Quantization Error In Pcm Formula
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Please help to improve this article by introducing more precise citations. (September 2011) (Learn how and when to remove this template message) Signal-to-Quantization-Noise Ratio (SQNR or SNqR) is widely used quality measure in analysing digitizing schemes such as PCM (pulse code modulation) and quantization noise in pcm multimedia codecs. The SQNR reflects the relationship between the maximum nominal signal strength and the quantization noise power formula quantization error (also known as quantization noise) introduced in the analog-to-digital conversion. The SQNR formula is derived from the general SNR (Signal-to-Noise signal to quantization noise ratio Ratio) formula for the binary pulse-code modulated communication channel: S N R = 3 × 2 2 n 1 + 4 P e × ( 2 2 n − 1 ) m m ( t ) 2 m quantization error formula p ( t ) 2 {\displaystyle \mathrm {SNR} ={\frac {3\times 2^{2n}}{1+4P_{e}\times (2^{2n}-1)}}{\frac {m_{m}(t)^{2}}{m_{p}(t)^{2}}}} where P e {\displaystyle P_{e}} is the probability of received bit error m p ( t ) {\displaystyle m_{p}(t)} is the peak message signal level m m ( t ) {\displaystyle m_{m}(t)} is the mean message signal level As SQNR applies to quantized signals, the formulae for SQNR refer to discrete-time digital signals. Instead of m ( t ) {\displaystyle m(t)} ,
How To Reduce Quantization Error
we will use the digitized signal x ( n ) {\displaystyle x(n)} . For N {\displaystyle N} quantization steps, each sample, x {\displaystyle x} requires ν = log 2 N {\displaystyle \nu =\log _{2}N} bits. The probability distribution function (pdf) representing the distribution of values in x {\displaystyle x} and can be denoted as f ( x ) {\displaystyle f(x)} . The maximum magnitude value of any x {\displaystyle x} is denoted by x m a x {\displaystyle x_{max}} . As SQNR, like SNR, is a ratio of signal power to some noise power, it can be calculated as: S Q N R = P s i g n a l P n o i s e = E [ x 2 ] E [ x ~ 2 ] {\displaystyle \mathrm {SQNR} ={\frac {P_{signal}}{P_{noise}}}={\frac {E[x^{2}]}{E[{\tilde {x}}^{2}]}}} The signal power is: x 2 ¯ = E [ x 2 ] = P x ν = ∫ x 2 f ( x ) d x {\displaystyle {\overline {x^{2}}}=E[x^{2}]=P_{x^{\nu }}=\int _{}^{}x^{2}f(x)dx} The quantization noise power can be expressed as: E [ x ~ 2 ] = x m a x 2 3 × 4 ν {\displaystyle E[{\tilde {x}}^{2}]={\frac {x_{max}^{2}}{3\times 4^{\nu }}}} Giving: S Q N R = 3 × 4 ν × x 2 ¯ x m a x 2 {\displaystyle \mathrm {SQNR} ={\frac {3\time
Data Conversion Website Quantization Error and Signal - to - Noise Ratio calculations The signal to noise ratio of a quantized signal is 2+6*(no of bits), as shown in the following table. Resolution and Signal to Noise Ratio for signals
Quantization Error Definition
coded as n bits bits, n levels, 2n Weighting of LSB, 2-n SNR, dB 1 quantization error example 2 0.5 8 2 4 0.25 14 3 8 0.125 20 4 16 0.0625 26 5 32 0.03125 32 6 64 0.01563 quantization of signals 38 7 128 0.00781 44 8 256 0.00391 50 9 512 0.00195 56 10 1024 0.00098 62 11 2048 0.00048 68 12 4096 0.00024 74 13 8192 0.00012 80 14 16384 0.00006 86 15 32768 0.00003 92 https://en.wikipedia.org/wiki/Signal-to-quantization-noise_ratio 16 65536 0.00001 98 These values are for a signal matched to the full-scale range of the converter. If a signal with a range of 5V is measured by an 8 bit ADC with a range of 10V then only 7 bits are effectively in use, and a signal to noise ratio of 44 rather than 50 will apply. Proof: Suppose that the instantaneous value of the input voltage is measured by http://www.skillbank.co.uk/SignalConversion/snr.htm an ADC with a Full Scale Range of Vfs volts, and a resolution of n bits. The real value can change through a range of q = Vfs / 2n volts without a change in measured value occurring. The value of the measured signal is Vm = Vs - e, where Vm is the measured value, Vs is the actual value, and e is the error. The maximum value of error in the measured signal is emax = (1/2)(Vfs / 2n) or emax = q/2 since q = Vfs / 2n The RMS value of quantization error voltage is whence The Signal to Noise Ratio (SNR) is defined as It is normally quoted on a logarithmic scale, in deciBels ( dB ). or The RMS signal voltage is then The error, or quantization noise signal is Thus the signal - to - noise ratio in dB. is since Vfs = 2n q, then which simplifies to N.B. This equation is true only if the input signal is exactly matched to the Full Scale Range of the converter. For signals whose amplitude is less than the FSR the Signal - to - Noise Ratio will be reduced. Download a .pdf file of the analysis of quantization error and signal to noise ratio
