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How can I find the Transfer Function having Magnitude(dB), Phase(degree) and Frequency(Hz)?
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I have 3 individual files which are Magnitude(dB), Phase(degrees) and Frequency(Hz) in excel. I need to find the transfer function with these data. After getting the transfer function, I need to plot back the graph (magnitude and phase) from transfer function to compare with my data.
I wish to get something like the picture attached below.

5 Comments
Star Strider
on 13 Dec 2021
Something is definitely wrong with these data!
They do not describe a frequency-response function —
Freq = readmatrix('https://www.mathworks.com/matlabcentral/answers/uploaded_files/833095/Frequency.xlsx')
Freq = 2001×1
100.0000
100.1000
100.2000
100.3000
100.4000
100.5000
100.6000
100.7000
100.8000
100.9000
Magn = readmatrix('https://www.mathworks.com/matlabcentral/answers/uploaded_files/833100/Magnitude.xlsx')
Magn = 2001×1
-43.9413
-43.8619
-43.7825
-43.7031
-43.6236
-43.5442
-43.4648
-43.3853
-43.3059
-43.2265
Phse = readmatrix('https://www.mathworks.com/matlabcentral/answers/uploaded_files/833105/Phase.xlsx')
Phse = 2001×1
111.3686
105.0197
98.6691
92.3168
85.9627
79.6068
73.2493
66.8899
60.5289
54.1661
cplxv = Magn .* exp(1j*deg2rad(Phse))
cplxv =
16.0107 -40.9206i
11.3669 -42.3634i
6.5992 -43.2823i
1.7667 -43.6673i
-3.0714 -43.5154i
-7.8555 -42.8298i
-12.5269 -41.6205i
-17.0287 -39.9038i
-21.3058 -37.7023i
-25.3064 -35.0444i
-28.9819 -31.9643i
-32.2879 -28.5010i
-35.1848 -24.6983i
-37.6381 -20.6041i
-39.6186 -16.2698i
-41.1034 -11.7495i
-42.0755 - 7.0997i
-42.5244 - 2.3783i
-42.4460 + 2.3563i
-41.8430 + 7.0453i
-40.7241 +11.6309i
-39.1050 +16.0565i
-37.0070 +20.2678i
-34.4576 +24.2132i
-31.4897 +27.8445i
-28.1415 +31.1175i
-24.4557 +33.9927i
-20.4791 +36.4355i
-16.2619 +38.4169i
-11.8574 +39.9135i
-7.3210 +40.9082i
-2.7095 +41.3902i
1.9193 +41.3548i
6.5078 +40.8043i
10.9987 +39.7469i
15.3364 +38.1974i
19.4671 +36.1767i
23.3398 +33.7114i
26.9069 +30.8337i
30.1248 +27.5810i
32.9543 +23.9949i
35.3611 +20.1215i
37.3163 +16.0102i
38.7969 +11.7132i
39.7858 + 7.2851i
40.2720 + 2.7817i
40.2511 - 1.7402i
39.7249 - 6.2236i
38.7016 -10.6124i
37.1955 -14.8517i
35.2273 -18.8886i
32.8231 -22.6729i
30.0145 -26.1578i
26.8383 -29.3004i
23.3357 -32.0620i
19.5518 -34.4090i
15.5354 -36.3132i
11.3379 -37.7519i
7.0128 -38.7085i
2.6154 -39.1723i
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18.3123 +31.7606i
21.7473 +29.4175i
24.8887 +26.7052i
27.6969 +23.6599i
30.1370 +20.3219i
32.1786 +16.7354i
33.7970 +12.9476i
34.9726 + 9.0079i
35.6919 + 4.9679i
35.9471 + 0.8802i
35.7367 - 3.2023i
35.0649 - 7.2268i
33.9421 -11.1413i
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30.4138 -18.4414i
28.0570 -21.7337i
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23.3752 +15.5448i
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12.5332 +16.8210i
