Question on Spotting Aliasing

[magomago]

I’ve been studying some signals recently (and will probably continue this for a while as I'm drawn to it) and came across this:

“To avoid aliasing when defining a decaying sinusoid such as x(t) = cos( w*t +b)* exp(-at) choose deltaT = pi/ (4*sqrt(a^2+w^2))
They quoted an example and said to verify this rule by plotting the following function with the following interval: x(t) exp(-.1*t).*cos(pi.*t) Intervals of .1,.5,1.5,2.

The computed minimum time interval was .2499 ~ .25

So I expected to see aliasing on steps of .5 , 1.5, and 2

Here is my output...should I have seen aliasing at .5? I don't see aliasing of the signal at all. I plotted it out to t=200, and even at an amplitude of 10^-8, it still looked fine. Clearly steps of 1.5 and 2 are just too big and completely alter the signal.

Anyone understand what I don't see? I'd love to understand why the minimum step is calculated as .25, yet I'm looking fine at steps of .5

 

[TecHNooB]

Nyquist for cos(pi*t) is 1 second. So the aliasing probably has to do with the exponential term. I'd elaborate more but I'm still in bed :| zzzzzzzzzzzzzz

 

[Matthiasa]

Could always do a Fourier transform of the sampled data system to see where the frequency bands fall.

Just because it's aliased doesn't necessarily mean that a large amount of data is lost.
Filtering tends to be used to remove those higher frequencies(the ones being aliased) due to frequency band limitations in real systems with minimal change.

 

[Mark R]

What happens if you offset the sampled time points?

So, instead of sampling at 0, t, 2t, 3t, 4t...nt, sample at 0.1, t+0.1, 2t+0.1...

What is the minimum t that doesn't show aliasing at any offset?

 

[Matthiasa]

Aliasing is independent of offset.

But for this question there is actually no T value that fully satisfies it.
The equation here has infinite bandwidth, though much at levels to low to matter.