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Solving discrete-time differential equations with Matlab

An LTI discrete system can also be described by a linear constant coefficient difference equation of the form:

This equation describes a recursive approach for computing the current output, given the input values and previously computed output values. In practice, this equation is computed forward in time, from n=-∞ to n=∞. Therefore, another form of this equation is:

A solution to equation (2) can be obtained in the form:

The homogeneous part yh[n] is given by:

Where zk, k=1,…,N are N roots (also called natural frequencies) of the characteristic equation:

This characteristic equation is important in determining the stability of the system. If roots zk satisfy the condition:

Then a casual system described by equation (2) is stable. The particular part of the solution, yp[n], is determined from the right-hand side of equation (1), where we will use z-transform for solving the difference equation.

Matlab solving

A function called filter in available in Matlab to solve Discrete-Time difference equations, given the input and the difference equation coefficients. In its simplest form it is invoked by:

y=filter(b,a,x)

Where b and a are the coefficient arrays from the equation (1), and x is the input sequence array. One must ensure that the coefficient a0 not be zero.

To compute and plot impulse response, Matlab also provides the function impz:

h=impz(b,a,n);

It computes samples of the impulse response of the filter at the sample indices given in n with numerator coefficients in b and denominator coefficients in a.

Example 1.

Given the following difference equation:

  1. Calculate and plot the impulse response h[n] at n=-20,…,100
  2. Calculate and plot the unit step response s[n] at n=-20,…,100
  3. Is the system specified by h[n] stable?

Solution:

  1. To determine h[n] we use the following script:

n=[-20:120];
a=[1,-1,0.9];
b=[1];
h=impz(b,a,n);
stem(n,h)
grid
xlabel(‘n’); ylabel(‘h[n]’)

The script yields:

Figure 1. Impulse response h[n] for example 1.
  1. 2. To determine s[n] we use the following script:

n=[-20:120];
a=[1,-1,0.9];
b=[1];
x=stepseq(0,-20,120);
s=filter(b,a,x);
stem(n,s)
xlabel(‘n’); ylabel(‘s[n]’);

The script yields:

Figure 2. Step response s[n] for example 1.
  1. Is the system specified by h[n] stable?

From Figure 1 we see that h[n] is practically zero n>120. Hence the sum:

Can be determined with the following script:

sum(abs(h))

This yields:

ans = 14.8785

That is to say:

This result implies that the system is stable. An alternate approach is to use the stability condition of the roots:

z=roots(a);
magz=abs(z)

magz =

0.9487
0.9487

Since the magnitude of both roots are less than one, the system is stable.

Actually, there are two forms of solving a linear constant coefficient difference equation: finding the particular and the homogeneous solutions; finding the zero-input and the zero-state responses. It is by using the z-transform that we can derive a method to obtain both.

Finding the particular and the homogeneous solutions

In construction…

Working with initial conditions – The zero-input and the zero-state responses.

In Matlab another form of the function filter can be used to solve for the differential equation, given its initial conditions.

In construction…

Related: Digital Filters – Moving Average Filters

Source:

  • Digital Signal Processing Using Matlab, 3erd ed
  • Fundamentos_de_Señales_y_Sistemas_usando la Web y Matlab
  • Oppenheim – Señales y Sistemas
  • Análisis de Sistemas Lineales Asistido con Scilab – Un Enfoque desde la Ingeniería Eléctrica.

Literature review by:

Prof. Larry Francis Obando – Technical Specialist – Educational Content Writer

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