14.4177 +15.1483i
16.0829 +13.2667i
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17.5846 - 9.9691i
16.2054 -11.9769i
14.5990 -13.7988i
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12.4635 +12.6028i
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17.1517 + 3.1267i
17.3473 + 1.0088i
17.2833 - 1.1106i
16.9622 - 3.2000i
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13.2872 -10.6564i
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6.6188 -15.4473i
4.6645 -16.0866i
2.6529 -16.4807i
0.6146 -16.6253i
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0.4817 + 1.4415i
0.6931 + 1.3513i
0.8884 + 1.2304i
1.0630 + 1.0815i
1.2131 + 0.9080i
1.3352 + 0.7141i
1.4266 + 0.5041i
1.4852 + 0.2829i
1.5097 + 0.0555i
1.4997 - 0.1729i
1.4553 - 0.3969i
1.3777 - 0.6115i
1.2687 - 0.8118i
1.1307 - 0.9933i
0.9671 - 1.1517i
0.7816 - 1.2835i
0.5784 - 1.3858i
0.3623 - 1.4561i
0.1382 - 1.4930i
-0.0887 - 1.4956i
-0.3133 - 1.4640i
-0.5304 - 1.3988i
-0.7350 - 1.3017i
-0.9224 - 1.1749i
-1.0885 - 1.0214i
-1.2293 - 0.8446i
-1.3418 - 0.6488i
-1.4233 - 0.4383i
-1.4721 - 0.2181i
-1.4870 + 0.0068i
-1.4678 + 0.2312i
-1.4150 + 0.4499i
-1.3298 + 0.6581i
-1.2141 + 0.8508i
-1.0708 + 1.0237i
-0.9032 + 1.1729i
-0.7150 + 1.2949i
-0.5107 + 1.3871i
-0.2950 + 1.4472i
-0.0728 + 1.4741i
0.1508 + 1.4670i
0.3705 + 1.4263i
0.5814 + 1.3529i
0.7787 + 1.2485i
0.9578 + 1.1156i
1.1146 + 0.9572i
1.2456 + 0.7771i
1.3478 + 0.5794i
1.4188 + 0.3686i
1.4572 + 0.1497i
1.4619 - 0.0723i
1.4331 - 0.2924i
1.3714 - 0.5054i
1.2782 - 0.7065i
1.1557 - 0.8910i
1.0069 - 1.0548i
0.8351 - 1.1941i
0.6443 - 1.3057i
0.4390 - 1.3872i
0.2239 - 1.4365i
0.0039 - 1.4528i
-0.2158 - 1.4356i
-0.4302 - 1.3853i
-0.6344 - 1.3033i
-0.8238 - 1.1914i
-0.9939 - 1.0522i
-1.1408 - 0.8889i
-1.2613 - 0.7055i
-1.3525 - 0.5060i
-1.4125 - 0.2952i
-1.4398 - 0.0779i
-1.4340 + 0.1409i
-1.3951 + 0.3561i
-1.3241 + 0.5628i
-1.2226 + 0.7563i
-1.0932 + 0.9320i
-0.9387 + 1.0860i
-0.7629 + 1.2147i
-0.5696 + 1.3153i
-0.3636 + 1.3854i
figure
subplot(2,1,1)
plot(Freq, Magn)
grid
subplot(2,1,2)
plot(Freq, Phse)
grid

frd = idfrd(cplxv, Freq, 1/(2*Freq(end)))
frd =
IDFRD model.
Contains Frequency Response Data for 1 output(s) and 1 input(s).
Response data is available at 2001 frequency points, ranging from 100 rad/s to 300 rad/s.
Sample time: 0.0016667 seconds
Status:
Created by direct construction or transformation. Not estimated.
figure
plot(Freq, imag(cplxv))

NrPoles = nnz(islocalmax(imag(cplxv)))
NrPoles = 44
sys_tf = tfest(frd, 2, 1)
sys_tf =
-19.56 s - 3578
-------------------------
s^2 + 0.9124 s + 1.296e04
Continuous-time identified transfer function.
Parameterization:
Number of poles: 2 Number of zeros: 1
Number of free coefficients: 4
Use "tfdata", "getpvec", "getcov" for parameters and their uncertainties.
Status:
Estimated using TFEST on frequency response data "frd".
Fit to estimation data: 2.943%
FPE: 188.8, MSE: 188.1
figure
compare(frd, sys_tf)
grid

.
