calculateTF

Noise and signal transfer function of delta-sigma modulator

Since R2026a

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Syntax

[ntf,stf] = calculateTF(ABCD,k)

Description

[ntf,stf] = calculateTF(ABCD,k) calculates the noise transfer function (NTF) and signal transfer function (STF) of a delta sigma modulator described by the ABCD matrix with a quantizer gain k.

Examples

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Open Live Script

Define delta-sigma modulator parameters.

order = 5;      % Modulator order
OSR = 32;       % Oversampling ratio
N = 8192;       % Number of simulation points
f = 85;         % Input signal frequency bin
amp = 0.5;      % Input signal amplitude (should be less than 1)
fB = ceil(N/(2*OSR));

Create a sinusoidal input signal.

u = amp * sin(2*pi*f/N * (0:N-1)) % Create a sine wave input
u = 1×8192

         0    0.0326    0.0650    0.0972    0.1289    0.1601    0.1906    0.2203    0.2491    0.2768    0.3034    0.3286    0.3525    0.3748    0.3956    0.4147    0.4320    0.4475    0.4611    0.4727    0.4823    0.4899    0.4953    0.4987    0.5000    0.4991    0.4961    0.4911    0.4839    0.4746    0.4634    0.4502    0.4350    0.4181    0.3993    0.3789    0.3568    0.3332    0.3082    0.2819    0.2544    0.2258    0.1963    0.1659    0.1348    0.1032    0.0711    0.0387    0.0061   -0.0264

Synthesize the noise transfer function (NTF).

H = synthesizeNTF(order, OSR, 1)
H =
 
         (z-1) (z^2 - 1.997z + 1) (z^2 - 1.992z + 1)
  ----------------------------------------------------------
  (z-0.7778) (z^2 - 1.613z + 0.6649) (z^2 - 1.796z + 0.8549)
 
Sample time: 1 seconds
Discrete-time zero/pole/gain model.
Model Properties

Realize the NTF into coefficients for a CRFB modulator structure.

[a, g, b, c] = realizeNTF(H, 'CRFB')
a = 1×5

    0.0007    0.0084    0.0550    0.2443    0.5579


g = 1×2

    0.0028    0.0079


b = 1×6

    0.0007    0.0084    0.0550    0.2443    0.5579    1.0000


c = 1×5

     1     1     1     1     1

Assemble the final ABCD matrix.

ABCD = stuffABCD(a, g, b, c, 'CRFB')
ABCD = 6×7

    1.0000         0         0         0         0    0.0007   -0.0007
    1.0000    1.0000   -0.0028         0         0    0.0084   -0.0084
    1.0000    1.0000    0.9972         0         0    0.0633   -0.0633
         0         0    1.0000    1.0000   -0.0079    0.2443   -0.2443
         0         0    1.0000    1.0000    0.9921    0.8023   -0.8023
         0         0         0         0    1.0000    1.0000         0

Run the simulation.

[v,xn,xmax,y] = simulateDSM(u, ABCD)
v = 1×8192

     1    -1    -1     1     1    -1     1    -1     1     1    -1     1     1     1    -1     1     1    -1     1    -1     1     1     1     1     1    -1     1    -1     1     1     1     1     1    -1     1     1    -1     1    -1     1    -1     1     1     1    -1    -1     1     1    -1     1


xn = 5×8192

   -0.0007    0.0000    0.0007    0.0001   -0.0005    0.0003   -0.0002    0.0006    0.0001   -0.0004    0.0005    0.0000   -0.0004   -0.0008    0.0001   -0.0003   -0.0007    0.0003   -0.0000    0.0009    0.0006    0.0003   -0.0001   -0.0004   -0.0008    0.0002   -0.0001    0.0009    0.0006    0.0002   -0.0002   -0.0005   -0.0009    0.0001   -0.0004   -0.0008    0.0001   -0.0003    0.0006    0.0001    0.0009    0.0004   -0.0001   -0.0007    0.0001    0.0008    0.0002   -0.0005    0.0002   -0.0005
   -0.0084   -0.0002    0.0087    0.0017   -0.0055    0.0039   -0.0026    0.0075    0.0016   -0.0044    0.0062    0.0010   -0.0045   -0.0100    0.0010   -0.0038   -0.0087    0.0029   -0.0013    0.0111    0.0075    0.0036   -0.0004   -0.0047   -0.0092    0.0028   -0.0013    0.0112    0.0075    0.0035   -0.0008   -0.0056   -0.0108    0.0004   -0.0046   -0.0100    0.0008   -0.0047    0.0061    0.0006    0.0112    0.0054   -0.0010   -0.0081    0.0009    0.0102    0.0031   -0.0049    0.0032   -0.0053
   -0.0633   -0.0068    0.0604    0.0126   -0.0407    0.0269   -0.0202    0.0544    0.0147   -0.0294    0.0484    0.0125   -0.0276   -0.0720    0.0058   -0.0302   -0.0701    0.0124   -0.0185    0.0735    0.0525    0.0281   -0.0001   -0.0323   -0.0690    0.0161   -0.0128    0.0803    0.0595    0.0341    0.0038   -0.0320   -0.0738    0.0046   -0.0330   -0.0772   -0.0019   -0.0431    0.0349   -0.0040    0.0761    0.0390   -0.0061   -0.0600    0.0032    0.0741    0.0261   -0.0316    0.0269   -0.0348
   -0.2443   -0.0490    0.2066    0.0423   -0.1585    0.0888   -0.0833    0.1978    0.0647   -0.0984    0.1934    0.0734   -0.0745   -0.2534    0.0218   -0.1157   -0.2814    0.0102   -0.1075    0.2386    0.1820    0.1070    0.0104   -0.1113   -0.2619    0.0435   -0.0623    0.2931    0.2423    0.1688    0.0682   -0.0641   -0.2330    0.0452   -0.0982   -0.2807   -0.0191   -0.1825    0.0998   -0.0416    0.2637    0.1457   -0.0142   -0.2231    0.0006    0.2749    0.1164   -0.0948    0.1221   -0.1046
   -0.8023   -0.2752    0.5256    0.0641   -0.5804    0.1557   -0.3792    0.4995    0.1453   -0.3567    0.5640    0.2627   -0.1730   -0.7752    0.0252   -0.4171   -1.0153   -0.1975   -0.6057    0.4545    0.3477    0.1701   -0.1011   -0.4921   -1.0330   -0.1531   -0.4965    0.6285    0.5829    0.4586    0.2274   -0.1435   -0.6917    0.1447   -0.2887   -0.9159   -0.1781   -0.7326    0.0971   -0.3451    0.6185    0.3323   -0.1304   -0.8189   -0.1851    0.7052    0.3034   -0.3277    0.3557   -0.3216


