 a smooth spectral density $S_X(\omega;\boldsymbol{\theta})$ of the latent process $\{X_t\}$, were it to be directly observed, could be obtained by smoothing the periodogram across frequencies~\citep[p. 235--253]{SpectralAnalysis}, as long as $S_X(\omega;\boldsymbol{\theta})$ is continuous.


	For the modulated process $\{\Y_t\}$, the latent time series $\{X_t\}$ is not observed, so we instead compute the periodogram of the modulated (and observed) process itself, $\hat{S}_{\Y}\sN(\omega)$, and we define the expected periodogram to be
	\[
		\overline{S}_{\Y}^{(N)}(\omega;\boldsymbol{\theta}) = \E\left\{\hat{S}_{\Y}\sN(\omega);\; \boldsymbol{\theta}\right\}, \ \ \omega\in\R.
	\]
Note that this quantity is also $2\pi$-periodic.
It is necessary to understand how modulation in the time domain will affect the expected periodogram.
Proposition \ref{prop=freqPropSbar} gives more insight on how $\overline{S}_{\Y}\sN(\omega;\boldsymbol{\theta})$ relates to the modulating sequence $g_t$ and the spectral density $S_X(\omega;\boldsymbol{\theta})$ of the latent stationary process $\{X_t\}$.

\begin{proposition}[Expectation of the periodogram of a modulated time series]
	\label{prop=freqPropSbar}
	The expectation of the periodogram of the modulated time series takes the form
	\begin{equation}
		\label{eq=Sbar1}
		\overline{S}_{\Y}\sN(\omega;\boldsymbol{\theta}) 
		= 2\pi\int_{-\pi}^\pi{S_X(\omega-\lambda;\boldsymbol{\theta})S_g\sN(\lambda)d\lambda}, \ \forall\omega\in\R,
	\end{equation}
	which is a periodic convolution.
	Here $S_g\sN(\lambda)$ is the squared value of the Fourier Transform 
	of the finite sequence $\{g_t\}_{t=0,\cdots,N-1}$ i.e.
	\[
		S_g\sN(\lambda) = \frac{1}{2\pi N}\left|\sum_{t=0}^{N-1}{g_t e^{-i\lambda t}}\right|^2,
	\]
	defined for $\lambda\in\R$ and which is $2\pi$ periodic.
\end{proposition}
\begin{proof}
	The proof for this proposition, which is a well-known result, can be found in~\citet[p.~562]{Dunsmuir1981}.
\end{proof}
When $g_t=1$ everywhere, which corresponds to observing the stationary latent process directly, the quantity $S_g\sN(\lambda)$ is the usual Féjer kernel \citep{Bloomfield2000} defined by,
\begin{equation}
\label{eq=FejerKernel}
	\mathcal{F}^{(N)}(\lambda) = \frac{\sin^2\left(\frac{N\lambda}{2}\right)}{2\pi N\sin^2\left(\frac{\lambda}{2}\right)}, \ \ \forall\lambda\in\mathbb{R}\setminus\Omega_N,
\end{equation}
which behaves asymptotically (as $N$ tends to infinity) as a Dirac delta-function centred at zero. This explains why the periodogram is, asymptotically, an unbiased estimator of the spectral density of a stationary process up to a multiplicative factor of $2\pi$~\citep{BrockwellDavisTimeSeries}.

When $g_t$ is such that the modulated process is asymptotically stationary, \citet{Dunsmuir1981b} approximate $\frac{1}{2\pi}\sum_{\tau=-\infty}^\infty{\gamma(\tau)e^{i\omega\tau}}$, where $\gamma(\tau) = R_g(\tau)c_X(\tau)$ using the notation of~\eqref{asymptotic}, for $\omega$ at Fourier frequencies by,
\begin{equation}
	\label{eq=DunsmuirSpectrum}
	\widetilde{S}_{\Y}^{(D)}(\omega;\btheta) = \frac{2\pi}{N}\sum_{\lambda\in\Omega_N}{S_X(\omega-\lambda;\btheta)S_g^{(N)}(\lambda)}.
\end{equation}


