Complementary Intermittently Nonlinear Filtering for Mitigation of Hidden Outlier Interference

The outlier interference can be further obscured by the signal of interest itself, especially when the typical signal amplitude is comparable with, or larger than the typical amplitude of the interference outliers. Examples of particularly challenging waveforms that severely obscure low-amplitude outlier noise include broadband chirp signals used in radar, sonar, and spread-spectrum communications, and “bursty,” high crest factor signals such as those used in OFDM systems. This challenge can be addressed by Complementary Intermittently Nonlinear Filtering (CINF) introduced in [1, 4]. This approach capitalizes on the “excess band” observation of wideband outlier noise for its efficient in-band mitigation. This significantly extends the mitigation range, in terms of both the rates of the outlier generating events and the mitigable signal-to-noise ratios (SNRs), in comparison with the mitigation techniques focused on the apparent in-band effects of outlier interference. Here, in Figs. 4 through 7 we provide a detailed illustration of such a complementary filtering arrangement for mitigation of outlier noise affecting a broadband chirp signal.

First, Fig. 4 illustrates the impulse and frequency responses of the linear filters used in the example of Fig. 6. The bandwidth of the front-end filter should be large relative to the bandwidth of the signal of interest, so that a sufficiently wide “excess band” is available, and its time-bandwidth product should be sufficiently small so that the combined impulse response of the front-end filter cascaded with the bandstop filter contains a distinct outlier. The amount of excess bandwidth that can be allocated for outlier noise mitigation depends on the particular requirements and constraints placed on a system, and the excess bandwidth availability affects both the “mitigable rates” (e.g. in terms of the rates of outlier generating events) and “mitigable SNRs” (e.g. in terms of outlier-to-thermal noise powers) of the outlier interference. For example, for a random pulse train, when the ratio of the bandwidth and the pulse arrival rate becomes significantly smaller than the time-bandwidth product of a filter, the resulting signal becomes effectively Gaussian due to the so-called “pileup effect” [5, e.g.], making the impulsive noise completely disappear. The time-bandwidth product of a lowpass Bessel filter is approximately that of a Gaussian filter, ${2\log_{2}(2)/\pi}$, and thus $\lambda_{\rm c}\approx 2.27B_{0}$ is the “pileup threshold” rate of the front-end Bessel filter with the 3 dB bandwidth $B_{0}$. For outlier arrival rates significantly above $\lambda_{\rm c}$ the outlier noise can no longer be efficiently mitigated.

Due to high slew rates, the higher-frequency portion of the chirp signal is the most effective in obscuring low-amplitude broadband outliers. Therefore the bandstop complement of the bandpass filter should significantly reduce these high frequencies, in order for the outlier interference affecting the high-frequency portion of the chirp to become more conspicuous. On the other hand, the stopband of the bandstop filter should remain sufficiently narrow, so that the outlier in the combined impulse response of the front-end filter cascaded with the bandstop filter remains distinct. Thus, in the examples of Figs. 4 through 7, the high-frequency edge $f_{\rm c}$ of the stopband is chosen at approximately the highest frequency of the chirp signal, and the low-frequency edge is placed at approximately $f_{\rm c}/5$. While such a bandstop filter reduces the average slew rate of a linear chirp by about an order of magnitude, its stopband remains relatively narrow in comparison with the passband of the front-end filter, and it will mainly preserve the outlier structure of the noise. This is illustrated in Fig. 5, where $w(t)$ is the impulse response of the bandpass filter with a constant group delay $\Delta{t}$, and $\delta(t)$ is the Dirac $\delta$-function [6].

Fig. 6 illustrates a CINF structure employed for mitigation of wideband outlier noise affecting a linear chirp signal. Let us first note that the intermittently nonlinear filter used in this example is a feedback-based Analog Differential Clipper (ADiC) described in [1, 2] and in Section II-B, and thus we shall call this particular CINF a Complementary ADiC Filter (CAF).

