9. Butterworth Lowpass: This filter is characterized by the property that its magnitude response is flat in both passband and stopband. The magnitude-squared response of an N-th order lowpass filter is given by: Here, N is the order of the filter and  c is the cutoff frequency in rad/sec Plot of the magnitude-squared response is as depicted below:

10. for all N.

13. N=3; gc=0.5; k=0:5; pk = gc*exp(i*(pi/(2*N)*(2*k+N+1)); pk = Columns 1 through 5 -0.2500 + 0.4330i -0.5000 + 0.0000i -0.2500 - 0.4330i 0.2500 - 0.4330i 0.5000 - 0.0000i Column 6 0.2500 + 0.4330i