Causal filtering

We note in passing that, when there is a finite number of filter coefficients, the filter is called a finite impulse response (FIR) filter. Another commonly used filter is a causal filter. Here the filter coefficients with negative indices are zero, that is, U = 0 for i < 0, we say that the filter is causal (h. could also have been used in the definition of causal). The remaining discussion will consider filters that are both FIR and causal. The notation N/ will be used to denote the number of finite filter coefficients. 

Causality requires that the filter response at time n be computed on the basis of present and past information and not require knowledge of either the future input or output. Thus, the computation of zm (n) involves only present and past (values of the) inputs and only past outputs. 

Once the causal factors have been identified, the factors are analyzed using a root cause analysis tool, such as 5-AVhys or predefined trees. See Chapter 9 for a more detailed discussion of Barrier Analysis (sometimes called hazard-barrier-target analysis or HBTA) and Change Analysis (also referred to as Change Evaluation/Analysis or CE/A). In essence, these tools act as a filter to limit the number of factors, which are subjected to further analysis to determine root causes. 

The identification of causal factors points us to the key areas that need to he examined further for why that factor existed. It acts as a filter to limit the number of areas that are subjected to further analysis to determine root causes. This critical activity must be performed diligently and systematically to identify every causal factor applicable to the specific incident. If a causal factor is missed, one or more root causes will likely be omitted as well, which could lead to similar incidents in the future. 

In an efficient digital simulation, lumped loss factors of the form Gk (0)) are approximated by a rational frequency response Gk(c,mT). In general, the coefficients of the optimal rational loss filter are obtained by minimizing I Ylk (go) - Gk d r ) I with respect to the filter coefficients or the poles and zeros of the filter. To avoid introducing frequency-dependent delay, the loss filter should be a zero-phase, finite-impluse-response (FIR) filter [Rabiner and Gold, 1975], Restriction to zero phase requires the impulse response g.k(n) to be finite in length (i.e., an FIR filter) and it must be symmetric about time zero, i.e., ) k(-n) = gk(n). In most implementations, the zero-phase FIR filter can be converted into a causal, linear phase filter by reducing an adjacent delay line by half of the impulse-response duration. 

And model-based methods which are composed of quantitative model-based methods (such as analytical redundancy (Chow and Willsky, 1984), parity space (Gertler and Singer, 1990), state estimation (Willsky, 1976), or fault detection filter (Franck, 1990)) and qualitative model-based methods (such as causal methods digraphs (Shih and Lee, 1995), or fault tree (Venkatasubramanian, et ah, 2003)). 

The constructive model P(A(M) SC C )xN) acts as before, as a filter on the prior P(S( -C)x I A) output from the cosmological theory. In this respect, the anthropic bias is indistinguishable from the pruning rule generated by Hoyle but as it is driven by incomplete theoretical specification rather than mere intractability, the posterior distribution is not provisional and cannot act as a constraint on reductionist refinements of N or X It is not a substitute for a theory of particular initial conditions, because it adds nothing axiomatic to the structure of the causal theories represented by... 

Decompose the measured data within a window of dyadic length using a causal boundary corrected wavelet filter. 

Sensitisation may be produced by non-occupational environmental exposure. For example, three subjects were shown to be TDI sensitive by the simulated occupational test, where the causal exposure was from the exhaust fumes from an adjoining factory which had been sucked into the ventilating system of their warehouse. TDI was found in the air filters (Carroll et al. 1976). Sensitisation at home by two-can do it yourself polyurethane-TDI foam aerosol cans (Peters and Murphy 1971) and other forms of domestic exposure have been reported. Patients of this sort would, in the absence of histories of sensitivity or reactions to the common allergens, and the adult (late) onset of their asthma, be classified as having cryptogenic (intrinsic) asthma. The extent to which sensitivity of this sort to chemical environmental pollutants may be responsible for some cases of cryptogenic asthma will obviously need to be determined. 

The output of the simple moving average filter is the average of the M -F 1 most recent values of x(n). Intuitively, this corresponds to a smoothed version of the input, but its operation is more appropriately described by calculating the frequency response of the filter. First, however, the z-domain representation of the filter is introduced in analogy to the s- (or Laplace-) domain representation of analog filters. The z transform of a causal discrete-time signal x(n) is defined by... 

Compared to an FIR filter, an HR filter requires a much lower order than an FIR filter to achieve the same requirement of the magnitude response. Flowever, whereas an FIR filter is always stable, an HR filter can be unstable if the coefficients are not properly chosen. Assuming that the system (8.27) is causal, then it is stable if all of the poles lie inside the unit circle on the z plane. Since the phase of a stable causal HR filter cannot be made linear, FIR filters are chosen over HR filters in applications where linear phase is essential. 

If a system is not causal, then it is noncausal. An ideal filter which will filter in real time aU frequencies present in a signal/(t) requires knowledge of /(t) x > t and is an example of a noncausal system. 

A filter is causal if its output at a certain time moment depends only on the input data at that moment or earlier. If the output also relies on future input data, then the filtering is acausal. 

Boore DM, Akkar S (2003) Effect of causal and acausal filters on elastic and inelastic response spectra. Earthq Eng Struct Dyn 32(11) 1729-1748. doi 10.1002/ eqe.299... 

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