Alias frequency Formula I'm taking a multimedia systems class in my MSc Computer Science, and I'm having some trouble understanding the formula for the alias frequency - this could stem from my misunderstanding of the alias signal.

My understanding of an alias signal is that if you undersample your input signal (i.e. sample at a rate which is less than twice the maximum frequency) then we can get aliasing because we're not sampling frequently enough to capture the high frequency details.  The aliasing signal is the result of taking these sample values and joining them with a smooth curve.  

Therefore, the resulting signal has a frequency of half the sampling frequency, since a pure sinusoid will need two samples per oscillation (1 for each turning point) - this would mean that the alias frequency should just be a function of the sampling frequency.

The formula for the alias frequency is the absolute difference of the signal frequency and the closest integer multiple of the sampling frequency - can someone explain this to me?  Thanks in advance!
Suppose that the sampling is done at a rate of $1000$ Hz, one sample every millisecond. Suppose also that the signal being sampled is at $3200$ Hz,
the first sample is at the peak of the sinusoid. The next sample,
will be taken one millisecond later during which time the sinusoid will 
have gone through $3.2$ periods, and so the next sample will have the 
same value as if the sinusoid had gone through $0.2$ periods, not $3.2$ 
periods. The one after that will be $0.4$ periods away from the peak, and
so on.  This is exactly the same set of samples that we would gave gotten
if we had been sampling a $200$ Hz sinusoid.  In one millisecond it would have
progressed through $0.2$ of its period of $5$ milliseconds and so on.
In other word, just by looking at the samples alone we cannot tell whether
the samples came from a $3200$ HZ signal or from a $200$ Hz signal.

If the signal being sampled was at $2800$ Hz, then we would get samples
corresponding to $0$, $-0.2$ of the period, $-0.4$ of the period and
so on.  But because sinusoids look the same in either direction in time,
these samples also look like they are the result of sampling a $200$
Hz signal.  This is the reason why the formula you are given, viz.


  Aliased frequency is the absolute difference between the actual signal frequency
  and the nearest integer multiple of the sampling frequency.


works to give you the right answer.