Aliasing, Continuous sinusoids and discrete sinusoids....

[fog37]

TL;DR

Understand aliasing in relation to continuous and discrete sinusoids for computer simulations

Hello,

I understand that continuous sinusoids can have any arbitrary frequency $f$ and are always periodic with period $T=1/f$. A continuous sinusoid looks like this: $$x(t)= sin(2\pi f t+\theta_0)$$

On the other hand, discrete-time sinusoids are not always periodic. They are periodic only of an integer number $N$ exist such that $x(n)=x(n+N)$.

My dilemma: I understand that sampling a continuous sinusoid $x(t)= sin(2\pi f t+\theta_0)$ at a sampling rate $F_s=> 2f$ produces a discrete sinusoid $x(n)$ with special merits, i.e. its samples can be used to uniquely reconstruct, through frequency filtering and sinc interpolation, the original continuous sinusoid $x(t)$.

However, in many computer simulations, we are not concern with reconstructing the original continuous signal. We simply start with a discrete sinusoid (or other discrete signal), pass it through a filter/transformation/system to obtain an output discrete signal. The purpose of the simulation is to simulate the working of a physical system and its effect on an input signal to obtain a discrete output signal that is a good approximation of the actual continuous output signal.
That said, how does aliasing (which seems to be essential for the process of signal reconstruction) play a role in the context of computer simulation of signal processing? We are not trying to reconstruct signals in computer simulations...Does avoiding aliasing with the input signal prevent from processing a discrete input signal that is a bad approximate representation of the actual input signal to avoid obtaining an incorrect discrete representation of the actual and correct output signal?

 

[pbuk]

However, in many computer simulations, we are not concern with reconstructing the original continuous signal. We simply start with a discrete sinusoid (or other discrete signal), pass it through a filter/transformation/system to obtain an output discrete signal. The purpose of the simulation is to simulate the working of a physical system and its effect on an input signal to obtain a discrete output signal that is a good approximation of the actual continuous output signal.

That said, how does aliasing (which seems to be essential for the process of signal reconstruction) play a role in the context of computer simulation of signal processing? We are not trying to reconstruct signals in computer simulations...

I am not sure that I understand the difference between 'simulate a good approximation' and 'reconstruct'. Put simply, the maths tells us that sampling at a frequency of $2\omega$ preserves details with a frequency of $\le \omega$ so if that is enough detail then it is a good approximation.

Does avoiding aliasing with the input signal prevent from processing a discrete input signal that is a bad approximate representation of the actual input signal to avoid obtaining an incorrect discrete representation of the actual and correct output signal?

I'm not sure what you are asking, there are too many negatives and other qualifiers in one sentence (avoiding...prevent...bad...approximate...avoid...incorrect...). Note that as well as referring to the approximation of an input signal, the term 'aliasing' can also be used to refer to inaccuracies (specifically artifacts) introduced by that approximation (and anti-aliasing is the removal of these artifacts). I am not sure which meaning you are using.

 

[fog37]

Thank you pbuk,

Apologies for the poorly written question :)

Let me try to clarify. For example, let's say we intended to simulate how a continuous sinusoid $x(t)$ of frequency $f=3 Hz$ is processed by a linear system to determine the output signal $y(t)$. To do so, we would start with a discrete version $x(n)$ of the input sinusoid $x(t)$ and obtain a discrete output signal $y(n)$.

Do we need to worry about the concept of aliasing (and ensure that it is not happening and affecting our signals) for the input and output discrete signals? How do we ensure that aliasing is not present in this simulation? Simply by creating a discrete input signal whose samples are separated by a time interval $\Delta t < 1/6$ where the sampling rate is $F_s= 6$?

Thanks!

 

[pbuk]

Do we need to worry about the concept of aliasing (and ensure that it is not happening and affecting our signals) for the input and output discrete signals? How do we ensure that aliasing is not present in this simulation? Simply by creating a discrete input signal whose samples are separated by a time interval $\Delta t < 1/6$ where the sampling rate is $F_s= 6$?

OK, so you are using aliasing to mean aliasing artifacts. Think about what a 3Hz sine wave sampled at 6Hz could look like: if you are in phase the samples could all be zero! Or 1/12 s out of phase and you will get a square wave. In either case, nothing like the input - the aliasing process has created artificial information, hence the term aliasing artifact. Correcting this to get something back that is a sufficient approximation for a particular application is not trivial and is not my area of expertise - you might be better off in the Electrical Engineering forum asking 'Digital Signal Processing - how do I correct for sampling artifacts?'.