Design of an M-Band Graphic Equalizer
for Audio Signals
Design an M-band graphic equalizer for modifying (mono or stereo) audio signals.
Implementation where x is the vector of input audio samples, y is the vector of output audio
samples, M is the number of sub-bands, fs is the audio sampling rate, gain is a vector of length
M consisting of
design in Matlab as a function of the form
y = graphic(x, M , fs , gain, parameters),
the scaling gains of the sub-bands, ordered from lowest to highest frequency band, and expressed in
units of decibels with each gain in the range [-20, +20] dB
Abstract :
This project describes the design and implementation of an M-band-Graphic Equalizer using filter
banks.
Filter banks are a group of band-Pas filters that are connected in parallel. Each Parallel connection form
a channel to isolate a certain frequency-band from the input signal. By doing so we can post or
attenuate any band without affecting the other bands , this is done since band are isolated buy
individual channels. There are many methods to realize a filter bank , in this project uses the cosine
Modulated Filter Bank by modulate a low-pas filter , it shifts the Bas-band of a Law-Pas filter to a higher
frequencies to cover the entire band of frequencies of the input signal.
Filter bank structure :
If the bandwidths of each filter in the filter bank are equals then it’s called uniform filter bank , and if
not then it’s a non-uniform filter banks . in this paper the filter bank uses an octave structure where the
bandwidth of the current filter is twice the size of the size of the filter below as follow :
If ℎ0, ℎ1, ℎ2𝑎𝑛𝑑 ℎ3 are analysis filters then bandwithe of ℎ2 is twice the size of that of ℎ1 and ℎ3 is
twice the size of ℎ2 , while ℎ𝑜 𝑎𝑛𝑑 ℎ1 have the same bandwidth. The Octave bands can be formed by
merging the uniform bandwidth, so if 𝑘 is the number of uniformed bands ,then the number of nonuniformed octave bands 𝑀 = 1 + log(2) 𝑘 . If 𝑏0, 𝑏1, 𝑏2 𝑎𝑛𝑑 𝑏3 are the sup-uniform bands , then a 3
band sup-bands can be formed as : 𝐵0 = 𝑏0, 𝐵1 = 𝑏1, 𝑎𝑛𝑑 𝐵2 = 𝑐𝑜𝑚𝑏𝑖𝑛𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑏2 𝑎𝑛𝑑 𝑏3. The
uniform filter-bank consists of a parallel connection of the 𝑘 sup-band filter, the impulse responses,
merging bands is accomplished by adding the respective impulse responses .
Analysis Filters :
Filters formed using a cosine modulated low pass filter, designed using dolph windowfunction.
Multiplying the filter coefficients with the cosine function described previously, decided the shift of the
pass band of the low pass filter. This can be performed in Matlab by multiplying the low pass
filtersequence with the sequence formed by the cosine function. The bandwidth of each analysis filter is
formed using the octave structuring . In Matlab, octave bands are formed by according adding the
impulse responses or2𝑀−1 filters of the 2𝑀−1 band uniform filter bank. The synthesis filters have
impulse response that are matched with the respective analysis filters, but of opposite phase, so that we
can get a perfect reconstruction of the input signal. In Matlab, the order of the filter coefficients of each
analysis filter is reversed and then assigned to the respective synthesis filters. The output sequence from
each of the synthesis filters is multiplied with a scalar, which specifies the gain. In Matlab, The values of
the gain for respective channels is entered in to a gain vector during the execution of the program.
The function should do the following. :
1. Check that each gain is in the correct range, and adjust any out-of-range input values to
the closest in-range value:
The user gain input is of an 8 elements since we have 8 channels , every channel have a unique
gain , this matlab code supports the first part :
file = input('\nPlease Enter the audio file name for 8 band graphic
equalizer: \n','s');
[xi, Fs] = audioread(file);
%% her the user will inbut the wav file of choice and this function will a
signed FS sampling frequency(Fs) , and the data(xi) tyo be processed , her
the range of data and Fs is computed using this function instead of interring
a direct Fs or xi values.
-------------------------------------------------------
This part of the code will give the user 2 options , a defult gain adjustment
or a an input by the user for a gain of 8 frequancies between [0,1 :10] as
spesefied at the project
choice = 100;
k = 0;
val = 0;
while val == 0
if k > 0
fprintf('\nInvalid input')
end
choice = str2double(input('\nPlease press "0" to use the default gains or
press "1" to enter user defined gains: ','s'));
if choice == 0 || choice == 1
val = 1;
end
k = k +1;
end
------------------------------------------------------------------1: this will allow the user to use a 4 defaults gain adjustments based on 4
cases as:
%// the defult agustments a user can input //
fprintf(' Availabl_Adjustments : A.treble B.bass C.full treble D.full
bass\n ');
preset = input('Input an Adjustment: ','s');
switch preset
case 'A'
sgain = [0.5, 0.5, 0.5, 0.5, 0.5, 1.5, 2,3];
case 'B'
sgain = [ 3,2, 1.5, 0.5, 0.5, 0.5, 0.5, 0.5];
case 'C'
sgain = [0.1, 0.1, 0.1, 0.1, 0.5, 0.5, 2,3];
case 'D'
sgain = [3, 2, 0.5, 0.5, 0.1, 0.1, 0.1, 0.1];
end
----------------------------------------------------------------------2: if the user desided to inter his 8 elements gain , this is the part that
support it :
flag = 0;
k = 0;
sgain = 0;
if choice == 1 % this part will suport choice number 1, user will input 8
elemnts of gains between [0.1 :10].
while length(sgain) ~= 8 || flag == 0
if k > 0
fprintf('\nInvalid input')
end
sgain = input('\nPlease Enter gain as a 8 element array[# # # # # # #
#] with values between 0.1 - 10: ');
minimum = min(sgain);
maximum = max(sgain);
if minimum >= 0.1 && maximum <= 10
flag = 1;
end
k = k+1;
end
the reference I used to understand the equalizer design uses a window function called Dolph –
Chebyshev window to control the attenuation . Its equation is given as below:
𝜋𝑘
𝑊(𝐾) = ((−1)𝑘 cos [𝑁𝑐𝑜𝑠 −1 [𝛽 cos ( 𝑁 )]) /(cosh[𝑁𝑐𝑜𝑠ℎ−1 (𝛽)])) .
1
Β=𝛽 = cos[𝑁𝑐𝑜𝑠ℎ−1 (10𝛼 )], and 𝑁 her is the length of the sequence.
the level of the side lobes can be controlled by the parameter α. If α = 5, then there will be an
attenuation of 100dB on the side lobes. There is a tradeoff between the size of the main lobe and the
attenuation of side lobes. If the attenuation of the side lobe is very large, then there will be an increase
in the main lobe, and thus deviating from the ideal characteristics. The length of the filter also has an
effect on the widow characteristics. If the length is small, then the side lobe would be large enough,
deviating hugely from the ideal characteristics .
---------------------------------------------------------------------------
2. Using the M octave-structured sub-bands, so that, for example, if M = 5, then sub-bands
S1 , S2 .S3 , S4 , S5 have respective bandwidths B, B, 2B, 4B, and 8B, where the total number of
bands satisfies
f s
2 = B ×2M −1 . However, it is also permissible to use a different
(user-defined), more general selection of bandwidths for the M sub-bands, provided that the
octave-structured configuration is among the user options.
Filter Bank:
Filter bank separates the input-signal in to different components . each component caries signal
frequency sup-component of the input signal. These different components are divided in to 2 groupsAnalysis filters and syntheses filters . where at the analysis side down-sampling is done and at the
synthesis side up sampling is done. It is very important to add a signal processing unit between the 2
filters to avoid aliasing . the figure below illustrate the filter bank components.
Figer 1: Structure of a filter bank.
The octave filter I designed have an M=8 to meet the specifications
The cosine modulation :
Each filter of the analysis part has an impulse response of a cosine modulated version of a prototype a
low-pas filter with a cutoff frequency of a
𝑓𝑠
𝑀2
where 𝑀 is the number of Bannds in the filter bank and the
and 𝑓𝑠 is the sampling frequency of the signal .The length of the impulse-response of each filter is 𝑙 =
2𝑀 .
