The filter transfer function

We will now examine the frequency-domain description or frequency response of the LTI filter by means of the z-transform. The HR difference Equation (10.55), expanded to a few terms, is 

By using the notation 2. to denote the z-transform, we can take the z-transform of both sides as follows  

The coefficients are not affected by the z-transform, so, using the linearity of the z-transform, this simplifies to 

The z-transform time-delay property (Equations (10.35) and (10.46)) states that 

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A linear filter performs a convolution of the input function with the Fourier transform of the filter transfer function. According to the convolution theorem (cf. Section 4.2.3) application of a filter in one domain corresponds to multiplication of the Fourier transform of the function to be filtered with the filter-transfer function. To filter a backprojection image, eqn (6.1.3) is Fourier transformed,... 

The effect a filter exerts on a signal is represented by the filter transfer function. For linear systems, the filter output is the product of the transfer function and the filter input (cf. eqn (4.2.14), where Xi(transfer function). The output signal is high for those parameters for which the transfer function is high and vice versa. For a qualitative... 

A magnetization filter consists of modulated rf excitation, for example, from a sequence of nonselective rf pulses with given flip angles and given pulse separations. These parameters of the pulse sequence are adjustable, and they determine the characteristics of the filter, that is, they can be used to tune the filter transfer function. Therefore, they are referred to as extrinsic contrast parameters. They must be discriminated from the intrinsic contrast parameters, which are the NMR parameters specific of the sample under investigation and are related to the material properties [Manl]. Therefore, the parameter vectorp in the weight factor of eqn (7.1.2) is separated into two parts, a vector Pe of extrinsic parameters and a vectorPj(r) of intrinsic parameters. 

The echo maxima are weighted by a function of both T and T2. Similarly, the stimulated echo (Fig, 7.2.1(c)) can be used as a combination filter to introduce T and T2 weights. The echo time (tf2 in Fig. 7.2.19(c)) determines the T2 weight and the mixing time between the second and the third pulses (tn in Fig. 7.2.19(c)) the T weight. Note that the filter transfer functions for T) contrast by saturation recovery and the stimulated echo are inverted (cf. Fig. 7.2.1 (a) and (c)), so that both combination filters introduce different contrasts (cf. eqn (7.2.3)). 

The filter transfer function Hiw) is a ramp in frequency space with a high-frequency cutoff W = 1/2t Hz (Fig. 26.20). In the physical space Hiw) has the impulse response h r) =... 

An important class of magnetization filters are mobility filters which select magnetization based on the time scale of segmental motions ((19), and references therein). The parameters for discrimination are the amplitude and characteristic frequency or the correlation time tc of molecular motions. The effect a filter exerts on a NMR signal can be represented by the filter transfer function. Examples are given in Figure 30 (163,164) with transfer function for filters, which select magnetization based on the time scale of molecular motion. 

Hi p) is the filter transfer function of the vibration sensor corresponding to a bandpass filter between 30 and 800 Hz ... 

Figure 2b. Profiles of the modulation transfer function (MTF), its inverse and Wiener inverse-filter.

Vq is called the cut-off frequency. H v) is referred in this context as a filter transfer function. 

In this case we assume the disturbance term, e , is not white noise, rather it is related to through the following transfer function (noise filter)... 

So if we cannot attain perfect control, what do we do From the IMC perspective we simply break up the controller transfer function C( ) into two parts. The first part is the inverse of. The second part, which Morari calls a filter, is chosen to make the total physically leahzable. As we will show below, this second part turns out to be the closedloop servo transfer function that we defined as S(,j in Eq. (11.64). 

Here the frequency spectra of the output and input signals are given by Fc(co) and V (co), respectively, and the complex filter transfer function is given by... 

A chief advantage of all the filters described in Section IV is convenience. We have written the Fourier transfer function of each one. Certainly it is possible to perform the filtering in the Fourier space. It is also possible, however, and often even more convenient, to convolve the data with the filter function itself. This is especially true if the filter can be adequately approximated by a convolution kernel that vanishes except over a relatively small domain. 

Fig. 7.1.3 [Blii2] NMR-timescale of molecular motion and filter transfer functions of pulse sequences which can be utilized for selecting magnetization according to the timescale of molecular motion. The concept of transfer functions provides an approximative description of the filters. A more detailed description needs to take into account magnetic-field dependences and spectral densities of motion. The transfer functions shown for the saturation recovery and the stimulated-echo filter apply in the fast motion regime.

Fig. 7.2.1 Pulse sequences for T and related magnetization filters, typical evolution curves of filtered magnetization components, and schematic filter transfer functions applicable in the slow motion regime. Note that the axes of correlation times start at Tc = Wo (a) Saturation recovery filter, (b) Inversion recovery filter, (c) Stimulated echo filter.

The inverse filter transfer function is obtained for the stimulated-echo filter (Fig. 7.2.1(c), cf. Fig. 2.2.10(c)). It consists of three 90° pulses. The second pulse generates longitudinal magnetization, which is modulated in amplitude by the precession phases accumulated during the evolution time t /2 between the first two pulses. The filter time ff is the time between the second and the third pulse. Here the modulated components relax towards thermodynamic equilibrium with the longitudinal relaxation times Ti(r), and the memory of the initial two pulses is lost as tf increases. Therefore, the amplitude of the stimulated echo is given by (cf. eqn (2.2.39))... 

Fig. 7.2.4 [Rom2] Pulse sequence for the Tip filter and schematic filter transfer function.

The spectroscopist should also pay attention to the question whether the sampling interval Js, the speed v of the movable mirror and the time constant T of the electronic data recording system are in a proper relation with each other. It is well known (see also Section 5.4) that a low pass filter with time constant t has the complex transfer function (amplitude and phase )... 

The terms in parentheses are the noise transfer functions from the respective source to the output Vo of the circuit and B is the bandwidth of the filter. Since the noise is proportional to the bandwidth, B should be chosen as small as permitted by the application. 

In a buck, there is a post-LC filter present. Therefore this filter stage can easily be treated as a cascaded stage following the switch. The overall transfer function is then very easy to compute as per the rules mentioned in the previous section. However, when we come to the boost and buck-boost, we don t have a post-LC filter — there is a switch/diode connected between the two reactive components that alters the dynamics. However, it can be shown, that even the boost and buck-boost can be manipulated into a canonical model in which an effective post-LC filter appears at the output — thus making them as easy to treat as a buck. The only difference is that the original inductance L (of the boost and buck-boost) gets replaced by an equivalent (or effective) inductance equal to L/(l—D)2. The C remains the same in the canonical model. 

The first-order (RC) low-pass filter transfer function can be written in different ways as... 

The combined transfer function r(tn) for ETDAS and low-pass filtering is displayed in Fig. 5.6(b). As in Fig. 5.6(a), there are notches at multiples of the frequency of the UPO, which become narrower for increasing R. The amplitudes of frequencies larger than the cut-off frequency a are reduced and thus are only minor contributions to the feedback response. This is important to notice in order to understand how the low-pass filter improves the controllability of the system. 

Equation (29.5) yields the discrete transfer function of a first-order digital filter. The noise-free signal (the output of the filter) is given by... 

What is a discrete transfer function, and what is it needed for Develop the discrete transfer function for (a) a proportional control algorithm, (b) the velocity form of a PI control algorithm, and (c) a second-order digital filter. 

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