Fourier Transforms

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Fourier Transforms

The Fourier transform defines a relationship between a signal in the
time domain and its representation in the frequency domain. Being a
transform, no information is created or lost in the process, so the
original signal can be recovered from knowing the Fourier transform,
and vice versa.

The Fourier transform of a signal is a continuous complex valued
signal capable of representing real valued or complex valued
continuous time signals.

The tool allows you to view these complex valued
signals as either their real and quadrature (also known as imaginary)
components separately, or by a magnitude and phase
representation. You may switch between these two representations at
any point. Mathematically switching between the two representations
for a given complex value can be expressed as

and

or equivalently,

and

where

and

are the magnitude and phase of the complex number, and

and

are the real and quadrature components of the complex number. In this
tool, the magnitude is plotted on a dB scale. Select a few
signals, such as unit pulses and sine waves, and view them using the
two methods to see how they are related.

The Fourier transform itself is defined by the equation

where

is the Fourier transform of

Frequency is measured in Hertz, with

as the frequency variable.

Fourier transform of signals

Using the tool, display the Fourier transform of a 4ms unit
pulse. You will observe that the frequency response is a continuous
signal with a maximum at 0 Hz, and some periodicity. The frequency
response is zero at every multiple of 250Hz. Compare this with the
frequency response of a unit pulse of 8ms in duration. Here the
general shape of the signal is the same, but the zero crossings are at
a spacing of 125Hz. These figures are the reciprocals of the pulse
duration, indicating that there are inverse relationships between
time and frequency. Generally, longer time periods relate to smaller
frequency spans.

The formula for the frequency response of a unit pulse may
be calculated directly from the Fourier transform equation as

where

is the duration of the pulse. You can observe the changes in magnitude
caused by the different values of

, as well as the changes in the spacing of the zero crossings, a
function of the sin component.

Sinusoids and cosinusoids are signals that by definition contain only
one frequency of signal. The tool has two examples of these
with frequencies 333Hz and 500Hz. The time domain and frequency
transform of a 500Hz cosine wave is given by the following
equations

Delaying a 500Hz cosine wave by 0.5ms results in a sine wave signal,
and its transform can be seen to be

As this change is made, by adding the delay, you will observe that the
phase of the frequency transform changes, but the magnitude remains
the same. Alternatively, using the real and quadrature representation,
components that were purely real before becoming imaginary after the
delay.

Delay and phase change

Any of the signals can be advanced or delayed by a number of
predefined delays of up to 4ms. Alternatively, you can delay a signal
by an arbitrary amount by clicking and dragging the graph whilst
holding down a key on the keyboard. In the frequency domain this
relates to alteration of the phase of the signal, thus no difference
will be observed when viewing the Fourier transform magnitude plot,
but will be evident when viewing the phase of the transform or the
real and imaginary parts together.

Try this out for various types of signal.

Take particular note of the scaled unit impulse as without any delay
it results in a purely real transform of height 0.004 (the scaling
factor). When this signal is delayed, the transform becomes a
cosinusoid in the real component and a sinusoid in the imaginary. The
formula for this is

where is the
value of the delay.

This implies that a delay of a specific amount in the time domain
equates to multiplication by a phasor in the frequency domain. Set
the delay for the scaled unit impulse to 0.5ms as was done for the
500Hz cosine waveform in the previous section. Now note the values of
the real and imaginary parts of the transform at 500Hz and -500Hz. Now
switch the input signal to the 500Hz cosine and you should be able to
explain how the purely real transform of the undelayed waveform
relates to the purely imaginary transform of the delayed
signal.

Not only can the time domain signal be delayed, but the frequency
transform can be shifted, resulting in a phase change in the time
domain. Experiment with this observing the time domain signal as
magnitude and phase, and as real and quadrature to see the effects
that can be obtained. Try shifting the frequency response of a
cosinusoid, or a sinusoid, so that one of the frequency samples is set
to 0Hz. The result will be a complex phasor, consisting of a
cosinusoid and sinusoid in the real and imaginary components of the
time domain plus a DC offset from the 0Hz component.

Multiplication and convolution

Using the tool, review the transforms of the unit
pulse function and the cosine function. For the moment it is best to
view these using the magnitude and phase representation of the
frequency domain.

Now switch to one of the 8ms segment of a cosine or sine
waveforms. You should observe that the frequency domain plot is some
form of combination of the two types of signal. Strictly speaking, the
time domain signal is the multiplication of a unit pulse of 8ms
duration delayed by 4ms, and a cosinusoid or sinusoid waveform of the
selected frequency. The frequency domain transform is then the
addition of two sa functions which have been shifted in
frequency. Notice where the highest peaks are and you should observe
that these correspond with the frequency of the sine or cosine
signal. What has happened is that in the frequency domain the sa
function from the unit pulse and the two impulses from the sine or
cosine function have been convolved together. This is an example of
the general rule that multiplication in the time domain equates to
convolution in the frequency domain.

