Filter Characteristics of Linear Systems | Signals and Systems - Electronics and Communication Engineering (ECE) PDF Download

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

Filter Characteristics of Linear Systems

  • 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,
    Filter Characteristics of Linear Systems | Signals and Systems - Electronics and Communication Engineering (ECE)
    Or, 
    Filter Characteristics of Linear Systems | Signals and Systems - Electronics and Communication Engineering (ECE)
    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.

Question for Filter Characteristics of Linear Systems
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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 −

Types of Filters

Types of FiltersTypes of Filters

  1. 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).
  2. 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).
  3. 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).
  4. When the LTI system rejects only a particular band of frequencies and allows all other frequency components. The system is called band rejection filter (BRF).

Question for Filter Characteristics of Linear Systems
Try yourself:Which one is LPF Filter?
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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.

Response Curves of Filters

  • Response curves are utilized to depict the behavior of a filter. A response curve is essentially a graphical representation that illustrates the relationship between the attenuation ratio (VOUT / VIN) and frequency (as depicted in Figure below). 
  • Attenuation is frequently quantified in decibels (dB), while frequency can be represented in two ways: angular form (ω) measured in radians per second (rad/s) or the more widely used form, denoted as 'f,' which is measured in Hertz (Hz) or cycles per second. These two representations are linked by the equation ω = 2πf. 
  • Ultimately, filter response curves can be presented in three formats: linear-linear, log-linear, or log-log. The most commonly adopted approach is to display decibels on the vertical (y) axis and logarithmic frequency on the horizontal (x) axis.

Filter Characteristics of Linear Systems | Signals and Systems - Electronics and Communication Engineering (ECE)

Please note: A notch filter is a type of bandstop filter characterized by a narrow bandstop bandwidth. Notch filters are employed to reduce the amplitude of a specific range of frequencies.

Here are some common technical terms used in describing filter response curves:

  • -3dB Frequency (f3dB): This term, pronounced as "minus 3dB frequency," corresponds to the input frequency at which the output signal's amplitude decreases by -3dB compared to the input signal. The -3dB frequency is also known as the cutoff frequency. It's the point where the output power is reduced by half, often called the "half-power frequency." This frequency is where the output voltage equals the input voltage multiplied by 1/√2. In low-pass and high-pass filters, there is a single -3dB frequency. However, band-pass and notch filters have two -3dB frequencies, typically referred to as f1 and f2.
  • Center frequency (f0): The center frequency, used in band-pass and notch filters, is a central frequency situated between the upper and lower cutoff frequencies. The center frequency is typically calculated as the arithmetic mean or geometric mean of the lower and upper cutoff frequencies.
  • Bandwidth (β or B.W.): Bandwidth refers to the width of the passband, which is the range of frequencies that experience minimal attenuation when passing through the filter from input to output.
  • Stopband frequency (fs): This denotes a specific frequency at which the attenuation reaches a predefined level.
    For low-pass and high-pass filters, frequencies beyond the stopband frequency are termed the stopband.
    For band-pass and notch filters, two stopband frequencies exist, and the frequencies in between are also known as the stopband.
  • Quality factor (Q): The quality factor of a filter describes its damping characteristics. In the time domain, damping corresponds to the amount of oscillation in the system's step response. In the frequency domain, higher Q values result in more prominent peaking (either positive or negative) in the system's magnitude response. For band-pass or notch filters, Q represents the ratio between the center frequency and the -3dB bandwidth (the difference between f1 and f2).
    For both band-pass and notch filters:
    Q = f0 / (f2 - f1)
The document Filter Characteristics of Linear Systems | Signals and Systems - Electronics and Communication Engineering (ECE) is a part of the Electronics and Communication Engineering (ECE) Course Signals and Systems.
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FAQs on Filter Characteristics of Linear Systems - Signals and Systems - Electronics and Communication Engineering (ECE)

1. What are the different types of filters used in electrical engineering?
Ans. In electrical engineering, there are several types of filters commonly used. Some of the main types include: - Low-pass filter: This type of filter allows low-frequency signals to pass through while attenuating higher-frequency signals. - High-pass filter: This filter does the opposite of a low-pass filter, allowing high-frequency signals to pass through while attenuating lower-frequency signals. - Band-pass filter: This filter allows a specific range of frequencies to pass through while attenuating frequencies outside that range. - Band-stop filter: Also known as a notch filter, it attenuates a specific range of frequencies while allowing all other frequencies to pass through. - All-pass filter: This filter allows all frequencies to pass through but alters the phase relationship between the input and output signals.
2. What are the characteristics of linear systems in filter design?
Ans. Linear systems in filter design possess several important characteristics. Some key characteristics include: - Linearity: A linear system obeys the principle of superposition, meaning that the output response is directly proportional to the input response. This property enables easier analysis and design of filters. - Time-invariance: A linear system's response remains constant over time. This allows for consistent performance of the filter under varying input signals. - Additivity: The output response of a linear system can be obtained by adding the individual responses to each input component. This property is crucial in the analysis and design of filters. - Homogeneity: A linear system exhibits constant scaling behavior, meaning that scaling the input signal results in a proportional scaling of the output signal. - Stability: A linear system is considered stable if its output response remains bounded for any bounded input signal. Stability is a critical characteristic for reliable filter operation.
3. How do low-pass filters work in electrical engineering?
Ans. Low-pass filters are commonly used in electrical engineering to allow low-frequency signals to pass through while attenuating higher-frequency signals. They work based on the principle of frequency selectivity. A low-pass filter is designed with a cutoff frequency, which is the frequency at which the filter begins to attenuate the input signal. Frequencies below the cutoff frequency are passed through with minimal attenuation, while frequencies above the cutoff are progressively attenuated. The specific characteristics of a low-pass filter, such as the rate of attenuation beyond the cutoff frequency, are determined by the filter design parameters. These parameters may include the filter order, which determines the steepness of the cutoff, and the filter response, which can be Butterworth, Chebyshev, or other types. Low-pass filters find applications in various domains, such as audio processing, communication systems, and image processing, where it is essential to remove high-frequency noise or unwanted signals.
4. What is the purpose of a band-pass filter in electrical engineering?
Ans. A band-pass filter is a type of filter used in electrical engineering to allow a specific range of frequencies to pass through while attenuating frequencies outside that range. It is commonly used for signal processing applications where only a particular band of frequencies is of interest. The band-pass filter consists of a combination of a low-pass filter and a high-pass filter. The low-pass filter attenuates frequencies above the desired passband, while the high-pass filter attenuates frequencies below the desired passband. The combined effect results in a band-pass characteristic. Band-pass filters are utilized in various applications, such as radio and television receivers, wireless communication systems, audio equalizers, and biomedical signal processing. They enable the extraction or isolation of a specific frequency range from a complex signal, facilitating the analysis and manipulation of signals in specific frequency bands.
5. What is the significance of stability in filter design?
Ans. Stability is a crucial aspect in filter design as it ensures the reliable and consistent operation of the filter. A stable filter is one that produces a bounded output response for any bounded input signal. In the context of filter design, stability ensures that the filter does not introduce unwanted oscillations, overshoot, or instability in the output response. Unstable filters can lead to unpredictable behavior, distortion, or even damage to the system. The stability of a filter can be analyzed using various techniques, such as the transfer function, difference equations, or the Bode plot. Stability analysis helps determine the appropriate filter design parameters and ensures that the filter meets the desired specifications without compromising its stability. In practical applications, stability is of utmost importance to maintain the integrity of the filtered signals and to prevent any adverse effects on subsequent stages or systems that rely on the filter output.
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