This document contains an in-depth look into filters and equalisers.

**The Low-cut and High-cut filters:**

This filters is usually used in the begining of the signal path to filter out any unwanted humming and hiss.

Figure 1 - First order filter

The transfer function of this filter is as follows:

The time constant of R_{1}C_{1} determines the low-cutoff
frequency while the time constant of R_{2}C_{2} determines
the high cutoff frequency. The pass-band gain, A_{vpass}=R_{2}/R_{1}.

Figure 2 shows the standard topology used for an second order filter.

Figure 2 - Second-order lowpass or highpass filter

The transfer function of this circuit is as follows:

with

For a low-cut filter, that is a high-pass filter, Z_{a} must
be capacitors and Z_{b} resistors. By substituting Z_{a}=1/sC
and Z_{b}=R into above equation, the following equation is obtained:

For a high-cut filter, that is a low-pass filter, Z_{a} must
be resistors and Z_{b} capacitors. By substituting Z_{a}=R
and Z_{b}=1/sC into above equation, the following equation is obtained:

For both circuits the following equations apply:

, and

The best choice for the Q-factor is 0.707. This results in the sharpest curve, without any overshoot. If this Q-factor is selected, the value of K must be 1.586. Note that this is also the pass-band gain. The signal should therefore be attenuated if constant signal level is desired. If a slope steeper than 12db/oct is desired, a higher order filter can be used. It is best to use the Butterworth response for audio, because of its flat response.

**The shelving EQ circuit:**

The simplest type of equaliser is the low and high shelving equalisers.
This is usually called Bass and Treble on Hi-fi sets. Figure 3 shows the
basic topology for such an equaliser. The potentiometer, R_{p},
sets the boost cut ratio. The impedance of Z(s) must be high at the frequencies
to boost/cut and low at other frequencies, effectively shorting out R_{p}.

Figure 3 - Basic shelving equaliser circuit

The transfer function of this circuit is as follows:

If a capacitor is substituted for Z(s), a low shelving equaliser is
realised, while an inductor is needed for a high shelving equaliser. Figure
4 shows a circuit for a combined Lo and Hi shelving equaliser as used in
conventional hi-fi sets and mixing consoles. As can be seen, the inductor
in parallel with R_{treble} is omitted, C_{2} is instead
inserted. At the high frequencies, R_{bass} is shorted out by C_{1}.
The higher the frequency becomes, the lower C_{2}?s impedance become,
giving the gain setting on R_{treble} higher priority over the
unity gain network formed by the lo section.

Figure 4 - Lo and Hi shelving equaliser circuit

**Basic variable Q graphical EQ circuit:**

The circuit in Figure 5 shows the basic circuit for a graphical equaliser.
The frequency dependant impedance, Z(s), determines the centre frequency
of the equaliser. The impedance of Z(s) must be low at the centre frequency,
while high at other frequencies. By replacing Z(s) by a short it can be
easily seen that the left part of R_{p} forms an attenuator feeding
the input signal to the non-inverting operational amplifier, which gain
is controlled by the right half of R_{p}. If Z(s) is replaced by
an open-circuit, R_{p} is only connected between the two input
terminals of the op-amp, which is at the same potential and R_{p}
does not effect the circuit. Multiple bands can be implemented by placing
more than one pot in parallel, each with its wiper connected to a different
Z(s) circuit.

Figure 5 - Basic graphic equaliser circuit

If it is assumed that Z(s) has zero impedance at the centre frequency, the transfer function of the filter will be as follows:

It can be seen that if b<0.5, that is
R_{p}?s wiper is moved toward the left, the gain is below unity,
while if b>0.5, that is R_{p}?s wiper
is moved toward the right, the gain is above unity. The maximum boost/cut
ratio is controlled by the impedance of Z(s), which effectively limits
the value of b so that a<b<(1-a)
where a=Z_{pass}/R_{p}.