You can reconstruct the two constituent waveforms by shifting the
frequency response of the 8ms unit pulse to 500Hz, and to -500Hz.You
should find that the real component of the two shifted signals are the
same, but that the quadrature components are the complement of each
other. Thus when they are summed together, the result is a signal with
a real component and a zero quadrature component.

In fact an equivalent rule also holds that
convolution in the time domain equates to multiplication in the
frequency domain. Thus, for example, a complex phasor in the frequency
domain multiplied by a given signal's transform produces a time domain
function where an impulse is convolved with signal. This is precisely
what is happening when the delay value is being altered.

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Correlation

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Correlation

Correlation is a technique that is very similar in mechanism to
convolution, with one subtle difference.

The correlation integral is given by

While this may appear to be similar to equation (1) with some changes
of variable, unlike convolution where

is time reversed, in correlation this is not the case. The integral
thus no longer represents the output of a filter driven by an
input signal. Rather it is a tool used to measure the similarity
between two signals. A large correlation value (positive or
negative) represents a strong similarity between the two
signals, while a value near zero represents little
similarity. The delay value,

allows this comparison to be made on two signals at
different delay separations.

Using the correlation tool

compare a number of signals with a unit pulse shape, noting the
maximum values that you can obtain, and the delay at which these
values occur. For all signals that you correlate, except for
those signals incorporating impulses, the maximum value that you
will obtain will be found by those signals similar in shape to
the one you are correlating with. This is because correlation
identifies the similarity between the two signals.

If a signal is correlated with itself, a process known as
autocorrelation, then the maximum value of the correlation can
be found at a time shift of 0. Verify this yourself for any
signals that you wish to test out. For the correlation between
two different signals, known as cross-correlation, this is not
necessarily the case.

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The Application of Discrete Time Convolution

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The Application of Discrete Time Convolution

Discrete time convolution can be used to determine the output of a
sampled data system from its input and pulse response. A
commonly used discrete system is a finite impulse response (FIR)
filter.

This tool presents you with a schematic of a FIR Filter. The boxes
marked

represent delay elements whose output is the previous value of
their input. Using the single step button you can observe the
input signal,

as its samples work their way through the delay elements.

There are a number of multiplication symbols in the filter with
weights, which are the system response, associated with them.The
product of these weights and the input signal samples are summed
together to give the output of the filter,

You can view the values at various points in the filter as
either a bar denoting the size of the value, by a number, or by
both means. The colours of the values or bars correspond to the
colours of the graphs. Observing both windows simultaneously,
the mechanism by which the convolution is performed can be seen.
The system response is coded in the multiplication weights, the
product of the input signal and the system response is
calculated at the output of the multipliers, and the final sum
is calculated by the summation unit driving the filter output.

Experiment with setting various system responses and signals
such as setting a single value of the input to 1 to convince
yourself that the system response values of the filter define
the unit pulse response of the filter.

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Convolution

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Convolution

Convolution is the term given to the mathematical technique for
determining a system output given an input signal and the
system impulse response. It is written as

(1)

When you start the convolution tool

you will find that you can select signals, run animations, and control
the display properties. The current input signal and system
impulse response are displayed - you can change the data in
these graphs by clicking and dragging with the mouse or you can
select pre-defined signals using the menus. It is helpful if you
ensure that the signal and system response are coloured differently.

The other graphs show the time-reversed input signal and the system
response, the product of the time-reversed input signal
(due to the

in equation (1)) and the system response, and their convolution. The
product and convolution data will change as the time-reversed
input signal is moved relative to the system impulse response.

When the system impulse response is a unit pulse and the input is a
ramp function, you can test out the convolution process by
clicking on the

symbol in the combined time-reversed system impulse response and
signal graph and dragging the time reversed input signal
backwards and forwards through the system impulse response.
You can also let the tool do this for you by clicking on the
double arrow button.

As the ramp signal is shifted through the unit pulse, the product
graph clearly shows that the convolution is calculating the area
under a selected portion of the ramp signal. The convolution
graph plots the increasing, and then decreasing area of this
product. The system with this impulse response is known as a
finite time integrator as it performs integration over a finite
length of the input signal. Experiment with other input signals
such as sine waves, other unit pulses, impulses, etc.

