synthesizeNTF

Noise transfer function for delta-sigma modulator

Since R2026a

Syntax

ntf = synthesizeNTF(order,osr,opt,H_inf,f0)

Description

ntf = synthesizeNTF(order,osr,opt,H_inf,f0) synthesizes the noise transfer function (NTF) for a delta-sigma modulator.

Examples

Design and Simulate Delta-Sigma Modulator from Parameters

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;

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

order — Order of NTF

3 (default) | real scalar

Order of the NTF, specified as a real scalar.

Data Types: double

osr — Oversampling ratio

real scalar

Oversampling ratio of the delta-sigma modulator, specified as a real scalar. It is the ratio of the delta-sigma modulator sampling rate to the Nyquist rate.

Data Types: double

opt — NTF zero optimization

0 (default) | 1 | 2 | 3 | z

NTF zero optimization, specified as 0, 1, 2, 3, or z.

• 0 — Put all NTF zeros at the band center and do not optimize zeros.

• 1 — Optimize the NTF zeros according to the high OSR limit.

• 2 — Put at least one zero at the band center and optimize the rest.

• 3 — optimizes zeros using Optimization Toolbox™ techniques.

• z — Specify zero locations in complex form.

Data Types: double

H_inf — Maximum out-of-band gain of NTF

real scalar

Maximum out-of-band gain of the NTF, specified as a real scalar.

Lee’s rule states that H_inf < 2 yields a stable modulator with a binary quantizer. Reducing H_inf increases the likelihood of success, but reduces the magnitude of the attenuation provided by the NTF and the theoretical resolution of the modulator.

Data Types: double

f0 — Center frequency of binary delta-sigma modulator

real scalar

Center frequency of the binary delta-sigma modulator, specified as a real scalar.

Data Types: double

Output Arguments

ntf — Noise transfer function of delta-sigma modulator

object

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

Version History

Introduced in R2026a