Signals: Terms and basic concepts.

The following definitions are kept relatively brief and are intended to form the basis for practical tasks: each concept should be visualised with practical examples.

1. Signal.
   Intuitively, a signal is a physical quantity that varies with time, space, or any other independent variable or variables.
   Formally, a signal is modelled as a function of one or more independent variables.
2. Examples ...
   Signals in time: Speech, music, other noises; atmospheric pressure, rainfall, hours of sunshine; heart beat rate measured by an electrocardiogramm, electrical brain activity measured by an electroencephalogramm breathing rate, blood sugar level; stock exchange rates, unemployment ratio; height of the sun, position of stars; ...
   Signals in space: Pictures, written text, input to scanner or fax; maps; height of terrain; temperature distribution over an area; atmospheric pressure over an area; ...
3. Continuous-time signals.
   If the physical quantity is known for arbitrary values of the independent variable, the signal is a continuous-time signal. Natural, or physical, signals vary continuously in time and space at very different speeds.
4. Discrete-time signals.
   If the physical quantity is only known at separated intervals or points in time, then the signal is a discrete-time signal.
5. Dimensionality of signals.
   If a signal is a function of one variable, such as time, it is one-dimensional. Speech signals are generally modelled as one-dimensional signals. This is an idealization, because the properties of acoustic signals, for examples, are dependent on the position and speed in space of the source relative to the observer.
6. Periodic and non-periodic signals.
   A periodic signal is a signal which repeats itself over contiguous intervals of the independent variable. Periodic signals are modelled by a function
   

   

   for all t where T is some positive value. A periodic acoustic signal, for example, will repeat itself during consecutive intervals of length T. If T = 200 msec, the signal will repeat itself 50 times per second.
   A non-periodic signals is called a noise. The term noise is also used to refer to unwanted components of a useful signal; the ration of signal level to noise level in this meaning of the term is referred to as the signal to noise ratio.

7. Simple and complex signals.
   A simple signal is a signal which can be modelled by a sine (or cosine) function; for this reason, a simple signal is called sinusoid. Signals which cannot be modelled by a sine function are complex. Ripples on a pond or on a cup of coffee, and the sound produced by a tuning fork, are approximately sinusoid.
   Complex signals are signals which cannot be modelled by a sine function. Complex signals can be approximated by regarding them as the sum, for any value of the independent variable, of the values of different sine functions. The values of the component sine functions which model components of the complex signal represents the spectrum of the complex signal.
8. Properties of simple one-dimensional signals.
   Simple signals are modelled with sine and cosine functions and are termed sinusoid.
   1. Period.
      The period of a simple periodic signal is the value of T, the interval over which the signal is repeated. For example: a sinusoid signal with a period of 5 msec will repeat itself 500 times per second. The number of repetitions of a sinusoid signal per second is its frequency, counted in Hertz (Hz). The period and the frequency of a signal are related as follows:

      

      A sinusoid signal with a period of 2 sec has a frequency of 0.5 Hz, and one with a frequency of 2 Hz has a period of 0.5 sec. Likewise, a sinusoid signal with a period of 2 msec has a frequency of 0.5 kHz, and one with a frequency of 2 kHz has a period of 0.5 msec.

   2. Phase.
      The phase of a simple periodic signal is the argument of the sine function in degrees or radians. For example:

      

      e.g. , or
      

      

      e.g. 1.57 rad
      In more practical terms, a sinusoid signal starts at value zero, after a quarter of the period it has reached its positive peak (maximum) value, after half the period it has reached zero again, after three-quarters of the time it has reached its negative peak (minimum) value, and after the full period it has again reached zero. A point at which a signal has the value zero is called a zero crossing; if the signal moves from a positive to a negative value, it is a negative-going zero crossing, and if the signal moves from a negative to a positive value, it is a positive going zero crossing.
      Imagine walking across sand with a stick, and tracing a line very slowly moving the stick evenly back and forth from left to right. The resulting trace is approximately sinusoidal, the zero-crossing is when the stick is immediately in front of you, the positive peak being the furthest distance on the left (say), and the negative peak being the furthest distance on the right.

   3. Amplitude.
      The amplitude of the signal is the value of the quantity being measured at a given value of the independent variable, and varies between the positive and negative peak values.

9. Properties of complex periodic one-dimensional signals.
   A complex signal is the sum of the amplitudes of its component signals for each value of the independent variable. These amplitudes are related to the frequency and the phase of the signal. In the following equation, is the amplitude of signal i at time t, is the frequency of signal i at time t, and is the phase of signal i at time t; the resulting values (the instantaneous amplitudes) of each signals i from 1 to N at any time t are added, and the value of the complex signal at time t is the result.

   

   1. The values of the signals may be systematically related. If the frequencies can be modelled as integer multiples of the lowest frequency (i.e. 1f, 2f, ... nf,i.e. a harmonic series, then the signal is a harmonic signal. A special kind of harmonic series is the Fourier series, in which the amplitude of the harmonics varies with the number of the harmonic.
   2. A square wave is approximated by a series including only the odd-numbered harmonics.
   3. A sawtooth (triangular) waveform is approximated by a series including only the even-numbered harmonics. The waveform produced at the glottis is approximately a sawtooth.
10. Signal transmission.
    Acoustic signals pass through different media (gases, liquids and solids) at different speeds. For instance, they pass through gases which are lighter than air (such as helium) faster than they pass through air; if helium is breathed (e.g. in the air mixture used by divers), then speech sounds higher-pitched than in air). The distance in space between positive-going zero-crossings of a sinusoid signal is its wavelength, abbreviated by .