Mathieu NOE
on 14 Dec 2021
hello @Star Strider
I believe Magn is given in dB (like in the plot)
so first think is to convert back to linear magnitude
Magn = 10.^(Magn/20) , and then
cplxv = Magn .* exp(1j*deg2rad(Phse))
Accepted Answer
Mathieu NOE
on 14 Dec 2021
hello
I tried a few options , IIR or FIR filters fit.
As the phase plot shows , there is a quite significant phase roll rate, idicating the presence of a huge delay in the system.
I assumed a sampling rate of Fs = 1000 hz and found out that more or less we can fit either a FIR or a IIr high pass filter in series with almost 200 samples of delay (hudge !!)
maybe there are more powerful tools then the simple invfreqz (not giving any good results here) or the manual fit I am doing here
FIR fit plot :

IIR fit plot :

clc
clearvars
Freq = readmatrix('Frequency.xlsx');
Magn_dB = readmatrix('Magnitude.xlsx');
Phse = readmatrix('Phase.xlsx');
figure(1)
subplot(211),plot(Freq,Magn_dB);
subplot(212),plot(Freq,Phse);
Magn = 10.^(Magn_dB/20) ;
%% high pass filter model (IIR)
Fs = 1000; % ? to be confirmed
N = 8; % filter order
dc_gain = Magn(end); % asymptotic value
[val,ind] = min(abs(Magn_dB - Magn_dB(end) + 5)); % - 5dB (vs dc_gain) cut off frequency index search
fc = Freq(ind); % - 5dB (vs dc_gain) cut off frequency
[b,a] = butter(N,2*fc/Fs,'high');
b = b*dc_gain; % apply dc gain on numerator
[g,p] = dbode(b,a,1/Fs,2*pi*Freq);
% adding delay due to sampling
nd = 200; % delay (samples)
rpd = -360*nd*Freq/Fs;
p = p+rpd; % adding filter phase to samples delay phase
p = mod(p,360);
p = p -180; % polarity correction
figure(1)
subplot(211),plot(Freq,Magn_dB,Freq,20*log10(g));
title('IIR model fit')
ylabel('Modulus (dB)');
subplot(212),plot(Freq,Phse,Freq,p);
xlabel('Frequency (Hz)');
ylabel('Phase (°)');% return
%% high pass filter model (FIR)
Fs = 1000; % ? to be confirmed
N = 20;
dc_gain = Magn(end); % asymptotic value
[val,ind] = min(abs(Magn_dB - Magn_dB(end) + 3)); % - 3dB (vs dc_gain) cut off frequency index search
fc = Freq(ind); % - 3dB (vs dc_gain) cut off frequency
[b,a] = fir1(N,2*fc/Fs,'high');
b = b*dc_gain; % apply dc gain on numerator
[g,p] = dbode(b,a,1/Fs,2*pi*Freq);
% adding delay due to sampling
Fs = 1000; % ? to be confirmed
nd = 200; % delay (samples)
rpd = -360*nd*Freq/Fs;
p = p+rpd; % adding filter phase to samples delay phase
p = mod(p,360);
p = p -180; % polarity correction
figure(2)
subplot(211),plot(Freq,Magn_dB,Freq,20*log10(g));
title('FIR model fit')
ylabel('Modulus (dB)');
subplot(212),plot(Freq,Phse,Freq,p);
xlabel('Frequency (Hz)');
ylabel('Phase (°)');
%% Id with invfreqz (FIR)
h = Magn .* exp(1j*180/pi*(Phse));
nb = 40+nd;
na = 1;
iter = 1000;
[bb,aa] = invfreqz(h,pi*Freq/Fs,nb,na,[],iter); % stable approximation to system
[g,p] = dbode(bb,aa,1/Fs,2*pi*Freq);
p = mod(p,360);
figure(3)
subplot(211),plot(Freq,Magn_dB,Freq,20*log10(g));
subplot(212),plot(Freq,Phse,Freq,p);
10 Comments
Liang Kar Yan
on 14 Dec 2021
Hi @Mathieu NOE, thank you for your help. Is there any method to get the transfer function equation from the graph?