Figure 6 shows a practical circuit for Z(s). On the left side is the
classic series RLC circuit, while the circuit on the right side uses a
gyrator to replace the inductor. To simplify the analysis of the circuit
on the right, it is assumed that the impedance of C_{1}, R_{1}
is much higher than the impedance of C_{2}, R_{2}.

Figure 6 - Frequency dependent impedance circuit

The impedance of the circuit is described by the following equations:

A shelving equaliser can also be realised with this circuit topology. For a Lo shelving equaliser the circuit in Figure 6 can be replaced by a single capacitor in series with an resistor. For a Hi shelving equaliser, the circuit in Figure 6 is used as it is, except that the high frequency roll-off of the impedance is above the audio spectrum. The same scheme can also be used for a Lo shelving equaliser. The advantage of this is that the circuits would not boost any unwanted noise beyond the audio spectrum.

**Parametric equaliser circuit:**

Figure 7 shows a parametric equaliser circuit which is used in most older mixing consoles. In this case Z(s) is a bandpass filter. The transfer function of this circuit is as follows:

For the equations following, Z(s) must be positive and its bandpass gain must be below 0.5.

Figure 7 - Basic parametric equaliser circuit

The drawback of this circuit is that it is not a constant Q design. Another drawback is that the filter, Z(s), is always in the circuit, even if the boost/cut control is flat. The following circuit takes care of these two problems:

Figure 8 - Main signal flow circuit for a constant Q parametric equaliser

The above circuit forms the main signal flow path of a constant Q parametric
or graphical equaliser. The output, V, is connected the the input of a
bandpass filter. This bandpass filter determines the centre frequency and
Q-factor of the equaliser. The output of the bandpass filter is then send
to a variable gain section. The output of this section can either be sent
to V_{c}, to cut frequencies, or V_{b} to boost frequencies.
Note that constant Q operation is achieved by only sending a signal to
V_{c} or V_{b}, but not both.

All the resistors except R3 and R5 has the same values. The value of R3 determines the cut ratio, while the value of R5 determines the boost ratio.

If V_{c }= aZ(s)V and V_{b}
= bZ(s)V then

where Z(s) is the transfer function of the bandpass filter. To boost the mid frequencies, a must be 0 while b is adjusted. The opposite is true to cut mid frequencies. Note that for constant-Q operation, the one which is not used, must be zero.

One way to implement the variable gain stage and distributor is two
use a linear potentiometer with a centre tap. A resistor is connected from
the output of the banpass filter to the wiper of the potentiometer. This
resistor determines the maximum boost/cut ratio. The one end of the pot
is connected to V_{c}, while the other end is connected to V_{b}.

If a centre tap potentiometer is not available, a standard dual-gang
linear potentiometer can be used. The one half is used to control the gain
of special symmetrical gain stage, while the other half drives a comparator
which controls analogue switches which either connects the output of the
gain stage to V_{c} or V_{b}. A standard comparator in
combination with a SPDT analogue switch can be used to switch between the
V_{c} and V_{b} inputs. It is advisable to use hysterises
in the comparator.

The circuit in Figure 9 shows a circuit for a symmetrical gain stage. This gain stage gives maximum gain at the end of the boost/cut control?s travel, while giving zero gain at its centre.

Figure 9 - Symmetrical variable gain circuit

The gain of this circuit is as follows:

where R_{T}=(R+gR_{p})||(R+(1-g)R_{p})
and g the position of R_{p} is.

The maximum value of R_{T}, R_{Tmax} is when g=0.5,
which would also result in A_{vmin}. The minimum value of R_{T},
R_{Tmin} is when g=0 or g=1,
which would also result in A_{vmax}. Here is the equations for
R_{Tmax} and R_{Tmin}:

R_{Tmax} = (2R+R_{p})/4

R_{Tmin} = (R+R_{p})||R

If g=0.5, that is, a flat response, A_{v}
must be 0. To accomplice that R_{1} must equal R_{Tmin},
therefore

For maximum linearity of the boost/cut control, R must not be too small
compared to R_{p}. Note that the larger R is made, the smaller
the maximum gain of the gain stage becomes. This circuit shows a deadband
in its response about the centre of R_{p}. This is actually an
advantage, for it gives adequate time for the analogue switches to switch
from V_{c} to V_{b} and vice-versa. It is even possible
to give the switches a dead-band, where the filter and gain section is
totally cut-out if R_{p} is in its centre position.

