Complexity of the IR Spectra of Polyatomic Molecules:

In the light of the discussion above we could say that the infrared spectrum of a polyatomic molecule would consist of an absorption band at every of the (3N - 6) or (3N - 5) fundamental frequencies for nonlinear and linear molecules, correspondingly along with the corresponding overtones. Therefore, further, we might also have weak bands because of combination of two fundamentals vibrations known as combination bands or difference of two fundamentals called difference bands. More so, while sometimes a fundamental and an overtone have frequencies extremely close to each other, these might resonate leading to the appearance of signals at frequencies higher and lower associative to the original. This phenomenon is known as Fermi resonance.

As the number of atoms within a molecule increases, the spectrum becomes quite complex. Let us take instance of a larger molecule say n-decane, C₁₀H₂₂, having 32 atoms. A number of fundamental vibrational modes would be 3(32) - 6 = 96 -   6 = 90. Along with the corresponding overtones and combination bands the spectrum is expected to have a few hundred signals. Thus, the actual spectrum has much fewer of them. This is so since all the vibrations might not be IR active. You would recall in which for a vibration mode to be IR active it must be related with a change in the dipole moment. Other purpose for not observing so several signals could be that either they are too weak or are outside the range of the instrument.