Quantisation Error Power
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the original analog signal (green), the quantized signal (black dots), the signal reconstructed from the quantized signal (yellow)
Quantization Noise Power Formula
and the difference between the original signal and the reconstructed quantization noise in pcm signal (red). The difference between the original signal and the reconstructed signal is the quantization error
Quantization Error Formula
and, in this simple quantization scheme, is a deterministic function of the input signal. Quantization, in mathematics and digital signal processing, is the process of mapping quantization error definition 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. Quantization also forms the core of essentially what is quantization error 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 is an inherently non-linear and irreversible process (i.e., because the same output value is
into a discrete digital representation, there is a range of input values that produces the same output. That range
How To Reduce Quantization Error
is called quantum ($Q$) and is equivalent to the Least Significant quantization of signals Bit (LSB). The difference between input and output is called the quantization error. Therefore, the quantization
Quantization Error Example
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 https://en.wikipedia.org/wiki/Quantization_(signal_processing) 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 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 http://www.onmyphd.com/?p=quantization.noise.snr 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 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
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