Frequency Spacing between DFT/FFT Samples

If the sampling interval is t seconds, and the first (k = 0) data sample is at 0 seconds, then the kth (k > 0, k integer) data sample is at kt seconds. Similarly, if the frequency resolution is f Hz (), then the kth sample of the DFT occurs at a frequency of kf Hz. Actually, as you will soon see, this is valid for only up to the first half of the frequency components. The other half represent negative frequency components.

Depending on whether the number of samples, N, is even or odd, you can have a different interpretation of the frequency corresponding to the kth sample of the DFT. For example, suppose N is even and let . The following table shows the frequency to which each format element of the complex output sequence X corresponds.

Note that the pth element, X[p], corresponds to the Nyquist frequency. The negative entries in the second column beyond the Nyquist frequency represent negative frequencies.

For example, if N = 8, p = N/2 = 4, then

X[0]DC
X[1]f
X[2]2f
X[3]3f
X[4]4f (Nyquist freq)
X[5]–3f
X[6]–2f
X[7]f

Here, X[1] and X[7] will have the same magnitude, X[2] and X[6} will have the same magnitude, and X[3] and X[5] will have the same magnitude. The difference is that whereas X[1], X[2], and X[3] correspond to positive frequency components, X[5], X[6], and X[7] correspond to negative frequency components. Note that X[4] is at the Nyquist frequency.

The following illustration represents this complex sequence for N = 8.

Such a representation, where you see both the positive and negative frequencies, is known as the two-sided transform.

Note that when N is odd, there is no component at the Nyquist frequency.

For example, if N = 7, p = (N–1)/2 = (7–1)/2 = 3, and you have

X[0]DC
X[1]f
X[2]2f
X[3]3f
X[4]–3f
X[5]–2f
X[6]f

Now X[1] and X[6] have the same magnitude, X[2] and X[5] have the same magnitude, and X[3] and X[4] have the same magnitude. However, whereas X[1], X[2], and X[3] correspond to positive frequencies, X[4], X[5], and X[6] correspond to negative frequencies. Because N is odd, there is no component at the Nyquist frequency.

The following illustration represents the preceding table for N = 7.

This is also a two-sided transform, because you have both the positive and negative frequencies.