Quantization Error 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 original signal and the reconstructed signal (red). The difference between the
Quantization Error Definition
original signal and the reconstructed signal is the quantization error and, in this quantization error formula simple quantization scheme, is a deterministic function of the input signal. Quantization, in mathematics and digital signal processing, is the
Quantization Noise Power
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 quantization noise in pcm 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 quantizer. Contents 1 how to reduce quantization error 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 is the application of quantization
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
Quantization Error Example
Least Significant Bit (LSB). The difference between input and output is called the quantization error in pcm quantization error. Therefore, the quantization error can be between $\pm Q/2$. Any value of the error is equally likely,
Quantization Error In Analog To Digital Conversion
so it has a uniform distribution ranging from $-Q/2$ to $+Q/2$. Then, this error can be considered a quantization noise with RMS: $$ v_{qn} = \sqrt{\frac{1}{Q}\int_{-Q/2}^{+Q/2}x^2dx}=\sqrt{\frac{1}{Q}\left[\frac{x^3}{3}\right]_{-Q/2}^{+Q/2}} = \sqrt{\frac{Q^2}{2^3 3} + \frac{Q^2}{2^3 https://en.wikipedia.org/wiki/Quantization_(signal_processing) 3}} = \frac{Q}{\sqrt{12}}$$ 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 frequency. Due to the sampling step of an ADC, these harmonics get folded to the Nyquist band, http://www.onmyphd.com/?p=quantization.noise.snr 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: $$SNR = 6.02N + 1.76 (dB)$$ 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 h
Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About http://dsp.stackexchange.com/questions/2897/how-does-the-quantization-error-generate-noise Us Learn more about Stack Overflow the company Business Learn more about hiring developers or posting ads with us Signal Processing Questions Tags Users Badges Unanswered Ask Question _ Signal Processing Stack Exchange is a question and answer site for practitioners of the art and science of signal, image and video processing. Join them; it only takes a minute: Sign up Here's how it works: Anybody quantization error can ask a question Anybody can answer The best answers are voted up and rise to the top How does the quantization error generate noise? up vote 9 down vote favorite 3 I'm learning about sampling and DSP on my own. I have a hard time to understand how the quantization error results in noise. I think I miss a fundamental understanding but can't tell what it quantization error in is. So how does the quantization error generate noise? noise sampling share|improve this question asked Jul 15 '12 at 19:29 Jan Deinhard 1485 It's more distortion than noise. It depends on the signal, and is not random. –endolith Jul 15 '12 at 20:48 endolith, I think what I don't understand is how the error results in frequencies. –Jan Deinhard Jul 15 '12 at 21:04 2 distortion always produces additional frequencies. if you distort a sine wave, it becomes a different repetitive waveform. any repetitive waveform other than a sine wave is made up of multiple frequencies. –endolith Jul 15 '12 at 22:58 1 As @endolith has mentioned, let us assume you have a very bad ADC, such that you give it a pure tone, but get a signal that looks like a sine but has big steps in it. (So now your signal looks like a staircase that is going up and down with the original sine.) Now, you know intuitively that a step is composed of many frequencies. This is how an ADC will add frequencies as you are asking. It is a non-linear operation btw. If it was
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