The first step in designing this circuit is to select a value for R_{p}.
The best choice for R is the same as R_{p}. This gives the most
linear curve, without losing to much gain. After the values of R_{p}
and R have been selected, the value for R_{1} is calculated.

Next the values the of A_{vmax} and R_{2} must be calculated.
The value of R_{2} is calculated as follows:

The value of R_{2} cannot be negative, therefore

R_{T}-R_{1}-A_{vmax}R_{T} > 0 or

After the value of A_{vmax} has been selected, the value of
R_{2} can be calculated. It is best to compose R_{1} of
a fixed resistor and preset resistor in series. The circuit can then be
balanced to give zero gain at zero boost/cut.

**Bandpass filters used in equalisers:**

The filter in Figure 10 is a simple passive network. This circuit is
usually used in sweepable mid sections of older mixing consoles. Note that
a buffer is usually used to buffer the voltage from the wiper of R_{p}
and also to set the correct banpass gain. Let?s call this gain a.

Figure 10 - Passive bandpass filter network

The following formulas describes this bandpass filter:

If C_{1}=C_{2}=C and R_{1}=R_{2}=R like
in the practical circuit where R_{1} and R_{2} are part
of a dual-gang pot for a sweepable frequency, the equations is like this:

Figure 11 shows a circuit for an active bandpass filter. This circuit is used if the centre frequency and Q-factor is fixed, like in a graphical equaliser.

Figure 11 - Active single op-amp bandpass filter

The transfer function of this circuit is as follows:

It is rather complicated to design this filter, the following equations
simplify that. The input to the equations if f_{c}, Q, A_{vpass}
and an arbitrary selection of C.

The most flexable filter is a state variable filter. This type of bandpass filter's centre frequency and Q-factor can be independently adjusted. That is the reason why this type of filter is used in most modern sweepable and parametric equaliser circuits. Due to the complexity of the filter and the huge amount of op-amps, this filter have the drawback of being more noisy than the above filters. Figure 12 shows a circuit of a state-variable filter. The op-amps U1 and U2 form summing amplifiers, while U3 and U4 form two integrators. The output of U3 is the bandpass output, while the outputs of U2 and U4 is highpass and lowpass outputs respectively.

Figure 12 - State-variable active filter

The transfer function of the filter is as follows:

The Q-factor of the filter is adjusted by VR1, while the centre-frequency is adjusted by VR2. Note that for Q values below 1, the amplitude of the hp and lp outputs is 1/Q times higher than the bp output. If such low values of Q is needed, the input signal needs to be attenuated to prevent clipping of the hp and lp outputs. Taking this into account, it may be better to put the variable gain stage before the filter instead of after it. The drawback of putting the stages this way is that the noise generated by the filter will always be present on the output of the equaliser.

The following three equations is used to design the filter. It can be
seen that the design equations is quite simple. The centre frequency is
made sweepable by making R_{3} a dual-gang pot, while is the Q-factor
is made adjustable by making R_{2} adjustable.

By using the lowpass or highpass outputs, this filter can be used in a advanced shelving equaliser system. The only problem is that the lowpass and highpass gains is dependent on Q:

A_{lppass} = A_{hppass} = 1/Q

The remedy for this is to make R_{2} fixed and rather make the
resistor between V_{out} and the inverting input of U1 variable.
Take note that for low values of Q, the lowpass and highpass outputs might
clip for large signal levels. The input signal to the filter should therefore
be attenuated if low values of Q is going to be used.