Convolution with an impulse

Change the system response to a single impulse at 2ms. As you perform
the convolution, as before, you will find that the output is a
copy of the input signal, except that it is delayed by 2ms. In
this instance, the impulse is performing a sampling function of
the input signal as described by

Examination of the product will reveal that the integrand,

, is also an impulse. In fact, the impulse is a scaled unit impulse,
the scaling resulting in an impulse of area

, hence the final convolution result of

Thus, analytically we find that the system response of a unit impulse
results in the output of the system being a copy of the input
signal, with the only difference being a time shift.

As a heuristic argument as to why this should be the case, think of
the system response as the system output when the input signal
is an impulse function at time

When the system response is also an impulse, then the input signal
shape doesn't alter. If that impulse is at a non-zero time
value, then there is a time shift between the input and the
output of the system. (For real systems any time shift must
represent a delay, otherwise the system would predict its own
input signal)!

A system that sums two delayed versions of the input signal is
represented by two impulses in the system response. Experiment
with the effects that you can obtain by examining system
responses consisting of impulses.

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Signal Processing Fundamentals Dennis Bohn, Rane Corporation RaneNote 134 written 1997; last revised 5/04 Loudness Crossovers Equalizers Dynamic Controllers Screaming To Be Heard In space, no one can hear you scream ... because there is no air or other medium for sound to travel. Sound needs a medium; an intervening substance through which it can travel from point to point; it must be carried on something. That something can be solid, liquid or gas. They can hear you scream underwater ... briefly. Water is a medium. Air is a medium. Nightclub walls are a medium. Sound travels in air by rapidly changing the air pressure relative to its normal value (atmospheric pressure). Sound is a disturbance in the surrounding medium. A vibration that spreads out from the source, creating a series of expanding shells of high pressure and low pressure ... high pressure ... low pressure ... high pressure ... low pressure. Moving ever outward these cycles of alternating pressure zones travel until finally dissipating, or reflecting off surfaces (nightclub walls), or passing through boundaries, or getting absorbed -- usually a combination of all three. Left unobstructed, sound travels outward, but not forever. The air (or other medium) robs some of the sound's power as it passes. The price of passage: the medium absorbs its energy. This power loss is experienced as a reduction in how loud it is (the term loudness is used to describe how loud it is from moment to moment) as the signal travels away from its source. The loudness of the signal is reduced by one-fourth for each doubling of distance from the source. This means that it is 6 dB less loud as you double your distance from it. [This is known as the inverse square law since the decrease is inversely proportional to the square of the distance traveled; for example, 2 times the distance equals a 1/4 decrease in loudness, and so on.] How do we create sound, and how do we capture sound? We do this using opposite sides of the same electromagnetic coin. Electricity and magnetism are kinfolk: If you pass a coil of wire through a magnetic field, electricity is generated within the coil. Turn the coin over and flip it again: If you pass electricity through a coil of wire, a magnetic field is generated. Move the magnet, get a voltage; apply a voltage, create a magnet ... this is the essence of all electromechanical objects. Microphones and loudspeakers are electromechanical objects. At their hearts there is a coil of wire (the voice coil) and a magnet (the magnet). Speaking causes sound vibrations to travel outward from your mouth. Speaking into a moving-coil (aka dynamic) microphone causes the voice coil to move within a magnetic field. This causes a voltage to be developed and a current to flow proportional to the sound -- sound has been captured. At the other end of the chain, a voltage is applied to the loudspeaker voice coil causing a current to flow which produces a magnetic field that makes the cone move proportional to the audio signal applied -- sound has been created. The microphone translates sound into an electrical signal, and the loudspeaker translates an electrical signal into sound. One capturing, the other creating. Everything in-between is just details. And in case you're wondering: yes; turned around, a microphone can be a loudspeaker (that makes teeny tiny sounds), and a loudspeaker can be a microphone (if you SHOUT REALLY LOUD). Crossovers: Simple Division Loudspeaker crossovers are a necessary evil. A different universe, a different set of physics and maybe we could have what we want: one loudspeaker that does it all. One speaker that reproduces all audio frequencies equally well, with no distortion, at loudness levels adequate for whatever venue we play. Well, we live here, and our system of physics does not allow such extravagance. The hard truth is, no one loudspeaker can do it all. We need at least two -- more if we can afford them. Woofers and tweeters. A big woofer for the lows and a little tweeter for the highs. This is known as a 2-way system. (Check the accompanying diagrams for the following discussions.) But with two speakers, the correct frequencies must be routed (or crossed over) to each loudspeaker. Passive At the simplest level a crossover is a passive network. A passive network is one not needing a power supply to operate -- if it has a line cord, or runs off batteries, then it is not a passive