How to generate Moving Average model - matlab

Question

In order to generate Autoregressive model, we have the aryule() command and we can also use filtersEstimating AR model. But how do I generate MA model? For instance, can somebody please show how to generate MA(20) model? I could not find any appropriate technique to do so. The noise is generated from a nonlinear map
epsilon(1) = 0.01;
for i =1 : N
epsilon(i+1) = 4*epsilon(i)*(1-epsilon(i));
end
So, the MA model will regress over epsilon terms.
Q1: Shall be extremely helpful if the code and functional form of an MA model is shown preferably MA(20) using the above noise model.
Q2: This is how I generated an AR(20) using random noise but don't know how to use the above equation as the noise instead of using rand for both MA and AR
%Generate sine wave = A*sin(2*pi*f*t + phi)
t = linspace(0,1,1000);
A = 5;
f = 2;
phi = pi/8;
sinewave = A*sin(2*pi*f*t + phi);
noisy_sine=sinewave+0.5*randn(size(t));
subplot(1,2,1);
plot(t, sinewave)
hold on;
subplot(1,2,2);
plot(t,noisy_sine);
%Generate AR model(20)
order =20;
ARCoeff = aryule(noisy_sine,order);

Answer 1

I had a same problem. The following links can help :
https://www.mathworks.com/examples/econometrics/mw/econ-ex18477389-simulate-an-ma-process
Simulate an MA Process
This example shows how to simulate sample paths from a stationary MA(12) process without specifying presample observations.
Contents
Specify a model.
Generate sample paths.
Plot the simulation variance.
Step 1. Specify a model.
Specify the MA(12) model
where the innovation distribution is Gaussian with variance 0.2.
model = arima('Constant',0.5,'MA',{0.8,0.2},...
'MALags',[1,12],'Variance',0.2);
Step 2. Generate sample paths.
Generate 200 sample paths, each with 60 observations.
rng('default')
Y = simulate(model,60,'NumPaths',200);
Step 3. Plot the simulation variance.
figure
plot(Y,'Color',[.85,.85,.85])
hold on
h = plot(mean(Y,2),'k','LineWidth',2)
legend(h,'Simulation Mean','Location','NorthWest')
title('MA(12) Process')
hold off

Answer 2

You can do that with one line of code using filter.
Let us talk about AR and MA in the Laplace domain. Say that your have a transfer function H = B/A, where B and A are polynomials. B have the coefficients of your MA and A will have your AR coefficients. In equations:
Thus, given your input signal x, you can use Matlab's y=filter(B,A,x) to generate your AR, MA and ARMA processes. The documentation about filter is here. For your specific case of a pure MA model, just make B=1 and your A as a vector with 20 elements.
You probably already know that, but just in case and to help future readers here is a link with a review on how to transform your difference equations to Laplace domain. TL;DR: