What is a filter?

A filter is a device or process which removes some of the unwanted features or components like frequencies from a signal in signal processing. The feature of filters is the complete or partial suppression of some part of the signal, and filtering is a class of signal processing. This most often means removing some frequency bands or some frequencies. In the frequency domain, the filters do not exclusively act, especially in the image processing field many other targets for filtering exist. Without having to act in the frequency domain, the correlations can be removed for certain frequency components and not for others. The filters are widely used in electronics and telecommunication, in television, radio, radar, audio recording, music synthesis, control systems, computer graphics, and image processing.

Linear continuous time filters

In the signal processing world, the linear continuous-time circuit is one of the most common meanings for a filter, and simply filter is taken often synonymous. These circuits are usually designed to remove certain types of frequencies while allowing others to pass. Those circuits which perform this function are usually linear in response or approximately so. The output signal containing frequency components not present in the input signal would result potentially if there is any non-linearity. The network synthesis stands for the modern design methodology for linear continuous-time filters.

Some of the important filter families are listed below:

• Chebyshev filter: For a specified order and ripple it has the best approximation of the ideal response.
• Butterworth filter: It is the type that has a flat frequency response.
• Bessel type filter: It has a flat phase delay.
• Elliptic filter: For a specified order or ripple it has the steepest cutoff of any filter.

Among these filter families, the difference is that they use all different polynomial functions to approximate the response of an ideal filter. This results in different transfer functions of each filter.

Terminology

There are a few terms that are used to describe and classify linear filters which are:

• Into several band forms, the frequency response can be classified describing which frequency bands the filters passes called as the passband and which it rejects called the stopband.
  • Low-pass filter: Here only low frequencies are passed and high frequencies are attenuated.
  • High-pass filter: Here high frequencies are passes whereas low frequencies are attenuated.
  • Band-pass filter: The frequencies which are there in a frequency band are only passed.
  • Band-stop/band-reject filter: The frequencies which are only at the frequency band are attenuated.
  • Notch filter: It rejects just one specific frequency. It is an extreme band-stop filter.
  • Comb filter: It has multiple spaced passbands which are narrow and which give the band form a comb-like appearance.
  • All-pass filter: Here all the frequencies are passed but the output phase is modified.
• The cutoff frequency is the type of frequency in which beyond this frequency the filter will not pass signals.
• Roll-off is the rate in which beyond the cutoff frequency the attenuation increases.
• The transition band is usually the narrow band of frequencies that is in between a passband and a stopband.
• Ripple is termed as the variation of the filter's insertion loss in the passband.
• The degree of approximating polynomial is the order of the filter and it corresponds to the number of elements required to build it in passive-type filters.

Technologies

Using a number of different technologies, the filters can be built. In several different ways, the same transfer function can be realized, that is, the mathematical properties are the same in filters but the physical properties are different. The components, often in different technologies are proportional to each other, and in the respective filters fulfill the same role. For instance, in electrical/electronics, the resistors, inductors, and capacitors are similar to the dampers, masses, and springs in mechanics.

The transfer function

In the domain of complex frequencies, the transfer function of a filter is most defined. To/from this domain, the back and forth passage is operated by the Laplace transform and its inverse. The transfer function H(s) of a filter is given by the ratio of output signal Y(s) to the input signal X(s) as a function of the "s" which is the complex frequency:

$H\left(s\right) = \frac{Y\left(s\right)}{X\left(s\right)} with s = \sigma + j\omega$

For the filters which are constructed by discrete components or lumped elements,

• Their transfer function is the ratio of polynomials in "s" that is a rational function of "s". Encountered in either the numerator or the denominator polynomial the order of the transfer function will be the highest power of "s".
• The transfer function polynomials will have all real coefficients and so the poles and zeroes of the transfer function will either be in the real form or can occur in complex conjugate pairs.
• The real parts of all poles will be negative since the filters are assumed to be stable that is they will lie in the left half-plane side in complex frequency space.

The Laplace transform is used to construct the transfer function and therefore the null initial conditions are assumed because $L\left\{\frac{df}{dt}\right\} = s . L \left\{f\right(t\left)\right\} - f\left(0\right)$ and when f(0) = 0 then we can get rid of the constant and can use the general expression which is $L\left\{\frac{df}{dt}\right\} = s. L\left\{f\right(t\left)\right\}$.

Context and Applications

This topic is significant in the professional exams for graduate and postgraduate courses:

• Bachelors in Electrical Engineering
• Bachelors in Electronics and Telecommunication
• Masters in Electrical Engineering
• Masters in Electronics and Telecommunication

Practice Problems

Q1. Which filter has the flat frequency response?

1. Chebyshev filter
2. Butterworth filter
3. Bessel filter
4. Elliptic filter

Answer: Option b

Explanation: Butterworth filter is the type that has a flat frequency response.

Q2. Which type of filter passes only low frequencies and attenuates the higher one?

1. Low-pass filter
2. High-pass filter
3. All-pass filter
4. Band-reject filter

Answer: Option a

Explanation: In the low-pass filter, only low frequencies are passed and high frequencies are attenuated.

Q3. Which filter passes the high frequencies and attenuates the low frequencies?

1. Low-pass filter
2. High-pass filter
3. Band-reject filter
4. All-pass filter

Answer: Option b

Explanation: In a high pass filter, high frequencies are passed and low frequencies are attenuated.

Q4. Which type of filter passes all the frequencies?

1. Low-pass filter
2. High-pass filter
3. All-pass filter
4. Band-reject filter

Answer: Option c

Explanation: In an all-pass filter, all frequencies are passed but the output phase is modified.

Q5. What is the ratio in a transfer function?

1. The ratio of output to input
2. The ratio of input to output
3. The ratio of input to the square of the output
4. The ratio of output to the square of the input

Answer: Option a

Explanation: The transfer function H(s) of a filter is given by the ratio of output signal Y(s) to the input signal X(s) as a function of the "s" which is the complex frequency.

Related Concepts

• Electronic filter topology
• Lifter
• Noise reduction
• Smoothing