DFT Calculation

In this topic, you will see the exact frequencies to which the N samples of the DFT correspond. For the present discussion, assume that X[0] corresponds to D.C., or the average value, of the signal. To see the result of calculating the DFT of a waveform with the use of the following equation,



(1)

for k = 0, 1, 2, …, N – 1

Consider a D.C. signal having a constant amplitude of +1 V. Four samples of this signal are taken, as shown in the illustration below.

Each of the samples has a value +1, giving the time sequence x[0] = x[1] = x[3] = x[4] = 1

Using Equation (1) to calculate the DFT of this sequence and making use of Euler's identity, exp(–i) = cos() – jsin(), you get

Therefore, except for the DC component, X[0], all the other values are zero, which is as expected. However, the calculated value of X[0] depends on the value of N (the number of samples). Because you had N = 4, X[0] = 4. If N = 10, then you would have calculated X[0] = 10. This dependency of X[ ] on N also occurs for the other frequency components. Thus, you usually divide the DFT output by N, so as to obtain the correct magnitude of the frequency component.