# Glossary

alias
  An error appearing in a sampled signal when the bandwidth of the signal is greater than half the sampling frequency (that is, when the sampling frequency is lower than the Nyquist frequency). Such effects are also referred to as artefacts or ghosts.

anti-aliasing filter
  A low-pass filter that is able to remove aliasing in sampled signals by cutting all the spectral components that are greater than or equal to half the sampling frequency.

Bode plot
  Loosely, a graph of the frequency response of a device or system. Strictly, a pair of graphs showing frequency response and phase response over the same span of frequencies.

convolution
  A mathematical operation that combines two signals to produce a third signal. When two signals are convolved, the resultant third signal expresses how the shape of one signal is modified by the other.

decibel
  A logarithmic way of expressing a power ratio. For powers P1 and P2, their ratio in decibels is defined as 10 log10 (P1/P2). The symbol for decibels is dB. Strictly the decibel is not a unit, as any ratio must be a pure (that is, dimensionless) number. However, it is often regarded as a unit.

difference equation
  An equation in which all variables have been sampled at fixed intervals, and these variables are multiplied by some coefficient.

differential equation
  A mathematical equation in which one or more terms contains a mathematically differentiated variable.

digital signal processor (DSP)
  A semiconductor device similar to a microprocessor. Whilst a microprocessor is a general-purpose device, a DSP has been optimised to carry out the computations used for processing discrete signals.

first-order
  As applied to a filter, the simplest type of filter, having in its passive form a single reactive element (a capacitor or an inductor) and a roll-off of 20 dB/decade, or 6 dB/octave. As applied to a differential equation, such an equation in which the main variable is differentiated once. Any system that can be modelled with such a differential equation would be referred to as a first-order system.

Fourier transform
  A transformation that extends the concept of the Fourier series to non-periodic signals. It allows us to estimate the spectrum of a signal and perform a frequency analysis.

frequency response
  The response of a system (e.g. a filter) when we input sine waves at different frequencies (but equal amplitude). It tells us how the system will modify the spectrum of any input signal we feed to the system.

gain
  In amplification, a measure of how many times the input signal amplitude is increased. It is generally measured as the ratio between the input signal amplitude and the output signal level. If a gain value is given as just a number (i.e. with no units), the gain is likely to be a ratio of voltages; if the value is given in decibels, it is a ratio of powers. See also voltage gain and power gain.

octave
  The span of frequencies covered by a doubling of frequency, or by a halving of frequency. For example, the frequency span from 500 Hz to 1000 Hz is an octave, as is the span from 500 Hz to 250 Hz. In music, the eight notes of a diatonic scale (that is, doh, re, me … ti, doh) cover an octave; hence the name ‘octave’ for this span of frequencies.

operational amplifier
  A general-purpose analogue amplifier intended to be used as a component in other electronic circuits, and usually supplied as a multi-pin integrated-circuit device with two inputs and a single output. Typically an op-amp is a differential amplifier (that is, it amplifies the difference between its two inputs) and has an unusably high gain and extremely high input impedance. To give useful and predictable behaviour, external feedback circuitry must be applied. This circuitry determines essential parameters such as input impedance, output impedance, overall gain and frequency response, and also whether the circuit operates as a single-input amplifier or a differential amplifier.

order
  A numerical classification for filters (e.g. ‘first order’, ‘second order’, ‘third order’, etc.). The order is determined by the differential equation of the filter. For a first-order filter, the highest differential coefficient in the equation is first-order (e.g. dv/dt); for a second-order filter, the highest differential coefficient is second-order (e.g. d2v/dt2). The higher the order, the steeper the roll-off and the sharper the cut-off between passband and stop band. Increasing the order by one adds 20 dB/decade to the filter’s roll-off.

passband
  The band or bands of frequencies passed by a filter with least attenuation, or no attenuation. Frequencies outside the passband are cut off, or stopped, by the filter. Passband is the counterpart of stop band.

power gain
  The ratio of output power to input power. It is usually expressed in decibels (dB). A power gain of 0 dB means that the output power is the same as the input power. A power gain of 3 dB (or, more exactly, 3.0103 dB) means that the output power is double the input power.

quantisation
  Conversion of an analogue quantity, which could take any value within a range, to one of a set of discrete values.

roll-off
  The steepness of a filter’s attenuation in a stop band. Also, the steepness of the attenuation of any device that produces attenuation (for example, a linear amplifier at the extremes of its operating-frequency range).

stop band
  The band or bands of frequencies stopped, or cut off, by a filter. The counterpart of the passband.

taps
  In a digital filter, the number of taps is the number of terms in the mathematical expression that describes the filter. This expression is given in the form of a difference equation. In digital filter design, the maximum number of taps to be used in the implementation is required as part of the design specification.

voltage gain
  For a sinusoidal input and output, voltage gain is the ratio of the output voltage’s amplitude to that of the input voltage. It has no units.

window function
  In signal processing, a mathematical function that is zero-valued outside some chosen interval. When a signal is convolved with a window function, the resultant signal is also zero-valued outside the chosen interval, so it is the original signal viewed through the window function.

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