Sinusoidal signals and filtering

Any sinusoidal time signal can be expressed as $x(t)=X_0+M\;\sin(\omega t+\phi)$ containing:

• DC (direct current) component $X_0$
• amplitude $M$
• frequency $\omega=2\pi f$
• phase $\phi$

The AC (alternative current) component of the signal can be decomposed into the sum of sine and cosine signals without phase shift:

$$M\;\sin(\omega t+\phi)=A\;\sin (\omega t) + B\;\cos (\omega t)$$

$M$ and $\phi$ can be represented by the coefficients $A$ and $B$ as shown below. The expression above can be rewritten as:

$$A\;\sin (\omega t) + B\;\cos (\omega t) =\sqrt{A^2+B^2}[\fra... ...^2}} \sin (\omega t)+\frac{B}{\sqrt{A^2+B^2}} \cos (\omega t)]$$

If we let

$$\frac{A}{\sqrt{A^2+B^2}}=cos \phi,\;\;\;\;\frac{B}{\sqrt{A^2+B^2}}=sin \phi$$

then the expression above becomes

$$A\;\sin (\omega t) + B\;\cos (\omega t) =\sqrt{A^2+B^2} [ \c... ...n (\omega t)+\sin \phi \cos (\omega t)] =M \sin(\omega t+\phi)$$

It is seen that coefficients $A$ and $B$ are related to amplitude and phase by:

$$M=\sqrt{A^2+B^2},\;\;\;\;\;\;\;\phi=\tan^{-1} B/A$$

Moreover, according to Fourier theory, any signal $x(t)$ can be expressed as a linear combination of many sinusoidal components of different frequency, amplitude, and phase:

$$x(t)=X_0+\sum_{k=1}^\infty [ A_k \cos(k\omega_0 t)+ B_k \sin(k\omega_0 t) ]$$

where $X_0$ is the average of the signal $x(t)$, also called DC (direct current) component, and $k\omega_0$ ( $k=1, 2, \cdots,$) is the frequency of the kth component (also called the kth harmonic). For example, the signal below contains two frequency components:

$$x(t)=4 \sin(t) + \sin(10 \; t)$$

Sometimes the low frequency component may be caused by undesired noise (e.g., 60 Hz interference) and the high frequency is the desired signal, while some other times the low frequency may be the signal of interest and the high frequency is the noise. Depending on the specific application, the signal $x(t)$ may need to be filtered to keep the desired signal while removing the undesired noise, by a low-pass, high-pass, band-pass or a band-stop filter, as shown below: