Cascading a filter with itself to form a higher-order filter? I've been given an audio signal with some high frequency noise. The professor gave me the system function of a lowpass filter that gives me control over the cutoff frequency:

$$H(z)=\frac{1-\alpha}{2}\frac{1+z^{-1}}{1-\alpha z^{-1}}$$

where $\alpha=\frac{1-\sin \omega_c}{\cos \omega_c}$ and $\omega_c$ is the cutoff frequency in radians/sec.

The professor claims that if the filter is cascaded with itself $L$ times, it becomes a filter of order $L$.

My questions are the following:


What does increasing the order of the filter do?
Why would I want to increase the order of the filter?
If I cascade the filter with itself $L$ times, won't I lose the ability to control the cutoff frequency? How could I correct for this?


edit:

Here's what I get for the frequency response of the filter when the cutoff frequency is set to 1000 Hz. Sampling frequency is 16000 Hz.
Replace your $\alpha$ with the following to see if it helps:

fs = 16000;
fc = 3500;
wc = 2*pi*fc/fs;

L = 5;            % order of the cascade
B = 2^((L-1)/L);
C = cos(wc);
M = B-C-1;
N = 1+C-B*C;
alfa = ( -N + sqrt( N*N - M*M ) )/M;


I've got the following plots, compared to your posted $\alpha$ calculation.