# MATLAB Code to calculate intermodulation and Third Order Intercept (of an Amplifier)

My well annotated MATLAB code and plot is attempting to calculate the third order intercept of an amplifier.

1. I have raw data of a non linear transfer curve, Voltage in (array) and the amplified voltage out (array)
2. I generate two sinusoids and add them together to make a waveform.
3. The waveform is normalised so its constrained within the positive and negative of a single voltage input value from the transfer curve array.
4. This is looped for all voltage input values in the array. (A Master Loop).
5. Each time the loop progresses through the voltage input values, the waveform is amplified by the transfer curve (a polynomial fit equation)
6. In other words, its a power sweep across the input range of the amplifier, where the waveform is progressively increasing in power.
7. A Fourier Transform is calculated each time the input voltage is increased (giving higher amplification) which reveals the amplitudes of intermodulation product frequencies.
8. Finally, a plot is produced of the intermod at each voltage_in/voltage out_point on the transfer curve and the third order intercept (TOI) calculated. The intermod gradient is 3:1 ona log-log plot and the linear gain is 1:1 on a log log plot (which is included below and on the code).

I know the Third order intercept of this amplifier is -6.8 dBW. But I get 0.78 dBW.

Please can someone help, I've been at this on and off for over 3 months and its driving me crazy.

The script is annotated well, runs in less than a second and has excellent plots.

clear all
format long

% ==== Section 1  Raw Data ==================

%Raw Data in DB (To start : Power in was given in dBm, Power out was dBW)
p_in_tc = [-53.07   -52.66  -52.25  -51.83  -51.42  -51.01  -50.60  -50.18  -49.77  -49.36  -48.95  -48.53  -48.12  -47.71  -47.30  -46.88  -46.47  -46.06  -45.65  -45.23  -44.82  -44.41  -44.00  -43.58  -43.17  -42.76  -42.35  -41.94  -41.52  -41.11  -40.70  -40.29  -39.87  -39.46  -39.05  -38.64  -38.22  -37.81  -37.40  -36.99  -36.57  -36.16  -35.75  -35.34  -34.92  -34.51  -34.10  -33.69  -33.27  -32.86  -32.45]';
p_in_tc = p_in_tc - 30;  % 30dbM = 0dBW Converted to dBw
p_out_tc = [-36.31  -35.91653   -35.53542   -35.08876   -34.68977   -34.24591   -33.87132   -33.41216   -33.03855   -32.63183   -32.21841   -31.79861   -31.40797   -30.98433   -30.58884   -30.16253   -29.77254   -29.35253   -28.9475    -28.53066   -28.13344   -27.73318   -27.32436   -26.91283   -26.51466   -26.09394   -25.69517   -25.2961    -24.8958    -24.49461   -24.10408   -23.69662   -23.29808   -22.91381   -22.51055   -22.11622   -21.73276   -21.33714   -20.94937   -20.56428   -20.19364   -19.80442   -19.44888   -19.06418   -18.71527   -18.34826   -17.99767   -17.65278   -17.29772   -16.97233   -16.65949]';

%Input Data to Volts
p_in_watts = 10.^((p_in_tc)/10);
v_in_rms = (p_in_watts .* 50) .^(1/2);
v_in_peak = v_in_rms' .* sqrt(2); %Use this Input

%Output Data to Volts
p_out_watts = 10.^((p_out_tc)/10);
v_out_rms = (p_out_watts .* 50) .^(1/2);
v_out_peak = v_out_rms' .* sqrt(2) ; %Use this Output

% ===== Section 2 - Generate Waveform ==============
power = 3;
exponent = 1*10^power;
freq_spacing = 1;
freq_range = (3:freq_spacing:4)*exponent;
Fs =10*max(freq_range);
Ts = 1/Fs;
end_time = 10*10^(-power);
n = 0 : Ts : end_time-Ts;

for c=1:length(v_in_peak)  %Master Loop (Power sweep across all input voltage values, creating a waveform for each input voltage).

%Make the number of waves in the frequency range
for a = 1:length(freq_range)
random_phase(a) = 0;
y_exp(a,:) = (exp(i*(2*pi .* freq_range(a) .* n + random_phase(a))));
end
waveform = sum(y_exp);

% ==== Section 3 Normalisation (Keep waveform within input voltage ==================

low = -v_in_peak(c); high = v_in_peak(c); mini = min(real(waveform)); maxi = max(real(waveform));
low2 = -v_in_peak(c); high2 = v_in_peak(c); mini2 = min(imag(waveform)); maxi2 = max(imag(waveform));
for a=1:length(waveform)
wave_recon_real(a) =  low + ( (real(waveform(a)) - mini) * (high-low) ) / (maxi-mini);
wave_recon_imag(a) =  low2 + ( (imag(waveform(a)) - mini2) * (high2-low2) ) / (maxi2-mini2);
wave_recon_normalised(a) = wave_recon_real(a) + (i*wave_recon_imag(a));
waveform(a) = wave_recon_normalised(a);
end