circuit. The simplest passive crossover network consists of only two components: a capacitor connecting to the high frequency driver and an inductor (aka a coil) connecting to the low frequency driver. A capacitor is an electronic component that passes high frequencies (the passband) and blocks low frequencies (the stopband); an inductor does just the opposite: it passes low frequencies and blocks high frequencies. But as the frequency changes, neither component reacts suddenly. They do it gradually; they slowly start to pass (or stop passing) their respective frequencies. The rate at which this occurs is called the crossover slope. It is measured in dB per octave, or shortened to dB/octave. The slope increases or decreases so many dB/octave. At the simplest level, each component gives you a 6 dB/octave slope (a physical fact of our universe). Again, at the simplest level, adding more components increases the slope in 6 dB increments, creating slopes of 12 dB/oct, 18 dB/oct, 24 dB/oct, and so on. The number of components, or 6 dB slope increments, is called the crossover order. Therefore, a 4th-order crossover has (at least) four components, and produces steep slopes of 24 dB/octave. The steeper the better for most drivers, since speakers only perform well for a certain band of frequencies; beyond that they misbehave, sometimes badly. Steep slopes prevent these frequencies from getting to the driver. You can combine capacitors and inductors to create a third path that eliminates the highest highs and the lowest lows, and forms a mid-frequency crossover section. This is naturally called a 3-way system. (See diagram) The "mid" section forms a bandpass filter, since it only passes a specific frequency band. Note from the diagram that the high frequency passband and low frequency passband terms are often shortened to just high-pass and low-pass. A 3-way system allows optimizing each driver for a narrower band of frequencies, producing a better overall sound. So why not just use passive boxes? Problems The single biggest problem is that one passive cabinet (or a pair) won't play loud enough and clean enough for large spaces. If the sound system is for your bedroom or garage, passive systems would work just fine -- maybe even better. But it isn't. Once you try to fill a relatively large space with equally loud sound you start to understand the problems. And it doesn't take stadiums, just normal size clubs. It is really difficult to produce the required loudness with passive boxes. Life would be a lot easier if you could just jack everyone into their own cans amp -- like a bunch of HC 4 or HC 6 Headphone Amps scattered throughout the audience. Let them do the work; then everyone could hear equally well, and choose their own listening level. But life is hard, and headphone amps must be restricted to practice and recording. Monitor speakers on the other hand most likely have passive crossovers. Again, it's a matter of distance and loudness. Monitors are usually close and not overly loud -- too loud and they will feed back into your microphone or be heard along with the main mix: not good. Monitor speakers are similar to hi-fi speakers, where passive designs dominate ... because of the relatively small listening areas. It is quite easy to fill small listening rooms with pristine sounds even at ear-splitting levels. But move those same speakers into your local club and they will sound thin, dull and lifeless. Not only will they not play loud enough, but they may need the sonic benefits of sound bouncing off close walls to reinforce and fill the direct sound. In large venues, these walls are way too far away to benefit anyone. Figure 1. Passive 2-Way Crossover Figure 2. Passive 3-Way Crossover So why not use a bunch of passive boxes? You can, and some people do. However, for reasons to follow, it only works for a couple of cabinets. Even so, you won't be able to get the high loudness levels if the room is large. Passive systems can only be optimized so much. Once you start needing multiple cabinets, active crossovers become necessary. To get good coverage of like-frequencies, you want to stack like-drivers. This prevents using passive boxes since each one contains (at least) a high-frequency driver and a low-frequency driver. It's easiest to put together a sound system when each cabinet covers only one frequency range. For instance, for a nice sounding 3-way system, you would have low-frequency boxes (the big ones), then medium-sized mid-frequency boxes and finally the smaller high-frequency boxes. These would be stacked or hung, or both -- in some sort of array. A loudspeaker array is the optimum stacking shape for each set of cabinets to give the best combined coverage and overall sound. You've no doubt seen many different array shapes. There are tall towers, high walls, and all sorts of polyhedrons and arcs. The only efficient way to do this is with active crossovers. Some smaller systems combine active and passive boxes. Even within a single cabinet it is common to find an active crossover used to separate the low- and mid-frequency drivers, while a built-in passive network is used for the high-frequency driver. This is particularly common for super tweeters operating over the last audio octave. At the other end, an active crossover often is used to add a subwoofer to a passive 2-way system. All combinations are used, but each time a passive crossover shows up, it comes with problems. One of these is power loss. Passive networks waste valuable power. The extra power needed to make the drivers louder, instead boils off the components and comes out of the box as heat -- not sound. Therefore, passive units make you buy a bigger amp. A couple of additional passive network problems has to do with their impedance. Impedance restricts power transfer; it's like resistance, only frequency sensitive. In order for the passive network to