An equalizer is a circuit which is capable of amplifying or attenuating a certain frequency range, and to leave others unaltered. Now we have enough knowledge to interpret the curve which describes the behaviour of an equalizer. It is a graph, plotted on an amplitudefrequency diagram, which gets multiplied by the input signal in order to obtain the output signal (we remind once more that X(f), Y(f) and H(f) are expressed in the frequency domain).
Two preliminary examples will further clarify this concept for us.
H(f) = constant and in particular equals to 1 on the entire spectrum. We'll have, according with the latter statement, the following formula:
Y(f) = X(f)
In other words the input signal isn't affected by the circuit and therefore reaches the output unaltered.
H(f) = 1 in a particular frequency range, and elsewhere 0:
The result Y(f) is given by multiplying X(f) and H(f). When H(f) is equal to 0 we get Y(f) = 0. When H(f) = 1, we get Y(f)=X(f). This is a basic example of a bandpass ideal filter. Although this topic shall be explored in greater detail further on in the course, we can already begin to see how a transfer function of this kind allows us to extract only a certain frequency range from the input signal (between 5 KHz and 10 KHz on the abovementioned diagram).
The filter is called ideal because in reality it is impossible to create circuits that have transfer functions which such sudden transitions. In actual fact these transitions get smoothed and we'll see how the slope rate increases as the complexity, and therefore also the cost, of the circuit in question increases. There are different kinds of equalizers, and we'll be analyzing the most important ones in the next sections.
Its transfer function is shaped as illustrated in the following diagram:
This kind of equalizer has 3 controls:
Gain (reduction/amplification  cut/boost)
Acts upon amplitude A of the bell which can be both positive (amplification) and negative (attenuation). Maximum amplification is a parameter which depends on the quality of the circuit: to reach 15 dB gain without added distortions requires sophisticated technology. Generally we find this kind of EQ on mixing desk channels. The more professional the quality of the mixer, the more its EQ peaks allow high gain levels without added distortion. In medium quality mixers, the gain levels are usually of about 12 dB (bearing in mind that between 12 dB and 15 dB we have a doubling of the signal in electrical terms, and therefore a considerable difference).
Cutoff frequency
This is the frequency which has the highest (or lowest) gain on the bell. It is generally controlled by a potentiometer thus allowing the bell to be centered precisely in the frequency zone we intend to manipulate.
Qfactor
This is a parameter which measures the bell's amplitude, in other words, the amplitude of the frequency range that is being amplified (or attenuated). It is calculated using the following formula:
Q= f_{c}/(relative bandwidth)
where the relative bandwidth is measured at 3 dB below its peak (see previous diagram). The Qfactor works independently from the frequency zone in question. This becomes clear with a numeric example, bearing in mind that the frequency scale is logarithmic. Between 20 Hz and 100 Hz the width of the relative band is 80 Hz. If we then move to higher frequencies, i.e 10000 Hz, our bell would extend over the 9960 and 10040 Hz range. In other words we'd have an extremely tight bell indeed (which is in fact impossible to reproduce, for plain physical reasons).
So, if we keep constant the relative range of the bell and if, with the control of the central frequency, we drag the filter along the entire frequency spectrum, the bell will tighten as we reach the higher frequencies and widen as we reach the lower ones. Seeing that once they have been set, we'd like the relative range of the bell to remain constant along the entire spectrum, we will add the central frequency to the Qfactor formula as a normalizing factor.
Let's now see in practice with numbers how the quantities involved vary (w = relative bandwidth):
if f_{c} = 100 Hz and w = 40 Hz which means that the bell will have a relevant influence on the range 80 Hz  120 Hz
we'll have: Q=100/40=2.5
if f_{c} = 10000 Hz and Q = 2.5
we'll have: w = 10000/2.5 = 4000Hz which means the range going from 8000 Hz to 12000 Hz.
Here it becomes clear that w must vary if we want the bell to maintain a constant shape along the frequency spectrum (seeing that we have put a constant Qfactor). Since f_{c} has been doubled, maintaining the same Qfactor, the range has also been doubled, and this way the bell's shape has remained intact (we mustn't forget that frequencies are represented logarithmically in order to give a more precise representation of the way the human ear perceives sound. With low frequencies a difference of 20 Hz is relevant, whereas with high frequencies, a difference of 200 Hz becomes relevant).
This type of equalizer is used to control the far extremes of the audible frequency spectrum. It has two standard controls:
Cutoff frequency (rolloff): calculated at the point where the gain curve drops by 3 dB in relation to its maximum value.
Gain: applies an amplification or an attenuation to the signal above the cutoff frequency.
Completely parametric: it is possible to modify all three quantities that characterize the bell equalizer: central frequency (f_{c}), gain (A), Qfactor (Q). Professional mixers have a 4band parametric equalizer on each channel.
Semi parametric: fixed Qfactor. The bell's shape is fixed (generally Q is set at a value of approximately 1.5).
Peak: f_{c} and Q values are fixed and we can only act upon the gain. These EQs are the cheapest on the market and are installed on low quality mixers.
The following diagrams compare the equalizing section of a low quality mixer and a high quality mixer. We can see how the maximum applicable gain is 12 dB in the first, and 15dB (or even 18 dB) in the second. Moreover, the frequency spectrum is sub divided into 3 bands (lows, middles, highs) in the low quality mixers, whereas in the high quality ones we have 4 bands (lows, middlelows, middlehighs, highs). Finally, in the high quality mixers, the gain curve of lows and highs can take the form of a bell or of a shelf equalizer, thus allowing even greater versatility.
A graphic equalizer consists in a series of single bell equalizers. The width of the bell varies depending on the context the equalizer has been designed for.
Table 5.1. Classification of graphic equalizers

Passive equalizers only use passive components which don't require current and therefore cannot bring about a real increase in gain. Generally when gain is at its highest, the signal's amplitude doesn't change, whereas it gets attenuated when gain is decreased through a potentiometer or a cursor. Their main defect is that they introduce a slight signal drop due to the loss of passive components. Active equalizers use active electronic components, such as transistors [Transistor ] , and allow a real gain increase to take place. However, as a result of active circuiting higher levels of distortion and noise occur, although this only tends to happen with low quality equalizers. The next diagram compares the gain levels of active and passive equalizers:
Example of an ideal filter