# Computations on the Spectrum

When you have the amplitude or power spectrum, you can compute several useful characteristics of the input signal, such as power and frequency, noise level, and power spectral density.

## Estimating Power and Frequency

If a frequency component is between two frequency lines, the frequency component appears as energy spread among adjacent frequency lines with reduced amplitude. The actual peak is between the two frequency lines. You can estimate the actual frequency of a discrete frequency component to a greater resolution than the Δf given by the FFT by performing a weighted average of the frequencies around a detected peak in the power spectrum, as shown in the following equation.

| (A)

where j is the array index of the apparent peak of the frequency of interest.

The span j ± 3 is reasonable because it represents a spread wider than the main lobes of uniform, Hanning, Hamming, Blackman-Harris, Exact Blackman, Blackman, and Flat Top smoothing windows.

You can estimate the power in Vrms2 of a discrete peak frequency component by summing the power in the bins around the peak. In other words, you compute the area under the peak. You can use the following equation to estimate the power of a discrete peak frequency component.

| (B)

Equation B is valid only for a spectrum made up of discrete frequency components. It is not valid for a continuous spectrum. Also, if two or more frequency peaks are within six lines of each other, they contribute to inflating the estimated powers and skewing the actual frequencies. You can reduce this effect by decreasing the number of lines spanned by Equation B. If two peaks are within six lines of each other, it is likely that they are already interfering with one another because of spectral leakage.

If you want the total power in a given frequency range, sum the power in each bin included in the frequency range and divide by the noise power bandwidth of the smoothing window.

## Computing Noise Level and Power Spectral Density

The measurement of noise levels depends on the bandwidth of the measurement. When looking at the noise floor of a power spectrum, you are looking at the narrowband noise level in each FFT bin. Therefore, the noise floor of a given power spectrum depends on the Δf of the spectrum, which is in turn controlled by the sampling rate and the number of points in the data set. In other words, the noise level at each frequency line is equivalent to the noise level obtained using a Δf Hz filter centered at that frequency line. Therefore, for a given sampling rate, doubling the number of data points acquired reduces the noise power that appears in each bin by 3 dB. Theoretically, discrete frequency components have zero bandwidth and therefore do not scale with the number of points or frequency range of the FFT.

To compute the signal-to-noise ratio (SNR), compare the peak power in the frequencies of interest to the broadband noise level. Compute the broadband noise level in Vrms2 by summing all the power spectrum bins, excluding any peaks and the DC component, and dividing the sum by the equivalent noise bandwidth of the window.

Because of noise-level scaling with Δf, spectra for noise measurement often are displayed in a normalized format called power or amplitude spectral density. The power or amplitude spectral density normalizes the power or amplitude spectrum to the spectrum measured by a 1 Hz-wide square filter, a convention for noise-level measurements. The level at each frequency line is equivalent to the level obtained using a 1 Hz filter centered at that frequency line.

You can use the following equation to compute the power spectral density.

| (C)

You can use the following equation to compute the amplitude spectral density.

| (D)

The spectral density format is appropriate for random or noise signals. The spectral density format is not appropriate for discrete frequency components because discrete frequency components theoretically have zero bandwidth.