Question 10.13: Extend the 10-point DFT calculation of Example 9.2, to illus......

Part (a):
Table 10.7 repeats the values of j, t  \mathrm{and}  x_{j} from Table 9.1 of Example 9.2.
The value of x^{2}_{j} is not strictly required, but has been added to enable the mean square value to be calculated directly for subsequent checks. For later reference, the mean square value of the input is seen to be 0.710.
Table 10.8 repeats the values of k, f, Re(X_{k})  \mathrm{and}  Im(X_{k}) from Table 9.2, of Example 9.2. The values of X^{*}_{k}X_{k} are calculated, and in the last column the PSD values are given by Eq. (10.85a ):

with T = 0.5. The ‘spurious’ values above k = 5 cannot be used for calculating the PSD, but X^{*}_{k}X_{k} is retained at full length, N, for subsequent use.
As a check, the mean square value of the input time function should be given by summing the power in the PSD function, as given in the last column. The width of each band, \delta f, is 2 Hz, so the mean square value is 0.355  \delta f = 0.710, agreeing with the mean square value from Table 10.7. Note that the sum of the X^{*}_{k}X_{k} values is also 0.710.

Part (b)
The ACF is now given by applying the IDFT in the form of Eq. (10.86a):

which for the purposes of this example is written as:

Equation(C) is evaluated for the single value j  =  3 in Table 10.9. It is seen that the real part of R_{x}(j  =  3)  \mathrm{is}  -0.2268. The imaginary part is zero, as must be the case.
Repeating Table 10.9 for all the other values of j gives Table 10.10.

The values of R_{x} corresponding to values of j larger than 5 are spurious, and are not used in the final output. Thus R_{x} is output for positive values of \tau from 0 to 0.25 s.