% ==== Section 4 Amplification ==================

p = polyfit(v_in_peak,v_out_peak,3);

for a=1:length(waveform)

%waveform_amp(a) = polyval(p,abs(waveform(a))) / abs(waveform(a)); %Amplify the current phasor and divide by the current phasor length to cancel out phasor in waveform in next step
%waveform_amp(a) = waveform_amp(a) * waveform(a) ; %Now multiply original wave so just the amplified summed current phasor is left

%Gives straight line intermods
waveform_amp_real(a) = polyval(p,real(waveform(a))) ;
waveform_amp_imag(a) = polyval(p,imag(waveform(a))) ;
waveform_amp(a) = waveform_amp_real(a) + (i*waveform_amp_imag(a));

end

%Setup Fast Fourier Transform
N=length(n);
freq_domain = (0:N/2); %Show positive frequency only
freq_domain = freq_domain * Fs / N;

%Fast Fourier Transform of Amplified wave
ft2 = fft(waveform_amp)/N;
ft2_spectrum = 2*abs(ft2); % 2* to compensate for negative frequency energy
ft2_spectrum = ft2_spectrum(1:N/2+1); %Show positive Frequency Only

%Bin Spacing
Bins_per_freq= (N/Fs);

%Intermods (third order only)
intermod_1 = 2*freq_range(2) - freq_range(1);
intermod_2 = 2*freq_range(1) - freq_range(2);
freq_range_im3 = [intermod_1 intermod_2];

%Get the bin numbers of the intermods
im_bins = round(Bins_per_freq * (freq_range_im3) +1); %Get the intermod bin Numbers, Domain starts at 0, so add 1

%Intermod amplitudes
im = ft2_spectrum(im_bins);

%Store one intermod for each power sweep, the master loop variable is c
im_sweep = im(1) ;
intermod_power(c) = 10*log10( (((im_sweep* (1/sqrt(2))) .^2 ) /50 ));

%The Frequencies in the waveform
freq_bins = round(Bins_per_freq * (freq_range) +1);  %Get the frequency bin Numbers, Domain starts at 0, so add 1
freq_bin_amplitudes = ft2_spectrum(freq_bins);
single_frequency = freq_bin_amplitudes(1);
%Store one frequency power for each power sweep, the master loop variable is c
single_frequency_power(c) = 10*log10( (((single_frequency* (1/sqrt(2))) .^2) /50 ));

% ==== Section 5  Final power sweep, make the plots and caulcate TOI ==================

if c == length(v_in_peak)  %Final input voltage value

%Obtain linear line from polynomial
for a=1:length(v_in_peak)
v_out_linear(a) = p(3)*((v_in_peak(a))^1);
p_out_linear(a) = 10*log10(((v_out_linear(a)/sqrt(2))^2)/50);
end

% -- Transfer Curve Linear Line (From linear line equation) --
%  Simple y=mx+c Algebra to make a straight line
x1_tc = p_in_tc(1);
x2_tc = p_in_tc(2);
y1_tc = p_out_linear(1);
y2_tc = p_out_linear(2);
m_tc = 1;
c_tc = y1_tc - (m_tc*x1_tc);
eqn_x_tc = p_in_tc;
eqn_y_tc = (m_tc .* p_in_tc) + c_tc;

% -- Intermod Linear Line --
%  Simple y=mx+c Algebra to make a straight line
x1_im = p_in_tc(end-10);
x1_im_idx = nearestpoint(x1_im,p_in_tc);
x2_im = p_in_tc(end-5);
x2_im_idx = nearestpoint(x2_im,p_in_tc);
y1_im = intermod_power(x1_im_idx);
y2_im = intermod_power(x2_im_idx);
m_im = 3;
c_im = y1_im - (m_im*x1_im);
eqn_x_im = p_in_tc;
eqn_y_im = (m_im .* p_in_tc) + c_im;

%Find when lines intersect
x_intersect = (c_im - c_tc) / (m_tc - m_im);
y_intersect = m_im * x_intersect + c_im

