Abstract
The first part of this chapter examines the properties of frequency-selective filters that are LTI stable systems with a real and causal impulse response and a rational transfer function. Their frequency response is classified according to four prototypes: the lowpass, highpass, bandpass, and bandstop ideal filters. Ideal filters have real frequency response with jump discontinuities at band edges and are not computationally realizable: they must be approximated by continuous complex functions. The conditions for realizability are discussed in terms of magnitude and phase of the frequency response. The phase of a realizable filter presents jump discontinuities that are eliminated passing to a continuous-phase representation. Linear phase (LP) and generalized linear phase (GLP) filters are then studied, which do not cause phase distortion between input and output waveforms. Only FIR filters of four types can have exactly LP/GLP, and their impulse response must satisfy precise symmetry conditions. The second part of the chapter deals with implementing digital filtering, by arranging the difference equation into the most convenient structure. Finally, applications using downsampling before filtering are described.
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Notes
- 1.
This theorem was enunciated by Raymond Paley (1907–1933) and Norbert Wiener (1894–1964) and yielded a method for determining whether or not a causal impulse response exists for a given magnitude frequency response (Paley and Wiener 1933, 1934).
The Paley-Wiener Theorem (see, e.g., Paarmann 2001) states that given a magnitude frequency response that is square-integrable, then a necessary and sufficient condition for it to be the magnitude frequency response of a causal filter is that the following inequality be satisfied:
$$\begin{aligned} \int _{-\infty }^{+\infty }\frac{\left| \log \left| H(\mathrm {e}^{\mathrm {j}\omega })\right| \right| }{1+\omega ^2} \mathrm {d}\omega \le \infty . \end{aligned}$$As a consequence of the Paley-Wiener Theorem, it can be shown that causal filters, having impulse response constrained to be identically zero over the whole negative half n-axis, cannot have continuous bands where the frequency response has zero amplitude.
- 2.
As we shall see in Chap. 8, some filters can be nearly flat in the passband.
- 3.
Modulated signals are important in telecommunication engineering. Suppose we want to transmit a narrow-band lowpass signal through a network. This network only lets signals with frequency close to a certain \(\omega _0\) pass through (with \(\omega _0\) not close to zero) and filters out the rest. If we properly modulate the lowpass signal, we can shift its spectral content to the vicinity of \(\omega _0\) and allow signal transmission. Later we will be able to perform the inverse operation and recover the original signal.
- 4.
The worst case for arithmetic errors occurs when calculating the difference between two similar values.
- 5.
Let us define an impulse-train signal as
$$\begin{aligned} y_{kdec}[n]= \sum _{i=-\infty }^{+\infty }\delta [n-k_{dec} i] = {\left\{ \begin{array}{ll} 1 &{} \text{ for }\quad n\in k_{dec}\mathbb {Z}, \\ 0 &{} \text{ otherwise }, \end{array}\right. } \end{aligned}$$where \(\mathbb {Z}\) indicates the set of integer numbers. This sequence is 1 at one out of every \(k_{dec}\) samples, and zero everywhere else. Equivalently, the impulse train can be written as
$$\begin{aligned} y_{kdec}[n] = \frac{1}{k_{dec}} \sum _{i=0}^{k_{dec}-1} \mathrm {e}^{\mathrm {j}2 \pi i {n/k_{dec}}} = {\left\{ \begin{array}{ll} \frac{1}{k_{dec}} \sum _{i=0}^{k_{dec}-1} 1 = 1 &{} \text{ for }\quad n \in k_{dec} \mathbb {Z}, \\ \\ \frac{1}{k_{dec}} \frac{1-\mathrm {e}^{\mathrm {j}2 \pi i n}}{1-\mathrm {e}^{\mathrm {j}2 \pi i n/k_{dec}}} = 0 &{} \text{ otherwise }, \end{array}\right. } \end{aligned}$$where for the second case we used the expression for the finite sum of a geometric series and the fact that \(\mathrm {e}^{\mathrm {j}2 \pi i n}=1\). Now, let us calculate the z-transform of \(x_d[n]=x_f[k_{dec} n]\), i.e.,
$$\begin{aligned} X_d(z)=\sum _{n=-\infty }^{+\infty }x_f[k_{dec} n] z^{-n}. \end{aligned}$$We apply the substitution \(m=n k_{dec}\), keeping in mind that this makes the summation run only over indexes m that are integer multiples of \(k_{dec}\):
$$\begin{aligned} X_d(z)=\sum _{m \in k_{dec} \mathbb {Z}}x_f[m] z^{-m/k_{dec}}. \end{aligned}$$We can now use the above impulse train sequence \(y_{kdec}[n]\) to rewrite this as a summation over all integers:
$$\begin{aligned} X_d(z)= & {} \sum _{n=-\infty }^{+\infty }y_{kdec}[n] x_f[n] z^{-n/k_{dec}} = \sum _{n=-\infty }^{+\infty }\left( \frac{1}{k_{dec}}\sum _{i=0}^{k_{dec}-1}\mathrm {e}^{\mathrm {j}2 \pi i n/k_{dec}}\right) x_f[n] z^{-n/k_{dec}} =\\= & {} \frac{1}{k_{dec}}\sum _{i=0}^{k_{dec}-1} \sum _{n=-\infty }^{+\infty } \mathrm {e}^{\mathrm {j}2 \pi i n/k_{dec}}x_f[n] z^{-n/k_{dec}} = \\= & {} \frac{1}{k_{dec}}\sum _{i=0}^{k_{dec}-1} \sum _{n=-\infty }^{+\infty } x_f[n] \left( \mathrm {e}^{-\mathrm {j}2 \pi i/k_{dec}} z^{1/k_{dec}}\right) ^{-n} = \frac{1}{k_{dec}}\sum _{i=0}^{k_{dec}-1}X_f\left( \mathrm {e}^{-\mathrm {j}2 \pi i/k_{dec}} z^{1/k_{dec}}\right) . \end{aligned}$$This is the formula for the z-transform of the downsampler.