In Fig. 6, the output I of the wideband front-end filter consists of the chirp signal of interest $x(t)$ and the wideband noise with non-outlier and outlier components $n(t)$ and $i(t)$, respectively. Since the stopband of the bandstop filter is relatively narrow in comparison with the passband of the front-end filter, the outlier structure of the noise will be mainly preserved in the output III of the bandstop filter. However, the bandstop filter significantly reduces the average slew rate of the chirp signal, making the outlier component $i(t)$ more distinguishable and facilitating its efficient mitigation by an ADiC. As the result, the CAF output V (that is the sum of the ADiC output IV and the output II of the bandpass filter) will consist of the effectively unmodified signal $x(t)$ and the wideband noise with a reduced outlier component, $n(t)\!+\!\delta{i}(t)$, both delayed by the group delay $\Delta{t}$ of the bandpass filter. In Fig. 6, the traces marked by “$\Delta$” show the respective differences between the filtered signal+noise mixtures and the delayed input signal $x(t\!-\!\Delta{t})$ (without noise).

Further, Fig. 7 compares the outcomes of a bandpass filter applied to a chirp signal affected by outlier noise, with and without a CAF deployed ahead of the bandpass filter. The traces marked by “$\Delta$” show the respective noises in the passband, illustrating the effectiveness of CAF-based outlier interference mitigation.

A robust time-varying range ${\left[\alpha_{-}(t),\alpha_{+}(t)\right]}$ that excludes outliers of a signal can be obtained as a range between Tukey’s fences [12] constructed as linear combinations of the 1st ($Q_{[1]}$) and the 3rd ($Q_{[3]}$) quartiles of the signal in a moving time window:

$$[\alpha_{-},\alpha_{+}]\!=\!{\big{[}}Q_{[1]}\!-\!\beta\left(Q_{[3]}\!-\!Q_{[1]}\right)\!,Q_{[3]}\!+\!\beta\left(Q_{[3]}\!-\!Q_{[1]}\right)\!{\big{]}},$$

(1)

where $\alpha_{+}$, $\alpha_{-}$, $Q_{[1]}$, and $Q_{[3]}$ are time-varying quantities, and $\beta$ is a scaling parameter of order unity (e.g. $\beta=1.5$). In practical analog and/or real-time digital implementations, approximations for the time-varying quartile values can be obtained by means of Quantile Tracking Filters (QTFs) described in detail in [13, 4, 2]. In brief, the signal $Q_{q}(t)$ that is related to a given input $y(t)$ by the equation

$$\frac{{\mathrm{d}}}{{\mathrm{d}}{t}}\,Q_{q}=\frac{A}{T}\,\left[{\mathrm{sgn}}(y\!-\!Q_{q})+2q-1\right],$$

(2)

where $A$ is a parameter with the same units as $y$ and $Q_{q}$, and $T$ is a constant with the units of time, can be used to approximate (“track”) the $q$-th quantile of $y(t)$ for the purpose of establishing a robust range ${[\alpha_{-},\alpha_{+}]}$. (See [14, 15] for discussion of quantiles of continuous signals.) From equation (1), for a signal with symmetrical amplitude distribution the range that excludes outliers can also be obtained as ${[\alpha_{-},\alpha_{+}]=[-\alpha,\alpha]}$, where $\alpha$ is given by

$$\alpha=(1+2\beta)Q^{\star}_{[2]}\,,$$

(3)

and where $Q^{\star}_{[2]}=Q^{\star}_{1/2}$ is the 2nd quartile (median, $q=1/2$) of the absolute value of the signal.

Due to their high crest factor, for OFDM signals the scaling parameter $\beta$ in (3) should be relatively large (e.g. $\beta=3$) to ensure that the outliers in the signal itself are not clipped. In addition, since the OFDM symbols of duration $T_{\rm s}$ are separated by the guard intervals of duration $T_{\rm g}$, the quantile value $q=1/2$ in (3) should be increased to ${q={\frac{1}{2}}(T_{\rm s}+2T_{\rm g})/(T_{\rm s}+T_{\rm g})}$ to account for the guard intervals. Further, for a balanced compromise between robustness to outliers and the response speed, the QTF slew rate parameter $\mu=A/T$ can be chosen to be relatively large, e.g. ${\mu=\frac{1}{10}V_{\rm c}/(T_{\rm s}+T_{\rm g})}$, and the average QTF output $\overline{Q^{\star}_{q}}(t)$ can then be obtained in a moving time window of width $(T_{\rm s}\!+\!T_{\rm g})n$, where $n$ is an integer. Fig. 9 illustrates QTF outputs convergence to steady states for different initial conditions (different black lines) and signal+noise mixtures, for ${\mu=\frac{1}{10}V_{\rm c}/(T_{\rm s}+T_{\rm g})}$ and $Q^{\star}_{q}(t)$ averaging in a moving time window of width $T_{\rm s}\!+\!T_{\rm g}$. In the upper panel, the QTF input corresponds to an OFDM signal affected by the thermal noise only, with 30 dB SNR in the OFDM passband, and the signal is not being clipped. In the middle and the lower panels, outlier interference with different compositions and intensities is added to the OFDM signal, and the excessively strong outliers are limited to the range $\pm V_{\rm c}$.

