Forecasting with ARMA models, from a filter point of view ARMA
 models are afaik just filters with transfer function
$ {MA(z) \over AR(z)}  \equiv {FIR(z) \over IIR(z)} $ .
However forecasters of stock prices, market trends ...
seem to be mainly statisticians, with their own vocabulary and culture.
For example, "signal-to-noise ratio" is rarely mentioned;
for another, differencing must increase noise.
Can anyone suggest either


textbooks or introductory courses on ARMA forecasting
from a filter or signal processing point of view
websites with real time series and running code to ARMA-model them ?


(I'm interested in ARMA models for prediction, not in spectral analysis as such.
ARMA models may well be wrong for prediction from short, noisy data --
what will the economy do next year ? --
hence the need for real examples.)


Added: Some 40 years ago, R.W. Hamming wrote
in Digital Filters:


  ... We have a predicting filter without finding, or even talking about, the transfer function.
  Statisticians often do this,
  and neglect to examine the corresponding transfer function of the formula,
  which can often shed some light on the whole system.
ARMA divides the signal into two parts and that models the two parts.  

Financial time series are corrupted by different types of correlated and uncorrelated noises with definite functions that allow modeling and others more difficult as per having to use aproximations. In addition financial or economic times series may exhibit long memory processes (Black noise) or may have other long range dependencies or other non stationary issues. It is for this reason that besides the MA part it is hard to tell if a series is stochastic or deterministically chaotic or a mix between these.

The two terms of ARMA include an autoregresive term and a moving average term. The deterministic part of time series is modeled with ARIMA, ARFIMA etc family of models and the stockastic part using ARCH and GARCH family of models (usually volatility is modeled this way as a large portion of the stockastic part is white noise which can not be forecasted but its variance can).    

Diferenciation as you mention may eliminate some of the signal, however is necesary for regression in many cases as to make a time series weak sense stationary which is a requirement to have a meaningfull and valid regression. 

Signal to noise ratio is hard to guess for any unknonw noise+signal composition. However there are some ways to aproximate this by determining what noises affect your signal and what weights of noise correspond to each signal as well.

I would sugest from a signal procesing point to google / take a look at "financial / non stationary time series signal procesing" papers. There are many methods and ways as to extract information from a signal. 

I would sugest from a modeling point of view that you look at ARIMA or GARCH presentations from academia or papers from leading economic schools (LSE etc).

I will also sugest taking a look / google data preprocesing for regression (including the terms of unbalanced data set, stationarity, normality, linearity, aditivity and serial correlation, homocedasticity, spurious regression, cointegration, transformation of variables for financial time series).