Interchanging two linear operators?

[rsashwinkumar]

In general if two operators, say d1 and d2, are linear, then d1(d2(f(x)) = d2(d1(f(x)), assuming convergence for f(x). So as long as the two operators are linear, the operators can be interchanged (as far as i understood). Now consider two linear operators linear Low Pass Filtering (mathematically represented by the convolution integral) and sampling (represented mathematically by multiplication with a periodic dirac delta train of impulses). Now here though the two operators are linear, it is not correct to interchange the two. As generally Low pass filtering needs to be done before sampling to avoid aliasing, and the opposite (first sample and then do filtering) does not provide the same result. What is the point that I am missing? Please help me out...

 

[albbg]

Sampling isn't a linear operation.

 

[rsashwinkumar]

If x(t) sampled at every Ts gives x(nTs), a*x(t) gives a*x(nTs) and a linear combination of two inputs x1 and x2 too gives the same linear combination at the output. So Going by definition of linearity, sampling is linear i think, but it is not time invariant.

 

[albbg]

The sampling process is not LTI since it generates infinite replica of the spectrum

 

[rsashwinkumar]

Yes agreed. But sampling is still a linear operation. And as far as i read about in mathematics, two "linear" operators can be interchanged; i couldnt find any reference to time invariance/variance of an operator.

 

[albbg]

I don't agree with you. Only non-linear operators can generate frequencies other than fundamental.
The aim of the filterig is to eliminate the frequencies falling outside the Nyquist BW that the sampling otherwise will fold (that means a non-linear process) into the signal spectrum.

 

[rsashwinkumar]

A non-linear system generates only harmonics of the input tone (not considering the beat frequencies), but a periodically time varying system (still linear) generates tones at k*fs+fin (where fs is the fundamental frequency of the periodicity of the system). Since sampling does not generate harmonics of input tone but just modulates the spectrum, it is a Linear Periodically Time Varying System. So I would dare to call it a linear operation.

My question is that though we cant interchange sampling and filtering, mathematically since it is possible to define the two as linear operators and as far as I referred, linear operators can be interchanged and so it sounds like one can get the same functionality in the two cases. But I am missing something in this argument. I wanted to know what is it.

 

[albbg]

Yes but this definition of linearity only apply if you are operating on a band limited signal having the maximum frequency not violating the nyquist theorem, otherwise due to the spectrum folding, the linearity requirements are not fullfilled.
That means the system Filter+Ideal sampler is linear, ideal sampler operating on band limited signal (as said before) is still linear but and ideal sampler in general can be not-linear depending from the BW of the input signal.
If you have a limited BW signal you can swap filter <-> sampler