- 6.
Magnetoencephalography is technique aimed at detecting magnetic fields produced by the brain.
At the cellular level, individual neurons in the brain have electrochemical properties that result in the flow of electrically charged ions. Electromagnetic fields are generated by the net effect of this slow ionic current flow. While the magnitude of fields associated with an individual neuron is negligible, the effect of multiple neurons (for example, \(5\times 10^4-1 \times 10^5\) neurons) excited together in a specific area generates a measurable magnetic field outside the head. These neuromagnetic signals generated by the brain are extremely weak. Therefore, MEG scanners require superconducting sensors (SQUID, superconducting quantum interference device). The SQUID sensors are bathed in a large liquid helium cooling unit at approximately \(-269\,^{\circ }\)C. Due to low impedance at this temperature, the SQUID device can detect and amplify magnetic fields generated by neurons a few centimeters away from the sensors located on the patient’s head. MEG measurements span field magnitudes from about 10 femtoTesla (fT) for spinal cord signals to about several picoTesla (pT) for brain rhythms. Recall that the symbols micro (\(\mu \)), nano, (n), pico (p) and femto (f) in front of a measurement unit mean \(10^{-6}\), \(10^{-9}\), \(10^{-12}\), and \(10^{-15}\), respectively. To appreciate how small the MEG signals are, it should be recalled that Earth’s magnetic-field magnitude is about 0.5 \(\upmu \)T and the urban magnetic noise about 1 nT–1 \(\upmu \)T, or about a factor of 1 million to 1 billion larger than MEG signals. For this reason, the subject being studied and the MEG instrument must be in a magnetically shielded room, to mitigate external interference. MEG measurements are performed using magnetometers and/or gradiometers. The gradiometers values are expressed in T/m (Tesla/meter), while magnetometers measure the magnetic field in Tesla. As a rule of thumb, gradiometer measurements can be roughly converted into magnetometers units by multiplying them by the size of the sensor (typically, about 4 cm).
- 7.
References
Buzsáki, G.: Rhythms of the Brain. Oxford University Press, Oxford (2011)
Chen, W.K.: On flow graph solutions of linear algebraic equations. Society for industrial and applied mathematics (SIAM). J. Appl. Math. 15(1), 136–142 (1967)
Chow, Y., Cassignol, E.: Linear Signal Flow Graphs and Applications. Wiley, New York (1962)
Electroencephalogr. Clin. Neurophysiol. Neurophysiology: report of the committee on cessation of cerebral function. 37(5), 521 (1974)
Gustafsson, F.: Determining the Initial States in Forward-Backward Filtering. IEEE Trans. Sig. Process. 44(4), 988–992 (1996)
Jones, D.: Digital Filter Structures and Quantization Error Analysis. Available via OpenStax-CNX Web site. http://cnx.org/content/col10259/1.1/ (2005)
Mason, S.J., Zimmermann, H.J.: Electronic Circuits, Signals, and Systems. Wiley, New York (1960)
Markel, J.D., Gray, A.H. Jr: Linear Prediction of Speech. Springer, New York (1976)
Mitra, S.K.: Digital SignalProcessing, 2nd edn. McGraw-Hill, New York (2001)
Oppenheim, A.V., Schafer, R.W.: Discrete-Time Signal Processing. Prentice Hall, Englewood Cliffs (2009)
Oppenheim, A.V., Schafer, R.W., Buck, J.R.: Discrete-Time Signal Processing, 2nd edn. Prentice Hall, Upper Saddle River (1999)
Paarmann, L.D.: Design and Analysis of Analog Filters. A signal processing perspective. Kluwer, Dordrecht (2001)
Paley, R.E.A.C., Wiener, N.: Notes on the theory and application of fourier transforms. I-II. Trans. Am. Math. Soc. 35, 348–355 (1933)
Paley, R.E.A.C., Wiener, N.: Fourier Transforms in the Complex Domain, vol. 19. American Mathematical Society (SIAM) Colloquium Publications, New York (1934)
Proakis, J.G., Manolakis, D.G.: Digital Signal Processing: Principles, Algorithms and Applications. Prentice-Hall, Englewood Cliffs (2007)
Schlichthärle, D.: Digital Filters: Basics and Design. Springer, New York (2011)
So, H.C.: Digital Signal Processing: Foundations, Transforms and Filters, with Hands-on MATLAB Illustrations. McGraw-Hill, New York (2010)
Steriade, M., Gloor, P., Llinás, R., da Silva, F.H.L.: Basic mechanisms of cerebral rhythmic activity. Electroencephalogr. Clin. Neurophysiol. 76(6), 481–508 (1990)
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Alessio, S.M. (2016). Digital Filter Properties and Filtering Implementation. In: Digital Signal Processing and Spectral Analysis for Scientists. Signals and Communication Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-25468-5_6
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DOI: https://doi.org/10.1007/978-3-319-25468-5_6
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