The basic concept of an ADiC as a particular type of an intermittently nonlinear filter can be briefly described as follows [1]: First, we establish a robust range that excludes noise outliers while including the signal of interest; then, we replace the values that extend outside of the range with those in mid-range. When we are not constrained by the needs for either analog or wideband, high-rate real-time digital processing, in the digital domain these steps can perhaps be accomplished by a Hampel filter [16] or by one of its variants [17]. In a Hampel filter the “mid-range” is calculated as a windowed median of the input, and the range is determined as a scaled absolute deviation about this windowed median. However, Hampel filtering cannot be performed in the analog domain, and/or it becomes prohibitively expensive in high-rate real-time digital processing. Unlike Hampel filtering, ADiCs rely on continuous-time operations that allow wideband analog and/or high-rate digital implementations in real time.

Fig. 10 presents a feedback-based ADiC variant that has a number of practical advantages and is well suited for mitigation of hidden outlier noise [1]. As the diagram in the upper left of the figure shows, the ADiC output $y(t)$ can be described as

$$\left\{\begin{array}[]{cc}\!\!y(t)=\chi(t)+\tau\dot{\chi}(t)\\[2.84526pt] \!\!\displaystyle\dot{\chi}(t)=\frac{1}{\tau}\,{{{B}}^{\boldmath\alpha_{+}}_{\boldmath\alpha_{-}}}\left(x(t)\!-\!\chi(t)\right)\end{array}\right.,$$

(4)

where $x(t)$ is the input signal, $\chi(t)$ is the differential clipping level (DCL), the blanking function ${{{B}}^{\boldmath\alpha_{+}}_{\boldmath\alpha_{-}}}(x)$ is a particular type of an influence function [16] that is defined as

$${{{B}}^{\boldmath\alpha_{+}}_{\boldmath\alpha_{-}}}(x)=\left\{\begin{array}[]{cc}\!\!x&\mbox{for}\quad\alpha_{-}\leq x\leq\alpha_{+}\\ \!\!0&\mbox{otherwise}\end{array}\right.,$$

(5)

and where $[\alpha_{-},\alpha_{+}]$ is a robust range for the difference signal ${x(t)-\chi(t)}$ (the blanking range). Thus such an ADiC is an intermittently nonlinear filter that outputs the DCL $\chi(t)$ only when outliers in the difference signal are detected, performing outlier noise mitigation without modifying the input signal otherwise. For the range fences such that ${\alpha_{-}\leq x(t)\!-\!\chi(t)\leq\alpha_{+}}$ for all $t$, the DCL $\chi(t)$ is the output of a 1st order linear lowpass filter with the 3 dB corner frequency $1/(2\pi\tau)$. However, when an outlier of the difference signal is encountered, the rate of change of ${\chi(t)}$ is zero and the DCL maintains its previous value for the duration of the outlier.

As discussed in Section I, apparent outliers in the interference can disappear and reappear due to various filtering effects, including fading and multipass, as the signal propagates through media and/or the signal processing chain. Although various combinations of linear filters can be used to increase the difference between the temporal and/or amplitude structures of the interference and the signal of interest, enhancing the outlier components of the interference and facilitating its mitigation by intermittently nonlinear filtering, such an approach may not be easily accomplished in practice. For example, it may require a sufficient a priori knowledge of the interference structure (e.g. the presence of adjacent channel interference), and/or employing various machine learning and optimization-based approaches. The “interference enhancement” may be especially difficult to accomplish in complex, highly nonstationary interference scenarios, e.g. in mobile and cognitive communication systems where the transmitter positions, powers, signal waveforms, and/or spectrum allocations vary dynamically. Thus one of the main requirements for the CINF-based interference mitigation is its ability to operate under the “no harm” constraint such that nonlinear filtering does not degrade the resulting signal quality, as compared with the linear filtering, for any signal+noise mixtures. That is, while providing resistance to outlier noise, in the absence of such noise CINFs must behave effectively linearly, avoiding the detrimental effects, such as distortions and instabilities, often associated with nonlinear filtering.

