a. Filter usually removes information - operates in a domain. Filters can be used to processing both signals and images. It is a 1-D filter when operating on 1-D signals, and a 2-D filter when operating on 2-D images.

b. Time domain- an example is a filter than averages every two digitized points of a signal:

Average = (s₁+s₂) / 2

This '2-point' average, for example, could be used to replace both points in the signal. The averager then acts as a smoothing function.

c. Spatial domain - say we have information being recorded from many points on the heart surface. The electrical signals are called electrograms. We can take any arbitrary axis along the heart surface and use the values of each recorded electrogram along that axis, for one instance in time, as a spatial vector. It can be filtered in the same way as the time series vector.

d. Frequency domain - this usually refers to the Fourier frequency domain, i.e., that domain whose basis is a series of sinusoids that are integer multiples of some fundamental frequency. Generally we speak of filtering in the frequency domain such that certain frequencies are removed. One can remove high frequencies, low frequencies, middle frequencies (called notch), and highest and lowest simultaneously (called bandpass).

e. Some filters might add information. This implies that information is available a priori that governs how information is added to a signal. For example, say we have a template which is a series of digitized points that represents one cardiac cycle of a blood pressure curve. Say we have an input which is a new blood pressure signal. Sometimes there is a loss of transmission of the new signal. If we compare the template to the new signal, we would fill in the gaps by making those digitized points the same as those at the corresponding position on the template.

a. A 'basis'. The basis vectors are sinusoids. A sine wave is completely described by the equation:

Where A is the amplitude, phi is the phase shift, omega determines the time duration, and B is the average level or bias. (Perhaps these parameters sound familiar from our discussions of the normalization of signals).

b. Example of another basis - Karhunen-Loeve (basis vectors are the eigenvectors).

a. Continuous - we can use integration methods to determine the continuous spectrum of frequencies of which a continuous signal is composed.

b. Discrete (DFT) - when the signal is discrete (digitized), so too will be the frequency transform. This is what we mean when we say that the sinusoidal basis is composed of integer multiples of some fundamental frequency.

c. Fast (FFT) - the result is the same as the DFT but the mathematics to get there are simplified. Basically, the FFT is geared toward processing by digital computer in the sense that the number of mathematical operations 'multiply' needed to compute the transform is vastly reduced compared to DFT. Whereas, the number of addition and subtraction operations has increased. Since 'multiply' operations require thrice or more computing time compared with 'adds' and 'subtracts' the overall computation time is markedly reduced.

4. Passive Filters - these are elements that are used in an electric circuit.

a. resistors - an element that acts to resist the flow of electric current. In the process, heat is produced. If the heat produced by current passing through the resistor is intense, it can generate visible light (as in an incandescent light bulb). Resistance R is proportional to the length of the resistor and inversely proportional to the cross-sectional area. When current flows through a resistor the result is a voltage drop (potential difference). This is governed by Ohm's Law:

V = I * R

Where V is the voltage and I is the current. For example, if the current is 1 amp and the resistance is 1.5 ohm, then the voltage drop across the resistor is 1.5 volts. If the initial voltage level was 4V, then after passing through the resistor the voltage level becomes 2.5V.

b. capacitors - capacitors impart impedance to an electric circuit. The impedance of a capacitor X_c is given by:

For any given signal that encounters a capacitance as it flows through a circuit, the low frequency components will be attenuated.

c. inductors - the impedance is given by:

d. high pass filter - removes low frequency components of the signal (for example, motion artifact).

e. low pass filter - removes high frequency components of the signal (for example, electronic noise)

a. active elements (op amps) - DC power supply is needed. (see Webster reference).

b. high pass filter - we'll take a look at an op-amp circuit (see Webster reference).

c. low pass filter - we'll take a look at an op-amp circuit (see Webster reference).

B. Adaptive Filtering

We'll discuss this topic the following week.