Why spectral analysis is based on autocovariance function instead of oringinal sequence? I have some questions about the spectral analysis in time series. For a zero-mean and covariance stationary time series $\{X_t\}$ with autocovariance function $\gamma(h)=Cov(X_{t+h},X_t)$. The $\textbf{spectral density}$ is defined to be the Discrete-time Fourier transform of $\gamma(h)$
$$f(\lambda)=\frac{1}{2\pi}\sum_{h=-\infty}^\infty \gamma(h)e^{-ih\lambda}, -\infty<\lambda<\infty.$$
Then many studies or research are based on this spectral density formula. However, I am still confused about some conceptual problems.
First question: The autocovariance function can not represent a unique time series. Why do we use Fourier transform in the autocovariance function instead of the original time series? Wouldn't it be more straightforward and convenient to analyze the original sequence?
Second question: The original sequence can be transferred to the frequency domain by FFT. Then we do denoising, forecasting, or find seasonal cycles. Can these methods using in spectral density and eventually achieve the same effect as using FFT in the original sequence? If not, what can we do in spectral density?

Spectral analysis can be based on both the ACF (computing its Fourier transform) or the observations (computing the Fourier transform and taking the modulus squared). Both approaches are equivalent. Historically, the first was perhaps easier a faster before the introduction of the Fast Fourier Transform (FFT).
First question: True, an ACF does not uniquely determine a time series. Nor does the spectral density. The ACF contains enough information to compute the spectral density $f(\lambda)$. Many different time series can share the same spctral density.
Second question:  Not quite clear to me what is the question here. Yes, transferring the time series to the frequency domain we can do all the things you mention and more. The spectral density is not something that enables us "to do something to the series", but rather a way to summarize its harmonic properties. Based on what we see in the spectral density, we can design filters, pre-process de series, etc.
