Derivation Of 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 quantization error formula between the original signal and the reconstructed signal is the quantization error and, in quantization error adc this simple quantization scheme, is a deterministic function of the input signal. Quantization, in mathematics and digital signal processing, is quantization error definition 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
Quantization Error Example
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 quantization error matlab 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, 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
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Quantization Error In Analog To Digital Conversion
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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 error 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, pushing the total noise power into the Nyquist band and with an approximately white spectrum (equally quantization error in 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 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 Sh