# Shouldn't the Sampling Theorem imply that there should be no information loss at all after a signal is processed?

I am very, very new to signal processing (only started a few days ago) so please bear with me because I may be missing the big picture.

Consider an arbitrary signal $x(t)$ with varying frequency to be processed by a computer.

Given the right sampling rate w.r.t the frequency, shouldn't the Sampling Theorem essentially imply that no information will be lost at all after the sampled signal is converted back to its continuous form, since there will be a sufficient number of discrete points for which the signal is described?

• That's true as long as the signal $x(t)$ is band-limited, and as long as the sampling frequency satisfies $f_s>2B$, where $B$ is the bandwidth of $x(t)$. – Matt L. Jan 6 '15 at 17:26
• To expand a bit on @MattL.'s comment: the assumption that $x(t)$ is band-limited implies that its duration is infinite; that is, there is no time before which (and/or after which) the signal is 0. This means that, in practice, you always lose something in the transition from analog to digital and back. – MBaz Jan 6 '15 at 17:37
• And to expand even a bit more, since there is no ideal reconstruction filter when going from digital to analog, you again lose something. And we still haven't mentioned amplitude quantization, which is an essential part of digitizing an analog signal. The sampling theorem only talks about time-discretization, but not about quantization. – Matt L. Jan 6 '15 at 18:00
• @MattL. Yes. The reason I singled out the band-limited requirement is that I think it's special: even if you assume everything is ideal, when sampling any signal of practical interest, something is going to be lost in translation. I know it's just a minor point, but I've seen many students who don't get it, which is why I like to bring it up. – MBaz Jan 6 '15 at 18:21
• @MattL. I see! I understand. So why does a band-limited signal imply that its duration is infinite as MBaz pointed out? – edgaralienfoe Jan 7 '15 at 14:23