4.6  Mean-square (MS) spectrum

The PSD is very useful especially when dealing with random signals and the power of the signal over a frequency range is obtained by integrating the PSD over that range. However, in some cases it is desired to use a function that allows to directly infer the average power of sinusoid components of a periodic signal, without the integration step. In these cases, the so-called MS or power spectrum is more convenient.⁹ However, in applications characterized by a mixed signal in which there is a deterministic signal of interest that is contaminated by random noise (such as sinusoids contaminated by AWGN), the PSD representation is often more convenient than the MS spectrum.

While in continuous-time the PSD $S(f)$ unit is watts/Hz, the mean-square spectrum 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 $N$ samples.

If the FFT size $N$ is chosen to be equal to the period $N_0$, i. e. $N=N_0$, 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 $k(2\pi ∕N)=k(2\pi ∕N_0)$. In case $N\ne N_0$, 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 ${\mathrm{\Omega }}_k=k(2\pi ∕N)$.

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

and

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