# minimum frequency in periodic components

Suppose we have the following signal model

$y(t) = A_1\sin(\omega_1 t+\phi_1) + A_2\sin(\omega_2 t+\phi_2) + ... + A_p\sin(\omega_p t+\phi_p) + z(t)$

When we are sampling at sampling frequency $f_\mathrm s$ we know that the sampling frequency must be at least 2 times the maximum frequency, but what about a condition for the minimum frequency (largest period)? Is there any reasonable limit below which it is useless existence of frequency? Suppose that the sampling frequency is $100\,\text{Hz}$, then the maximum frequency can be $50\,\text{Hz}$, but what about the minimum frequency?

• Why would there be a minimum frequency? – ThP Aug 15 '14 at 7:00
• no it is just curious question – dato datuashvili Aug 15 '14 at 7:05

No, there is no lower bound to the sampling theorem. Any signal $x(t)$ sampled at rate $f_\mathrm s$ can be reconstructed from its sampled version if it contains only spectral components lower than (1) $f_\mathrm{s}/2$. I.e. the frequencies contained in $x(t)$ are allowed to be as low as 0 Hz. Note however, that perfect reconstruction of $x(t)$ is only possible if an infinite number of samples $N$ has been drawn from $x(t)$.

When it comes to spectral analysis of $x(t)$ using its sampled version $x_n = x(n/f_\mathrm s)$ the number of acquired samples $N$ plays an important role because it determines the frequency resolution of the DFT. In order to be still distinguishable the lowest frequency cannot be arbitrarily low. This is intuitively understandable as the lower the frequency the longer the period and thus the signal should be measured for a sufficently long time. Here is some further reading about DFT frequency resolution and spectral leakage

Number of FFT points required for a specific frequency resolution for an oversampled signal

FFT of sine wave not coming as expected i.e single point

(1) Actually $x(t)$ must not contain a sine wave with exact frequency $f_\mathrm s$. For an interesting discussion here on dsp.se see Nyquist Frequency Phase Shift

• by the way noise is always affects right frequencies,that means if frequencies are 100 and 200,we can't exactly recovery these numbers right because of noise? – dato datuashvili Aug 15 '14 at 8:18
• Noise can make spectral analysis harder. For example it can cover the frequency peaks that should be detected or create frequency peaks that aren't present in the signal. It will never shift the signal frequencies though – Deve Aug 15 '14 at 15:56
• when you are estimating frequencies,then you will not get exact values of frequencies,but close to existed frequencies – dato datuashvili Aug 15 '14 at 15:57
• that is clear,but i need your help in dsp.stackexchange.com/questions/17789/… could you help me please? – dato datuashvili Aug 15 '14 at 16:01
• Generally not - due to spectral leakage. But if the frequency you're trying to detect happens to fall on the frequency grid of the DFT the exact frequency can be detected – Deve Aug 15 '14 at 16:03