} bits. The probability distribution function (pdf) representing the distribution of values in x {\displaystyle x} and can be denoted as f ( x ) {\displaystyle f(x)} . The maximum magnitude value of any x {\displaystyle x} is denoted by x m a x {\displaystyle x_{max}} . As SQNR, like SNR, is a ratio of signal power to some noise power, it can be calculated as: S Q N R = P s i g n a l P n o i s e = E [ x 2 ] E [ x ~ 2 ] {\displaystyle \mathrm {SQNR} ={\frac {P_{signal}}{P_{noise}}}={\frac {E[x^{2}]}{E[{\tilde {x}}^{2}]}}} The signal power is: x 2 ¯ = E [ x 2 ] = P x ν = ∫ x 2 f ( x ) d x {\displaystyle {\overline {x^{2}}}=E[x^{2}]=P_{x^{\nu }}=\int _{}^{}x^{2}f(x)dx} The quantization noise power can be expressed as: E [ x ~ 2 ] = x m a x 2 3 × 4 ν {\displaystyle E[{\tilde {x}}^{2}]={\frac {x_{max}^{2}}{3\times 4^{\nu }}}} Giving: S Q N R = 3 × 4 ν × x 2 ¯ x m a x 2 {\displaystyle \mathrm {SQNR} ={\frac {3\times 4^{\nu }\times {\overline {x^{2}}}}{x_{max}^{2}}}} When the SQNR is desired in terms of Decibels (dB), a useful approximation to SQNR is: S Q N R | d B = P x ν + 6 ν + 4.8 {\displaystyle \mathrm {SQNR} |
Thursday, May 26, 2011 Pulse Code Modulation: Quantization The discrete signal i.e. sampled values produced as a result of sampling (discussed in the previous post) has to be quantized, hence the process called Quantization. In simple words, this is the process in which a value http://atif-razzaq.blogspot.com/2011/05/pulse-code-modulation-quantization.html from a given set of values is assigned to each sample of the discrete signal. The number of values in the set actually is the number of quantization levels to which samples of discrete signals are assigned to. Digital communication is based on bits and bytes, the number of bits used identify the number of quantization levels, hence, in this case when the samples are binary encoded that means they are essentially being quantized in one of the fixed number of quantization levels.In the figure below, the process has been explained where quantization error an input discrete signal s(t) has been quantized into a signal sq(t). The input signal s(t) moves between low peak amplitude AL to high peak amplitude AH . This range from AL to AH is divided into M intervals (an interval represent a quantization level, also referred as quantization interval) each of size L such that L = (AH - AL)/M. In the figure, eight intervals have been shown i.e. M=8. Let As be the peak-to-peak amplitude of s(t) i.e. total range between high and low peaks i.e. As = AH - AL pcm quantization error hence L = As /M. The value of each quantization interval is taken as the center of the interval, in figure, shown as a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 . The range of values for each interval has been shown as l0, l1, l2 , l3 , l4 , l5 , l6 , l7 . Moreover, the midpoints between consecutive quantization intervals are shown by A01 , A12 , A23 , A34 , A45 , A56 , A67 . Each of these midpoints is at a distance of L/2 from its corresponding intervals values e.g. A67 is the midpoint between two quantization intervals whose values are a6 and a7 and is located at a distance of L/2 from a6 and a7. The distance between two consecutive midpoints and also between two interval values is L. It is, in fact, these mid points which define when the value of the quantized signal sq(t) changes. It should also be noted that the peak values i.e. AH and AL are at a distance of L/2 from their corresponding quantization levels. This results in affecting the peak-to-peak amplitude of sq(t) shown as Aq and has a value of (M-1)L.As shown in the figure, when the signal being quantized i.e. s(t) is in the range of l7 , the quantized signal sq(t) maintains a constant value of a7 . As soon as s(t) gets in the range of l6 , sq(t) makes a jump to the value of a6 and maintains this value as long as s(t) assumes a value in the range of l6 . The sudden jump is mad
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