The impulse-response of each analysis filter is given by :
ℎ, 𝑘(𝑛) = 2ℎ1𝑝 cos((2𝐾 + 1) /(4𝑀(2𝑛 + 1 − 𝑀)𝑝𝑖) , where ℎ1𝑝 is the prototype of a law pasfilter and = (1,2,3, … . . 𝑀 − 1) , the synthesis filter response are matched to that of the Analysis filter by
this equation : 𝑔𝑘(𝑛) = ℎ𝑘(𝐿 − 1 − 𝑛) , where 𝑛 = (1,2,3, , , , 𝐿 − 1). Because of this matching using
the cosine modulation, the total frequency bandwidth of the input signal is covered by all the
bandwidths of all the analysis filters , each filter have a bandwidth of 1/𝑀 of the total bandwidth of the
input signal .
Graphic equalizer :
The main idea of a graphic equalizing is to to attenuate different bands of frequancies based on the user
input. The graphic equalizers used in music industry or recording companies are the M-band octave
equalizer. The user can either boosts the gain or attenuate the selected band .
Design of M-Band Octave Structured :
The filters are designed using an FIR Low Pass filters, which are cosine modulated. This FIR Low Pass
filters are linearly phased , the filter coefficients are usually symmetrical. The ideal Low Pass filter with
cutoff frequency 𝑓𝑐 Hz has the following impulse response.
𝐿−1
)) , where
2
ℎ(𝑖𝑑𝑒𝑎𝑙) = 2𝑓𝑐𝑇𝑠𝑖𝑛𝑐(2𝑓𝑐𝑇(𝑛 − (
𝐿−1
) Samples
2
the filter is delayed by (
, where 𝐿 is
the length of the filter and 𝑇 is the sampling period.
This impulse response is non-causal and infinite in duration. Therefore we multiply this one with a
rectangular window function to restrict the number of samples to the length of the window :
W= (𝑛) = {
1, 𝑛 = 1,2,3, … . 𝑀 − 1
0 , 𝑛 = 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
we need to use windows whose values gradually change from 1 to 0, from both sides. There are several
windows and each have their merits and demerits. I found a window function called Dolph –Chebyshev
window
. The attenuation factor R is a multiple of the length L and thereby making it dynamic. The scalar that is
multiplied is decided by opservation .
This part of the code will support the FIR filter design and the 8M-pand Octav filter
With the requaiared bandwidth and the total number of bands satisfy the nedded Fs value :
function Y = graphic(x,Fs, M, sgain)
fprintf('Computing ...');
%Read the input WAV file
lx = length(x);
%//////////////////////////FIR filter design//////////////////////////
K = 2^(M-1); % ad to support part2 of the project also add the graph for part
2
L = 2*K;
%Cutoff frequency
B = Fs/(2*K);
%Normalized cutoff frequencies
fcn = B/(Fs/2);
if L > 100
slripple = 100;
else
slripple = 2*L;
end
bhlp = fir1(L-1, fcn, chebwin(L, slripple));
%Cosine modulated Octave filter bank have 2 componants :
%Analysis filters : down-sampling
for p = 1:K
for kb = 1:L
b(p, kb) = 2*bhlp(kb)*cos(((2*(p-1) + 1)/(4*K))*(2*(kb-1) + 1 - K)*pi);
end
end
i1 = 3;
for ka =
if ka <
h(ka+1,
else
i2 = i1
h(ka+1,
i1 = i2
end
end
0:M-1
2
1:L) = b(ka+1, 1:L);
+ 2^(ka-1) - 1;
1:L) = sum(b(i1:i2, 1:L));
+ 1;
%Synthesis filters : up-sampling
for ks = 1:M
g(ks, 1:L) = h(ks, L:-1:1);
end
%DC gain estimation
sumc = [zeros(1, 2*L - 1)]; %Initial
for kd = 1:M
sumc = sumc + conv(h(kd, 1:L), g(kd, 1:L));
end
a = sum(sumc);
%Output of the analysis filters
for kw = 1:M
w(kw, 1:(lx + L - 1)) = conv(x, h(kw, 1:L));
end
%Output of the synthesis filters
for kv = 1:M
v(kv, 1:(lx + 2*L - 2)) = conv(w(kv, 1:(lx + L - 1)), g(kv, 1:L));
end
%Gain adjustment
for ky = 1:M
y(ky, 1:length(v(ky, 1:(lx + 2*L - 2)))) = sgain(ky)*v(ky, 1:(lx + 2*L 2));
end
%///////////////////////////////////////////////////////////////////////
%//////////////////////////////////Output//////////////////////////////
Y = sum(y);
%Removing DC gain
Y = Y/a;
%///////////////////////////////////////////////////////////////////////
fprintf('Done\n');
%///////////////////PLOTTING: 8 Bands Filter Bank
if M == 8
H1 = freqz(h(1, 1:L), 1);
mag1 = 20*log10(abs(H1));
H2 = freqz(h(2, 1:L), 1);
mag2 = 20*log10(abs(H2));
H3 = freqz(h(3, 1:L), 1);
mag3 = 20*log10(abs(H3));
H4 = freqz(h(4, 1:L), 1);
mag4 = 20*log10(abs(H4));
H5 = freqz(h(5, 1:L), 1);
mag5 = 20*log10(abs(H5));
H6 = freqz(h(6, 1:L), 1);
mag6 = 20*log10(abs(H6));
H7 = freqz(h(7, 1:L), 1);
mag7 = 20*log10(abs(H7));
H8 = freqz(h(8, 1:L), 1);
mag8 = 20*log10(abs(H8));
q = 0:1/length(mag1):1 - 1/length(mag1);
plot(q, mag1, q, mag2, q, mag3, q, mag4, q, mag5,q,mag6,q,mag7,q,mag8), grid
on;
xlabel('Frequency:(Hz)');
ylabel('Magnitud_ Response:(dB)');
end
3. Play the original audio signal, followed by the graphic equalizer audio output.
As demonstrated in the matlab function the function first will play the input file , then it will play
the computed output file for 2 cases Stereo or Mono ,
The part of the code to support task 3 : is
%///////////////////////////Playing Input//////////////////////////////
fprintf('Playing Input Audio file...');
wavplay(xi, Fs);
fprintf('Done\n');
then playing the computed output :
%///////////////////////////Playing Output//////////////////////////////
fprintf('Playing Output...');
wavplay(Y', Fs);
fprintf('Done\n'); ,
notice that Y is the processed output .
incased of the Monofile , we have on input Vs one output .
incase of the sterio we have 2 inputs Vs 2 outputs
-----------------------------------------------------------------------------
comments to document the Matlab code.
The matlab program is well commented and easy to follow .
For example : in the matlab code , the part that deal with the Octav filter design is commented ,
and the tow combonants of the filter , the analysis filter where down-sampling is computed is
comented and the 2nd part where the Synthesis filter where up-sampling is computed is also
commented , the down code will give this part as follows :
%Cosine modulated Octave filter bank have 2 componants :
And the code is :
%Analysis filters : down-sampling
for p = 1:K
for kb = 1:L
b(p, kb) = 2*bhlp(kb)*cos(((2*(p-1) + 1)/(4*K))*(2*(kb-1) + 1 - K)*pi);
end
end
i1 = 3;
for ka = 0:M-1
if ka < 2
h(ka+1, 1:L) = b(ka+1, 1:L);
else
i2 = i1 + 2^(ka-1) - 1;
h(ka+1, 1:L) = sum(b(i1:i2, 1:L));
i1 = i2 + 1;
end
end
%Synthesis filters : up-sampling
for ks = 1:M
g(ks, 1:L) = h(ks, L:-1:1);
end
%DC gain estimation
sumc = [zeros(1, 2*L - 1)]; %Initial
for kd = 1:M
sumc = sumc + conv(h(kd, 1:L), g(kd, 1:L));
end
a = sum(sumc);
%Output of the analysis filters
for kw = 1:M
w(kw, 1:(lx + L - 1)) = conv(x, h(kw, 1:L));
end
%Output of the synthesis filters
for kv = 1:M
v(kv, 1:(lx + 2*L - 2)) = conv(w(kv, 1:(lx + L - 1)), g(kv, 1:L));
end
%Gain adjustment
for ky = 1:M
y(ky, 1:length(v(ky, 1:(lx + 2*L - 2)))) = sgain(ky)*v(ky, 1:(lx + 2*L 2));
end
4. Modifying the design to handle a stereo input signal. Automatically determine if the
input signal, x, is mono or stereo, and process accordingly. (Uses the .wav audio file
format, so that the signal is a column vector if mono, and a n by 2 array if stereo.)
This part of the project was challenging and took some time and some intuition to implement ,
it was easy to read the Mono file and play it using the wave play function and compute it as a
single input VS a single output , but the Stereo wav file was computed by considering tow inputs
VS 2 outputs since the file have 2 sets of data to be computed .
