I've got a question regarding an adaptive filter for interference calcellation:

Here, the interference is a periodic signal: $$x[n] = \cos(\pi/4\cdot n + \varphi_1) + \cos(3\pi/4\cdot n + \varphi_2)$$
where $$\varphi_1$$ and $$\varphi_2$$ are independent and uniformly distributed between 0 and $$2 \pi$$.
The signal is filtered by an IIR filter with impulse response: $$h[n] = (\frac{1}{3})^n$$

I should design an adaptive filter which minimizes the error in a MSE sense.

Now my first question is the following: what is the maximum filter length N I can choose for my adaptive filter? Here I am not quite sure how to answer this question and I would be very happy for some help! : )

• Can you please clarify the diagram a bit? Is this a "homework" type question? Also, how comfortable are you with the "ideal" low pass filter in the time domain? – A_A Feb 11 at 9:26
• Yes it is a homework. I know that the ideal lowpass filter in time domain is a sinc-function... – Phinie Feb 11 at 9:34
• If I have a single sinusoid as an input signal, my autocorrelation matrix is only invertible for N <= 2. We have learned in class that the autocorrellation matrix of signals with discrete lines (#lines = L) in frequency spectrum is only invertible for a filter order N <= L. Therefore the maximum filter order Nmax in my case is 4, as I have 2 sinusoidal functions as an input signal? – Phinie Feb 11 at 9:44
• Matt's answer at this link may help you: dsp.stackexchange.com/questions/37902/… but typically the filter needn't be longer than the delay spread of the channel (and ends up adding more noise if too long). – Dan Boschen Feb 11 at 17:58