Filters

Filters

Signal filtering plays a fundamental role in electronics and communications.  Filters modify specific frequency components of time-domain signals and are used as a tool for signal quality improvement, information recovery and frequency separation [1].  Filters are a fundamental frequency domain tool and as a component in electronic circuits and digital signal processing allow us to:

There are many different analog and digital filter designs, with varying implementations and transfer functions.  However, the general idea of a filter is that its transfer function should attenuate the  magnitude of specific frequency components of a signal, or introduce a known phase-delay  to specific frequency components whilst leaving other frequency components of the signal unchanged.   Typically, in the communications industry, we are mostly interested in the amplitude-frequency effects of a filter.

The ideal filter

An ideal filter multiplies the passband frequency components by 1 (i.e does not change them in any way), and attenuates the noise (i.e signal we don’t care about) in the stopband frequencies by an infinite amount.  The transition from passband to stopband for an ideal filter is instantaneous.   The frequency response of such a filter is shown below, on the left.  The corresponding time domain impulse response of the filter is shown on the right.  Of course, this description is of an idealised sinc filter, shown below, and is not practically realizable.

http://idav.ucdavis.edu/~okreylos/PhDStudies/Winter2000/BoxFilter.gif
An ideal box or sinc filter (credit)

There are also some things that even idealized filters cannot do.  A filter cannot remove common mode or differential mode disturbances and interference in the passband.   That means if someone else is using the same frequencies as you are, there isn’t much that can be done to remove their interference!

Filter Response Types:

Low-Pass Filters

A low-pass filter allows all of the frequencies below the cut-off frequency to pass through it and it attenuates the higher frequency components of a signal.  As the frequency increases so the amount of attenuation increases.  Low pass filters are useful in suppressing high frequency noise, and limiting the bandwidth of analog signals.

High Pass Filters

A high-pass filter allows all of the frequencies above a certain value to pass through it and it attenuates the lower frequency components of a signal.  As the frequency decreases towards zero so the amount of attenuation increases.  High pass filters are useful for isolating equipment from DC currents and also from low frequency noise sources like AC power signals at 50Hz, or in the case of old telephone systems, the 20Hz ringing signal.

Band-Pass Filter

Constructed from the combination of a high-pass filter with a low cutoff frequency, and low-pass filter with a higher cut-off frequency.  Band-pass filters find widespread use in the RF front end of telecommunications equipment, predominantly in limiting power of transmissions to a specific frequency band and also in eliminating out-of-band noise from received signals.  Another increasingly common use of analog band-pass filters in telecommunications is in co-location or co-existence filters inside radio equipment or handsets.  Modern handsets have a slew of different radios simultaneously operating at different frequency bands and access technologies.  Ensuring that the radios can peacefully co-exist in such close quarters without degrading each others’ performance is a major design challenge!

Band-Stop / Rejection Filter

Band Stop or Rejection Filters work in exactly the opposite way to band-pass filters.  They are constructed by placing a high-pass filter  with a high cut off, in series with a low pass filter with a low cut-off frequency.  Band Rejection filters are useful for eliminating interference on specific frequencies.

Digital & Analog Filters

Filters can be implemented for analog or digital signals.  Digital Filters by comparison are implemented by digital signal processing and operate on digital information.

Analog Filters

Analog filters can be implemented in various forms depending on the application:

  • Passive electronic filters consisting of Resistors, Inductors and Capacitors.
  • Active electronic filters that use amplifiers which are very common.
  • Surface Acoustic Wave filters are often used for super heterodyne receivers at the intermediate frequency in digital receivers in radios and in television sets.
  • Cavity filters are mechanical boxes with a specific geometry that enables high-fidelity filtering of high power microwave signals.

Analog Filters are usually constructed out of a physical circuit and operate on analog, continuous time-domain signals.  Analog Filters play an extremely important role in communications, especially as an important step in signal conditioning prior to entering an analog to digital converter.  Analog filters also used to have a role to play in pulse shaping for older modulation types such as spread spectrum technologies (that require a high symbol rate).

