4.6  Mean-square (MS) spectrum

As discussed, the PSD is particularly useful when dealing with random signals, since the signal power over a given frequency range is obtained by integrating the PSD over that range.

In some situations, however, it is preferable to use a representation that allows direct inference of the average power of the sinusoidal components of a periodic signal, without requiring integration. In such cases, the so-called mean-square (MS) or power spectrum is more convenient.⁹

On the other hand, in applications involving mixed signals, where a deterministic component of interest is contaminated by random noise (e.g., sinusoids corrupted by additive white Gaussian noise, AWGN), the PSD representation is often more appropriate than the MS spectrum.

While in continuous-time the PSD $S(f)$ unit is watts/Hz, the mean-square spectrum $S_{\mathrm{ms}}[k]$ is given directly in watts. Assuming a signal with fundamental period $N_0$ samples, the mean-square spectrum corresponds to the squared magnitude of the corresponding DTFS:

with the property

where $P$ is the signal power.

Recall from the discussion associated to Eq. (2.49) that, if $x[n]$ is periodic with fundamental period $N_0$, its DTFS $\tilde{X}[k]$ can be obtained with an $N_0$-point FFT:

One often uses Eq. (4.37) even for a non-periodic $x[n]$, but then the result spectrum must be properly interpreted: as if the signal were a periodic version ${\sum <!--\; FUNCTION\; APPLICATION\; -->}_{p=-\infty }^{\infty }{x}_N[n-\mathit{pN}]$ of the windowed version $x_N[n]$ of $x[n]$ using $N=N_{\mathrm{\text{fft}}}$ samples.

If the FFT size $N_{\mathrm{\text{fft}}}$ is chosen to be equal to the period $N_0$, i. e. $N_{\mathrm{\text{fft}}}=N_0$, the $k$-th FFT value in Eq. (4.37) corresponds exactly to the $k$-th DTFS coefficient, that is, they both represent the same frequency $k(2\pi ∕N_{\mathrm{\text{fft}}})=k(2\pi ∕N_0)$. In case $N_{\mathrm{\text{fft}}}\ne N_0$, Eq. (4.36) is still a valid way to obtain the average power $P$, but the $k$-th bin frequency ${\mathrm{\Omega }}_k$ must be interpreted according to the FFT grid as ${\mathrm{\Omega }}_k=k(2\pi ∕N_{\mathrm{\text{fft}}})$.

In general, an estimate ${Ŝ}_{\mathrm{\text{ms}}}[k]$ of the MS spectrum of a discrete-time signal $x[n]$ can be obtained from its $N$-length windowed version $x_N[n]$ with

where $N_{\mathrm{\text{fft}}}=N$, and then

The “hat” in ${Ŝ}_{\mathrm{\text{ms}}}[k]$ indicates that in general, Eq. (4.38) is an estimate of the true MS spectrum.