Test: Digital Filters Design Consideration

Explanation: We know that the ideal low pass filter is non-causal. Hence, a ideal low pass filter cannot be realized in practice.

Explanation: At n=0, the equation for ideal low pass filter is given as h(n)=ω/π.
From the given figure, h(0)=0.25=>ω=π/4.
Thus the given figure represents the unit sample response of an ideal low pass filter at ω=π/4.

Explanation: If h(n) has finite energy and h(n)=0 for n<0, then according to the Paley-Wiener theorem, we have

Explanation: If |H(ω)| is square integrable and if the integral  is finite, then we can associate with |H(ω)| and a phase response θ(ω), so that the resulting filter with the frequency response H(ω)=|H(ω)|e^jθ(ω) is causal.

Explanation: One important conclusion that we made from the Paley-Wiener theorem is that the magnitude function |H(ω)| can be zero at some frequencies, but it cannot be zero over any finite band of frequencies, since the integral then becomes infinite. Consequently, any ideal filter is non-causal.

Explanation: Given h(n) is causal and h(n)= h_e(n)+h_o(n)
=>h_e(n)=1/2[h(n)+h(-n)] Now, if h(n) is causal, it is possible to recover h(n) from its even part he(n) for 0≤n≤∞ or from its odd component h_o(n) for 1≤n≤∞.
=>h(n)= 2h_e(n)u(n)-h_e(0)δ(n) ,n ≥ 0.

Explanation: Given h(n) is causal and h(n)= h_e(n)+h_o(n)
=>h_e(n)=1/2[h(n)+h(-n)] Now, if h(n) is causal, it is possible to recover h(n) from its even part h_e(n) for 0≤n≤∞ or from its odd component h_o(n) for 1≤n≤∞.
=>h(n)= 2h_o(n)u(n)+h(0)δ(n) ,n ≥ 1
since ho(n)=0 for n=0, we cannot recover h(0) from h_o(n) and hence we must also know h(0).

Explanation: . If h(n) is absolutely summable, i.e., BIBO stable, then the frequency response H(ω) exists and
H(ω)= H_R(ω)+j H_I(ω)
where H_R(ω) and H_I(ω) are the Fourier transforms of he(n) and ho(n) respectively.

Explanation: Since h(n) is completely specified by he(n), it follows that H(ω) is completely determined if we know HR(ω). Alternatively, H(ω) is completely determined from HI(ω) and h(0). In short, H_R(ω) and H_I(ω) are interdependent and cannot be specified independently when the system is causal.

Explanation: Pass band edge ripple is the frequency at which the pass band starts transiting to the stop band.