3.2 SPECTRAL ANALYSIS OF NONPERIODIC FUNCTIONS
A nonperiodic function of time is a function that is nonrepetitive over time. A stream of binary data as typically transmitted by digital communication systems is a stream of nonperiodic functions, each pulse having equal probability of being one or zero, independent of the value of other pulses in the stream. The analysis of the spectral properties of nonperiodic functions is thus an important component of the study of digital transmission.
3.2.1 The Fourier Transform
A nonperiodic waveform, v(t) say, may be represented in terms of its frequency characteristics by the following relationship:
The factor V(f) is the amplitude spectral density or the Fourier transform of v(t). It is given by
Because V(f) extends from ∞ to +∞ (i.e., it exists on both sides of the zero frequency axis) it is referred to as a two-sided spectrum.
An example of the application of the Fourier transform that is useful in the study of digital communications is its use in determining the spectrum of a nonperiodic pulse. Consider a pulse v(t) shown in Fig. 3.1(a), of amplitude V, and that extends from t = τ/2 to t = τ/2. Its Fourier transform, V(f), is given by
Figure 3.1 Rectangular pulse and its spectrum.
The form (sin x)/x is well known and referred to as the sampling function, Sa(x). The plot of V(f) is shown in Fig. 3.1(b). It will be observed that it is a continuous function. This is a common feature of the spectrum of all nonperiodic waveforms. We note also that it has zero crossings at ±1/ τ, ±2/ τ, . . .
The Fourier transform V(f) of an impulse of unit strength is also a useful result. By definition an impulse δ(t) has zero value except at time t = 0, and an impulse of unit strength has the property
Equation (3.5) indicates that the spectrum of an impulse δ(t) has a constant amplitude and phase and extends from ∞ to +∞.
A final example of the use of the Fourier transform is the analysis of what results in the frequency domain when a signal m(t), with Fourier transform M(f), is multiplied by a sinusoidal signal of frequency fc. In the time domain the resulting signal is given by
and its Fourier transform is thus
Figure 3.2 (a) The amplitude spectrum of a waveform with no special component beyond fm. (b) The amplitude spectrum of the waveform in (a) multiplied by cos 2πfct. (From Taub, H., and Schilling, D., Principles of Communication Systems, McGraw-Hill, 1971, and reproduced with the permission of the McGraw-Hill Companies.)
3.2.2 Linear System Response
A linear system is one in which, in the frequency domain, the output amplitude at a given frequency bears a fixed ratio to the input amplitude at that frequency and the output phase at that frequency bears a fixed difference to the input phase at that frequency, irrespective of the absolute value of the input signal. Such a system can be characterized by the complex transfer function, H(f) say, given by
where |H(f)| represents the absolute amplitude characteristic, and θ(2πf) the phase characteristic of H(f).
Consider a linear system with complex transfer function H(f), as shown in Fig.3.3, with an input signal vi(t), an output signal vo(t), and with corresponding spectral amplitude densities of Vi(f), and Vo(f). After transfer through the system, the spectral amplitude density of Vi(f) will be changed to Vi(f) H(f). Thus
Figure 3.3 Signal transfer through a linear system.
An informative situation is the one where the input to a linear system is an impulse function of unit strength. For this case, as per Eq. (3.5), Vi(f) = 1, and
Thus, the output response of a linear system to a unit strength impulse function is the transfer function of the system.
3.2.3 Energy and Power Analysis
In considering energy and power in communication systems, it is often convenient to assume that the energy is dissipated in a 1-ohm resistor, as with this assumption one need not keep track of the impact of the true resistance value, R say. When this assumption is made, we refer to the energy as the normalized energy and to the power as normalized power. It can be shown that the normalized energy E of a nonperiodic waveform v(t), with a Fourier transform V(f), is given by
The preceding relationship is called Parseval's theorem. Should the actual energy be required, then it is simply E [as given in Eq. (3.14)] divided by R.
The energy density, De(f), of a waveform is the factor dE(f)/df. Thus, by differentiating the right-hand side of Eq. (3.14), we have
For a nonperiodic function such as a single pulse, normalized energy is finite, but power, which is energy per unit time, approaches zero. Power is thus somewhat meaningless in this context. However, a train of binary nonperiodic adjacent pulses does have meaningful average normalized power. This power, P say, is equal to the normalized energy per pulse E, multiplied by fs, the number of pulses per second; that is,
If the duration of each pulse is τ, then fs = 1/τ. Substituting this relationship and Eq. (3.14) into Eq. (3.16), we get
The power spectral density, G(f), of a waveform is the factor dP(f)/df. Thus, by differentiating the right-hand side of Eq. (3.17), we have
To determine the effect of a linear transfer function H(f) on normalized power, we substitute Eq. (3.11) into Eq. (3.17). From this substitution we determine that the normalized power, Po, at the output of a linear network, is given by
Also, from Eq. (3.11), we have
Substituting Eq. (3.18) into Eq. (3.20), we determine that the power spectral density Go(f) at the output of a linear network is related to the power spectral density Gi(f) at the input by the relationship