%Extrapolate past the intersect point for visualisation on plot
eqn_x_tc_int = min(eqn_x_tc):1:x_intersect+4;
eqn_y_tc_int = interp1(eqn_x_tc,eqn_y_tc,eqn_x_tc_int,'linear','extrap');
eqn_x_im_int = p_in_tc(x1_im_idx):1:x_intersect+4;
eqn_y_im_int = interp1(eqn_x_im,eqn_y_im,eqn_x_im_int,'linear','extrap');

figure(1);clf;
bar(freq_domain,ft2_spectrum)
ax=gca; set(gca,'Fontsize',7,'Ticklength',[-0.005 0]); ax.XAxis.Exponent = power;
title('Output Waveform - Frequency','Fontsize',7);
xlabel('Frequency (MHz)','Fontsize',7);ylabel('Actual Amplitude (V)','Fontsize',7);
%xlim([((freq_range(1)-1)*exponent) (freq_range(end)+1)*exponent]);

figure(2);clf;
ax=gca; set(gca,'Fontsize',7)
hold on;grid on; grid minor;
title('RMS Input Power vs RMS Output Power ','Fontsize',7); xlabel('Input Power (dBW) ','Fontsize',7); ylabel('Output Power (dBW) ','Fontsize',7);

plot(p_in_tc,p_out_tc,'--b','DisplayName','Transfer Curve (Raw Data) - RMS');
plot(p_in_tc, single_frequency_power,'-.m*','DisplayName','Single Frequency Power - RMS','MarkerSize',2);
plot(p_in_tc, intermod_power,'r','DisplayName','Intermod Power - RMS');
plot(p_in_tc,p_out_linear,'*g','DisplayName','Poly Order 1 (linear gain)');
plot(eqn_x_tc_int,eqn_y_tc_int,'k','linestyle','--','DisplayName','Linear Gain Extrapolated');
plot(eqn_x_im_int,eqn_y_im_int,'k','linestyle','--','DisplayName','Intermods Extrapolated');

legend('toggle','Location','northwest')
legend('boxoff')
hold off

end
%Final MASTER LOOP
clearvars -except p_in_tc p_out_tc phase_tc v_in_peak v_out_peak power exponent freq_spacing freq_range Fs Ts end_Time n c freq_domain intermod_power single_frequency_power
end


EDIT: Here is the figure, blue dotted line is the measured transfer characteristic, grey dashed lines are the extrapolated linear fits to the data.

Its a log-log plot. The intermod gradient is 3, giving 3:1. The linear extrapolation is from the polynomial using the order 1 term. You can clearly see the TOI is too high.

• It's always a good start when a MATLAB script starts with warning off! Apr 20, 2018 at 17:10
• How do you know that the third order intercept is -6.8 dBW? Where has this data come from? Apr 20, 2018 at 17:12
• Your code is missing nearestpoint(), is it from this file exchange? Apr 20, 2018 at 17:14
• From the figure you didn't add to your post it is clear that a linear fit is insufficient to model the 3rd order intercept. Try a higher order polynomial. Apr 20, 2018 at 17:33
• Also. I know its -6.8dBW because its been measured and the manufacturer states this. The linear fit, 1:1 is plotted alongside the intermods which is a slope 3:1. The poly nomial is of order 3, and third order intermods appear with the cubed term of the polynomial. Thankyou for your input, put its already present in the script. Apr 21, 2018 at 8:07

Run your test points over a 20 or 30 or 40 dB input range, and take log10 of the input and the output values. Take data every 5 or 10 dB. Discard data within 10 dB of the Compression points.

Expect to see exactly!! 1:1 input/output slope for the Vin/Vout of fundamental.

Expect to see exactly!! 3:1 input/output slope for the Vin/Vout of the 3rd order energy.

• What is your input power range? Go for 20 or 30 or 40 dB. Apr 22, 2018 at 1:54

I'm just going to preface this answer by saying that you should treat people who are helping you for free with more respect. Nobody here owes you an answer.

Looking at what is your presumably measured data of the transfer function, it is obvious that a linear fit is not accurate as the measured data is not linear.

Therefore instead of extrapolating using linear interp1, you should be using fit.

Below I have plotted second and third order polynomials fit on your measured data. The second order has an intersect around -3.8 dB, the third order around -8 dB.

As you have said that you expect -6.8 dB exactly, what you actually need are better measurements. Extrapolating to such a removed point will always give you inaccuracy. You should do what analogsystemrf has said and get data to within 10 dB.

Here is the code I added to fit higher order polynomials.

poly2Fit = fit(p_in_tc, p_out_tc, 'poly2');
poly3Fit = fit(p_in_tc, p_out_tc, 'poly3');
plot(poly2Fit,'g')
plot(poly3Fit,'b')


EDIT: In case you absolutely must use a linear fit, you should use the gradient from the end of the curve and not the start, e.g.

 startPoint = 0.95;
endpoly1Fit = fit(p_in_tc(ceil(startPoint*end):end),...
p_out_tc(ceil(startPoint*end):end),...
'poly1');


This code only uses the data from 95% of the way along the measured curve, and results in an intersect around -5 dB:

• Hi. Wasnt being rude with my replies. Thanks for answering. Apr 22, 2018 at 19:40
• @NatalieJohnson I got it wrong then! Apologies for not being able to help. I would say that if you are confident in your extrapolation, it lies somewhere in the polynomial you are using to model the distortion characteristic. Apr 24, 2018 at 13:54