Fourier Spectral Analysis for Addition, Multiplication, Convolution, Differentiation, and Integration
We all know that for Fourier spectral analysis, you will get an impulse of significant magnitude at the frequency values of sine or cosine waves. Here, we will discuss the spectral analysis or frequency components when we add, subtract, multiply, convolve, integrate or differentiate signals, etc.
For sinusoidal signals
Addition
When we add two sinusoidal signals, the frequency components of the combined signal will be the frequencies of the individual signals.
Subtraction
Again, for subtraction, the frequency components of the combined signal will be the frequencies of the individual signals. But if the amplitudes and periods for both signals are the same, then the combined signal will be null.
Multiplication
The frequency domain components for the time domain multiplication of two sinusoidal signals will be f1 ± f2, where f1 is the frequency of the first sinusoidal signal and f2 is for the other.
Convolution
The frequency domain components for the time-domain convolution of two sinusoidal signals will be centered at the original sinusoidal signal frequencies.
Differentiation
For a signal's differentiation, the signal's frequency components will be the same as the original signal, but its amplitude will be scaled by j.ω.
Integration
For integration of a signal, the frequency components of the signal will be the same as the original signal, but its amplitude will be scaled by 1 / j.ω