Signals sampling

Question

I have a simple question, but sadly I'm kind of "noob" in signals theory. A signal having 4 harmonics at the following frequencies: 1 kHz, 2 kHz, 3.5kHz and 4.2 kHz. (How can a signal have harmonics "without" a fundamental frequency?) Find the minimum sampling frequency so that the signal can be completely recovered from its sampled version.

I think I should fint $f_s = 2 f_{max}$ but I do not know how to find $f_{max}$.

Please somebody help me.

Answer 1

This exercise is a verbatim copy from 5.C.1. in document Lab 5: SAMPLING OF SIGNALS, dating from 27/11/2017. This document apparently contains a "lecture" part, that "could" help you finding the answer. Unfortunately, some assertions in this document belong to folk signal processing, such as (5.4):

The sampling frequency ($f_p$) must be greater than twice the highest frequency of $x(t)$: $f_s > 2 f_{\max}$

There is a typo in the text, as $f_p$ and $f_s$ refer to the same sampling frequency concept. The "must" part is not true. Anyway, there are three levels of possible answers. To start with, an harmonic is a common name for a sinusoidal signal defined by a frequency, an amplitude and a phase. However, a signal with harmonics, i.e. a signal containing multiples of a fundamental frequency $f_0$, i.e. $kf_0$, $k\in \mathbb{N}^*$ can exist without the fundamental at $k=1$ (if I remember well). You can read (and hear) more at Physics of music, notes: The missing fundamental or The Well-Tempered Timpani, In Search of the Missing Fundamental: The Missing Fundamental.

Now, let us proceed with the answers.

1. The most probably expected answer is twice the maximum frequency among the four harmonics, which is the basic Nyquist
2. A more involved form considers the frequency span, between 1 kHz and 4.2 kHz, as you can further reduce the rate using an $f_{\min}$-$f_{\max}$ diagram, eluded to in Confusion regarding Nyquist Sampling Theorem or here, from Vaughan et al., 1991, The theory of bandpass sampling:
3. The last I can think of further use the fact than only four harmonics are considered, thus the signal is sparse in the Fourier domain, and compressive sensing can offer even lower rate sampling.