# Equalizer

An equalizer adjusts the magnitudes of selected frequency intervals in a signal.

An equalizer can, for example, increase or decrease the magnitude of high frequencies while leaving all the magnitude of other frequencies (low and mid-range ones) the same.

## An example two-band digital equalizer

To create a two-band digital equalizer, we will combine a low pass filter and a high pass filter. The low pass filter will pass only low frequencies, below some cutoff frequency. The high pass filter will pass only high frequencies, above the same cutoff frequency. If we want to increase the magnitude of high frequencies, for example, we can add some gain to the high pass filter but not to the low pass filter and then combine the signals coming out of the two filters.

Take the sampling frequency 2000 Hz and create two filters of length 201. Use the cutoff frequency 40 Hz. Suppose that we add gain of 2 dB to the high pass filter, which means that we should multiply the coefficients of the high pass filter by 102/20 = 1.2589. Suppose that we use the Bartlett-Hann window, the magnitude response of which is somewhat smooth. Then the magnitude response of the total filter (the two-band equalizer) would be the one shown in figure below. This is an appropriate magnitude response for the desired equalizer.

The detailed steps to creating the equalizer are as follows.

1. Create a finite impulse response low pass filter at 40 Hz (see Low pass filter).
2. Create a finite impulse response high pass filter of the same length at 40 Hz or create a filter of different length and pad the shorter of the two filters with zeroes (see High pass filter).
3. Multiply the coefficients of the high pass filter with the desired gain. The desired gain is 2 dB or 102/20 = 1.2589.
4. Add the two filters together to obtain the equalizer.

If aL(k) are the coefficients of the low pass filter, aH(k) are the coefficients of the high pass filter, and g is the gain, then the total filter a(k) of the equalizer would be as follows.

$$a(k)=a_L(k)+g\,a_H(k)$$

The amount of gain applied to individual filters translates to the same amount of gain in the magnitude response of the total filter.

This is a two-band equalizer consisting of a low pass and a high pass filter and is quite simple. Multiband equalizers are just as simple. A three-band equalizer would need a low pass filter for the low frequency band, a band pass filter for the mid frequency band, and a high pass filter for the high frequency band. A ten-band equalizer would use one low pass filter, eight band pass filters, and one high pass filter.

To choose the appropriate cutoff frequencies for the filters, note that the human year interprets the frequency spectrum exponentially (e.g., the top of an octave is a note with frequency that is two times the frequency at the bottom of the octave). We should, split the frequency spectrum of a fixed band equalizer exponentially. If, for example, we are working with the frequency range of human hearing, say 20 Hz to 20000 Hz, we could create a ten-band equalizer by splitting this frequency range in 10 bands exponentially. The cutoff frequencies of the respective filters would be: 20 Hz = 20 * (20000 / 20)0/10, 39.9 Hz ≈ 20 * (20000 / 20)1/10, 79.6 Hz ≈ 20 * (20000 / 20)2/10, and so on until 20,000 Hz = 20 * (20000 / 20)10/10.

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