Signals may be represented mathematically in different ways. The 10.11. basis for many waveforms is the sinusoidal form and thus sine and cosine representation is a convenient mathematical method of de- (10.11) scribing signals. A second method, often preferred for mathematical convenience, is the representation of a pair of real value signals in complex It is important to distinguish between the real signal and the mathematics. A complex signal consists of a real signal and an quantitative representation, which is necessarily an approximation. imaginary signal, which may be visualised as two voltages across The amount of error in the approximation depends on the complex- two resistors. Complex valued signals are processed just as real ity of the signal, with simple waveforms, such as the sinusoid, valued signals, except the rules of complex arithmetic are used. having less error than complex waveforms. Conventionally j is taken to represent the imaginary part. A non-periodic or aperiodic signal is one for which no value of T The relationship between the sine and cosine representation and satisfies Equation 10.11. In principle this includes all actual signals the complex representation is given in Equation 10.12, where w/o is since they must start and stop at finite times. However, aperiodic given by Equation 10.13. signals can be presented quantitatively in terms of periodic signals. Examples of periodic signals include the sinusoidal signals and (10.12) periodically repeated non-sinusoidal signals, such as the rectangular pulse sequences used in radar. Non-periodic signals include speech waveforms and random sig- (10.13) nals arising from unpredictable disturbances of all kinds. In some cases it is possible to write explicit mathematical expressions for non-periodic signals and in other cases it is not. In addition to periodic and non-periodic signals are those signals 10.6.3 Fourier series representation that are the sum of two or more periodic signals having different periods. T will not be satisfied in Equat