Signals and Systems – Filter Characteristics of Linear Systems

Linear System – A system for which the principle of superposition and the principle of homogeneity is valid is called a linear system.

Filter Characteristics of Linear System

For a given linear system, an input signal 𝑥(𝑡) produces a response signal 𝑦(𝑡). Therefore, the system processes the input signal 𝑥(𝑡) according to the characteristics of system. The spectral density function of the input signal 𝑥(𝑡) is given by 𝑋(𝑠) in s-domain or 𝑋(𝜔) in frequency domain. Also, the spectral density function of the response signal 𝑦(𝑡) is given by 𝑌(𝑠) in s-domain and 𝑌(𝜔) in frequency domain. Therefore,

$$\mathrm{Y\left ( s \right )=H\left ( s \right )\cdot X\left ( s \right )}$$

Or,

$$\mathrm{Y\left ( \omega \right )=H\left ( \omega \right )\cdot X\left ( \omega \right )}$$

Where, 𝐻(𝑠) or 𝐻(𝜔) is the transfer function of the system.

Thus, the system modifies the spectral density function of the input signal. The linear system acts as a filter for various frequency components, i.e., some frequency components are amplified and some frequency components are attenuated. Also, some frequency components may remain unaffected.

Similarly, each frequency component suffers a different amount of phase shift in the process of transmission. Hence, the system modifies the spectral density function of the input signal according to its filter characteristics. This modification is performed according to the transfer function 𝐻(𝑠) or 𝐻(𝜔) of the system. The transfer function represents the response of the system for various frequency components. The transfer function 𝐻(𝜔) acts as a weighted function or spectral shaping function to the different frequency components in the input signal. Therefore, an LTI system acts as a filter.

Depending upon the response of the system for various frequency components of the input signal, i.e., filter characteristics, a given linear system can act as following types of filter −

• When the LTI system allows the transmission of only low frequency components and blocks all the high frequency components. Then, the system is called the low-pass filter (LPF).

• When the LTI system allows the transmission of only high frequency components and blocks all the low frequency components, the system is called the high-pass filter (HPF).

• When the system allows the transmission of only a particular band of frequencies and blocks all other frequency components. Then, the system is called the band-pass filter (BPF).

• When the LTI system rejects only a particular band of frequencies and allows all other frequency components. The system is called bandrejection filter (BRF).

The band of frequency components which is allowed by the filter is called passband and the band of frequency components which is not allowed to pass through the filter is called stop-band or rejection-band.