✓ Passband - |f| <= 8kHz |H(f)| ^ 2 > - 2dB
Stopband - If|≥ 24kHz, |H(f)| ^ 2 < - 25dB
For the first problem, you will design a LPF by constructing a transfer function with no zeros and n repeated real poles at some frequency omega_{o} The resulting transfer function (in the Laplace domain) would be
H(s) = (omega_{o} ^ n)/((s + omega_{o}) ^ n)
a) Find an expression for the magnitude response, |H(f)| (or |H(f)| ^ 2 if you prefer).
b) Prove that it is not possible to find a combination of n and omega_{o}(or*f_{o}) such that both the
passband and stopband specifications can be met.
c) c) In order to make an achievable filter design, suppose we relax the stopband frequency. That is, suppose we replace f_{z} = 24kHz with something larger. How large must we make f_{2} so that the design specifications are achievable with a n = 3 order filter.
d) d) Alternatively, in order to make an achievable filter design, suppose we relax the stopband attenuation level. That is, suppose we replace alpha_{2} = - 25dB with something larger. How large must we make alpha_{2} (in dB) so that the design specifications are achievable with n = 3 order filter.
Note: In this problem, the transfer function was given in terms of omega_{o} which is in angular frequency (rad/sec) while the passband and stopband specifications were gien in terms of frequency in Hz. Don't forget to account for these differences.