What does the amplitude in a frequency response of a signal signify?

Well according to my understanding it tells the concentration of signals at a particular frequency but what does the number really mean? Suppose in a frequency response graph i have such a relation :: "a1"(amplitude) at "f1" (frequency) a2 at f2 , a3 at f3. What do i infer from it?

• "Concentration" is too imprecise: it'd be better to give a precise quantity, such as voltage, energy or power. Try this yourself: calculate the FT (or the FS) of $A\cos(2\pi ft)$, and see what is the amplitude of the coefficients you get. – MBaz Oct 15 '18 at 21:59
• Signals don't have frequency responses, they have frequency spectra; it is systems that have frequency responses. For a a linear time-invariant system system, if the frequency response has amplitude $A$ at frequency $f_0$ (note that $A$ is a (possibly negative) real number), then it tells you that if the input to the system is the signal $\cos(j2\pi f_0t)$, the output signal is $A\cos(j2\pi f_0t + \theta)$, that is, the system has a gain of $|A|$ at frequency $f_0$. – Dilip Sarwate Oct 16 '18 at 1:13

If you have a time series and compute the DFT, this will give you the concentration of energy at each discrete frequency point you have chosen. When you compute the PSD of a signal in MATLAB (using pwelch or cpsd), you'll get units of power/frequency. This describes the amount of energy or power in that specific frequency range (bin). If instead your goal is to relate the amplitudes of sinusoids in a time series directly to the values that you're reading off the peaks in the PSD plot, you need to ensure that your FFT is appropriately scaled. In the MATLAB functions cpsd() or pwelch(), the FFT is scaled such that the $$\textit{total energy}$$ in the signal is preserved (done by dividing by the FFT by the sampling period, $$t_s$$). If you want to recover the original amplitudes of a signal (e.g., $$x=A_1 sin(\omega_1*pi*t)+A_2 sin(\omega_2*pi*t))$$, you would want to ensure that the FFT is scaled by the duration of the signal, $$T$$.