Quantization Error Quantization Noise
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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
Quantization Error Definition
original signal and the reconstructed signal (red). The difference between the original quantization error formula signal and the reconstructed signal is the quantization error and, in this simple quantization scheme, is a
Quantization Noise Power
deterministic function of the input signal. Quantization, in mathematics and digital signal processing, is the process of mapping a large set of input values to a (countable) smaller quantization error example set. 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. The difference between an input value and its quantized value (such as how to reduce quantization error 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 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
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Quantization Noise In Pcm
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Quantization Of Signals
a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top What is “Maximum Quantization Error”? up vote 2 down vote favorite 1 I have https://en.wikipedia.org/wiki/Quantization_(signal_processing) an formula for this "Maximum Quantization Error" but i dont know what it is based in. Its just thrown in my study material without further explanation. It is defined as: $$Q = \dfrac {\Delta x}{2^{N+1}}$$ where $N$ is the number of bits used for quantization in a analog to digital conversion, and $\Delta x$ is, in portuguese "Faixa de Excursão do Sinal", I don't know what would be the correct translation, but I bet on something like "Signal Excursion Band". I know, http://dsp.stackexchange.com/questions/15925/what-is-maximum-quantization-error its a strange name. Can someone help me with this? What is this $\Delta x$? Sorry for my bad english, it isnt my native language. adc quantization share|improve this question edited Apr 29 '14 at 17:07 jojek♦ 6,71041444 asked Apr 29 '14 at 15:19 Diedre 20115 Evidently you are learning the basics. Speaking as a retired EE; real designs are a lot more complicated. The answer below is idealized for discussion. While not wrong, there are large confounding terms in physical implementation. –rrogers Dec 30 '15 at 14:42 add a comment| 1 Answer 1 active oldest votes up vote 4 down vote accepted When you quantize a signal, you introduce and error which can be defined as $$q[n] = x_q[n]-x[n]$$ where $q[n]$ is the quantization error, $x[n]$ the original signal, and $x_q[n]$ of the quantized signal. The maximum quantization error is simply $max(\left | q \right |)$, the absolute maximum of this error function. Dx in this definition seems to be the range of the input signal so we could rewrite this as $$Q = \frac{max(x)-min(x)}{2^{N+1}}$$ Let's look at a quick example. Let's assume you have a signal that's uniformly distributed between -1 and +1 and you want to quantize this with 3 bits. You have a total 8 of quantizaton steps which would map to [-1 -.75 -.5 -25 0 .25 .5 .75]. The difference between steps is 0.25. If you round during quantization the maximum error will be half of th
into a discrete digital representation, there is a range of input values that produces the same output. That range is called quantum ($Q$) and is equivalent to the Least Significant Bit (LSB). The difference http://www.onmyphd.com/?p=quantization.noise.snr between input and output is called the quantization error. Therefore, the quantization error can be between $\pm Q/2$. Any value of the error is equally likely, so it has a uniform distribution ranging from $-Q/2$ to $+Q/2$. Then, this error can be considered a quantization noise with RMS: $$ v_{qn} = \sqrt{\frac{1}{Q}\int_{-Q/2}^{+Q/2}x^2dx}=\sqrt{\frac{1}{Q}\left[\frac{x^3}{3}\right]_{-Q/2}^{+Q/2}} = \sqrt{\frac{Q^2}{2^3 3} + \frac{Q^2}{2^3 3}} = \frac{Q}{\sqrt{12}}$$ What is the frequency spectrum of the quantization quantization error noise? We know the quantization noise power is $v_{qn}^2$, but where is it concentrated or spread in the frequency domain? The quantization error creates harmonics in the signal that extend well above the Nyquist frequency. Due to the sampling step of an ADC, these harmonics get folded to the Nyquist band, pushing the total noise power into the Nyquist band and with an approximately white spectrum (equally spread across quantization error quantization all frequencies in the band). How does the Signal-Noise Ratio (SNR) relates to the number of bits in the digital representation? Assuming an input sinusoidal with peak-to-peak amplitude $V_{ref}$, where $V_{ref}$ is the reference voltage of an N-bit ADC (therefore, occupying the full-scale of the ADC), its RMS value is $$V_{rms} = \frac{V_{ref}}{2\sqrt{2}} = \frac{2^NQ}{2\sqrt{2}}.$$ where $N$ is the number of bits available for discretization. The relation $V_{ref} = 2^NQ$ comes from the fact that the range $V_{ref}$ is divided among $2^N$ steps, each with quantum $Q$. To calculate the Signal-Noise Ratio, we divide the RMS of the input signal by the RMS of the quantization noise: $$SNR = 20\log\left(\frac{V_{rms}}{v_{qn}}\right) = 20\log\left(\frac{\frac{2^NQ}{2\sqrt{2}}}{\frac{Q}{\sqrt{12}}}\right) = 20\log\left(\frac{2^N\sqrt{12}}{2\sqrt{2}}\right)$$ $$ = 20\log\left(2^N\right) + 20\log\left(\frac{\sqrt{6}}{2}\right) = 6.02N + 1.76 (dB).$$ In fact, the equation: $$SNR = 6.02N + 1.76 (dB)$$ generalizes to any system using a digital representation. So, a microprocessor representing values with N bits will have a SNR defined by the above formula. If I helped you in some way, please help me back by liking this website on the bottom of the page or clicking on the link below. It would mean the world to me! Tweet Contents | Resources || Print Show your love: Tweet
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