Power Spectrum VI
- Updated2025-07-30
- 5 minute(s) read
Computes the Power Spectrum, Sxx, of the input sequence X. Wire data to the X input to determine the polymorphic instance to use or manually select the instance.
The Power Spectrum Sxx(f) of a function x(t) is defined as
Sxx(f) = X*(f)X(f) = |X(f)|²,where X(f) = F{x(t)}, and X* (f) is the complex conjugate of X(f).
The Power Spectrum VI uses the FFT and DFT routines to compute the power spectrum, which is given by
where Sxx represents the output sequence Power Spectrum, and n is the number of samples in the input sequence X.
When the number of samples, n, in the input sequence X is a valid power of 2
n = 2mfor m = 1, 2, 3, …, 23,
the Power Spectrum VI computes the fast Fourier transform of a real-valued sequence using the a fast radix-2 FFT algorithm and efficiently scales the magnitude square. The largest Power Spectrum the VI can compute using the FFT is 223 (8,388,608 or 8M).
When the number of samples in the input sequence X is not a valid power of 2 but is factorable as the product of small prime numbers, the Power Spectrum VI computes the discrete Fourier transform of a real-valued sequence using an efficient DFT algorithm and scales the magnitude square. The largest Power Spectrum the VI can compute using the fast DFT is 222 – 1(4,194,303 or 4M – 1).
Let Y be the Fourier transform of the input sequence X and n be the number of samples in it. You can show that
|Yn –i|² = |Y–i²|.You can interpret the power in the (n – 1)th element of Y as the power in the –ith element of the sequence, which represents the power in the –ith harmonic. You can find the total power for the ith harmonic (DC and Nyquist component not included) using
Power in the ith harmonic = 2|Yi|² = |Yi|² + |Yn – 1|² 0 < i < n/2The total power in the DC and Nyquist components are |Y0|² and |Yn/2|² respectively.
If n is even, let 
. The following table shows the format of the output sequence Sxx corresponding to the Power Spectrum.
| Array Element | Interpretation |
|---|---|
| Sxx0 | Power in DC component |
| Sxx1 = Sxx(n – 1) | Power at frequency Δf |
| Sxx2 = Sxx(n – 2) | Power at frequency 2Δf |
| Sxx3 = Sxx(n – 3) | Power at frequency 3Δf |
| ⋮ | ⋮ |
| Sxx(k – 2) = Sxxn – (k – 2) | Power at frequency (k – 2)Δf |
| Sxx(k – 1) = Sxxn – (k – 1) | Power at frequency (k – 1)Δf |
| Sxxk | Power in Nyquist harmonic |
The following illustration represents the preceding table information.
Notice that the power spectrum is symmetric about the Nyquist frequency as the following illustration shows.
If n is odd, let 
. The following table shows the format of the output sequence Sxx corresponding to the Power Spectrum.
| Array Element | Interpretation |
|---|---|
| Sxx0 | Power in DC component |
| Sxx1 = Sxx(n – 1) | Power at frequency Δf |
| Sxx2 = Sxx(n – 2) | Power at frequency 2Δf |
| Sxx3 = Sxx(n – 3) | Power at frequency 3Δf |
| ⋮ | ⋮ |
| Sxx(k – 2) = Sxxn – (k – 2) | Power at frequency (k – 2)Δf |
| Sxx(k – 1) = Sxxn – (k – 1) | Power at frequency (k – 1)Δf |
| Sxxk = Sxxn – k | Power at frequency kΔf |
The following illustration shows that when n is odd, the power spectrum is symmetric about the Nyquist frequency, but the Nyquist frequency does not fall on a frequency bin.
The format described in the preceding tables is an accepted standard in digital signal processing applications.
Examples
Refer to the following example files included with LabVIEW.
- labview\examples\Signal Processing\Spectral Analysis\DC Centered Spectrum.vi
- labview\examples\Signal Processing\Transforms\Generalized Fourier Spectrum.vi
- labview\examples\Signal Processing\Transforms\FFT and Power Spectrum Units.vi