422 Analysis and Application of Analog Electronic Circuits

(quantization noise) nq

(analog integrator) (one-bit quantizer)

(equivalent gain 1)

FIGURE 10.24

A heuristic frequency domain block diagram of a first-order – modulator.

input, Vx , and the internal quantization noise, nq (see the following section). Assuming a linear system:

Thus the 1-bit quantization noise in the output is boosted at high frequencies so Vs′ jω nq/Ki. The signal component in Vs′ rolls off at −6 dB/octave above ω = Ki r/s. At low frequencies, the noise in Vs′ is negligible and Vs′ Vx. Because the – modulator is really a sampled system at frequency fc, the noise in Vs and Q has the root power density spectrum shown in Figure 10.25. Note that the broadband quantization noise is concentrated at the upper end of the root spectrum. The range from 0 ≤ f ≤ fb is relatively free of noise. Operation in this range is achieved by first having the FIR LPF operate on the data stream from Q at clock rate, fc, and then decimation, so fb = fo/2, the Nyquist frequency of the decimation frequency. Finally, the counter counts the decimated data for 2N fo clock periods.

10.6 Quantization Noise

This section describes an important source of noise in signals periodically sampled and converted to numerical form by an ADC. The numerical samples can then be digitally filtered or processed, and then returned to analog form, vy(t), by a digital-to-analog converter. Such signals include modern digital audio, video, and images. The noise created by converting from analog to digital form is called quantization noise (QN) and it is associated with the fact thatwhen a noise-free analog signal, vx(t), (with almost infinite resolution) is sampled and digitized, each vx*(nT) is described by a binary number of finite length, introducing an uncertainty between the original vx(nT) and vx*(nT). This uncertainty gives rise to the quantization noise, which

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Bandwidth of interest

FIGURE 10.25

A typical root power density spectrum of signal and quantization noise in a first-order – modulator.

can be considered to be added to vy(t) or to vx(t). (A paper by Kollár (1986) has a comprehensive review of quantization noise. Also, Chapter 14 in the text by Phillips and Nagle (1994) has a rigorous mathematical treatment of QN.)

When a Nyquist band-limited analog signal is sampled and then converted by an N bit ADC to digital form, a statistical uncertainty in the digital signal amplitude exists that can be considered to be equivalent to a broadband quantization noise added to the analog signal input before sampling. In the quantization error-generating model of Figure 10.26, a noise-free Nyquist-limited analog signal, vx(t), is sampled and digitized by an N-bit ADC of the round-off type. The ADC’s numerical output, vx*(nT), is the input to an N-bit DAC. The quantization error, e(n), is defined at sampling instants as the difference between the sampled analog input signal, vx(nT), and the analog DAC output, vy(nT). (From now on, the shorter notation, x(n), y(n), etc. will be used.) The ADC/DAC channel has unity gain. Thus the quantization error is simply:

Figure 10.27 illustrates a bipolar rounding quantizer function relating analog sampler output, x(n), to the binary DAC output, x*(n). In this example, N = 3. (This uniform rounding quantizer has 2N levels and (2N – 1) steps.) When y(n) is compared to the direct path, the error, e(n), can range over

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FIGURE 10.26

A quantization noise error-generating model for an ideal N-bit ADC driving an N-bit ideal DAC.

FIGURE 10.27

A 3-bit rounding quantizer I/O function.

±q/2 in the center of the range, where q is the voltage step size of the ADC/DAC. It is easy to see that for full dynamic range, q should be:

V

q = ( xMAX ) volts (10.50) 2N − 1

where VxMAX is the maximum (peak-to-peak) value of the input, vx(t), to the ADC/DAC system.

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FIGURE 10.28

The rectangular probability density function generally assumed for quantization noise.

For example, if a 10-bit ADC is used to convert a signal ranging from −5 to +5 V, then by Equation 10.50, q = 9.775 mV. If x(t) has zero mean and its probability density function (PDF) has a standard deviation σx > q, then it can be shown that the PDF of e(n) is well modeled by a uniform (rectangular) density, fe(e), over e = ±q/2. This rectangular PDF is shown in Figure 10.28; it has a peak height of 1/q. The mean-squared error voltage is found from the expectation:

Thus, it is possible to treat quantization error noise as a zero-mean, broad-band noise with a standard deviation of σq = q/ 12 V, added to the ADC/DAC input signal, x(t). The QN spectral bandwidth is assumed to be flat over ±fs/2, where fs is the sampling frequency, i.e., the Nyquist range.

In order to minimize the effects of quantization noise for an N-bit ADC, it is important that the analog input signal, vx(t), use nearly the full dynamic range of the ADC. In the case of a zero-mean, time-varying signal that is Nyquist band limited, gains and sensitivities should be chosen so that the peak expected x(t) does not exceed the maximum voltage limits of the ADC.

If x(t) is an SRV and has a Gaussian PDF with zero mean, the dynamic range of the ADC should be about ±3 standard deviations of the signal. Under this particular condition, it is possible to derive an expression for the mean-squared signal-to-noise ratio of the ADC and its quantization noise. Let the signal have an RMS value σx volts. From Equation 10.50, the quantization step size can be written:

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From which it can be seen that σx > q for N ≥ 3. Relation 10.52 for q can be substituted into Equation 10.51 for the variance of the quantization noise. Thus, the mean-squared output noise is:

Thus, the mean-squared signal-to-noise ratio of the N-bit rounding quantizer is:

sense stationary white-noise process, where each sample, e(n), is uniformly distributed over the quantization error. The error sequence is also assumed to be uncorrelated with the corresponding input sequence, x(n). Furthermore, the input sequence is assumed to be a sample sequence of a stationary random process, {x}. Note that e(n) is treated as white sampled noise (as opposed to sampled white noise). The auto-power density spectrum of e(n) is assumed to be flat (constant) over the Nyquist range: −π/T ≤ ω ≤ π/T r/s. (T is the sampling period.) e(n) propagates through the digital filter; in the time domain this can be written as a real discrete convolution:

m= −•

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