Filter coefficients

Rows 1-8 are the approximation filter coefficients and rows 9-16 represent the detail filter coefficients. At each next row the two coefficients are moved two positions (shift b equal to 2). This procedure is schematically shown in Fig. 40.43 for a signal consisting of 8 data points. Once W has been defined, the a wavelet transform coefficients are found by solving eq. (40.16), which gives ... 

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. 

It is easy to assume that the filter coefficients in equation 5.15 (a / and b j]) are constant. In fact, there are many instances when this is not the case. For example, in real-time audio processing a user moves a slide potentiometer (either physical or possibly on the display) this is digitized and the host processor must change the coefficients in various filters. 

The filter coefficients must be updated at a regular interval. If the number of filters coefficients is large (for example, a high order LPC filter) and/or a large number of filters must operate in parallel, then this may interfere with the computation. Moorer called it the parameter update problem [Moorer, 1981],... 

The memory organization of DSPs are also different from ordinary processors because (1) Memory is typical static RAM and virtual memory support is totally absent (2) Several machines separate data and instruction streams (Harvard Architecture) (at the cost of extra pins). Additionally, modular arithmetic address modes have been added to most processors. This mode finds particular utility in filter coefficient pointers, ring buffer pointers and, with bit reversed addressing, FFTs. One further difference is the use of loop buffers for filtering. Although often called instruction caches by the chip manufacturers, they are typically very small (for example, the AT T DSP-16 has 16 instructions) and furthermore, the buffer is not directly interposed between memory and the processor. 

Time Domain Analysis. Perhaps the simplest and most traditional use of a DSP is filtering. DSPs are designed to implement both Finite Impulse Response (FIR) and Infinite Impulse Response (HR) filters as fast as possible by implementing (a) a single cycle multiply accumulate instruction (b) circular addressing for filter coefficients. These two requirements can be found in all modem DSP architectures. 

The set of coefficients which are chosen is determined at run time, rather than being fixed as in traditional polyphase filter design. Also, Smith and Gosset describe a method for interpolating between coefficients to reduce the size of the table holding the filter coefficients. Performing this interpolation between coefficients adds additional incentive to keep the coefficients in their original non-reordered form to simplify the coefficient interpolation calculation. 

If a large filter coefficient table is practical, then the interpolation of coefficients that Smith and Gosset describe is not needed, and an approach closer to Crochiere and Rabiner becomes more preferable. Basically, that approach is to create a prototype filter table for a large value of L and stored the coefficients in a re-ordered form as... 

Figure 8.9 A sampling playback oscillator using high order interpolation. Every output sample is a vector dot product of fFinput samples and one of the filter coefficient vectors, stored re-ordered from the original prototype filter. The fractional phase address selects the filter coefficient vector used.

ARMA filters to modify waveforms. The filters allow interpolation of stored filter coefficient sets under user control to modify the spectrum of the recorded signals. 

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. 

Digital filter coefficient Temperature coefficient of resistance Resistance thermometer parameter Ratio of specific heats... 

The degree of smoothing is determined by the filter coefficient a, with a = 1 being no smoothing and a = 0 being infinite smoothing (no effect of new measurements). The filter coefficient a is related to the filter time constant xF and the sampling interval At by... 

Several studies have been conducted on particle capture by collectors of different shapes. Among the first studies were those of Albrecht (6), and Kaufman (7), who investigated the capture of airborne particles flowing past simple collector geometries. In packed columns, particle capture can be quantified in terms of the filter coefficient, X, defined by ... 

The relationship between the filter coefficient and the single collector efficiency is obtained by combining the preceeding two equations ... 

With the system and the latexes used, is always smaller than 0.09, so equation 9 can be applied to the HDC system, and hence theoretical values of the filter coefficient can be calculated by using equation 3. Evaluation of the sedimentation term was found to be negligible in our system. 

Figure 9 Theoretical and Experimental Values of the Filter Coefficient for a surfactant concentration of 10 mM SLS and a superficial velocity of 2 cm/min.

The effectiveness of deep-bed filters in removing suspended particles is measured by die value of die filter coefficient which in turn is related to the capture efficiency of a single characteristic grain of the bed. Capture efficiencies are evaluated in the present paper for nil cases of practical importance in which London forces and convective-diffusion serve to transport particles to the surface of a spherical collector immersed in a creeping How field. Gravitational forces are considered in some cases, but the general results apply mainly to submicron or neutrally buoyant particles suspended in a viscous fluid such as water. Results obtained by linearly superimposing the in-... 

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