Digital filters

Digital filters have sever key advantages over analog filters in that they are not affected by tolerances in component values, manufacturing processes, temperature differences and aging. The performance of digital filters is also vastly superior to that of analog filters achieving much higher stop band rejection, smaller transition bands, low passband distortion and linear or even zero phase delay!

Digital filters are useful for processing almost any form of digital information!  In communications equipment, digital filtering is used for conditioning digital signals prior to modulation or after being converted from analog to digital values.  Digital filters find applications in pulse shaping and can also be used to remove higher frequency noise components from over sampled digital signals.  This whitepaper and tutorial details an example of using oversampling and a digital filter for pulse shaping of a transmitted digital signal to enhance spectral efficiency!

Properties of Filters

Pass Band & Stop Band

The pass band is a term that collectively refers to the range of frequencies that a filter allows through it.  The stop band is the term used to collectively describe the range of frequencies that are sufficiently attenuated by the filter for us to ignore.  The amount of attenuation required in the stop band is called the stop band attenuation. The frequencies at which the passband stops are called the cut-off or edge frequencies.  The cut-off frequencies in analog filters are widely accepted to be the frequencies at which the amplitude of the frequency response is attenuated by -3dB.  Digital filters are less standardized, the attenuation level that determines the cut-off frequency is usually specified.  Common values are 99%, 90%, 70,7% and 50% [reference]

Transition Band

The transition band refers to the range of frequencies between the pass band and stop band that are not sufficiently attenuated by the filter for us to ignore.  All practical filters have a finite rate at which they can transition from the passband to the stop band.  Some filter implementations are capable of achieving very high frequency roll-off, minimizing the size of the transition band.  Some digital filters are capable of roll-off rates as high as -36dB/Hz!

Frequency response of a low pass filter, showing the passband, transition band and stopband a. (credit)

 

Passband & Stop Band Ripple

Some filter implementations like the Chebyshev and Elliptical filters can introduce a “ripple” in the passband and/or the stop band of the signal, causing the signal to be distorted.  The maximum tolerable Passband ripple of a filter is generally specified in the design requirements.

Phase-Delay & Phase-Response

The phase delay measures the amount by which a single frequency component is delayed when traveling through the filter.  This short time delay has the effect of delaying the phase of the sinusoidal wave relative to where we were expecting it.

Quick Note: It is important to realize that phase-delay is actually dependent on units of time and is converted to an angular measurement by multiplying by the frequency.  Thus, for a constant time delay through a system, as the frequency increases, so the phase-delay angle will increase too!  You can prove this in your head by thinking of a wave of 1 Hz going through a system with a time delay of 0.25 seconds.  We know that the phase will be shifted back by -90 degrees or π/2 radians.  Imagine we had a wave of 2Hz going through the same system.  The same time delay results in a phase-delay of -180 degrees or π radians!

The phase-delay as a function of frequency is shown as the phase-response of a filter. Here is an image of a phase response:

http://www.analog.com/en/analog-dialogue/articles/phase-relations-in-active-filters.html
Phase response of a first order, low pass filter. (credit)

Filters can be designed to have zero phase, linear phase or non-linear phase responses.  A zero phase response system is one that does not change the phase of the signal at all, which implies that it introduces no delay to the time domain signal at any frequency.  A linear phase response system is one that introduces a constant delay to all frequencies, like in the thought experiment above.

Obviously, if designing a control system or time-delay sensitive system,  we would prefer zero phase response (zero delay) in our signal, which would imply instantaneous measurement or control, but often we have to settle for a linear phase-response if we are dealing with real-time systems.

Systems with a non-linear phase response have a phase-response that changes with frequency!  A non-linear phase response can cause distortion of the time domain signal as different frequency components will now arrive at their peak amplitude at different times relative to each other.  This kind of distortion either speeds up or slows down the rate of change of a time domain signal and is referred to as ringing.   Non-linear phase response is a concern in systems design where accurate replication of the time domain signal is a key requirement such as digital receivers (another reference here).