work exactly right, the source impedance (the amplifier's output plus the wiring impedance) must be as close to zero as possible and not frequency-dependent, and the load impedance (the loudspeaker's characteristics) must be fixed and not frequency-dependent (sorry, not in this universe; only on Star Trek). Since these things are not possible, the passive network must be (at best), a simplified and compromised solution to a very complex problem. Consequently, the crossover's behavior changes with frequency -- not something you want for a good sounding system. One last thing to make matters worse. There is something called back-emf (back-electromotive force: literally, back-voltage) which further contributes to poor sounding speaker systems. This is the phenomena where, after the signal stops, the speaker cone continues moving, causing the voice coil to move through the magnetic field (now acting like a microphone), creating a new voltage that tries to drive the cable back to the amplifier's output! If the speaker is allowed to do this, the cone flops around like a dying fish. It does not sound good! The only way to stop back-emf is to make the loudspeaker "see" a dead short, i.e., zero ohms looking backward, or as close to it as possible -- something that's not gonna happen with a passive network slung between it and the power amp. All this, and not to mention that inductors saturate at high signal levels causing distortion -- another reason you can't get enough loudness. Or the additional weight and bulk caused by the large inductors required for good low frequency response. Or that it is almost impossible to get high-quality steep slopes passively, so the response suffers. Or that inductors are way too good at picking up local radio, TV, emergency, and cellular broadcasts, and joyfully mixing them into your audio. Such is life with passive speaker systems. Figure 3. Active 2-Way Crossover Figure 4. Active 3-Way Crossover Active Active crossover networks require a power supply to operate and usually come packaged in single-space, rack-mount units. (Although of late, powered loudspeakers with built-in active crossovers and power amplifiers are becoming increasingly popular.) Looking at the accompanying diagram shows how active crossovers differ from their passive cousins. For a 2-way system instead of one power amp, you now have two, but they can be smaller for the same loudness level. How much smaller depends on the sensitivity rating of the drivers (more on this later). Likewise a 3-way system requires three power amps. You also see and hear the terms bi-amped, and tri-amped applied to 2- and 3-way systems. Active crossovers cure many ills of the passive systems. Since the crossover filters themselves are safely tucked away inside their own box, away from the driving and loading impedance problems plaguing passive units, they can be made to operate in an almost mathematically perfect manner. Extremely steep, smooth and well-behaved crossover slopes are easily achieved by active circuitry. There are no amplifier power loss problems, since active circuits operate from their own low voltage power supplies. And with the inefficiencies of the passive network removed, the power amps more easily achieve the loudness levels required. Loudspeaker jitters and tremors caused by inadequately damped back-emf all but disappear once the passive network is removed. What remains is the amplifier's inherent output impedance and that of the connecting wire. Here's where the term damping factor comes up. [Note that the word is damp-ing, not damp-ning as is so often heard; impress your friends.] Damping is a measure of a system's ability to control the motion of the loudspeaker cone after the signal disappears. No more dying fish. Siegfried & Russ Active crossovers go by many names. First, they are either 2-way or 3-way (or even 4-way and 5-way). Then there is the slope rate and order: 24 dB/oct (4th-order), or 18 dB/oct (3rd-order), and so on. And finally there is a name for the kind of design. The two most common being Linkwitz-Riley and Butterworth, named after Siegfried Linkwitz and Russ Riley who first proposed this application, and Stephen Butterworth who first described the response in 1930. Up until the mid `80s, the 3rd-order (18 dB/oct) Butterworth design dominated, but still had some problems. Since then, the development (pioneered by Rane and Sundholm) of the 4th-order (24 dB/oct) Linkwitz-Riley design solved these problems, and today is the norm. What this adds up to is active crossovers are the rule. Luckily, the hardest thing about an active crossover is getting the money to buy one. After that, most of the work is already done for you. At the most basic level all you really need from an active crossover are two things: to let you set the correct crossover point, and to let you balance driver levels. That's all. The first is done by consulting the loudspeaker manufacturer's data sheet, and dialing it in on the front panel. (That's assuming a complete factory-made 2-way loudspeaker cabinent, for example. If the box is homemade, then both drivers must be carefully selected so they have the same crossover frequency, otherwise a severe response problem can result.) Balancing levels is necessary because high frequency drivers are more efficient than low frequency drivers. This means that if you put the same amount of power into each driver, one will sound louder than the other. The one that is the most efficient plays louder. Several methods to balance drivers are always outlined in any good owner's manual. Equalizers You may have heard it said that equalizers are nothing more than glorified tone controls. That's pretty accurate and helps explain their usefulness and importance. Simply put, equalizers allow you to change the tonal balance of whatever you are controlling. You can increase (boost) or decrease (cut) on a band-by-band