The outcomes for this module are for the student to understand the following.
A NOTE RE: APPLICABILITY OF THIS INFORMATION
This information is most relevant to those working with acoustic signals: audiologists and those who make measurements of the human voice. Filters are also encountered in electrophysiologic measurements, and have applications in any area where electrodes are placed on a patient.
THE PURPOSE OF FILTERS
The purpose of a filter is rather simple. It blocks the passage of some frequency energy, while allowing energy at other frequencies to pass.
Passive filters don't use an operational amplifier. They are designed using resistors and capacitors only.
Active filters use resistors, capacitors and the operational amplifier. The operational amplifiers use power, and therefore, (e.g. in hearing aids) contribute to battery current drain. While passive filters can only provide limited filtering, active filters can be designed to provide very precise filtering.
Digital filtering is, in theory, unlimited in its flexibility. The signal, in its mathematical (binary) state, is manipulated. A digital filter can imitate the design of any passive or active filter. The only catch is that it takes time to make the calculations necessary to implement a digital filter. With faster microprocessors, this is no longer a serious limitation in most designs.
TERMINOLOGY OF FILTER TYPES - they only sound like fashion statements
High pass / low cut - Either term means the same thing. This type of filter is designed to allow frequencies higher than a certain point (its cut-off) to go through the circuit unimpeded. Frequencies below the cut-off are attenuated (reduced).
Low pass / high cut - The opposite, the low frequencies aren't affected, while the higher frequencies are attenuated in amplitude.
Band pass - A band-pass filter allows a certain range of frequencies
to remain. Frequencies above a certain point are rejected, and
those lower than a certain point are rejected. The width of the
band pass region can vary. Traditional band pass regions are an
octave, or some increment of an octave (2 octaves, 1/3 octave
etc.). Filtering a wide-band (white) noise with a band-pass filter
results in narrow-band noise, which is of course the traditional
masking stimulus.
Band reject - Rather than allowing frequencies within a certain
range to pass, a band reject filter attenuates one certain frequency
range. The most common frequency range is near 60 Hz, which minimizes
the 60 cycle hum. (But, if the 60 cycle hum has harmonic distortion,
as often happens, just eliminating the 60 cycle may not eliminate
the noise contamination.) In electrophysiologic work, 60 cycle
"notch filters" (as they are often called) aren't necessarily
desirable. They cause a phase delay which alters the latency of
the waveform. For filtering speech or acoustic signals, the 60
Hz filter is seldom a problem.
REJECTION RATES
A filter is never perfect at cutting off the undesired frequencies. For instance, if you set a high-pass filter to 1000 Hz, the energy at 900 Hz won't be entirely gone. The amount of attenuation a filter provides is described by its "rejection rate" which is specified in decibels per octave.
The beginning point of the filter - its cut-off point - is specified by the frequency at which the filter attenuates the signal by 3 dB. In the figure below, the cut-off is at 1000 Hz.
Figure 1. Illlustration of a hypothetical low-pass filter. The cut-off is at 1000 Hz. Two filter slopes are shown: 3 and 12 dB/octave.
As illustrated above, the amount of attenuation for frequencies above 1000 Hz depends upon the filter slope. If you took white noise - noise that has equal energy at each frequency - and "passed it through" (ran it through) the 3 dB/octave filter, there would be 3 dB less energy at 2000 Hz. If the 12 dB per octave slope were used, the energy would be 12 dB less at 2000 Hz. Since it was already 3 dB down at 1000 Hz, there would be 15 dB less energy at 2000 Hz than at frequencies below 1000 Hz (e.g. 500 Hz).
Let's see if you have this concept. You put 100 dB per cycle (per frequency) noise into this 12 dB/octave filter. How much energy is at 250 Hz? 100 dB, right? This filter does not affect frequencies below 1000 Hz. How much energy would be at 8000 Hz? 61 dB (level per cycle). Why? There are 3 octaves between 1000 and 8000 Hz (1k to 2k, 2k to 4k, 4k to 8k), and 3 times 12 = 36. The signal was already down 3 dB at 1k, now it is down 36 more dB at 8k, yielding a total attenuation of 39 dB. And of course, 100-39 = 61.
So, as you see, filters don't typically provide a total block of energy above the designated cut-off frequency. Some filters, like a 60 Hz band reject, will be designed to have very sharply sloping filter "skirts".
DESIGNATING FILTER WIDTH
The starting point of a filter is when there is 3 dB reduction in energy. If a filter is one octave wide, centered at 1000 Hz, it will attenuate 707 and 1414 Hz by 3 dB. I'll get into the reason the cut-offs aren't 500 and 1500 in a bit.
Sometimes you will hear of Q factors, which are related to the filter width. This is usually used to describe a biological filter, like the human ear. Remember any psychoacoustics? Remember tuning curves? (How different frequency noise bands mask hearing of a pure tone.) Broad tuning curves are abnormal, and one way of describing the tuning curve is with a Q factor. You look at the width of the filter at, for example, 10 dB above the threshold for hearing the tone, in relation to the frequency, and that gives you the Q10 dB. It's just one of those things a doctoral-level audiologist should have at least heard of, in case you run into it in the literature.
