6. Digital Filter Properties and Filtering Implementation

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: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.

As we shall see in Chap. 8, some filters can be nearly flat in the passband.

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.

The worst case for arithmetic errors occurs when calculating the difference between two similar values.

Let us define an impulse-train signal aswhere \(\mathbb {Z}\) 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 aswhere 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.,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 \(k_{dec}\):We can now use the above impulse train sequence \(y_{kdec}[n]\) to rewrite this as a summation over all integers:This is the formula for the z-transform of the downsampler.

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).

Here we are referring to the classical categorization of brain waves (see Buzsáki 2011). This classification was introduced by International Federation of Societies for Electroencephalography and Clinical (1974). See also Steriade et al. (1990).