· Article Processing Charges · Articles in Press · Author Guidelines · Bibliographic Information · https://www.hindawi.com/journals/isrn/2011/731989/ Citations to this Journal · Contact Information · Editorial Board http://www.allsyllabus.com/aj/note/ECE/Digital%20Communication/unit3/Quantization%20Process.php · Editorial Workflow · Free eTOC Alerts · Publication Ethics · Reviewers Acknowledgment · Submit a Manuscript · Subscription Information · Table of Contents Abstract Full-Text PDF Full-Text HTML Full-Text ePUB Full-Text XML Linked References Citations to this quantization error Article How to Cite this Article ISRN Signal ProcessingVolume 2011 (2011), Article ID 731989, 7 pageshttp://dx.doi.org/10.5402/2011/731989Research ArticleApproximation Formula for Easy Calculation of Signal-to-Noise Ratio of Sigma-Delta ModulatorsValeri Mladenov,1 Panagiotis Karampelas,2,3 Georgi Tsenov,1 and Vassiliki Vita31Department of Theoretical Electrical Engineering, Technical University of Sofia, 8 Kliment Ohridski Street, 1000 Sofia, Bulgaria2IT Faculty, Hellenic American quantization error in University, 12 Kaplanon Street, 10680 Athens, Greece3Department of Electrical Engineering Educators, School of Pedagogical and Technological Education (ASPETE), N. Heraklion, 14121 Athens, GreeceReceived 29 October 2010; Accepted 1 December 2010Academic Editors: L.-M. Cheng and S. KalitzinCopyright © 2011 Valeri Mladenov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.AbstractThe signal-to-noise ratio (SNR) is one of the most significant measures of performance of the sigma-delta modulators. An approximate formula for calculation of signal-to-noise ratio of an arbitrary sigma-delta modulator (SDM) has been proposed. Our approach for signal-to-noise ratio computation does not require modulator modeling and simulation. The proposed formula is compared with SNR calculations based on output bitstream obtained by simulations, and the reasons for small discrepancies are explained. The p
Delta Modulation (DM) QUANTIZATION NOISE Adaptive Delta Modulation Coding Speech at Low Bit Rates Digital Multiplexers Light Wave Transmission Quantization Process The process of transforming Sampled amplitude values of a message signal into a discrete amplitude value is referred to as Quantization. The quantization Process has a two-fold effect: 1. the peak-to-peak range of the input sample values is subdivided into a finite set of decision levels or decision thresholds that are aligned with the risers of the staircase, and 2. the output is assigned a discrete value selected from a finite set of representation levels that are aligned with the treads of the staircase.. A quantizer is memory less in that the quantizer output is determined only by the value of a corresponding input sample, independently of earlier analog samples applied to the input. Types of Quantizers: 1. Uniform Quantizer 2. Non- Uniform Quantizer 0 Ts 2Ts 3Ts Time Analog Signal Discrete Samples ( Quantized ) In Uniform type, the quantization levels are uniformly spaced, whereas in nonuniform type the spacing between the levels will be unequal and mostly the relation is logarithmic. Types of Uniform Quantizers: ( based on I/P - O/P Characteristics) 1. Mid-Rise type Quantizer 2. Mid-Tread type Quantizer In the stair case like graph, the origin lies the middle of the tread portion in Mid –Tread type where as the origin lies in the middle of the rise portion in the Mid-Rise type. Mid – tread type: Quantization levels – odd number. Mid – Rise type: Quantization levels – even number. Quantization Noise and Signal-to-Noise: “The Quantization process introduces an error defined as the difference between the input signal, x(t) and the output signal, yt). This error is called the Quantization Noise.” q(t) = x(t) – y(t) Quantization noise is produced in the transmitter end of a PCM system by rounding off sample values of an analog base-band signal to the nearest permissible representation levels of the quantizer. As such quantization noise differs from channel noise in that i