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 . 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 2 x[ — ] =X z)z and so our equation beeomes  

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

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 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... 

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.

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. 

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))... 

Filter transfer functions and spectral densities of motion... 

Fig. 7.2.3 Pulse sequences for T2 filters and schematic filter transfer functions, (a) Hahn-echo filter, (b) CPMG filter.

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

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

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) =... 

The filtering of the data can be carried out both in spatial and frequency domains. The spatial filters are used as masks. The enhancement in the frequency domain involved the Fourier Transform of the image and its multiplication by a filter transfer function. The inverse transform of the result can be taken to produce enhanced images in the frequency domain. The most common spatial filters include lowpass, highpass, and median filters. 

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. 

But 1/A = Ts, and so Eq. (7.178) yields s instead of zero. The conclusion is that no active hlter section employing only a single op-amp (modeled by the one-pole roUofif model) can be made insensitive to the time constant of the op-amp. However, it is possible to devise a composite op-amp consisting of two individual op-amps that allows us to synthesize a filter transfer function T (s) that is first-order insensitive to the op-amp time constants. A circuit... 

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

The adjustment procedure is the following First the seismometer transfer function is determined by measuring the seismometer response with some initial parameters of the correcting circuit. After that, the transducer transfer function is calculated by dividing the seismometer transfer function by a product of the electronic filter transfer functions ... 

In case the electronics in the feedback has frequency-independent response, the transfer function of the instrument would be proportional to acceleration. The output filter transfer function //out(ffl) is used to shape the low- and high-cutoff frequencies. 

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