EE 2170: Design and Analysis of Signals and Systems

Design of a 10-band Graphic Equalizer

 

Instructor: Carlos Davila

Dept. of Electrical Engineering, Southern Methodist University

 

The object of this lab is to design a 10-band graphic equalizer. This is a two-week lab[1]. Graphic equalizers are used extensively in the music industry for recording, performance, and high-end audio applications. The following links give information about equalizers and bandpass filters:

 

 

In this lab, we will use Simulink to rapidly prototype a 10-band graphic equalizer. Most commercial equalizers use either 1/3 octave or 2/3 octave bandpass filters. We will design one octave bandpass filters. Following are the design specifications for the equalizer:

 

1. Using Matlab, design 10 different bandpass filters having center frequencies of 31.5 Hz, 63 Hz, 125 Hz, 250 Hz, 500 Hz, 1000 Hz, 2000 Hz, 4000 Hz, 8000 Hz, and 16000 Hz. These center frequencies correspond to the ISO (International Standards Organization) standard for graphic equalizer center frequencies.

 

2. The bandwidth for each filter is measured as the frequency difference f2 – f1 > 0 where f2 and f1 correspond to the frequencies where the gain is down 3 dB from the maximum gain (12 dB) at the center frequency. Choose f1 and f2 such that the center frequency, fc corresponds to the geometric mean of f1 and f2, i.e. fc = (f1f2)1/2.

 

3. The center frequency gain for each bandpass filter should vary between +12 dB and -12 dB (i.e. the filters have both “boost” and “cut”). Use a slider gain block to adjust the gain between these two settings. The filter gain in dB corresponds to:

 

 

4. The passband gain of the equalizer when all bandpass filters are at 0dB gain (each slider is in the center position) should vary by no more than 1 dB.

 

 

Each bandpass filter can be designed using the matlab command “butter”. The following comes from the Matlab help file:

 

“[b,a] = butter(n,Wn,'s') designs an order n lowpass analog Butterworth filter with angular cutoff frequency Wn rad/s. It returns the filter coefficients in the length n+1 row vectors b and a, in descending powers of s, derived from this transfer function: 

 

 

butter's angular cutoff frequency Wn must be greater than 0 rad/s. If Wn is a two-element vector with w1 < w2, butter(n,Wn,'s') returns an order 2*n bandpass analog filter with passband w1 < w < w2”

 

Implement each bandpass filter in Simulink and connect them in parallel and sum their outputs as shown in the following image:

The “triangles”are boost/cut aimplifiers. To test your design, plot the frequency response, both magnitude and phase, of each of the 10 bandpass filters using the Matlab function “freqs”. The frequency response is just H(s) with s = jW, where H(s) is the Laplace transform of the impulse response, h(t). To test your final design, use a sine-wave generator to find the frequency response of the equalizer when the gain on each bandpass filter is +12 dB, as was done in the previous lab. Alternately, you can find the equivalent parameters of the equalizer for the Matlab “freqs” command. Submit your Simulink model and a detailed description of your design process.

 

100% Extra Credit Project: Design a 10-band constant-Q graphic equalizer. To do this you will have to learn about s-functiosn in Simulink help. Each bandpass filter will have to alter its characteristics as its gain is modified in order to maintain a constant quality factor (Q):

 

 

 


[1] The grade for this lab will be twice that of 1-week labs.