Note: The phase of a frequency is not perceptible to the human ear, and so the phase response of high fidelity audio filters and amplifiers is theoretically of no concern to us.

Typically, digital filters can be designed with a zero or perfectly linear phase-response and this is not an issue, but unfortunately, physically realizable filters have a much poorer performance in this regard!

Group Delay

Group delay is defined as the derivative of the phase-response with respect to frequency and has units of time.  Group delay is also a measure of the non-linearity of the phase-response of a system.  A linear phase response system will have a constant group-delay.  A highly non-linear phase response will have a rapidly changing group-delay!

To think of group delay, remember the following:

  • Phase delays are caused by time delays in the system.
  • Phase-delay is calculated from time-delay by multiplying by the frequency of interest.  You could say that phase-delay is measured in frequency seconds (Hz.s) or (rad.s.s-1 = rad)
  • If everything has the same time delay, then the phase-delay will be linear with respect to frequency, and the gradient of the line will be equal to the time-delay of the system.  Negative gradient will indicate a time-delay).
  • If we take the derivative of phase-delay with respect to frequency, we will simply get the time delay of the system!

So, group-delay is actually just a measure of the time delay of the entire system with respect to frequency.  Imagine a square pulse arrives at the input of the system.  Group delay describes how each frequency component of that square pulse will be delayed through the system!

Group delay is a useful way to evaluate the normalized response of  a filter design.  That is the logical way to think about it.  Here is a picture of a Chebyshev filter showing frequency-response and group-delay.

Gain and group delay of a fifth-order type I Chebyshev filter with ε = 0.5 (credit)

Q-Factor

Quality-factor is actually not a term used specifically for filter design but for many applications in engineering and physics, including antennas and other forms of resonant systems.  The Q factor of a resonator or oscillator is the ratio of its central frequency to the bandwidth over which it works.  Example, we build a filter with a central frequency of 1850 Hz and a total bandwidth of 3100Hz.  The Q factor of such a filter would be approximately 0.6.

Q Factor gives us a way to understand the potential performance of a filter with regards to the shape of the frequency response and phase-response of  a filter.  Filters with a high Q factor, that work over a very small bandwidth, typically have a very rapid change in phase and amplitude with respect to frequency.

Insertion loss

All filters regardless of their type, digital or analog, will introduce some form of loss to the passband signal.  Insertion loss in telecommunications refers to the loss incurred by inserting a device into the path of the signal.  A good reference discussing the sources of insertion loss can be found here.

Transducers…

You should also be aware that the transducer you are using to create the analog signal itself also has a frequency range of operation and will also filter out frequencies outside of its own range.  The picture below is the frequency response chart of a Shure SM57 microphone.   As you can see from the chart it is much less sensitive to lower frequencies and much more sensitive to higher frequencies in the audible range.  This means that the microphone will distort the original signal by attenuating lower frequencies below 200 Hz and amplifying frequencies above 2kHz.  You can read more on microphones and their response charts here.

SM57 Frequency Response Chart
Frequency Response of a Shure SM57 Microphone (Image Credit)

Loss of Information

Whenever we use a transducer to create an analog signal, or a filter to limite the bandwidth of a signal, we must always accept that we are distorting the original input and losing information.  The question however is how much information is an acceptable amount to lose and do we care about the information we are losing?  For instance,  when capturing the sound of the human voice most of the energy is concentrated within the bands of 200 to 2000 Hz.  Band limiting the signal to 3400 Hz will result in a band limited baseband signal that still allows people to communicate clearly over a digital phone.  Similarly, audio destined for high quality musical playback can be band limited to 20Hz – 20kHz because  we cannot hear any of the higher or lower frequencies and it makes no discernible difference to us!  The same can be said of images and the color gamut that can be captured by a camera, supported by a video codec or displayed by a monitor!

Resources:

Here are some great resources I found on the topic of both Analog and Digital Filters.

Also, click on the links in the article!

That’s all for now!

Rob

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