stuffABCD

ABDC matrix from specified modulator topology

Since R2026a

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Syntax

ABCD = stuffABCD(a,g,b,c,form)

Description

ABCD = stuffABCD(a,g,b,c,form) calculates the ABCD matrix from the parameters of a specified modulator topology.

Examples

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Design and Simulate Delta-Sigma Modulator from Parameters

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

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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`a` — Feedback/feedforward coefficients
real-valued vector

Feedback coefficients from the quantizer or feedforward coefficients to the quantizer, specified as a real-valued vector.

Data Types: double

`g` — Resonator coefficients
real-valued vector

Resonator coefficients, specified as a real-valued vector.

Data Types: double

`b` — Feed-in coefficients from modulator input to integrator
real-valued vector

Feed-in coefficients from the modulator input to each integrator, specified as a real-valued vector.

Data Types: double

`c` — Integrator inter-stage coefficients
real-valued vector

Integrator inter-stage coefficients, specified as a real-valued vector. In unscaled modulators, c is all ones.

Data Types: double

`form` — Delta-sigma modulator topology
`CRFB` | `CRFF` | `CIFB` | `CIFF` | `CRFBD` | `CRFFD` | `CIFBD` | `CIFFD`

String specifying the modulator topology. The supported topologies are:

form	Modulator Topologu
CRFB	Cascade of resonators, feedback form
CRFBD	Cascade of resonators, feedback form, with quantizer delay
CRFF	Cascade of resonators, feedforward form
CRFFD	Cascade of resonators, feedforward form, with quantizer delay
CIFB	Cascade of integrators, feedback form
CIFBD	Cascade of integrators, feedback form, with quantizer delay
CIFF	Cascade of integrators, feedforward form
CIFFD	Cascade of integrators, feedforward form, with quantizer delay

Data Types: double

Output Arguments

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`ABCD` — Loop filter of delta-sigma modulator
matrix

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

Version History

Introduced in R2026a

stuffABCD

Syntax

Description

Examples

Design and Simulate Delta-Sigma Modulator from Parameters

Input Arguments

a — Feedback/feedforward coefficients real-valued vector

g — Resonator coefficients real-valued vector

b — Feed-in coefficients from modulator input to integrator real-valued vector

c — Integrator inter-stage coefficients real-valued vector

form — Delta-sigma modulator topology CRFB | CRFF | CIFB | CIFF | CRFBD | CRFFD | CIFBD | CIFFD

Output Arguments

ABCD — Loop filter of delta-sigma modulator matrix

Version History

See Also

`a` — Feedback/feedforward coefficients
real-valued vector

`g` — Resonator coefficients
real-valued vector

`b` — Feed-in coefficients from modulator input to integrator
real-valued vector

`c` — Integrator inter-stage coefficients
real-valued vector

`form` — Delta-sigma modulator topology
`CRFB` | `CRFF` | `CIFB` | `CIFF` | `CRFBD` | `CRFFD` | `CIFBD` | `CIFFD`

`ABCD` — Loop filter of delta-sigma modulator
matrix