# 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

## 1 Answer

For no information to be lost on conversion back to continuous form, the signal would first need to be perfectly band-limited, and you would need an ideal reconstruction filter. A perfectly band-limited signal is infinite in extent. Since you want this arbitrary signal to be processed by a computer, your computer would need infinite memory. You would also have to wait an infinite amount of time for your perfect reconstruction filter to settle.

Finite sampling time and filter length problems also completely ignores other potential lossy issues, such as quantization and sampling clock jitter, etc.

One might be able to process an infinite signal that can be represented in certain closed symbolic forms within finite time and resources, but there does not seem to be a method to convert any arbitrary signal into such a form.

Back in the real-world (stuff you can buy or make), one would normally accept an information loss from imperfect band-limiting, finite filter length, jitter (etc.) that is around or below the quantization and numeric noise floors. This allows the processing to happen using an amount of RAM that one can afford to hopefully finish within one's lifetime. Thus leading to information loss and imperfect reconstruction.

• But all analog devices have noise, too, so nothing is perfect anyway. – endolith Jan 6 '15 at 20:49