xmax = 5×1

    0.0014
    0.0167
    0.1147
    0.3735
    1.1084


y = 1×8192

         0   -0.7697   -0.2102    0.6227    0.1930   -0.4203    0.3463   -0.1588    0.7486    0.4221   -0.0533    0.8926    0.6152    0.2018   -0.3796    0.4399    0.0149   -0.5679    0.2635   -0.1330    0.9368    0.8375    0.6654    0.3977    0.0079   -0.5338    0.3431   -0.0054    1.1124    1.0575    0.9220    0.6776    0.2916   -0.2736    0.5440    0.0902   -0.5591    0.1551   -0.4244    0.3790   -0.0907    0.8443    0.5286    0.0355   -0.6841   -0.0820    0.7763    0.3421   -0.3216    0.3293

Analyze the output.

figure;
spec = fft(v .* ds_hann(N)) / (N/4);
plot(dbv(spec));
axis([0 N/2 -120 0]);
title('Spectrum of Modulator Output');
xlabel('Frequency Bin');
ylabel('dBV');
grid on;

Figure contains an axes object. The axes object with title Spectrum of Modulator Output, xlabel Frequency Bin, ylabel dBV contains an object of type line.

Calculate SNR.

snr = calculateSNR(spec(1:fB),f)
snr = 
82.5313

Calculate the NTF and STF of the modulator.

[ntf,stf] = calculateTF(ABCD,1)
ntf =
 
         (z-1) (z^2 - 1.997z + 1) (z^2 - 1.992z + 1)
  ----------------------------------------------------------
  (z-0.7778) (z^2 - 1.613z + 0.6649) (z^2 - 1.796z + 0.8549)
 
Sample time: 1 seconds
Discrete-time zero/pole/gain model.
Model Properties

stf =
 
  1
 
Static gain.
Model Properties

Map the ABCD matrix back to coefficients of the CRFB topology.

[a,g,b,c] = mapABCD(ABCD,'CRFB')
a = 1×5

    0.0007    0.0084    0.0550    0.2443    0.5579


g = 1×2

    0.0028    0.0079


b = 1×6

    0.0007    0.0084    0.0550    0.2443    0.5579    1.0000


c = 1×5

     1     1     1     1     1

Dynamically scale the ABCD matrix so that the state maxima are less than specified limit 1.

nlev = 2;
xlim = 1;
ymax = nlev+5;
[ABCDs,umax]=scaleABCD(ABCD,nlev,f,xlim,ymax,N)
ABCDs = 6×7

    1.0000         0         0         0         0   -0.0007    0.0007
    1.0000    1.0000   -0.0028         0         0   -0.0084    0.0084
    1.0000    1.0000    0.9972         0         0   -0.0633    0.0633
         0         0    1.0000    1.0000   -0.0079   -0.2443    0.2443
         0         0    1.0000    1.0000    0.9921   -0.8023    0.8023
         0         0         0         0   -1.0000    1.0000         0


umax = 
8192

Input Arguments

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State space description of the loop filter of the delta-sigma modulator, specified as a matrix.

Data Types: double

Gain of the quantizer, specified as a real scalar.

Data Types: double

Output Arguments

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Noise transfer function of the delta-sigma modulator in pole zero form, returned as a zpk object.

Signal transfer function of the delta-sigma modulator in pole zero form, returned as a zpk object.

Version History

Introduced in R2026a

See Also

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