Differencing Method makes timeseries stationary - acts as for detrending (but not always). White noise is the main example of stationary data (but not always). Dicky-Fuller test for stationarity can help to proove stationarity or not. Stationarity (zero-mean homoscedastic noise) is the main assumption for timeseries analysis - is achieved with detrending & seasonality extraction from data series (also weighted for VaR), leaving residuals as noise.
As for (smooting) FILTERING (10.6.2 Wavelets): windowed Fourier transforms is used for analysis of stationary data. Wavelet analysis can be used for non-stationary data analysis (is more difficult to analyse non-stationary data). Anyway they (FT & WT) give the opportunity to consider both time-domain & frequency domain (as for time & frequency function).
Whether to denoise or not - depends on goals of your timeseries analysis. If the goal is to explore speed & direction of approximated data - than you should (for approximation or smoothing or forecasting). If the goal is to find outliers in the noise (some extreme values) - then you don't need to denoise your data.
The main difficulty of all AR-family methods of TS-analysis is that they are hugely parametric (as well as GARCH & ARCH). You can try to find MachineLearning solutions (e.g. Autoencoder LSTM based) or create NeuralNetwork (with lstm or convolution layers) as non-parametric solution [as of Bayesian Structural Time Series]. [need huge data - much more than 1000, used when need to solve the problem of "curse of dimensionality"]
But the main idea is to increase Signal-to-Noise ratio to get pure signal - & problem of Signal-to-noise ratio is generally solved in PCA. So, getting 1st PC - you further work with data free of noise (as of orthogonal projection), I suppose... further removing noise again (with FT or WT) seems unsuitable for me
(if I'm mistaken - hope somebody will correct)
here: "It does not eliminate noise, but it can reduce noise"
P.S. the simplier - the better... Often simple Exponential smoothing gives better forecast than ARIMA