Using the Fourier representation of x(t), and the expression found for the Fourier coefficients we can write Equation 2.41. C_k = ck (2.44) As an example of this analysis, the Fourier series and power spectral density of the rectangular pulse train shown in Figure 2.2 will be given, and since the signal x(t) is even, we will select the integration interval from as in Equation 2.45 and 2.46, (2.41) for k = =1, = =2, and the power spectral density is obtained by squaring these quantities. (2.45) This relationship between the average power and the square of the Fourier coefficients is called Parseval's relation, as mentioned ear- lier in the context of function representation. The kth harmonic component of the signal has a power 1ckl2 and hence the total average power in a periodic signal is just the sum of the average (2.46) powers in all the harmonics. If 1ck 12 is plotted as a function of kfo, the resulting function is called the power density spectrum of the periodic signal x(t). Since the power in a periodic signal exists only at discrete values of frequencies i.e. Equation 2.42 is satisfied, the It is now important to introduce the concept of aperiodic signals signal is said to have a line spectrum, and the frequency spacing and transform methods, before dealing with discrete data. between two adjacent spectral lines is equal to the reciprocal of the fundamental period Tp. Fourier transforms (2.42) Consider an aperiodic signal x(t) which is of finite duration, Figure 2.3. A periodic signal xp(t) can be created from x(t) by translation The Fourier coefficients C are in general complex, and may be by fixed amounts, Tp, as shown. This new periodic signal, xp(t), represented as in Equation 2.43 and hence an alternative to the approaches x(t) in the limit as Tp approaches infinity, and the power spectrum representation may be obtained by plotting the spectrum of xp(t) should be obtainable from the spectrum of x(t) in magnitude spectrum I ck I and phase spectrum Ok as a function of the same limit. frequency. As befo