basis just the desired frequencies. Equalizers come in all different sizes and shapes, varying greatly in design and complexity. Select from a simple single-channel unit with 10 controls on 1-octave frequency spacing (a mono 10-band octave equalizer), all the way up to a full-featured, two-channel box with 31 controls on 1/3-octave frequency spacing (a stereo 1/3-oct equalizer). There are graphic models with slide controls (sliders) that roughly "graph" the equalizer's frequency response by the shape they form, and there are parametric models where you choose the frequency, amplitude, and bandwidth desired (the filter parameters - see diagram below) for each band provided. Far and away, the simplest and most popular are the 1/3- and 2/3-octave graphics. They offer the best combination of control, complexity and cost. In selecting graphic equalizers, the primary features to consider are the number of input/output channels, the number of boost/cut bands, the center-frequency spacing of each, and the accuracy of the output vs. the front panel settings. Up until the recent development of true response graphics, the front panel settings only approximated the equalizer's actual response. Prior to true response graphics, adjacent band interaction caused the actual output response to deviate from the front panel settings. Described as either constant-Q or variable-Q (see diagrams), the individual filter bandwidth behavior determined the interaction. In the early '80s, Rane developed the first constant-Q designs to preserve the same shape (bandwidth) over the entire boost/cut range. In contrast, variable-Q designs have varying bandwidths (the shape changes) as a function of boost/cut amount. Rane's constant-Q design offered a big improvement in output response vs. front panel settings and became the most popular design until Rane and others developed the first true response graphic equalizers. Now true response graphics offer the best response. Using Equalizers Equalizers can do wonders for a sound system. Let's start with loudspeaker performance. An unfortunate truth regarding budget loudspeakers is they don't sound very good. Usually this is due to an uneven frequency response, or more correctly a non-flat power response. An ideal cabinet has a flat power response. This means that if you pick, say, 1 kHz as a reference signal, use it to drive the speaker with exactly one watt, measure the loudness, and sweep the generator over the speaker's entire frequency range, all frequencies will measure equally loud. Sadly, with all but the most expensive speaker systems, they will not. Equalizers can help these frequency deficiencies. By adding a little here and taking away a little there, pretty soon you create an acceptable power response - and a whole lot better sounding system. It's surprising how just a little equalization can change a poor sounding system into something quite decent. The best way to deal with budget speakers -- although it costs more -- is to commit one equalizer channel for each cabinet. This becomes a marriage. The equalizer is set, a security cover is bolted-on, and forever more they are inseparable. (Use additional equalizers to assist with the room problems.) And now for the hard part, but the most important part: If you do your measurements outside (no reflections off walls or ceiling) and up in the air (no reflections off the ground) you can get a very accurate picture of just the loudspeaker's response, free from room effects. This gives you the room-independent response. This is really important, because no matter where this box is used, it has these problems. Of course, you must make sure the cost of the budget speaker plus the equalizer adds up to substantially less than buying a really flat speaker system to begin with. Luckily (or should this be sadly) this is usually the case. Again, the truth is that most cabinets are not flat. It is only the very expensive loudspeakers that have world-class responses. (Hmmm ... maybe that's why they cost so much!) The next thing you can do with equalizers is to improve the way each venue sounds. Every room sounds different -- fact of life -- fact of physics. Using exactly the same equipment, playing exactly the same music in exactly the same way, different rooms sound different -- guaranteed. Each enclosed space treats your sound differently. Reflected sound causes the problems. What the audience hears is made up of the direct sound (what comes straight out of the loudspeaker directly to the listener) and reflected sound (it bounces off everything before getting to the listener). And if the room is big enough, then reverberation comes into play, which is all the reflected sound that has traveled so far, and for such a (relatively) long time that it arrives and re-arrives at the listener delayed enough to sound like a second and third source, or even an echo if the room is really big. It's basically a geometry problem. Each room differs in its dimensions; not only in its basic length-by-width size, but in its ceiling height, the distance from you and your equipment to the audience, what's hung (or not hung), on the walls, how many windows and doors there are, and where. Every detail about the space affects your sound. And regretfully, there is very little you can do about any of it. Most of the factors affecting your sound you cannot change. You certainly can't change the dimensions, or alter the window and door locations. But there are a few things you can do, and equalization is one of them. But before you equalize you want to optimize how and where you place your speakers. This is probably the number one item to attend to. Keep your loudspeakers out of corners whenever possible. Remove all restrictions between your speakers and your audience, including banners, stage equipment, and performers. What you want is for most of the sound your audience hears