I managed to do the function in a way that will identify the user input as a mono or stereo t.
I used the fft function to plot the result of the computing as follow :
A: the mono file :
switch type % to suport the extra points , defininf if its Mono or Sterio and
plot input vs output
-------------------------------------------case 'mono'
Y = graphic(x,Fs,M,sgain);
figure(2)
Lx = length(x);
NFETx = 2^nextpow2(Lx);
wx = fft(x,NFETx)./Lx;
fx = Fs/2*linspace(0,1,NFETx/2 + 1);
plot(fx,2*abs(wx(1:NFETx/2+1)))
hold on
Ly = length(Y);
NFETy = 2^nextpow2(Ly);
wy = fft(Y,NFETy)./Ly;
fy = Fs/2*linspace(0,1,NFETy/2 + 1);
plot(fy,2*abs(wy(1:NFETy/2+1)),'k')
hold off
title('Mono FFT of input Vs FFT of output,,plue is the input black is
the output ')
xlabel('Frequency (Hz)')
ylabel('Magnitude')
this is the plot for the mono case using the default gain by the user :
B : the sterioe case :
The sterio case was computed by reading the wav file in to 2 inputs as :
x(1:0) and x(2:0) and the output Y [ Y1:Y2] , the n using the nfft function to to plot the
output vs the input , this is the code I used with the plots for a default gain :
case 'stereo'
Y1 = graphic(x(1,:),Fs,M,sgain);
Y2 = graphic(x(2,:),Fs,M,sgain);
Y = [Y1 ; Y2];
figure(2)
Lx1 = length(x(1,:));
NFETx1 = 2^nextpow2(Lx1);
wx1 = fft(x(1,:),NFETx1)./Lx1;
fx1 = Fs/2*linspace(0,1,NFETx1/2 + 1);
plot(fx1,2*abs(wx1(1:NFETx1/2+1)))
hold on
Ly1 = length(Y1);
NFETy1 = 2^nextpow2(Ly1);
wy1 = fft(Y1,NFETy1)./Ly1;
fy1 = Fs/2*linspace(0,1,NFETy1/2 + 1);
plot(fy1,2*abs(wy1(1:NFETy1/2+1)),'k')
hold off
title('Stereo FFT of channel 1 input Vs FFT of channel 1 output,,
plue is the input black is the output')
xlabel('Frequency (Hz)')
ylabel('Magnitude')
figure(3)
Lx2= length(x(2,:));
NFETx2 = 2^nextpow2(Lx2);
wx2 = fft(x(2,:),NFETx2)./Lx2;
fx2 = Fs/2*linspace(0,1,NFETx2/2 + 1);
plot(fx2,2*abs(wx2(1:NFETx2/2+1)))
hold on
Ly2 = length(Y2);
NFETy2 = 2^nextpow2(Ly2);
wy2 = fft(Y2,NFETy2)./Ly2;
fy2 = Fs/2*linspace(0,1,NFETy2/2 + 1);
plot(fy2,2*abs(wy2(1:NFETy2/2+1)),'k')
hold off
title('Stereo FFT of channel 2 input Vs FFT of channel 2 output,,plue
is the input black is the output ')
xlabel('Frequency (Hz)')
ylabel('Magnitude')
end
(Y1 = graphic(x(1,:),Fs,M,sgain);)
(Y2 = graphic(x(2,:),Fs,M,sgain);)
5. Conclusion :
This project helped to understand the basic concepts of filter banks in this project I used the(8M- pandas). There are several ways of realizing a filter bank and one way is to use the Cosine
Modulated filter bank. I observed the following points.
Instead of designing band-pass filters for different bands, the cosine modulated filter banks,
modulates a low-pass filter. By using the suitable parameters and the right frequency
In simple words, it simply shifts the pass band of the low pass filter towards higher frequencies,
and covering the entire band of frequencies of the input signal.
Filter banks have many applications in signal processing. one of the applications is Graphic
Equalizer, the user can ‘graphically’ modify different bands by intering the wanted frequencies
for different channels .
References :
Ch.Srinivasa Kumar et al. / International Journal on Computer Science and Engineering (IJCSE)
http://www.enggjournals.com/ijcse/doc/IJCSE11-03-08-116.pdf