filter design

Band-stop filter

In signal processing, a band-stop filter or band-rejection filter is a filter that passes most frequencies unaltered, but attenuates those in a range to very low levels. It is the opposite of a band-pass filter. A notch filter is a band-stop filter with a narrow stopband (high Q factor).

Other names include band limit filter, T-notch filter, band-elimination filter, and band-rejection filter.

Typically, the width of the stopband is less than 1 to 2 decades (that is, the highest frequency attenuated is less than 10 to 100 times the lowest frequency attenuated). In the audio band, a notch filter uses high and low frequencies that may be only semitones apart.

RF example 1: Non-linearities of power amplifiers For instance, when measuring non-linearities of power amplifiers a very narrow notch filter could be very useful to avoid the carrier so maximum input power will not be exceeded.

High-pass filter

A high-pass filter is a filter that passes high frequencies well, but attenuates (or reduces) frequencies lower than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a low-cut filter; the terms bass-cut filter or rumble filter are also used in audio applications. A high-pass filter is the opposite of a low-pass filter, and a bandpass filter is a combination of a high-pass and a low-pass.

It is useful as a filter to block any unwanted low frequency components of a complex signal while passing the higher frequencies. Of course, the meanings of low and high frequencies are relative to the cutoff frequency chosen by the filter designer.

Applications: Such a filter could be used to direct high frequencies to a tweeter speaker while blocking bass signals which could interfere with or damage the speaker. A low-pass filter, using a coil instead of a capacitor, could simultaneously be used to direct low frequencies to the woofer. See audio crossover.

High-pass and low-pass filters are also used in digital image processing to perform transformations in the frequency domain.

Most high-pass filters have zero gain (-inf dB) at DC. Such a high-pass filter with very low cutoff frequency can be used to block DC from a signal that is undesired in that signal (and pass nearly everything else). These are sometimes called DC blocking filters.

Low-pass filter

A low-pass filter is a filter that passes low frequencies well, but attenuates (or reduces) frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter when used in audio applications.

A high-pass filter is the opposite, and a bandpass filter is a combination of a high-pass and a low-pass.

The concept of a low-pass filter exists in many different forms, including electronic circuits (like a hiss filter used in audio), digital algorithms for smoothing sets of data, acoustic barriers, blurring of images, and so on. Low-pass filters play the same role in signal processing that moving averages do in some other fields, such as finance; both tools provide a smoother form of a signal which removes the short-term oscillations, leaving only the long-term trend.

Band-pass filter

A band-pass filter is a device that passes frequencies within a certain range and rejects (attenuates) frequencies outside that range. An example of an analogue electronic band-pass filter is an RLC circuit (a resistor-inductor-capacitor circuit). These filters can also be created by combining a low-pass filter with a high-pass filter.

An ideal filter would have a completely flat passband (e.g. with no gain/attenuation throughout) and would completely attenuate all frequencies outside the passband. Additionally, the transition out of the passband would be instantaneous in frequency. In practice, no bandpass filter is ideal. The filter does not attenuate all frequencies outside the desired frequency range completely; in particular, there is a region just outside the intended passband where frequencies are attenuated, but not rejected. This is known as the filter roll-off, and it is usually expressed in dB of attenuation per octave of frequency. Generally, the design of a filter seeks to make the roll-off as narrow as possible, thus allowing the filter to perform as close as possible to its intended design. However, as the roll-off is made narrower, the passband is no longer flat and begins to “ripple.” This effect is particularly pronounced at the edge of the passband in an effect known as the Gibbs phenomenon.

At the values of “s” for which this quadratic equation equals zero, the transfer function has theoretically infinite gain. These values, which establish the performance of each type of filter over frequency, are known as the poles of the quadratic equation. Poles usually occur as pairs, in the form of a complex number (a + jb) and its complex conjugate (a – jb). The term jb is sometimes zero.

The thought of a transfer function with infinite gain may frighten nervous readers, but in practice it isn’t a problem. The pole’s real part “a” indicates how the filter responds to transients, and its imaginary part “jb” indicates the response over frequency. As long as this real part is negative, the system is stable. The following text explains how to transfer the tables of poles found in many text books into component values suitable for circuit design.

Filter Types

The most common filter responses are the Butterworth, Chebyshev, and Bessel types. Many other types are available, but 90% of all applications can be solved with one of these three. Butterworth ensures a flat response in the passband and an adequate rate of rolloff. A good “all rounder,” the Butterworth filter is simple to understand and suitable for applications such as audio processing. The Chebyshev gives a much steeper rolloff, but passband ripple makes it unsuitable for audio systems. It is superior for applications in which the passband includes only one frequency of interest (e.g., the derivation of a sine wave from a square wave, by filtering out the harmonics).

The Bessel filter gives a constant propagation delay across the input frequency spectrum. Therefore, applying a square wave (consisting of a fundamental and many harmonics) to the input of a Bessel filter yields an output square wave with no overshoot (all the frequencies are delayed by the same amount). Other filters delay the harmonics by different amounts, resulting in an overshoot on the output waveform. One other popular filter, the elliptical type, is a much more complicated filter that will not be discussed in this text. Similar to the Chebyshev response, it has ripple in the passband and severe rolloff at the expense of ripple in the stopband.

Standard Filter Blocks

The generic filter structure (Figure 1a) lets you realize a highpass or lowpass implementation by substituting capacitors or resistors in place of components G1 through G4. Considering the effect of these components on the opamp feedback network, one can easily derive a lowpass filter by making G2/G4 into capacitors and G1/G3 into resistors. (Doing the opposite yields the highpass implementation.)

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Analog and digital filters

In signal processing, the function of a filter is to remove unwanted parts
of the signal, such as random noise, or to extract useful parts of the signal,
such as the components lying within a certain frequency range.

The following block diagram illustrates the basic idea.

There are two main kinds of filter, analog and digital. They
are quite different in their physical makeup and in how they work.

An analog filter uses analog electronic circuits made up from components
such as resistors, capacitors and op amps to produce the required filtering
effect. Such filter circuits are widely used in such applications as noise
reduction, video signal enhancement, graphic equalisers in hi-fi systems, and
many other areas.

There are well-established standard techniques for designing an analog
filter circuit for a given requirement. At all stages, the signal being filtered
is an electrical voltage or current which is the direct analogue of the physical
quantity (e.g. a sound or video signal or transducer output) involved.

A digital filter uses a digital processor to perform numerical calculations
on sampled values of the signal. The processor may be a general-purpose computer
such as a PC, or a specialised DSP (Digital Signal Processor) chip.

The analog input signal must first be sampled and digitised using an ADC
(analog to digital converter). The resulting binary numbers, representing
successive sampled values of the input signal, are transferred to the processor,
which carries out numerical calculations on them. These calculations typically
involve multiplying the input values by constants and adding the products
together. If necessary, the results of these calculations, which now represent
sampled values of the filtered signal, are output through a DAC (digital to
analog converter) to convert the signal back to analog form.

Note that in a digital filter, the signal is represented by a sequence of
numbers, rather than a voltage or current.