Mean Square Quantization Error Formula
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Please help to improve this article by introducing more precise citations. (August 2016) (Learn how and when to remove this template message) Mean square quantization quantization error definition error (MSQE) is a figure of merit for the process of
Uniform Quantization
analog to digital conversion. In this conversion process, analog signals in a continuous range of values are converted
Quantization Error In Pcm
to a discrete set of values by comparing them with a sequence of thresholds. The quantization error of a signal is the difference between the original continuous value and
Quantization Step Size Formula
its discretization, and the mean square quantization error (given some probability distribution on the input values) is the expected value of the square of the quantization errors. Mathematically, suppose that the lower threshold for inputs that generate the quantized value q i {\displaystyle q_{i}} is t i − 1 {\displaystyle t_{i-1}} , that the upper threshold is t i uniform quantization pdf {\displaystyle t_{i}} , that there are k {\displaystyle k} levels of quantization, and that the probability density function for the input analog values is p ( x ) {\displaystyle p(x)} . Let x ^ {\displaystyle {\hat {x}}} denote the quantized value corresponding to an input x {\displaystyle x} ; that is, x ^ {\displaystyle {\hat {x}}} is the value q i {\displaystyle q_{i}} for which t i − 1 ≤ x < t i {\displaystyle t_{i}-1\leq x the original analog signal (green), the quantized signal (black dots), the signal reconstructed from the quantized difference between uniform and nonuniform quantization signal (yellow) and the difference between the original signal and quantization example the reconstructed signal (red). The difference between the original signal and the reconstructed signal is mid rise and mid tread quantizer the quantization error and, in this simple quantization scheme, is a deterministic function of the input signal. Quantization, in mathematics and digital signal processing, https://en.wikipedia.org/wiki/Mean_square_quantization_error is 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 digital signal processing, as the process of representing a signal in digital form ordinarily involves rounding. https://en.wikipedia.org/wiki/Quantization_(signal_processing) 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 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 into a discrete digital representation, there is a range of input values that produces the same output. That range is called quantum ($Q$) http://www.onmyphd.com/?p=quantization.noise.snr and is equivalent to the Least Significant Bit (LSB). The difference 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 quantization error 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 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 mean square quantization 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 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 be down. Please try the request again. Your cache administrator is webmaster. Generated Thu, 20 Oct 2016 13:54:53 GMT by s_wx1157 (squid/3.5.20)