When $g_t$ is such that the modulated process $\Y_t$ has a significant correlation contribution, we derive the exact value of $\overline{S}_{\Y}\sN(\omega;\boldsymbol{\theta})$ by using the theoretical autocovariances of the latent model, in a similar fashion as in \citet{WhittleAdam} for stationary processes. This is the result of Proposition \ref{prop=expectationPeriodogram2}, which follows.
\begin{proposition}[Computation of the expected periodogram]
	\label{prop=expectationPeriodogram2}
	Let $\omega\in\R$.
	We have
	\begin{equation}
		\label{eq=computationOfPeriodogram}
		\overline{S}_{\Y}\sN(\omega;\boldsymbol{\theta}) = 
		2\mathcal{R}\left\{\sum_{\tau=0}^{N-1}{\overline{c}_{\Y}\sN(\tau;\boldsymbol{\theta})e^{-i\omega \tau}}\right\} 
		- \overline{c}_{\Y}\sN(0;\boldsymbol{\theta}),
	\end{equation}
	where $\overline c_{\Y}\sN(\tau;\bm\theta)$ is defined in~(\ref{eq=expectedAutocovSequence1}).
	By defining $\overline{c}_{\Y}\sN(-\tau;\boldsymbol{\theta}) = \overline{c}_{\Y}\sN(\tau;\boldsymbol{\theta})$ for $\tau=1, \cdots, N-1$ we can (equivalently) express this relationship as
	\[
		\overline{S}_{\Y}\sN(\omega;\boldsymbol{\theta}) =
		\sum_{\tau=-(N-1)}^{N-1}{\overline{c}_{\Y}\sN(\tau;\boldsymbol{\theta})e^{-i\omega \tau}},
	\]
\end{proposition}

\begin{proof}
The proof, which is standard~\citep[page 334]{BrockwellDavisTimeSeries}, follows directly from~\eqref{eq=defPeriodogram} and~\eqref{eq=expectedAutocovSequence1} in a few lines of algebra after aggregating along the diagonal of the covariance matrix.
\end{proof}
Therefore the expectation of the periodogram of $\bold{\Y}$ is the discrete Fourier transform of the expected sample autocovariance sequence. 
This is true even though we have not assumed stationarity; it is simply a consequence of the relation between the formal definitions of~\eqref{eq=expectedAutocovSequence1} and~\eqref{eq=defPeriodogram}.
Note that calculating the Fourier transform of the sequence $\overline{c}_{\Y}\sN(\tau;\theta)$ will always give a real-valued positive $\overline{S}_{\Y}\sN(\omega; \boldsymbol{\theta})$ for $\boldsymbol{\theta}\in\Theta$, as the latter is defined as the expectation of the squared modulus of the Fourier transform of the process. 


Proposition~\ref{prop=expectationPeriodogram2} can be used to compute the expected periodogram of an asymptotically stationary modulated process. In such cases, the difference between~\eqref{eq=DunsmuirSpectrum} and Proposition~\ref{prop=expectationPeriodogram2} is that~\eqref{eq=DunsmuirSpectrum} is a finite approximation of~\eqref{eq=Sbar1}, whereas Proposition~\ref{prop=expectationPeriodogram2} is exact. The difference occurs because~\eqref{eq=DunsmuirSpectrum} does not account for the bias of the periodogram that results from leakage (see \citet{WhittleAdam}), whereas these effects are naturally accounted for in Proposition~\ref{prop=expectationPeriodogram2}.


To justify the use of the expected periodogram in the setting of modulated processes with a significant correlation contribution, 
we now consider what conditions are required for the expected periodogram to \emph{carry enough information} so that the parameter vector is identifiable within the parameter set $\Theta$. 




\begin{proposition}[Identifiability of the expected periodogram]
\label{prop=identifiabilityViaPeriodogram}
If the modulated process has a significant correlation contribution, the expected periodogram is a one-to-one (i.e. injective) mapping from the parameter set $\Theta$ to the set of non-negative continuous functions on $[-\pi,\pi]$, for a large enough sample size. 
More specifically, for two distinct parameter vectors $\btheta$ and $\btheta'$, the expected periodograms $\overline{S}_{\Y}\sN(\omega;\btheta)$ and $\overline{