Amateur Radio Archive

Effective Bits Irom the DFT

One way to characterize the signal-to-noise ratio of a digitizer is to sample a quiet (low-noise) and spectrally pure full-scale sine wave and perform a discrete Fourier transform [DFT] on the resulting data. The dynamic range (in dB) from the peak of the fundamental to the noise floor of the DFT gives an idea of the low-level signals that can be resolved. The level of ihe noise floor depends on the number of frequency points (bins) in the DFT, and hence on the number of samples taken, since if the same noise

power is spread over more frequency bins, (here will be less noise power per bin.

The DFT spectrum can be used to produce an estimate of the signal-to-noise ratio of a digitizer by performing essentially the same measurement digitally that a distortion analyzer performs electronically. A distortion analyzer supplies a low-distortion sine wave as the input to a circuil under test. A notch filter is used to remove the fundamental frequency from the output signal. The power in the filtered signal is measured and a ratio is formed with ihe total output power of the circuit under test. A distortion analyzer measurement assumes that the power in the filtered output signal is dominated by harmonic terms generated by distortion in the circuit under test. In practice, however, the analyzer is unable to separate this power from the power contribution of wideband noise, and hence is actually-measuring the signal-to-noise ratio of the output signal.

An analogous operation can be performed on the DFT spectrum of a digitized pure sine wave. A certain number of frequency bins on either side of the fundamental peak are removed from Ihe DFT data. The data in each of the other frequency bins is squared (to yield a power term) and summed wilh similar results from the other frequency bins to calculate the total noise power. The data within the narrow band around the fundamental is squared and summed to give the total signal power. The ratio of these two lerms,

(a)

(0

Fig. 3. Linearity errors in an ADC (a) Integral linearity error (b) Differential linearity error, (c) Total linearity error.

Fig. 3. Linearity errors in an ADC (a) Integral linearity error (b) Differential linearity error, (c) Total linearity error.

expressed in dB, can be used to compute the number of effective bits of resolution of the digitizer-

Calculations of effective bits from DFT spectra will show variations if the test is performed repeatedly. This variation can be reduced if the spectral values from many independent trials are averaged point by point (as opposed to averaging the time-domain data). Spectral averaging will not reduce the level of the noise floor in the DFT data, but only the amount it varies. Therefore, if enough ensembles of spectral data are averaged, the number of effective bits calculated will converge to a single number.

Fig. 4 shows the DFT for 4096 samples of a mathematically generated ideal sine wave quantized to 16 bits (±32.767 counts). From this, we see that a perfect 16-bit digitizer will show a noise floor of about -127 dB when quantization error is the only source of noise, if the signal-to-noise ratio is calculated using the method described above, the result is 97.0 dB. or 16,0 effective bits, which is what we would expect.

Other types of digitizer errors can show up on a DFT plot. Distortion reveals itself as harmonic components at multiples of the fundamental input frequency. This can be distortion in the input signal, harmonic distortion in the input amplifier, or integral nonlinearity in the ADC. As mentioned before, integral linearity error can be approximated by a second-order or third-order term in the transfer function of the ADC. These higher-order terms generate spurious harmonic components in the DFT spectrum.

Other spurious signals can show up in the DFT spectrum besides harmonic distortion. Internal clock signals can produce unwanted signal components (spurs) either by direct cross talk or through inter modulation with the input signal. These effects are commonly grouped together into a single specification of spurious DFT signals.