Pulsing circuits generate EMI that can affect sensitive circuitry and adversely contribute to the spectral signature of the equipment. “Flicker-noise” concepts, derived from chaos theory, have been employed to efficiently pulse circuitry while generating a virtually undetectable spectral signature. Pure flicker pulsing requires that the components be driven with a set of uncorrelated pulses, with random heights, started at random times. Jensen, et. al. *(Phys. Rev. B40, 7425, 1989*) demonstrated that the power spectral density for such a signal generates (1/f)^{n}-type noise given by:

where v is the pulse rate and G(ô) is the weighted distribution of lifetimes, defined as:

where P(S,W) is the joint probability for a pulse to have area S and width W. In most circuitry, S, W, and the pulse spacing, T, are normally fixed, with a duty cycle given by W/(W+T).

The power spectral density (PSD) for a periodic signal with W = 122 ms, S = nW = 122 µV-s, and T = 1.83 µs is given in Figure 1. When S, W, and T are allowed to vary, the results are dramatically different. Figure 2 shows the PSD for a signal with W, S, and T given by uniform probability density functions with mean values: —W= 122 ms, —S = n —W= 122 µV-s, and —T= 1.83 ms. The PSD was obtained using Welch’s method of estimating the power spectrum from the time series data signal, using a Hanning window of length 256 and with a sampling frequency of FS = 820 kHz.

The analytic solution can be determined from Equation 1, assuming fixed pulse amplitudes and a probability density function P(W) of the form:

Assuming the pulses are all fixed amplitude, S = nW, requires

therefore

and

This results in a power spectral density given by:

For W_{MIN} = 0, S(f) reduces to:

S(f) corresponding to this pulse distribution, with n = 512 pulses/s and W_{MAX}=244 ms, is given in Figure 3. The analytic solution agrees with the PSD obtained from the time series data and also shows the origin of the (1/f)^{2} dependence.

In our application, we were required to improve the efficiency of LED drivers, without affecting the spectral signature of the system. Pulsing an LED is an effective means of attaining maximum light intensity with minimum power drain. A rule of thumb for most LEDs is that driving the LED in the I_{RMS} = 1- to 10-mA range is the most optically efficient mode of operation. Pulsing LEDs in this range can more than double the light intensity of continuous operation at the same effective current.

We were constrained by a low clock speed, so we fixed W_{f} to the minimum possible pulse width (122 ms) to achieve the lowest possible duty cycle. We also determined that the maximum possible pulse spacing, T, without introducing noticeable flicker was 1.852 ms. This resulted in a duty cycle of 6 %. Furthermore, we were forced to separate T into a fixed component, t_{f}, and a random component, t_{r} = T − t_{f}, to avoid excessive flicker. The random component, t_{r}, was given by a uniform probability density function with a mean value of —t_{r}.

The largest ratio —t_{r}/T that provided a uniform light intensity given these constraints was 46%. The PSD for a signal given these constraints is provided in Figure 1, and is shown on an expanded scale in Figure 4. Although both signals in Figure 1 have the same total energy, the PSD of the flicker-pulsed signal is reduced to background noise. The analytic solution, S(f), for a signal with W fixed at W_{f} and S fixed at S_{f}, with n = 1 and n = 512 pulses/s is shown in Figure 5. In this case, the joint probability density function becomes:

so that Equations 2 and 1 reduce to:

and

The degree to which T must vary to obtain the desired power at the fundamental frequency, f0, is shown in Figure 6. It gives the power at f0 relative to the power at f_{0} for fixed T, as a function of —t_{r}/T. It shows that an 80% reduction in the PSD at f_{0} can be obtained with a —t_{r}/T ratio of only 20%. A —t_{r}/T ratio of 46% reduces the fundamental peak more than 90%.

Flicker pulsing can significantly diminish the spectral signature of the circuitry. Although ideally the pulse duration, magnitude, and spacing should be random, a significant reduction in conspicuous PSD components can be achieved when imposing practical constraints. We were able to significantly reduce the dominant components of the power spectrum using fixed pulse duration and magnitude. We employed flicker pulsing with a PSD approaching (1/f)^{2} to drive our components more efficiently, resulting in a 40% increase in battery life. The contribution of the pulses to the spectral signature of the equipment appears only in the background noise of EMI detectors.