FIR Filters

The demodulation algorithm is very simple the DSP multiplies the received signal by each carrier, and then filters the result using a FIR filter. This kind of digital filter is phase linear, (constant group delay important for the EC combinations). Other filters may be programmed, other demodulation algorithms may be used. 

For an IIR filter, the parameter T in Eq. (9) tends to infinity. IIR filters can be represented as a function of previous filter outputs and often can be computed with fewer multiplications and reduced data storage requirements compared to a FIR filter. A popular example of an IIR filter is the exponentially weighted moving average (EWMA) or exponential smoothing, which is represented as... 

FIR and IIR filters both are able to meet a given filter specification and, in that sense, they are identical. However, FIR filters can be designed for... 

Response (FIR) filters have been implemented in the FPGA. Finally, the in-phase and quadrature signal pair is stored in a memory, also built inside the FPGA, according to the acquisition phase indicated by the pulse programmer. 

There are several other chemometric approaches to calibration transfer that will only be mentioned in passing here. An approach based on finite impulse response (FIR) filters, which does not require the analysis of standardization samples on any of the analyzers, has been shown to provide good results in several different applications.81 Furthermore, the effectiveness of three-way chemometric modeling methods for calibration transfer has been recently discussed.82 Three-way methods refer to those methods that apply to A -data that must be expressed as a third-order data array, rather than a matrix. Such data include excitation/emission fluorescence data (where the three orders are excitation wavelength, emission wavelength, and fluorescence intensity) and GC/MS data (where the three orders are retention time, mass/charge ratio, and mass spectrum intensity). It is important to note, however, that a series of spectral data that are continuously obtained on a process can be constructed as a third-order array, where the three orders are wavelength, intensity, and time. 

Figure 3.7 FIR filter cascaded with reverberator R(z)[Moorer, 1979],...

Modeling the early echoes using a sparse FIR filter results in an early response that can have an overly discrete sound quality, particularly with bright impulsive inputs. In practice it is necessary to associate some form of lowpass filtering with the early response to improve the sound quality. The simplest possible solution uses a single lowpass filter in series with the FIR filter [Gardner, 1992], where the filter response can be set empirically or by physical consideration of the absorptive losses. 

As can be seen, it requires memory (N+M+l locations) for the tapped delay lines as well as N+M+l filter coefficients. It also requires a multiplier with a result that is accumulated by the sum. It is important that the accumulator have a guard band of sufficient size to avoid overflow during accumulation. The FIR filter is similar except it lacks the feedback terns. 

Koilpillai and Vaidyanathan, 1991] Koilpillai, D. and Vaidyanathan, P. P. (1991). New Results on Cosine-Modulated FIR Filter Banks Satisfying Perfect Reconstruction. In Proc. IEEE Int. Conf. Acoust., Speech and Signal Proc, pages 1793 -1796. 

Hankel norm, 453 differentiators, 431 equation error, 453 group delay error, 454 integrators, 431 phase error, 453 Finite differences, 424, 430 Finite impulse response (FIR) filter, 87, 101-102, 128... 

When the room to be simulated doesn t exist, we can attempt to predict its impulse response based on purely physical considerations. This requires detailed knowledge of the geometry of the room, properties of all surfaces in the room, and the positions and directivities of the sources and receivers. Given this prior information, it is possible to apply the laws of acoustics regarding wave propagation and interaction with surfaces to predict how the sound will propagate in the space. This technique has been termed auralization in the literature and is an active area of research [Kleiner et al., 1993]. Typically, an auralization system first computes the impulse response of the specified room, for each source-receiver pair. These finite impulse response (FIR) filters are then used to render the room reverberation. 

Figure 3.5 Canonical direct form FIR filter with single sample delays.

Moorer proposed a slightly different structure, shown in figure 3.7, where the late reverb is driven by the output of the early echo FIR filter [Moorer, 1979], Moorer described this as a way of increasing the echo density of the late reverberation. The delays D and D2 can be adjusted so that the first pulse output from the late reverberator corresponds with the last pulse output from the FIR section. The gain g serves to balance the amount of late reverberation with respect to the early echoes. An important feature of this structure, apart from the early echo modeling, is the control... 

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. 

To discover the way in which FIR filters enhance data, consider the simple FIR filter given by the impulse-response equation... 

FIGURE 10.15 Transfer functions for FIR filters shown as a function of window size. Transfer functions for simple, moving-average filters with windows of 5 (...), 10 (-), and... 

Bialkowski, S.E., Real-time digital filters FIR filters, Anal. Chem., 60, 355A, 1988. 

Finite impulse response filter -> nonrecursive filter FIR filter -> nonrecursive filter... 

One can show that the relationship between high-pass and low-pass finite impulse response (FIR) filters and the corresponding wavelet and... 

Figure 6.7. The signal decomposition and reconstruction using FIR filters. Note that g and h are the dual filters and S is the reconstructed signal.

The signal band (formerly centered at, ) is thus shifted to dc. The following digital FIR filter with transmission zeros at, 2 fm, and 3 fm cancels out the dc offsets and 1/f noise of the analog front-end (now centered at fm) as well as some overtones of the original square wave abased into the Nyquist band of the signal sampled at 4 fm. 

In practice, the convergence parameter was chosen to be 25% of the upper bound. The above derivation assumes the use of a transversal filter and so is applicable to the adaptation of the finite impulse response (FIR) filter used in this work. 

Each of these algorithms was used to adaptively update a two-weight FIR filter and stabilize a Rijke-tube combustor through acoustic actuation. Both the gradient descent and pattern search methods proved quite effective and produced 40 to 50 dB of attenuation of the instability peak. For example, Fig. 18.6 shows the power spectral density of the uncontrolled tube and the system controlled with the Hooke and Jeeves algorithm. The steady-state results of all the... 

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. 

FIR filtering. Finite impulse response (FIR) filters are linear low-pass filters which can be represented as... 

FMHfiltering. A FIR-Median Hybrid Filter (FMH) is a median filter which has a preprocessed input from M linear FIR filters [3, 21]. Thus, the FMH filter output is the median of only M values, which are the outputs of M FIR filters applied to the original data. For an FMH filter of length 21 + 1 with three FIR substructures (M = 3), the data are split into three parts on which the FIR filters are applied. Then, the median operator is applied on the outputs of all FIR filters to obtain the output of the FMH filter. For this particular example, the three FIR filters used are ... 

Standard median and FMH filters have been widely used in non-Gaussian error elimination [21,27]. Standard median filters simply use the middle observation from data in a moving window, whereas FMH filters preprocess the data with FIR filters as discussed earlier. FMH filters are superior to the standard median filters due to their improved ability to preserve temporally localized features, while eliminating errors. However, proper selection of the FIR filters requires knowledge about the maximum duration of the outliers. When such knowledge is available, the length of the FIR filters used can be... 

Fig. 13 The effective FIR filter corre.tpomlmg to OLMS Hoar filtering at two scales (L — 2) by keeping all coefficients except the last wavelet coefficient at scale m = 2.

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