RIPPLE
No, its not cheap booze we're discussing. Ripple describes how
"flat" the filter really is within the pass band. In
Figure 1 you see an illustration of the perfect flat low frequency
pass region. In reality, it's never perfectly flat; the perturbations
are called "ripple". If ripple is not specified, it
is probably 3 dB. A really good filter has minimal ripple, such
as less than 1 dB.
OF INTEREST ONLY TO AUDIOLOGISTS, I SUSPECT
Programmable hearing aids with multiple bands allow us to customize the frequency response of a hearing aid - to a point. They use narrow-band filters which have a given dB/octave slope.
Let's assume that you have a hearing loss that slopes sharply from 1k to 2k Hz, and that you would like to have provide 30 dB of gain at 2k Hz, while only providing 5 dB of gain at 1k Hz. This is a 25 dB separation from the gain at 1k and 2k.
The blue line in Figure 3 illustrates what you wish to have happen. You want 25 dB less energy at 1k relative to 2k. If you set the filter (shown in green) to -25. The problem is that the filter with the least attenuation - in effect - wins. Look at the amount of reduction of 1k energy by the 2k filter. Only 15 dB reduction is achieved. If I transform this back into dB gain, if I set the volume of the hearing aid so that there is 30 dB of gain at 2k, I still have 15 dB of gain at 1k because of that "leaky" 2k filter.
So, the number of bands in a programmable hearing aid is important, as is your ability to adjust the high and low cut-offs, but if the rejection slope isn't steep enough, you don't buy much with the additional bands.
You may hear active filtering referred to
by "order", as in "this is a third-order filter".
An order refers to a 6-dB/octave filter, so a third-order filter
has 18 (3X6) dB per octave filter slopes.
SOUND LEVEL METER FILTER TYPES
One of my pet peeves is reading in the newspaper a statement about the hazards of sound, and having them refer to it as just dB. As in "Car stereo systems can put out as much as 100 dB." What TYPE OF DB???
What's a dB anyhow?
A decibel is just a ratio of one sound to another, a measurement of relative pressure. Anything you can put in a ratio format can be a decibel. Decibels sound pressure level are a mathematical transformation of a pressure reading, related to another pressure. The smallest sound pressure audible in an early psychoacoustics study was 0.0002 dynes per cm2. That is a reading of force on a diaphragm of a microphone This became the "reference" pressure for dB sound pressure level. Let's examine the scenario of a microphone sensing a pressure 0.2 dynes per cm2. Through the wonders of electronics, and the marvels of math, this becomes a decibel using the formula you probably last saw in your undergraduate speech science course:
I'll do the math for you.
= 20 X log (1000) Remember in logs, how you count the zeros to find a log 10…
= 20 X 3
= 60 dB SPL
Note that this decibel type is dB SPL. The only time you can say just dB (and be technically correct) is when you are saying higher or lower than another amount. So, it's OK to say 60 dB gain, or 20 dB attenuation, but not "This speaker puts out 80 dB."
Enough of a review. Audiologists are advised to brush up on this stuff before taking Chuck Berlin's course!
What are sound level meter dB scales?
When measuring sound using a sound level meter, you have several types of decibels to choose from. All of them are based on the decibel sound pressure level. You could read dB in an octave band, for instance. Or you could select the filter scale for dB types A, B or C. These filters reduce the energy of primarily low frequency energy.
So, if you are measuring a low frequency sound, like the bass sound from a stereo at 100 Hz, you will read 20 dB lower with the A filter setting selected on the sound level meter. So, as you can see, it does matter which type of decibel you use! That is one of my biggest pet peeves, it ranks right next to not specifying how far away one is from the sound source. (Duh, is it 100 dB A at the speaker, 3 inches from the speaker, 3 feet from the speaker…)
The A, B and C scales have different uses. The A scale is widely used for industrial noise measurement because it "devalues" low-frequency energy, which is not as damaging to the ear as high-frequency energy. The A scale also mimics how the ear responds to soft sounds. The C scale mimics the ear's response to loud sounds, and could be argued as being applicable for indicating the subjective loudness of very loud sounds.
Octave Calculations
I mentioned earlier that I would reveal the reason why a one octave filter, centered at 1000 Hz, would have 3 dB down points at 707 and 1414, rather than at 750 and 1500. I know you are so curious by know you're going to bust - but hold on, I'm about to explain why! WARNING: four-letter word about to be used: MATH.
Esoteric you say. Yes, but in my academic career I've used this formula for verifiable reasons dozens of times. Mostly in designing digital filters, I will admit.
Audiologists - one of the limitations of our doctoral program is that we don't have and advanced course in psychoacoustics. For those interested in loudness, the issue of octaves comes into play yet again. Loudness, for instance of two tones, is perceived differently depending upon whether the tones are within one "critical band" or are separated by a greater frequency than that. The loudness of a narrow-band masker changes, even when the overall SPL stays constant, once it is wider than a critical band width. The width of the critical band stays constant at around 100 Hz up to 500 or 1000 Hz, then is around 0.2 octaves (some would say 0.25 -- a quarter-octave) above this point. How this relates to hearing aid fitting is a nice topic for Happy Hour, hopefully with Gus Mueller!
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