Visualize a difference equation with Matlab

Question

I have a difference equation for a Single Pole Infinite Impulse Response Filter, defined on a discrete time-series:

$y[n]-(1-\alpha)*y[n-1]=\alpha*x_n$

While the []s brackets refer to a position n within the series. I'm looking for a visual way to represent this in order to get how this equation behaves. I have tried wolfram-alpha, MalLab... Is anyone me a pointer how I can make MatLab (e.g.) show me the plot for this function? Use-case is a DC offset filter, that uses this SPIIR filter with $\alpha=0,0004$. So it's mostly DSP related.

Best, Marius

Answer 1

Let's do a quick rewrite into more 'mathematical' notation:

$$y_n = \alpha x_n + (1-\alpha) y_{n-1}$$

By repeated substitution you can see that this is equal to:

$$ \begin{align} y_n & = \alpha \left( x_n + (1-\alpha) x_{n-1} + (1-\alpha)^2 x_{n-2} + \cdots \right) \\ & = \alpha \sum_{k=0}^\infty (1-\alpha)^k x_{n-k} \end{align}$$

So $y$ is an infinite sum of past values of $x$ (which is why it's called an infinite impulse response filter). One way to visualize this is to look at the weights

$$w_k = \alpha(1-\alpha)^k$$

as a function of $k$, which you can achieve in Matlab by

alpha = 0.2;
k = 0:20;
w = alpha .* (1-alpha).^k;
bar(k,w)

another way is to generate some data x and calculate y from it, and compare the two:

x = randn(30,1);
y = zeros(30,1);

y(1) = x(1);

for k = 2:30
  y(k) = alpha * x(k) + (1-alpha) * y(k-1);
end

plot(1:30, [x y])
legend({'x','y'})

Is this what you meant by 'visualize' the equation?