Although it is perhaps unrealistic to require that any “default” set of CINF parameters satisfies the “no harm” constraint while improving the signal quality for all conceivable interference conditions, this constraint can always be met, for any particular interference scenario, in an ADiC-based filtering. Indeed, in the limit of a wide ADiC’s blanking range such that ${\alpha_{-}\leq x(t)\!-\!\chi(t)\leq\alpha_{+}}$ for all $t$, a CAF becomes effectively an allpass filter with a constant group delay, and it will not degrade the resulting signal quality, as compared with the linear filtering, for any signal+noise mixtures. Note, however, that when a CAF does improve the signal quality, its performance can be further enhanced by optimizing its parameters.

Figs. 11 through 13 illustrate and quantify the effectiveness of the outlier interference mitigation provided by the CAF-based filtering arrangement outlined in Fig. 8. The signal of interest models a simplified OFDM-based signal and consists of QPSK-modulated symbols of duration $T_{\rm s}$ separated by guard intervals of duration ${T_{\rm g}\approx 0.46T_{\rm s}}$. The OFDM central frequency is $f_{\rm c}$, the bandwidth is $f_{\rm c}/3$, and the number of subcarriers is $256$. Various mixtures of wideband thermal and outlier noise are added to the signal, with the outlier interference such as Poisson noise with normally distributed amplitudes (Fig. 11), and periodic Gaussian bursts with different rates and duty cycles (Figs. 12 and 13). In the simulations, two base values of the thermal noise SNRs in the OFDM passband were used, 30 dB and 10 dB, and the power of the added outlier interference (measured in relation to the thermal noise power) varies in a 50 dB range, from $-20\,$dB to 30 dB.

The front-end wideband filter is a flat-group-delay (Bessel) filter with the nominal 3 dB passband $[f_{\rm c}/75,4f_{\rm c}]$, and its time-bandwidth product is approximately that of a Gaussian filter, ${2\log_{2}(2)/\pi}$. Thus $\lambda_{\rm c}\approx 9f_{\rm c}$ is the “pileup threshold” rate of the front-end filter [1]. As discussed in [1], for outlier arrival rates significantly above $\lambda_{\rm c}$ the outliers in the interference effectively disappear due to the pileup effect, and can no longer be mitigated.

As shown in Fig. 8, an analog clipper, along with a QTF-based gain control circuit, is employed ahead of the $\Delta\Sigma$ modulator to limit the magnitude of excessively strong outliers to the range $\pm V_{\rm c}$. As discussed earlier in this section, the QTF-based range control ensures that only large noise outliers are clipped, and not the outliers of the signal itself. This by itself provides resilience to high-power, low rate outlier interference, improving the resulting signal quality in the presence of such interference. Hence the comparison of the effects of the front-end clipping alone, without the CAF-based filtering, is included in the simulation results alongside the effects of the CAF-based filtering. Since, for the noise compositions used in the simulations, there is no need for the “interference enhancement” before the CAF, the wideband filter in the ADiC-based decimation filter of Fig. 8 is a simple 2nd order lowpass Bessel filter, and there are no modifications in the decimation filter.

Figs. 11, 12 and 13 show the comparative improvements in the passband SNRs and in the channel capacities, as functions of the outlier-to-thermal noise power in the OFDM passband, for different outlier noise compositions and moderate (10 dB) and high (30 dB) thermal noise SNRs. Since CAF-based filtering removes noise outliers, the passband noise after such filtering is effectively Gaussian, and the Shannon formula [18] can be used to calculate the limit on the channel capacity. However, the passband noise without the CAF in the signal chain may not be Gaussian, especially for low outlier rates and high outlier interference powers. Nevertheless, we still use the Shannon formula as a proxy measure for the capacity of the linear channel, to quantify the comparative signal quality improvement. In all simulations, the QTF, CAF, and linear filter parameters are constant, and no parameter optimization is performed.

Fig. 14 illustrates the comparative time domain appearances of the OFDM symbol and the noise traces, after the wideband front-end filter, at the approximate onset of the CAF-based mitigability of outlier interference. A further increase in the power of the outlier component will result in significantly larger relative improvement in the signal quality.