Mathieu NOE
on 14 Dec 2021
hello again
this is a method using ifft to fit a FIR model to a given complex FRF- it works on some examples like the one below
clc
clearvars
%
Fs = 1e3;
Freq = linspace(0,Fs/2,100);
b = fir1(48,[0.3 0.5]); % Window-based FIR filter design
frf = freqz(b,1,Freq,Fs);
if mod(length(frf),2)==0 % iseven
frf_sym = conj(frf(end:-1:2));
else
frf_sym = conj(frf(end-1:-1:2));
end
fir = real(ifft([frf frf_sym]));
frfid = freqz(fir,1,Freq,Fs);
figure(1)
subplot(211),plot(Freq,20*log10(abs(frf)),Freq,20*log10(abs(frfid)));
legend('FIR input model','identified FIR model');
subplot(212),plot(Freq,180/pi*angle(frf),Freq,180/pi*angle(frfid));
legend('FIR input model','identified FIR model');
but when I try to use it on your data , it fails ... ugh !
clc
clearvars
Fs = 1e3;
Freq = readmatrix('Frequency.xlsx');
Magn_dB = readmatrix('Magnitude.xlsx');
Phse = readmatrix('Phase.xlsx');
figure(1)
subplot(211),plot(Freq,Magn_dB);
subplot(212),plot(Freq,Phse);
Magn = 10.^(Magn_dB/20) ;
frf = Magn .* exp(1j*pi/180*(Phse)); % FRF complex
if mod(length(frf),2)==0 % iseven
frf_sym = conj(frf(end:-1:2));
else
frf_sym = conj(frf(end-1:-1:2));
end
fir = real(ifft([frf; frf_sym]));
frfid = freqz(fir,1,Freq,Fs);
figure(1)
subplot(211),plot(Freq,20*log10(abs(frf)),Freq,20*log10(abs(frfid)));
subplot(212),plot(Freq,180/pi*angle(frf),Freq,180/pi*angle(frfid));
Mathieu NOE
on 14 Dec 2021
I think we could get a better result if your data was available on broader frequency range (from 0 to Fs/2) is Fs = sampling rate
Liang Kar Yan
on 14 Dec 2021
Hi, thank you so much for your help. The frequency is actually at 100GHz to 300GHz.
Mathieu NOE
on 14 Dec 2021
are you measuring an analog or digital filter / circuit ?
seems very high frequency range for anything digital ....
Mathieu NOE
on 14 Dec 2021
converting my code from 'Hz' to GHz by factor 10^9 is just making a new scale factor on the frequency axis , but not any impact on the FIR model output by itself
clc
clearvars
Freq = readmatrix('Frequency.xlsx'); % in Hz
Freq = Freq*1e9;% now in GHz
Fs = 2.56*max(Freq);
Magn_dB = readmatrix('Magnitude.xlsx');
Phse = readmatrix('Phase.xlsx');
figure(1)
subplot(211),plot(Freq,Magn_dB);
subplot(212),plot(Freq,Phse);
Magn = 10.^(Magn_dB/20) ;
frf = Magn .* exp(1j*pi/180*(Phse)); % FRF complex
if mod(length(frf),2)==0 % iseven
frf_sym = conj(frf(end:-1:2));
else
frf_sym = conj(frf(end-1:-1:2));
end
fir = real(ifft([frf; frf_sym]));
% fir = fir(1:end/4);
frfid = freqz(fir,1,Freq,Fs);
figure(1)
subplot(211),plot(Freq,20*log10(abs(frf)),Freq,20*log10(abs(frfid)));
subplot(212),plot(Freq,180/pi*angle(frf),Freq,180/pi*angle(frfid));
Liang Kar Yan
on 14 Dec 2021
I'm actually designing a terahertz waveguide at 100GHz to 300GHz and want to observe the s parameters (for this data set is actually the s21 graph).
Liang Kar Yan
on 14 Dec 2021
I think the first code is what I want and I manage to get the transfer function as well, thank you so much.
Mathieu NOE
on 14 Dec 2021
this is maybe something for you
but beyong my limited time possibilities right now
all the best for the future
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