# Signals sampling

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 $$f_{max}$$.

• Do you need additional answers? – Laurent Duval Dec 30 '18 at 22:01

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.
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: