# Frequency Spacing between DFT Samples

If the sampling interval is Δt seconds and the first data sample (k = 0) is at 0 seconds, the kth data sample, where k > 0 and is an integer, is at kΔt seconds. Similarly, if the frequency resolution is Δf Hz, the kth sample of the DFT occurs at a frequency of kΔf Hz. However, 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, let N = 8 and p represent the index of the Nyquist frequency p = N/2 = 4. The following table shows the Δf to which each format element of the complex output sequence X corresponds.

X[p] | Δf
X[0] | DC
X[1] | Δf
X[2] | 2Δf
X[3] | 3Δf
X[4] | 4Δf (Nyquist frequency)
X[5] | –3Δf
X[6] | –2Δf
X[7] | –Δf

The negative entries in the second column beyond the Nyquist frequency represent negative frequencies, that is, those elements with an index value greater than p.

For N = 8, X[1] and X[7] have the same magnitude; X[2] and X[6] have the same magnitude; and X[3] and X[5] have the same magnitude. The difference is that X[1], X[2], and X[3] correspond to positive frequency components, while X[5], X[6], and X[7] correspond to negative frequency components. X[4] is at the Nyquist frequency.

The following figure illustrates the complex output sequence X for N = 8.

A representation where you see the positive and negative frequencies is the two-sided transform.

When N is odd, there is no component at the Nyquist frequency. The following table lists the values of Δf for X[p] when N = 7 and p = (N–1)/2 = (7–1)/2 = 3.

X[p] | Δf
X[0] | DC
X[1] | Δf
X[2] | 2Δf
X[3] | 3Δf
X[4] | –3Δf
X[5] | –2Δf
X[6] | –Δf

For N = 7, 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, X[1], X[2], and X[3] correspond to positive frequencies, while 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 figure illustrates the complex output sequence X[p] for N = 7.

The previous figure also shows a two-sided transform because it represents the positive and negative frequencies.