to come directly from the speakers. You want to minimize all reflected sound. If you have done a good job in selecting and equalizing your loudspeakers, then you already know your direct sound is good. So what's left is to minimize the reflected sound. Next use equalization to help with some of the room's more troublesome features. If the room is exceptionally bright you can beef up the low end to help offset it, or roll-off some of the highs. Or if the room tends to be boomy, you can tone-down the low end to reduce the resonance. Another way EQ is quite effective is in controlling troublesome feedback tones. Feedback is that terrible squeal or scream sound systems get when the audio from the loudspeaker gets picked-up by one of the stage microphones, re-amplified and pumped out the speaker, only to be picked-up again by the microphone, and re-amplified, and so on. Most often, this happens when the system is playing loud. Which makes sense, because for softer sounds, the signal either isn't big enough to make it to the microphone, or if it does, it is too small to build-up. The problem is one of an out-of-control, closed-loop, positive-feedback system building up until something breaks, or the audience leaves. Use your equalizer to cut those frequencies that want to howl; you not only stop the squeal, but you allow the system to play louder. The technical phrase for this is maximizing system gain before feedback. It's important to understand at the beginning that you cannot fix room related sound problems with equalization, but you can move the trouble spots around. You can rearrange things sonically, which helps tame excesses. You win by making it sound better. Equalization helps. Figure 5. Bandpass Filter Parameters Figure 6. Variable-Q Graphic Figure 7. Constant-Q Graphic Equalizers are useful in augmenting your instrument or voice. With practice you will learn to use your equalizer to enhance your sound for your best personal expression: deepen the lows, fill the middle, or exaggerate the highs ... whatever you want. Just as an equalizer can improve the sound of a poor loudspeaker, it can improve the sound of a marginal microphone, or enhance any musical instrument. Equalizers give you that something extra, that edge. (We all know where "radio voices" really come from.) Seeing Sound To make loudspeaker and sound system measurements easy, you need a real-time analyzer (RTA). An RTA allows you to see the power response, not only for the loudspeaker, but even more importantly, for the whole system. Stand-alone RTAs use an LED or LCD matrix to display the response. A built-in pink noise generator (a special kind of shaped noise containing all audible frequencies, optimized for measuring sound systems) is used as the test signal. A measuring microphone is included for sampling the response. The display is arranged to show amplitude verses frequency. Depending upon cost, the number of frequency columns varies from 10 on 1-octave centers, up to 31 on 1/3-octave centers (agreeing with graphic equalizers). Amplitude range and precision varies with price. With the cost of laptop computers tumbling, the latest form of RTA involves an accessory box and software that works with your computer. These are particularly nice, and loaded with special memory, calculations and multipurpose functions like also being an elaborate SPL meter. Highly recommended if the budget allows. Dynamic Controllers Dynamic controllers or processors represent a class of signal processing devices used to alter an audio signal based solely upon its frequency content and amplitude level, thus the term "dynamic" since the processing is completely program dependent. The two most common dynamic effects are compressors and expanders, with limiters and noise gates (or just "gates") being special cases of these. The dynamic range of an audio passage is the ratio of the loudest (undistorted) signal to the quietest (just audible) signal, expressed in dB. Usually the maximum output signal is restricted by the size of the power supplies (you cannot swing more voltage than is available), while the minimum output signal is fixed by the noise floor (you cannot put out an audible signal less than the noise). Professional-grade analog signal processing equipment can output maximum levels of +26 dBu, with the best noise floors being down around -94 dBu. This gives a maximum dynamic range of 120 dB (equivalent to 20-bit digital audio) -- pretty impressive number -- but very difficult to work with. Thus were born dynamic processors. Compressors Compressors are signal processing units used to reduce (compress) the dynamic range of the signal passing through them. The modern use for compressors is to turn down just the loudest signals dynamically. For instance, an input dynamic range of 110 dB might pass through a compressor and exit with a new dynamic range of 70 dB. This clever bit of processing is normally done using a VCA (voltage controlled amplifier) whose gain is determined by a control voltage derived from the input signal. Therefore, whenever the input signal exceeds the threshold point, the control voltage becomes proportional to the signal's dynamic content. This lets the music peaks turn down the gain. Before compressors, a human did this at the mixing board and we called it gain-riding. This person literally turned down the gain anytime it got too loud for the system to handle. You need to reduce the dynamic range because extreme ranges of dynamic material are very difficult for sound systems to handle. If you turn it up as loud as you want for the average signals, then along comes these huge musical peaks, which are vital to the punch and drama of the music, yet are way too large for the power amps and loudspeakers to handle. Either the power amps clip, or the loudspeakers bottom out (reach their travel limits), or both -- and the system sounds terrible. Or