# What is the difference between the PSD and the Power Spectrum?

Can any one explain the difference in the amplitude between the two and what does each one represents?

For example, when using the pwelch algorithm in MATLAB how do the two different options ('psd','power') affect the units of the output?

(I know the default is 'psd' and I can use 'power' to show something closer to the rms calculation, as stated here)

• Can I please ask you to rephrase the question so that it reflects exactly what it is? That would be the last two lines "What is the difference between the PSD and the Power Spectrum?". The way it is phrased now, it seems to be about MATLAB and help about specific software platforms is not the point of DSP.SE. – A_A Aug 30 '16 at 9:17
• Shortly speaking: $PS=PSD\cdot ENBW$, where $ENBW$ is the Equivalent Noise Bandwidth. Assuming that your signal units are in pascals ($\mathrm{Pa}$), then units are: $PSD [\mathrm{Pa^2/Hz}]$ and $PS [\mathrm{Pa^2}]$. – jojek Aug 30 '16 at 10:17
• @jojek Is ENBW a constant? How would one calculate it? is this enough? – havakok Sep 1 '16 at 8:50

The power spectrum is a general term that describes the distribution of power contained in a signal as a function of frequency. From this perspective, we can have a power spectrum that is defined over a discrete set of frequencies (applicable for infinite length periodic signals) or we can have a power spectrum that is defined as a continuous function of frequency (applicable for infinite length aperiodic signals).

In the first case, each discrete component has units of power (W, mW, etc.). In the second case, each point in the continuous spectrum has units of power per frequency (W/Hz, mW/Hz, etc.). In the second case, one must integrate over a band of frequencies to obtain units of power. It is meaningless to talk about the amount of power at a frequency in the second case, one must talk about the amount of power contained in a spectral band (an interval of frequencies) or in several such bands. It is meaningful however to talk about the amount of power spectral density (PSD) at a particular frequency. This is an indicator of how much weight this frequency will contribute to the overall power if included in one of the spectral bands.

This allows us to compare the distribution of power at various frequencies in a signal (as opposed to comparing the power directly). This distribution is defined over a small delta of frequency (delta going to zero in the calculus sense). An alternative to PSD, would be to divide up the signal into chunks of a finite size (say 10 Hz) and compute the power for each bin (one way to obtain this would be to integrate the PSD over each bin). If another person performed a similar experiment and generated their own plot using a larger bin size (say 20 Hz), then the overall shape of both plots would be similar, but the latter plot would seem coarser and its numerical values would be greater. This would make it difficult to compare the two signals. Using power spectral density, the amplitude and bin size ambiguity is eliminated. This allows a fair comparison between the power distribution of two different signals.

For more information, here is a link to a related Wikipedia article.

• +1 for a very nice and easy to understand explaination. – Anh Tuan Jun 25 '18 at 4:39

There are three ways to normalise the resulting spectrum, depending on how one wants to use the PSD:

1. to read signal values directly off the plot;
2. to read the noise power spectral density directly off the plot;
3. to quantitatively determine the power in any frequency band by adding the values of all bins in that band.

I think Matlab’s pwelch function implicitely returns a spectrum of the second type. The normalisation procedure is presented in this document.