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overview
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Although the rms value of the noise can be accurately approximated by q/√12, under certain conditions the frequency domain content can be highly correlated with the ac input signal. For example, low-amplitude periodic signals are more correlated than high-amplitude random signals. It is usually assumed that theoretical quantization noise behaves as white noise, uniformly distributed over the Nyquist bandwidth from dc to fs/2. However, this is not the case at all. In the case of strong correlation, the quantization noise is concentrated on the harmonics of the input signal, which is exactly what we don't want to see.
overall architecture process
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In most practical applications, the input to the ADC is a frequency (always summed with some unavoidable system noise), so the quantization noise tends to be random. However, in spectral analysis applications (or performing an FFT to an ADC using a spectrally clean sine wave as input), the degree to which the quantization noise is related to the signal depends on the ratio of the sampling frequency to the input signal
Explanation of technical terms
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ADC:
Analog-to-digital converter, or A/D converter, or ADC for short, usually refers to an electronic component that converts an analog signal into a digital signal. A common analog-to-digital converter converts an input voltage signal into an output digital signal. Since the digital signal itself has no practical significance, it only represents a relative size. Therefore, any analog-to-digital converter needs a reference analog quantity as a conversion standard, and the more common reference standard is the largest convertible signal size. The output digital quantity represents the magnitude of the input signal relative to the reference signal
technical details
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Note that this change in harmonic distortion seen from the ADC is an artifact of the sampling process, caused by the dependence of quantization error on the input frequency. In practical ADC applications, quantization error generally appears as random noise, due to the randomness of wideband input signals, and usually a small amount of system noise acts as a "perturbation" signal to further randomize the quantization error spectrum. It is very important to understand the above principles, because the single-tone sine wave FFT test of ADC is one of the recognized performance evaluation methods. In order to accurately measure the harmonic distortion of an ADC, steps must be taken to ensure that the test setup truly measures the ADC distortion and not the artifacts caused by quantization noise correlations. Therefore, the frequency ratio must be chosen correctly, and sometimes a small amount of noise (perturbation) must be added to the input signal. The same precautions should be taken when measuring DAC distortion with an analog spectrum analyzer.
The FFT output of an ideal 12-bit ADC is shown. Note that the average value of the FFT noise floor is about 107dB below full scale, but the theoretical SNR of the 12-bit ADC is 74dB. The FFT noise floor is not the SNR of the ADC because the FFT is like an analog spectrum analyzer with a bandwidth of fs/M, where M is the number of points in the FFT. Due to the processing gain of the FFT, the theoretical FFT noise floor is thus 10log10(M/2)dB lower than the quantization noise floor
For an ideal 12-bit ADC with an SNR of 74dB, a 4096-point FFT would produce a processing gain of 10log10(4096/2)=33dB, so the total FFT noise floor is 74+33=107dBc. In fact, the FFT noise floor can be further reduced by increasing the number of FFT points, just as the noise floor of an analog spectrum analyzer can be reduced by reducing the bandwidth. Therefore, when testing an ADC with an FFT, one must ensure that the FFT is large enough that the distortion is indistinguishable from the FFT noise floor itself. Averaging multiple FFTs cannot further reduce the noise floor, only the difference between the magnitudes of the individual noise spectral components.
summary
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For example:
This paper describes three distinct phases of the derivation process:
1. Ideal analog-to-digital converter (ADC) transfer function formulation and operation.
2. Root mean square (rms) derivation based on integral method. '
3. SNR formula derivation to obtain the value of SNR=6.02N+1.76dB. '
Ideal ADC transfer function. Digital (binary) output values are represented on the y-axis and analog inputs are represented on the x-axis. Diagonal steps represent quantized values of the analog input signal. A dashed line through the ladder indicates its midpoint. Represents the quantization noise of an ideal N-bit ADC with a ramping input signal. The 1LSB peak-to-peak quantization error can be approximated by an uncorrelated sawtooth waveform with a maximum peak-to-peak swing of q in the range q/2 to –q/2. Note that t1 and t2 are time points, which will be used later in the derivation stage. This signal is the difference between the quantized output signal (solid line) and the analog input signal (dashed line).