going the other way, if you set the system gain to prevent these overload occurrences, then when things get nice and quiet, and the vocals drop real low, nobody can hear a thing. It's always something. So you buy a compressor. Using it is quite simple: Set a threshold point, above which everything will be turned down a certain amount, and then select a ratio defining just how much a "certain amount" is. All audio below the threshold point is unaffected and all audio above this point is compressed by the ratio amount. The earlier example of reducing 110 dB to 70 dB requires a ratio setting of 1.6:1 (110/70 = 1.6). The key to understanding compressors is to always think in terms of increasing level changes in dB above the threshold point. A compressor makes these increases smaller. From our example, for every 1.6 dB increase above the threshold point the output only increases 1 dB. In this regard compressors make loud sounds quieter. If the sound gets louder by 1.6 dB and the output only increases by 1 dB, then the loud sound has been made quieter. Some compressors include attack and release controls. The attack time is the amount of time that passes between the moment the input signal exceeds the threshold and the moment that the gain is actually reduced. The release time is just the opposite -- the amount of time that passes between the moment the input signal drops below the threshold and the moment that the gain is restored. These controls are very difficult to set, and yet once set, rarely need changing. Because of this difficulty, and the terrible sounding consequences of wrong settings, Rane correctly presets these controls to cover a wide variety of music and speech -- one less thing for you to worry about. System overload is not the only place we find compressors. Another popular use is in the making of sound. For example when used in conjunction with microphones and musical instrument pick-ups, compressors help determine the final timbre (tone) by selectively compressing specific frequencies and waveforms. Common examples are "fattening" drum sounds, increasing guitar sustain, vocal "smoothing," and "bringing up" specific sounds out of the mix, etc. It is quite amazing what a little compression can do. Check your owner's manual for more tips. Figure 8. Gate/Expander/Compressor/Limiter Action Expanders Expanders are signal processing units used to increase (expand) the dynamic range of the signal passing through it. However, modern expanders operate only below the set threshold point, that is, they operate only on low-level audio. Operating in this manner they make the quiet parts quieter. The term downward expander or downward expansion evolved to describe this type of application. The most common use is noise reduction. For example, say, an expander's threshold level is set to be just below the quietest vocal level being recorded, and the ratio control is set for 2:1. What happens is this: when the vocals stop, the signal level drops below the set point down to the noise floor. There has been a step decrease from the smallest signal level down to the noise floor. If that step change is, say, -10 dB, then the expander's output attenuates 20 dB (i.e., due to the 2:1 ratio, a 10 dB decrease becomes a 20 dB decrease), thus resulting in a noise reduction improvement of 10 dB. It's now 10 dB quieter than it would have been without the expander. Limiters Limiters are compressors with fixed ratios of 10:1 or greater. Here, the dynamic action prevents the audio signal from becoming any bigger than the threshold setting. For example, say the threshold is set for +16 dBu and a musical peak suddenly comes along and causes the input to jump by 10 dB to +26 dB, the output will only increase by 1 dB to +17 dBu -- basically remaining level. Limiters find use in preventing equipment and recording media overloads. A limiter is the extreme case of compression. You will hear the term pumping used in conjunction with poorly designed or improperly set limiters. Pumping describes an audible problem caused by actually hearing the gain change -- it makes a kind of "pumping" sound. This is particularly a problem with limiters that operate too abruptly. Rest assured that Rane limiters are designed not to have any audible side-effects. Noise Gates Noise gates (or gates) are expanders with fixed "infinite" downward expansion ratios. They are used extensively for controlling unwanted noise, such as preventing "open" microphones and "hot" instrument pick-ups from introducing extraneous sounds into your system. When the incoming audio signal drops below the threshold point, the gate prevents further output by reducing the gain to "zero." Typically, this means attenuating all signals by about 80 dB. Therefore once audio drops below the threshold, the output level basically becomes the residual noise of the gate. Common terminology refers to the gate "opening" and "closing." A gate is the extreme case of downward expansion. Just as poorly designed limiters can cause pumping, poorly designed gates can cause breathing. The term breathing is used to describe an audible problem caused by being able to hear the noise floor of a product rise and lower, sounding a lot like the unit was "breathing." It takes careful design to get all the dynamic timing exactly right so breathing does not occur. Rane works very hard to make sure all of its dynamic processors have no audible funny business. Another popular application for noise gates is to enhance musical instrument sounds, especially percussion instruments. Correctly setting a noise gate's attack (turn-on) and release (turn-off) adds "punch," or "tightens" the percussive sound, making it more pronounced -- this is how Phil Collins gets his cool snare sound, for instance. "Signal Processing Fundamentals" This note in PDF.

 

© 1997 Rane Corporation

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